Bryan Lindsley

The Simple Guide to Problem Mapping (only 4 steps)

In this short post I’m going to show you how to create a visual model of your complex problem with a 4-step problem mapping method.

This type of modeling can be applied to any complex social problem, like homelessness, poverty or crime. Whether you’re a street activist or long-time systems change practitioner, the systems thinking method I share below can help you gain understanding about the structure and dynamics of your problem and increase your likelihood of making good decisions about solving it. 

These are the exact steps we use to start mapping in my online course about systems thinking and solving social problems .

In the post I’ll also talk about the different types of problem mapping, how mapping itself is a problem-solving process, and give an example of mapping the issue of illegal opioids.

It doesn’t matter what you call it

Let’s get this out of the way first. There’s a lot of methods for visualizing connected ideas or systems, including problem mapping, mind mapping, cognitive mapping and issue mapping. There’s also a variety of ways people refer to the results: mind maps, mental models, causal loop diagrams, solution maps, and problem solution maps.

When you’re just getting started, jargon and differences in mapping method don’t matter very much. My intent in this blog is to remove the somewhat artificial barriers that prevent people from regularly creating and using problem maps.

For now, I don’t want you to worry about any of this. Just begin with Step 1 below by putting your ideas about the problem and its causes on paper. You can easily adjusting your map later if you want it to reflect a certain mapping protocol.

Problem mapping as problem-solving process

Mapping is just one of many problem solving techniques. But, it is particularly suited to complex problems with many variables and interconnections. Whereas verbally describing a series of complex relationships is very difficult, a simple picture really can be worth a thousand words.

At the beginning, it’s important to recognize that the goal isn’t to create a perfect representation of reality. That’s not possible and it would be foolish to try. Rather, the process of creating the map is about gaining insight about problem in way that can’t be had with words or equations alone.

Along the way, you’ll make explicit many assumptions you have about the problem, as well as how they are connected. In other words, mapping allows you to simultaneously capture details about parts the problem while creating a representation of the “big picture.” This type of switching back and forth from reductive analysis to synthesis is an excellent complex problem-solving approach .

Mapping isn’t a choice

You might not think you need to formally model your problem, but the question is not whether you should model or not. Mental models are always used, even if only implicitly in our heads or baked into the assumptions we make. The choice is really about whether you want to use an implicit and vague model in your head or you want to use an explicit and detailed model.

I know you already have a rudimentary idea about your social problem and what causes it. This is your most basic mental model of the problem. The question is – how accurate is it? For most people who haven’t written it down, the model in their head is simple, perhaps a linear cause and effect model.

Complex problem mapping example: illegal opioids

As an example, let’s look at the issue of illegal opioids.

The two opioid models and policies I show bel0w are adapted from Narcotics and the Community: A System Simulation . Note that this is a simplified version for instruction purposes only and doesn’t reflect the intricacies of more recent opioid issues likes prescription opioids and fentanyl.

visual model problem solving

The logic of the model is straight-forward: with less illegal opioid supply, there will be less addiction and thus less addiction-related crime. It’s intuitive and easy to remember. Our minds are full of models like this, each fairly simple because the human brain isn’t able to hold more than a handful of variables in mind at the same time. 

While it is hard to disagree with the premise of the simple cause and effect model, mistakes can happen when we assume the effects of our actions will be similarly simple and linear. For example, one common policy recommendation to reduce opioid addiction has been to curb the supply of illegal opioids. Let’s examine the consequences of this policy with a slightly more detailed model that captures its additional complexity.

Adding variables, or nodes

Let’s add three additional variables (or nodes) not in the simple model.

The number of opioid addicts isn’t one static number (a fixed quantity, or “stock” as it’s referred to in a stock and flow diagram). Rather, the number is determined by rates of addiction and attrition (or “flows”). So let’s add “addiction rate” and “attrition rate” to the map.

Because a reduced opioid supply would make the price of illegal opioids rise, let’s add “opioid price” as well.

visual model problem solving

Causal feedback loops

In the simple cause and effect model, price isn’t considered and any reduction in the supply deterministically reduces addiction and crime. But, adding the price of opioids to the model gives us a counterintuitive feedback loop: because addicts need more money to get the same amount of opioids, reducing the supply could actually make crime worse.

On the other hand, the addition of price to the model also mediates the rate at which people become addicted (represented by addiction and attrition rate nodes). When the price is high, there are fewer new users, which is good.

The point of the more complex model is not to show that reducing the illegal supply of opioids is a good or bad policy. Like all actions aimed at social problems, it is a trade-off between benefits and costs.

Rather, my intent is to show that overly simple mental models of a problem, even when logically correct, can lead us to make decisions that lead to unintended consequences. Better models don’t necessarily help us find the right solution, but they can supply needed insight about how the problem works systemically and give a sense of how our actions may cascade through the larger system.

So, if you don’t already have an explicit model of your problem or if you’re just working from a simple linear model in your mind, here are the steps you should take to create your own.

Problem mapping in four steps

Step 1. brainstorm primary causes and concepts.

Think of a problem and spend some time brainstorming all of its causes, including any other relevant concepts. For now, concepts can be any important variables in your problem: actors, stocks (e.g. quantities) and flows (e.g. rates), or even abstract concepts (e.g. wealth, democracy, etc.). Keep these tips in mind:

  • Use nouns and avoid verbs, since actions will be represented in the map with arrows.
  • Try to pick things that can go up or down in quantity, strength or influence over time.
  • Be as specific as possible. When possible choose metrics over abstract concepts.

You might have a long list, and some causes and concepts may be more important that others. That’s OK.

For our purposes to get started, select the top 3-5 causes. These are the issues you believe are most fundamental in causing your problem.

My example below has only three primary causes for simplicity, but yours may have more. After you get through the steps there will be plenty of time for you to add or subtract variables.

visual model problem solving

Step 2. Brainstorm second-order causes

Most people think step 2 is about brainstorming solutions. But don’t do that yet! You don’t yet understand how the problem works, so solutions at this point will likely be similarly incomplete. To start getting a fuller picture of how your problem functions as part of a larger system, brainstorm second order causes. To put it simply, what causes each of your primary causes?

You can pull second-order causes from your initial brainstorming list, or brainstorm a new list for each primary cause.

Once you do that, your model may look something like this:

visual model problem solving

Step 3. Add interrelationships between causes

We’re getting closer to a comprehensive model – just two more steps to go!

In this step you operationalize the biggest insight about complex systems: the relationships are more important than the components . Right now, you still have a fairly simple, linear model. Every node leads directly to your problem, which makes it a kind of hierarchy.

I’m fairly certain that in real life your problem exists in a more complicated web of connections. In this step you connect causes that have a relationship to any other cause, even those that don’t lead directly to the problem.

For example, one of your second-order causes may be related to another second-order cause. That may sound confusing, but your task right now is just to draw connections between any two nodes that you feel might affect each other (see red lines in model below). 

visual model problem solving

Step 4. Define causality of each relationship

The final step is to characterize each of the relationships between nodes as increasing or decreasing (for example, Cause #1a increases Cause #1, or Cause #1a decreases Cause #1).

A plus sign represents increase and a negative sign represents decrease.

visual model problem solving

Use this question to help you determine the direction of the relationship:

When this component increases, does the other component increase or decrease?

Note that sometimes nodes will have a two-way relationship. For example, in the model below an increase in Cause #1 increases Cause #2a, but an increase in Cause #2a decreases Cause #1. This is a balancing feedback loop.

The first draft of your model is complete. Who-hoo!

Three problem mapping principles

Here are a few words of guidance to keep in mind as you start modeling.

#1. Always model a problem, never a system

Problems themselves dictate the necessary boundaries that formal systems (like the education system) do not.

#2. The map is not the terrain

Your model is only an abstraction of reality and should always be regarded with a degree of skepticism and knowledge that the terrain may change.

#3. All models are wrong and incomplete

The purpose of using a model isn’t to find the solution, but to increase your understanding of the problem and explore the effects of possible interventions.

The whole process is most valuable when you remember that mapping is more of an art than science.

What to do after you’ve created a first draft

Share it with others for feedback.

There’s a few directions you can go from here. The first is to talk through your model to a trusted colleague who also has some understanding of the problem, or encourage them to create their own model following these steps.

Differing perspectives can uncover different assumptions about the problem and lead to fruitful dialogue. Your model can be updated based on feedback, or you can work with your colleague to combine models, adding and subtracting nodes and relationships as you see fit.

Convert it into a fuzzy cognitive map

The other direction is to convert your model into a fuzzy cognitive map using computer software. This is super exciting because it allows you to run simulations of potential changes you could make and see resulting changes in the system as a whole.

For example, in the earlier opioid addiction model we could run a simulation of a policy that provides free legal supply of opioids to addicts (in an effort to reduce both crime and long-term addition), and calculate system-wide changes. This helps uncover feedback loops and potential unintentional consequences.

Creating a fuzzy cognitive map only requires a few additional steps. I go through them step-by-step plus how to run what-if scenarios in my next blog posts.

problem solving flowchart

Problem-Solving Flowchart: A Visual Method to Find Perfect Solutions

Lucid Content Team

Reading time: about 7 min

“People ask me questions Lost in confusion Well, I tell them there's no problem Only solutions” —John Lennon, “Watching the Wheels”

Despite John Lennon’s lyrics, nobody is free from problems, and that’s especially true in business. Chances are that you encounter some kind of problem at work nearly every day, and maybe you’ve had to “put out a fire” before lunchtime once or twice in your career.

But perhaps what Lennon’s saying is that, no matter what comes our way, we can find solutions. How do you approach problems? Do you have a process in place to ensure that you and your co-workers come to the right solution?

In this article, we will give you some tips on how to find solutions visually through a problem-solving flowchart and other methods.

What is visual problem-solving?

If you are a literal thinker, you may think that visual problem-solving is something that your ophthalmologist does when your vision is blurry. For the rest of us, visual problem-solving involves executing the following steps in a visual way:

  • Define the problem.
  • Brainstorm solutions.
  • Pick a solution.
  • Implement solutions.
  • Review the results.

How to make your problem-solving process more visual

Words pack a lot of power and are very important to how we communicate on a daily basis. Using words alone, you can brainstorm, organize data, identify problems, and come up with possible solutions. The way you write your ideas may make sense to you, but it may not be as easy for other team members to follow.

When you use flowcharts, diagrams, mind maps, and other visuals, the information is easier to digest. Your eyes dart around the page quickly gathering information, more fully engaging your brain to find patterns and make sense of the data.

Identify the problem with mind maps

So you know there is a problem that needs to be solved. Do you know what that problem is? Is there only one problem? Is the problem sum total of a bunch of smaller problems?

You need to ask these kinds of questions to be sure that you are working on the root of the issue. You don’t want to spend too much time and energy solving the wrong problem.

To help you identify the problem, use a mind map. Mind maps can help you visually brainstorm and collect ideas without a strict organization or structure. A mind map more closely aligns with the way a lot of our brains work—participants can bounce from one thought to the next defining the relationships as they go.

basic mind map

Mind mapping to solve a problem includes, but is not limited to, these relatively easy steps:

  • In the center of the page, add your main idea or concept (in this case, the problem).
  • Branch out from the center with possible root causes of the issue. Connect each cause to the central idea.
  • Branch out from each of the subtopics with examples or additional details about the possible cause. As you add more information, make sure you are keeping the most important ideas closer to the main idea in the center.
  • Use different colors, diagrams, and shapes to organize the different levels of thought.

Alternatively, you could use mind maps to brainstorm solutions once you discover the root cause. Search through Lucidchart’s mind maps template library or add the mind map shape library to quickly start your own mind map.

Create a problem-solving flowchart

A mind map is generally a good tool for non-linear thinkers. However, if you are a linear thinker—a person who thinks in terms of step-by-step progression making a flowchart may work better for your problem-solving strategy. A flowchart is a graphical representation of a workflow or process with various shapes connected by arrows representing each step.

Whether you are trying to solve a simple or complex problem, the steps you take to solve that problem with a flowchart are easy and straightforward. Using boxes and other shapes to represent steps, you connect the shapes with arrows that will take you down different paths until you find the logical solution at the end.

project development decision tree

Flowcharts or decision trees are best used to solve problems or answer questions that are likely to come up multiple times. For example, Yoder Lumber , a family-owned hardwood manufacturer, built decision trees in Lucidchart to demonstrate what employees should do in the case of an injury.

To start your problem-solving flowchart, follow these steps:

  • Draw a starting shape to state your problem.
  • Draw a decision shape where you can ask questions that will give you yes-or-no answers.
  • Based on the yes-or-no answers, draw arrows connecting the possible paths you can take to work through the steps and individual processes.
  • Continue following paths and asking questions until you reach a logical solution to the stated problem.
  • Try the solution. If it works, you’re done. If it doesn’t work, review the flowchart to analyze what may have gone wrong and rework the flowchart until you find the solution that works.

If your problem involves a process or workflow , you can also use flowcharts to visualize the current state of your process to find the bottleneck or problem that’s costing your company time and money.

manufacturing flow example

Lucidchart has a large library of flowchart templates to help you analyze, design, and document problem-solving processes or any other type of procedure you can think of.

Draw a cause-and-effect diagram

A cause-and-effect diagram is used to analyze the relationship between an event or problem and the reason it happened. There is not always just one underlying cause of a problem, so this visual method can help you think through different potential causes and pinpoint the actual cause of a stated problem.

Cause-and-effect diagrams, created by Kaoru Ishikawa, are also known as Ishikawa diagrams, fishbone diagrams , or herringbone diagrams (because they resemble a fishbone when completed). By organizing causes and effects into smaller categories, these diagrams can be used to examine why things went wrong or might go wrong.

cause-and-effect diagram example

To perform a cause-and-effect analysis, follow these steps.

1. Start with a problem statement.

The problem statement is usually placed in a box or another shape at the far right of your page. Draw a horizontal line, called a “spine” or “backbone,” along the center of the page pointing to your problem statement.

2. Add the categories that represent possible causes.

For example, the category “Materials” may contain causes such as “poor quality,” “too expensive,” and “low inventory.” Draw angled lines (or “bones”) that branch out from the spine to these categories.

3. Add causes to each category.

Draw as many branches as you need to brainstorm the causes that belong in each category.

Like all visuals and diagrams, a cause-and-effect diagram can be as simple or as complex as you need it to be to help you analyze operations and other factors to identify causes related to undesired effects.

Collaborate with Lucidchart

You may have superior problem-solving skills, but that does not mean that you have to solve problems alone. The visual strategies above can help you engage the rest of your team. The more involved the team is in the creation of your visual problem-solving narrative, the more willing they will be to take ownership of the process and the more invested they will be in its outcome.

In Lucidchart, you can simply share the documents with the team members you want to be involved in the problem-solving process. It doesn’t matter where these people are located because Lucidchart documents can be accessed at any time from anywhere in the world.

Whatever method you decide to use to solve problems, work with Lucidchart to create the documents you need. Sign up for a free account today and start diagramming in minutes.

About Lucidchart

Lucidchart, a cloud-based intelligent diagramming application, is a core component of Lucid Software's Visual Collaboration Suite. This intuitive, cloud-based solution empowers teams to collaborate in real-time to build flowcharts, mockups, UML diagrams, customer journey maps, and more. Lucidchart propels teams forward to build the future faster. Lucid is proud to serve top businesses around the world, including customers such as Google, GE, and NBC Universal, and 99% of the Fortune 500. Lucid partners with industry leaders, including Google, Atlassian, and Microsoft. Since its founding, Lucid has received numerous awards for its products, business, and workplace culture. For more information, visit lucidchart.com.

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The Ultimate Guide to Teaching Math with Visual Models

What role do visual models play in the modern math classroom? Are they something that’s nice to have…if we have the time? Are they a distraction from the ‘real math’ that we should be teaching students? Or are they an essential ingredient in the development of conceptual understanding?

The Evidence for Visual Models

A clue to the value of visual models can be seen in the impact of  Singapore Math .  Beginning in the early 1980’s, Singapore, a single city South Asian nation, redesigned their math curriculum to emphasize the use of visual models and physical manipulatives.

By the 1990s, Singapore had emerged as an international juggernaut of math education. Standardized tests such as  TIMMS  (Trends in International Mathematics and Science Study) and  PISA  (Programme for International Student Assessment) consistently  showed Singapore at the top . (They’ve recently fallen to #2, behind China).

The  US , on the other hand,  currently ranks 37th  out of 78 tested nations. While we came in just ahead of countries like Belarus (38) and Kazakhstan (54), we placed well behind rivals like Russia (22nd) and China (1st).

A  2005  American Institute for Research  study  compared math instruction in Singapore and the US, finding that Singapore placed a greater emphasis on conceptual understanding. Later, the designers of  Common Core Math Standards  relied heavily on Singapore’s approach.

But now, many states are pulling back from  Common Core ; partly  due to parent complaints , and partly  due to flat test scores . You might ask, are visual models overrated?

No. The issues with  Common Core  are complex, and largely a result of  poorly written textbooks  and  outdated approaches to professional development .

But the research is clear. A  2014 study found  that students who use visual models were  six times as likely  to correctly solve word problems. And a  2012 study found  that students with learning disabilities were less likely to use visual representations. But when they were explicitly taught to use them, their performance improved substantially.

Need a crash course on bringing  visual models and manipulatives  to your math classroom? Registration is now open for our online  Elementary School  and  Middle School  visual models workshops.

How Visual Models Changed my Math Teaching

Halfway through my teaching career, I experienced the power of visual models first-hand, in my own classroom. I had just switched from teaching English to math. (Weird, I know).

Early in the year, my 5th graders took the diagnostic test at the beginning of the textbook. IT was meant to review what they’d learned in 4th grade, but they  bombed it . The highest grade was around 40%.

I tried charging forward with Chapter 1. But they were completely lost. Fortunately, a colleague introduced me to  Khan Academy , which was a lifesaver. It gave my students differentiated lessons and provided me with detailed insights into what each student needed.

I also noticed that  Khan  used a lot of visual models. In fact, I completed a lot of the lessons myself, just so I could learn about arrays, area models, and so on. I’d never learned these things growing up, so I needed to play the student before I could teach them.

The lessons involving visual models were really having an impact. And most of my students really enjoyed them. Soon, I began incorporating visual models into my own lessons We even did a  visual models video project .

But some of my students resisted using visual models. Interestingly, most of the resistance came from my “advanced students.” The “struggling learners” were thriving. Arrays and number lines were helping them fill in gaps they had accumulated over years of procedural math.

The visual models were really leveling the playing field. Students who were convinced they were “no good at math,” began to thrive. And those who were great at memorizing times tables and ‘stacking’ were pushed, for the first time, to demonstrate conceptual understanding.

Visual Models, Representations, and Conceptual Understanding

Visual models play an important role in conceptual math. Not only are they effective tools for developing conceptual understanding, but they are a great way to assess such understanding.

visual model problem solving

One reason is somewhat obvious. Visual models, when drawn to scale, make clear  what numbers and operations mean . If you model ‘357’ with  base-10 blocks , it’s clear that the 3 hundreds are  much bigger  than the 7 ones. And if you create a  4 x 8 array , it’s easy to see why 4 times 8 is 32.

But the vast majority of math class is spent in the  symbolic  realm. Students  memorize times tables , perform operations by stacking, and divide fractions with ‘keep, change, flip.’ These procedures allow them to perform calculations without understanding the numbers they’re manipulating or the operations they’re performing.

This surface understanding is problematic for several reasons. For one, students don’t remember what they’ve learned. The brain is designed to weed out disconnected bits of information. In order for students to retain what they learn, they must connect it to something  meaningful.

Another problem is application. Every math teacher knows the agony of the  word problem . “They understood the math. But they just can’t do word problems.” The issue is not ‘literacy.’ The issue is that students can only apply math to real-world scenarios when they  actually understand it.

the five representations of mathematical ideas: physical, visual, symbolic, contextual, and verbal

John Van de Walle addressed  The Five Representations of Mathematical Ideas  in his seminal book,  Teaching Student-Centered Mathematics.  We can take any mathematical idea and represent it physically, visually, symbolically, verbally, or contextually. He explains that when students can translate from one representation to another, “there is a better chance of a concept being formed correctly and integrated into a rich web of ideas.”

Connecting Visual Models to Math Standards

To get the most out of visual models, it’s important not to think of them as ‘one more thing we have to cover.’ Models are a tool for helping students understand the math you’re  already teaching . Instead of worrying that you ‘need to teach times tables  and  arrays,’ use arrays to demonstrate the  meaning of multiplication .

Unfortunately, it’s not always easy to figure out which models to use, or how to connect them with your grade level standards.

While the modern math standards in the  US  are a vast improvement over the pre-Common Core standards, they can sometimes be confusing. If you’ve spent any time reading the  CC Standards , or your state, you’ve probably noticed that quite a few skills can be wrapped up in a single standard.

4-D Math simplifies math standards by breaking their concepts into four dimensions: numbers, operations, connections, and representations

To  break standards down into manageable pieces , I use a framework called  4-D Math.  I started noticing that the concepts embedded in math standards could be organized into 4 main categories:

  • Numbers: A quantity or measurement – the ‘nouns’ of math
  • Operations: Performing actions on numbers – the ‘verbs’ of math
  • Connections: Associations between numbers, such as comparisons and patterns – the ‘conjunctions’ of math
  • Representations: How a math idea is represented (visual, symbolic, etc)

I realized that the standards were really just a long, complicated description of how these dimensions combined to form a mathematical concept. By separating them back into the 4-dimensions, I could easily understand individual standards, see connections between standards, and come up with  conceptual approaches  for teaching each standard, (including visual models).

A 4-D Math Breakdown 

For example, take a look at this 3rd grade standard from the  Operations and Algebraic Reasoning  domain:

3rd grade standard 3.OA.D.8

I think this is a perfectly reasonable expectation for a 3rd grader. But there is a lot wrapped up in here. If, instead, we break it down into the 4-dimensions…

4-D Breakdown of 3.OA.D.8

Imagine you had a student who was struggling with this standard. By breaking it down, we can assess each element independently.

A 4-D breakdown can also help us decide which visual models are the best fit for the standard we’re teaching.

Building Numeracy with Visual Models

What is a number?

The answer to this question is constantly changing as our students progress through their math journeys. And at every stage, visual models remain a powerful way to extend their understanding.

Numeracy in the Early Grades

Students’ first exposure to math is through counting. Any uniform object or image can be useful, but  counting discs  are widely-used to teach number names and  cardinality .

By first and second grades, students’ develop two critical numeracy concepts,  Base-10  and ‘number-as-length.’

Modeling the progression of whole numbers with counters, ten frames, base-10 blocks, and number lines

Ten Frames  help students to organize counters into tens, and begin to appreciate the importance of ‘10’ in our number system. Then, they extend their understanding into larger numbers, counting by tens, and so on, through the use of  Base-10 Blocks .

At this age, students are also exposed to the idea of  measurement.  Previously, numbers were understood to be an ordered list (1, 2, 3…) or a collection of objects (quantity). By measuring objects and introducing  number lines,  students begin to see  length  as a type of number.

Numeracy in Upper Elementary

In grades 3-5, students continue to develop their understanding of Base-10 and measurement. But what makes this period unique is the idea that a whole number can be broken into parts.

We begin by using  Fraction Circles   to introduce halves, thirds, and quarters. Then, we move to  Rectangular Fraction Models  to represent a wider range of denominators, as well as decimals.

Modeling the progression of fractions and decimals with fraction circles, area models, base-10 blocks, and number lines

Once we’ve established the concept of a decimal, we  revisit  Base-10 Blocks , this time to extend understanding of place value to smaller decimals, such as hundredths and thousandths. Finally, we connect fraction and decimal understanding to the concept of measurement by representing fractions and decimals on a number line.

Numeracy in Middle School

In grades 6-8, the concept of a number becomes more abstract, with the introduction of negative numbers and variables.

Modeling integers and variables with number lines, integer discs, and algebra tiles

A  number line  is usually the easiest  introduction to negative numbers . As students should already be familiar with number lines, we simply extend them to the left of the zero. (Or below the zero in the case of  vertical number lines ).

And while students should have some exposure to variables in elementary, it’s not until middle school that they become central to the curriculum. Variables can be tough to represent visually, as we don’t know their exact size. But  algebra tiles  do a good job of making the abstract concept a bit more concrete.

Teaching Operations with Visual Models

Just as a surgeon performs operations on her patients, operations are actions that we perform on numbers.

There are a few critical understandings that students should develop around operations. One is the basic definition of an operation, described above. And as they learn each new operation, they should connect it conceptually to those they’ve learned. For example, teach subtraction as the opposite (or inverse) of addition, and multiplication as a repetition of addition.

Just as students’ understanding of number gradually becomes more sophisticated, so does their understanding of operations. But while new types of numbers are introduced almost every year, the four basic operations remain the main focus after grade 3. The learning consists of applying these operations to the new types of numbers students encounter.

Visual models play an important role in the introduction of operations, their interconnectedness, and their application to different number types.

When modeling operations, it’s important to show the  action  being performed , not just the result. For example, a model showing  3/4  is not the same as a model of  1/2 plus 1/4 . When modeling operations, be sure to make clear the starting value, the action performed, and the result. This can be done by drawing the three stages, using color coding, arrows, and so on.

Strategies to visualize operations in visual models, including using colors, drawing arrows, or making models in stages

Modeling Operations in the Early Grades

Students in the early grades are first exposed to addition and subtraction, as an extension of counting and cardinality. They can count to 5, then count three more, or combine two groups of apples and count the total. Eventually, they will learn four meanings of addition: counting on, combining, extending (measurements), and comparing. They will also learn that each of these performed in reverse is subtraction.

As addition and subtraction are so closely related to counting, the same visual models used to introduce numbers are also used for these operations. Counters can be used alone or with  part-part-whole organizers  and  ten frames.  Number lines are a good tool for addition, but especially useful for subtraction, as students can learn to think of subtraction as the distance (difference) between two numbers. This leads to ‘counting up’ as a mental math strategy for subtraction.

Modeling Operations in Upper Elementary

Operations become significantly more complex in grade 3, as students encounter multiplication and division. Eventually, they will learn all  Five Meanings of Multiplication  (and division),  but we typically start with repeated addition or equal groups.

Then, they  create arrays,  multiplying by arranging objects into rows and columns. This prepares them for area: rows and columns become lengths and widths, and counting objects becomes the measurement of a two-dimensional space.   Area models  can be used to reinforce the concept of area (grade 3), to multiply multi-digit numbers (grades 3-4), and to  multiplying fractions   (grades 4-5).

Modeling the progression of multiplication and multiplicative thinking with equal groups, arrays, area models, bar models, and double number lines

By grade 5, students extend their understanding of 2-D measurement (area) to 3-D measurement (volume), which can be modeled with unit cubes and  3-dimensional arrays .

Since division is the opposite (inverse) of multiplication, each of these modeling strategies can also be used for division. When modeling multiplication, we give students two factors, and they find the product. To model division, we provide the product (dividend) and one factor (divisor), and they find the other factor (quotient).

Modeling Operations in Middle School

By middle school, students have learned all 4 operations with whole numbers, fractions, and decimals, with one exception: dividing a fraction by a fraction. Fraction division can be represented with bar models, which will also be useful for a number of other middle school concepts.

For the most part, though, the focus will be on applying the four operations to negative numbers and variables. As the math concepts get more abstract, it takes some creativity to represent them with visual and physical models.  Creating zero pairs with integer discs  is a clever way to represent addition and subtraction of integers. And number lines are also effective for adding and subtracting integers and rational numbers.

To multiply integers, I create arrays with integer discs. First, I model one factor (yellow for positive, red for negative). If the second factor is negative, I’ll flip all the chips over. But for this type of modeling to be effective, students need to have a solid foundation in modeling concrete numbers and operations.

Algebra Tiles  are helpful for modeling operations with variables, especially when  combining like terms .

Modeling exponents (2^5) with arrays

Students are also exposed to two new operations in middle school: exponents and roots. The trick to visualizing exponents is modeling repeated multiplication. A square can simply be an array or actual square, and cubes can be shown with  unit cubes  or  3-dimensional arrays.  To show 2 5 , start with a 2 x 2 array (2 2 ), then stack another on top (2 3 ), and keep iterating until you’ve doubled 5 times (2 5 ).

Other exponents become more complex, so again, you’ll need to get creative. For negative exponents, you could work in two stages: first, model without the negative to find the denominator, then create a simple fraction model.

For simple square roots (fractional exponents), count out the radicand in discs. Then, try to create a square array. The number of rows or columns will be the root. You can also use this  Desmos activity  for modeling square roots.

Teaching Connections with Visual Models

One aspect of math instruction that is often overlooked is the importance of  connections . It’s easy to see the importance of numbers in math, as well as the operations we perform on those numbers. But most math in the real world involves numbers interacting with each other.

The clearest example is the concept of  equality.  Students in first grade are supposed to learn the concept of  equality as balance.  Yet many students make it to middle school and beyond thinking that the equal sign means “the answer goes here.” The National Council of Teachers of Mathematics ( NCTM) free online balance tool  has a that can help students visualize the meaning of equality. This idea also extends to the concept of inequality – just as equality can be visualized with a balanced scale, inequality can be shown as an unbalanced scale.

The progression of patterns and functions can be modeled with repeating shape patterns, growing shapes, bar graphs, line plots, and the coordinate plane.

Another important progression is that of  patterns and functions . Patterns are one of the first math concepts students encounter – often before they are even exposed to numbers. Eventually, the pattern concept evolves to the idea of a function: a connection between an independent and dependent variable.

The coordinate plane (x-y graph) is the current standard for representing functions visually. But though it is a representational (scale) model, many students work in the coordinate plane without actually understanding the meaning of what they are showing.

To develop conceptual their conceptual understanding, students should learn to  represent functions in multiple forms:  visually, symbolically, contextually, and so on.  Growing Shapes   are particularly useful for modeling the progression from pattern to function.

Proportions are another important connection, which also include aspects of operations (multiplicative comparison). Proportional reasoning plays a role in many middle school standards, including unit conversions, percentages, and rates. Proportions can be visualized using bar models, double number lines, and the coordinate plane.

Bringing Visual Models to Your Classroom

Visual models are one of the most powerful tools for increase student engagement and achievement. In this post, I’ve tried to give you the foundations you’ll need to make visual models a staple of your math classroom.

Of course, there’s only so much you can learn from a single blog post. So if you found this helpful, you’ll want to register for one of our upcoming workshops. In these hands-on, interactive workshops, you’ll collaborate with fellow teachers and come away with the strategies and resources you need to bring the learning back to your classroom.

  • Teaching with Visual Models in Grades 1-5
  • Teaching with Visual Models in Middle School

You’ll also find visual models activities for almost any math standard in our online store .

Finally, don’t forget to sign up for our  Educator’s Newsletter . You’ll receive our latest posts in your inbox each week, and you’ll be the first to know about new products, discounts, and special events.

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About the Author

Jeff Lisciandrello is the founder of Room to Discover and an educational consultant specializing in student-centered learning practices

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March 18, 2021, by Rupert Knight

Using visual models to solve problems and explore relationships in Mathematics: beyond concrete, pictorial, abstract – Part 1

This two-part blog series by Marc North explores some thinking and strategies for using representations in Mathematics lessons. Part 1 unpicks some of the key theoretical ideas around the use of representations and models and foregrounds how representations can be used to both solve problems and explore mathematical relationships. Part 2 will illustrate these theoretical ideas practically via a classroom based Maths activity.

Our brains don’t like abstract ideas!

There is little doubt that visual models are a key part of the learning and teaching of mathematics. One of the reasons for this is that while much of school mathematics involves abstract concepts that can be generalized across a range of topics and problems, our brains actually don’t like abstract ideas! Instead, many of us prefer to learn and think through concrete examples and deliberately look for concrete and practical examples to help to explain abstract concepts. See, for example, the Learning Scientists site .

Visual models provide a useful tool for ‘concretising’ complex and abstract ideas. The human brain responds positively to information packaged in creative and visual ways, which is why throughout our daily lives we are constantly bombarded with visual imagery and stimuli. It is often easier to remember information presented in picture form than as a string of words, and visual models provide succinct and organised summaries of information. Visual models can also demonstrate relationships between different items and, when shown dynamically, can show how those relationships change and evolve.

Multiple Representations in Teaching for Mastery

The emphasis on a ‘Teaching for Mastery’ approach in both Primary and Secondary schools has ushered in a clear priority for using representations to model and illustrate mathematical ideas and problems. As indicated in Figure 1, the ‘Big Idea’ of representation and structure provides students with access to mathematical concepts and supports them to visualise patterns and make connections both within and between concepts.

visual model problem solving

In the Teaching for Mastery approach, the Concrete-Pictorial-Abstract (CPA) framework provides the main approach that structures how teachers are encouraged to work with different representations. The CPA approach is a multi-sensory teaching model that introduces abstract concepts in a concrete and tangible way, by moving from concrete materials, to pictorial representations, to abstract symbols and problems.

visual model problem solving

Figure 2: Concrete-Pictorial-Abstract approach (Caroll, Pikul, Foust & Grodziak)

The Concrete dimension is the ‘doing’ stage, where children use physical resources (e.g. manipulatives) to model problems. This stage can also involve the use of concrete situations that are linked to real-life contexts. The Pictorial dimension is the ‘seeing’ stage, where pictures are used to model the concrete resources and the problems. The ‘abstract’ stage is the ‘symbolic’ stage, where abstract mathematical symbols are used to model problems. Many teachers have adopted this approach enthusiastically, operationalizing this practice in various ways:

visual model problem solving

Origins of the CPA approach – Bruner’s Representation Modes

The theoretical origins of this approach stem from Jerome Bruner’s wor k on different representation modes. As a social-constructivist, Bruner argues that children’s problem-solving skills are developed through inquiry and discovery, and also that to support deep learning subject matter should be represented and experienced by children in terms of how they will view and experience the world. This is facilitated by using different representation modes that model the stages of our learning and that reflect the ways in which humans store and encode knowledge and information in memory. The enactive stage (‘based on action’) (from birth to one year old) involves the encoding and storage of information through direct manipulation of objects – for example, think of a baby playing with a rattle. At this early stage, there is no clear internal representation of the object by the individual. The iconic stage (‘based on images’) (from one to six years old) involves an internal representation of external objects visually in the form of a mental image or icon – for example, a child being able to draw a picture of a tree without actually having a tree in front of them. The symbolic stage (‘based on language’) (seven years and up) is when information is able to be stored in the form of a code or symbol – for example, being able to describe a tree in words or through writing.

Potential challenges with the CPA approach

Although the CPA approach is based on Bruner’s work, there are some important distinctions – which also give rise to some potential challenges with this approach.

First, the CPA framework has adopted a theory that describes children’s development and learning through various age ranges into a sequence for instruction for children at all ages. This has resulted in some teachers using this approach in a strictly hierarchical way, always starting with the concrete and progressing to the abstract – and treating the abstract as the ultimate goal of the learning experience in Mathematics. Those students who are not able to demonstrate mastery of the abstract are then deemed to have a lower level of understanding (or no understanding), despite potentially still being able to demonstrate deep understanding wile engaging with concrete and pictorial representations. Some also use the CPA model as a differentiation tool, with lower-attaining students presented with tasks containing mainly concrete and pictorial representations and higher-attaining students encouraged to engage more quickly with abstract elements. Although Bruner’s representation modes are hierarchical in the sense that they map out children’s learning stages through various age ranges, by age 7 years the expectation is that ALL children are capable of creating and storing knowledge at a symbolic level. As such, learning experiences should offer all children opportunities to experience their learning through actions, images, and more formal symbolic means. The sequence in which different representations of knowledge are explored should be determined by the sequence that will lead to the most in-depth understanding of a concept. This could mean working symbolically first, then drawing a picture, then working concretely (e.g. by building a model), or engaging backwards and forwards with each representation mode concurrently while developing and refining understanding – which is precisely what architects and engineers do.

Second, it is problematic to associate the ‘abstract’ stage in the CPA framework exclusively with abstract knowledge and to think that it is only through engagement with formal mathematical symbols and calculations that abstract knowledge is developed. Abstract mathematical structures can also be engaged and represented through enacted activities and pictures or icons. For example, uni-fix cubes are commonly treated as an abstract representation of a concrete object (like apples). Similarly, a bar model inherently contains a degree of abstractness because it shows a standardized or generic model of a unique scenario.

visual model problem solving

There is a risk, then, that some teachers may not recognise how much abstractness their teaching and resources contain and may wonder why some students continue to be confused despite access to different representations. Encountering formal mathematical structures in enactive or iconic forms does not automatically reduce the degree of abstractness; rather, it merely presents these structures via a medium other than symbols, notation and language.

Although there are numerous classroom resources available that use the CPA approach (for example, see here ), what is less common are resources that help teachers understand which models are the most effective for illustrating certain concepts, why this is, how to build links between different models to support deep relational understanding, and how to use models to compare and contrast different methods. So, while some teachers are using a large number of different representations, they are not always able to give students insight into the decisions behind why specific models are prioritised over others, which makes it difficult for students to know which models to choose when working independently. Also, while models are most commonly used to describe problems and then aid with the solving of those problems, less common is the use of models to compare and contrast different ways of working and to explore mathematical relationships and structures.

It seems important to consider two agenda:

1. the importance of deliberateness when choosing and using models; 2. and, using models to compare and contrast different ways of working and to explore mathematical relationships and structures.

The discussion below draws out some key ideas that have framed these agendas.

Different purposes for mathematics models

While the CPA and Enactive-Iconic-Symbolic frameworks set out different types of models and representations, it is also helpful to think about the different purposes that these can serve. The Realistic Mathematics Education (RME) approach provides some useful thinking around this. RME, as explained here , was developed in the Netherlands as a specific approach to the teaching of subject Mathematics. This approach has also been used in the United Kingdom with GCSE-resit Mathematics students, as shown here . Three key features of RME are helpful for this discussion:

1. Use of realistic contexts 2. Different purposes for models 3. The ‘progressive formalisation of models’ principle

The first key feature is engagement with abstract mathematics contents in realistic contexts (1), where ‘realistic’ refers to contexts that students can imagine and relate to. The contexts provide an anchor in which to ground understanding of abstract contents, a reference point for structuring thinking about abstract ideas. In part, this reflects some similarity with the ‘concrete’ dimension of the CPA framework.

A second feature of RME is different purposes for working with models (2). ‘Models of’ mathematics are models developed to represent a scenario or problem, with the model bearing a close connection to the problem situation at hand – for example, using a picture of a pizza to represent a fraction of a whole. When (or if) these models are developed and generalised to represent, describe and investigate mathematical structures and relationships over a range of problem situations and even content topics, the model becomes a ‘Model for’ exploring and understanding mathematics. Arrays, bar models and double number-lines are example of models that can be used in this way to describe and investigate mathematical methods, structures and relationships across a range of problem types and situations. ‘Models for’ are powerful precisely because they allow us to investigate mathematical relationships and explore different ways of working.

From a RME perspective, when using representations, it is essential to choose models and representations that can easily be developed from models of a specific local situation to models for describing more general and abstract relationships. This progressive formalisation of models (3) helps students navigate a learning trajectory to abstract concepts and equips them with a small number of models that have applicability over a range of problem and content types.

visual model problem solving

A key distinction here with the CPA approach is that, from a RME perspective, there is no expectation for students to work through a hierarchy from concrete experiences and pictures to symbolic representations. Instead, the move is from experiences and representations that are bound to local situations towards experiences and representations that are generalizable across a range of situations. The focus is less on the format of the representation and more on how the representation can be molded and developed to explore mathematical relationship and structure.

What does this mean for classroom practice?

The discussion above has attempted to highlight the importance of thinking about both the formats of the representations we use in our teaching AND the purpose of those representations. Using different types of representations that are blended into a deliberate sequence (like the CPA sequence) is helpful for supporting students to think about mathematical concepts from different perspectives – like different pieces of a puzzle, with each piece giving some unique information about the whole picture. However it is also important to think about what we use representation for so that students don’t believe that the only purpose for different representations is to solve problems. A much richer understanding is that representations, when carefully chosen, allow us to explore mathematical relationships, to see connections between mathematical concepts, and – in so doing – to develop a deeper understanding.

Looking ahead to part 2

Part 2 illustrates these theoretical ideas practically via a classroom based Maths activity that explores the relationship between different methods for solving a problem involving a conversion rate (from miles to kilometers). Without giving away too many clues, the key questions that Part 2 explores are:

How are each of these different methods linked, what’s the same and what’s different about them, are there methods that haven’t been considered yet, and what is the most effective ‘model for’ exploring the similarities and differences between them?

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I found this a really helpful analysis Dr North. Thank you very much.

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35 problem-solving techniques and methods for solving complex problems

Problem solving workshop

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All teams and organizations encounter challenges as they grow. There are problems that might occur for teams when it comes to miscommunication or resolving business-critical issues . You may face challenges around growth , design , user engagement, and even team culture and happiness. In short, problem-solving techniques should be part of every team’s skillset.

Problem-solving methods are primarily designed to help a group or team through a process of first identifying problems and challenges , ideating possible solutions , and then evaluating the most suitable .

Finding effective solutions to complex problems isn’t easy, but by using the right process and techniques, you can help your team be more efficient in the process.

So how do you develop strategies that are engaging, and empower your team to solve problems effectively?

In this blog post, we share a series of problem-solving tools you can use in your next workshop or team meeting. You’ll also find some tips for facilitating the process and how to enable others to solve complex problems.

Let’s get started! 

How do you identify problems?

How do you identify the right solution.

  • Tips for more effective problem-solving

Complete problem-solving methods

  • Problem-solving techniques to identify and analyze problems
  • Problem-solving techniques for developing solutions

Problem-solving warm-up activities

Closing activities for a problem-solving process.

Before you can move towards finding the right solution for a given problem, you first need to identify and define the problem you wish to solve. 

Here, you want to clearly articulate what the problem is and allow your group to do the same. Remember that everyone in a group is likely to have differing perspectives and alignment is necessary in order to help the group move forward. 

Identifying a problem accurately also requires that all members of a group are able to contribute their views in an open and safe manner. It can be scary for people to stand up and contribute, especially if the problems or challenges are emotive or personal in nature. Be sure to try and create a psychologically safe space for these kinds of discussions.

Remember that problem analysis and further discussion are also important. Not taking the time to fully analyze and discuss a challenge can result in the development of solutions that are not fit for purpose or do not address the underlying issue.

Successfully identifying and then analyzing a problem means facilitating a group through activities designed to help them clearly and honestly articulate their thoughts and produce usable insight.

With this data, you might then produce a problem statement that clearly describes the problem you wish to be addressed and also state the goal of any process you undertake to tackle this issue.  

Finding solutions is the end goal of any process. Complex organizational challenges can only be solved with an appropriate solution but discovering them requires using the right problem-solving tool.

After you’ve explored a problem and discussed ideas, you need to help a team discuss and choose the right solution. Consensus tools and methods such as those below help a group explore possible solutions before then voting for the best. They’re a great way to tap into the collective intelligence of the group for great results!

Remember that the process is often iterative. Great problem solvers often roadtest a viable solution in a measured way to see what works too. While you might not get the right solution on your first try, the methods below help teams land on the most likely to succeed solution while also holding space for improvement.

Every effective problem solving process begins with an agenda . A well-structured workshop is one of the best methods for successfully guiding a group from exploring a problem to implementing a solution.

In SessionLab, it’s easy to go from an idea to a complete agenda . Start by dragging and dropping your core problem solving activities into place . Add timings, breaks and necessary materials before sharing your agenda with your colleagues.

The resulting agenda will be your guide to an effective and productive problem solving session that will also help you stay organized on the day!

visual model problem solving

Tips for more effective problem solving

Problem-solving activities are only one part of the puzzle. While a great method can help unlock your team’s ability to solve problems, without a thoughtful approach and strong facilitation the solutions may not be fit for purpose.

Let’s take a look at some problem-solving tips you can apply to any process to help it be a success!

Clearly define the problem

Jumping straight to solutions can be tempting, though without first clearly articulating a problem, the solution might not be the right one. Many of the problem-solving activities below include sections where the problem is explored and clearly defined before moving on.

This is a vital part of the problem-solving process and taking the time to fully define an issue can save time and effort later. A clear definition helps identify irrelevant information and it also ensures that your team sets off on the right track.

Don’t jump to conclusions

It’s easy for groups to exhibit cognitive bias or have preconceived ideas about both problems and potential solutions. Be sure to back up any problem statements or potential solutions with facts, research, and adequate forethought.

The best techniques ask participants to be methodical and challenge preconceived notions. Make sure you give the group enough time and space to collect relevant information and consider the problem in a new way. By approaching the process with a clear, rational mindset, you’ll often find that better solutions are more forthcoming.  

Try different approaches  

Problems come in all shapes and sizes and so too should the methods you use to solve them. If you find that one approach isn’t yielding results and your team isn’t finding different solutions, try mixing it up. You’ll be surprised at how using a new creative activity can unblock your team and generate great solutions.

Don’t take it personally 

Depending on the nature of your team or organizational problems, it’s easy for conversations to get heated. While it’s good for participants to be engaged in the discussions, ensure that emotions don’t run too high and that blame isn’t thrown around while finding solutions.

You’re all in it together, and even if your team or area is seeing problems, that isn’t necessarily a disparagement of you personally. Using facilitation skills to manage group dynamics is one effective method of helping conversations be more constructive.

Get the right people in the room

Your problem-solving method is often only as effective as the group using it. Getting the right people on the job and managing the number of people present is important too!

If the group is too small, you may not get enough different perspectives to effectively solve a problem. If the group is too large, you can go round and round during the ideation stages.

Creating the right group makeup is also important in ensuring you have the necessary expertise and skillset to both identify and follow up on potential solutions. Carefully consider who to include at each stage to help ensure your problem-solving method is followed and positioned for success.

Document everything

The best solutions can take refinement, iteration, and reflection to come out. Get into a habit of documenting your process in order to keep all the learnings from the session and to allow ideas to mature and develop. Many of the methods below involve the creation of documents or shared resources. Be sure to keep and share these so everyone can benefit from the work done!

Bring a facilitator 

Facilitation is all about making group processes easier. With a subject as potentially emotive and important as problem-solving, having an impartial third party in the form of a facilitator can make all the difference in finding great solutions and keeping the process moving. Consider bringing a facilitator to your problem-solving session to get better results and generate meaningful solutions!

Develop your problem-solving skills

It takes time and practice to be an effective problem solver. While some roles or participants might more naturally gravitate towards problem-solving, it can take development and planning to help everyone create better solutions.

You might develop a training program, run a problem-solving workshop or simply ask your team to practice using the techniques below. Check out our post on problem-solving skills to see how you and your group can develop the right mental process and be more resilient to issues too!

Design a great agenda

Workshops are a great format for solving problems. With the right approach, you can focus a group and help them find the solutions to their own problems. But designing a process can be time-consuming and finding the right activities can be difficult.

Check out our workshop planning guide to level-up your agenda design and start running more effective workshops. Need inspiration? Check out templates designed by expert facilitators to help you kickstart your process!

In this section, we’ll look at in-depth problem-solving methods that provide a complete end-to-end process for developing effective solutions. These will help guide your team from the discovery and definition of a problem through to delivering the right solution.

If you’re looking for an all-encompassing method or problem-solving model, these processes are a great place to start. They’ll ask your team to challenge preconceived ideas and adopt a mindset for solving problems more effectively.

  • Six Thinking Hats
  • Lightning Decision Jam
  • Problem Definition Process
  • Discovery & Action Dialogue
Design Sprint 2.0
  • Open Space Technology

1. Six Thinking Hats

Individual approaches to solving a problem can be very different based on what team or role an individual holds. It can be easy for existing biases or perspectives to find their way into the mix, or for internal politics to direct a conversation.

Six Thinking Hats is a classic method for identifying the problems that need to be solved and enables your team to consider them from different angles, whether that is by focusing on facts and data, creative solutions, or by considering why a particular solution might not work.

Like all problem-solving frameworks, Six Thinking Hats is effective at helping teams remove roadblocks from a conversation or discussion and come to terms with all the aspects necessary to solve complex problems.

2. Lightning Decision Jam

Featured courtesy of Jonathan Courtney of AJ&Smart Berlin, Lightning Decision Jam is one of those strategies that should be in every facilitation toolbox. Exploring problems and finding solutions is often creative in nature, though as with any creative process, there is the potential to lose focus and get lost.

Unstructured discussions might get you there in the end, but it’s much more effective to use a method that creates a clear process and team focus.

In Lightning Decision Jam, participants are invited to begin by writing challenges, concerns, or mistakes on post-its without discussing them before then being invited by the moderator to present them to the group.

From there, the team vote on which problems to solve and are guided through steps that will allow them to reframe those problems, create solutions and then decide what to execute on. 

By deciding the problems that need to be solved as a team before moving on, this group process is great for ensuring the whole team is aligned and can take ownership over the next stages. 

Lightning Decision Jam (LDJ)   #action   #decision making   #problem solving   #issue analysis   #innovation   #design   #remote-friendly   The problem with anything that requires creative thinking is that it’s easy to get lost—lose focus and fall into the trap of having useless, open-ended, unstructured discussions. Here’s the most effective solution I’ve found: Replace all open, unstructured discussion with a clear process. What to use this exercise for: Anything which requires a group of people to make decisions, solve problems or discuss challenges. It’s always good to frame an LDJ session with a broad topic, here are some examples: The conversion flow of our checkout Our internal design process How we organise events Keeping up with our competition Improving sales flow

3. Problem Definition Process

While problems can be complex, the problem-solving methods you use to identify and solve those problems can often be simple in design. 

By taking the time to truly identify and define a problem before asking the group to reframe the challenge as an opportunity, this method is a great way to enable change.

Begin by identifying a focus question and exploring the ways in which it manifests before splitting into five teams who will each consider the problem using a different method: escape, reversal, exaggeration, distortion or wishful. Teams develop a problem objective and create ideas in line with their method before then feeding them back to the group.

This method is great for enabling in-depth discussions while also creating space for finding creative solutions too!

Problem Definition   #problem solving   #idea generation   #creativity   #online   #remote-friendly   A problem solving technique to define a problem, challenge or opportunity and to generate ideas.

4. The 5 Whys 

Sometimes, a group needs to go further with their strategies and analyze the root cause at the heart of organizational issues. An RCA or root cause analysis is the process of identifying what is at the heart of business problems or recurring challenges. 

The 5 Whys is a simple and effective method of helping a group go find the root cause of any problem or challenge and conduct analysis that will deliver results. 

By beginning with the creation of a problem statement and going through five stages to refine it, The 5 Whys provides everything you need to truly discover the cause of an issue.

The 5 Whys   #hyperisland   #innovation   This simple and powerful method is useful for getting to the core of a problem or challenge. As the title suggests, the group defines a problems, then asks the question “why” five times, often using the resulting explanation as a starting point for creative problem solving.

5. World Cafe

World Cafe is a simple but powerful facilitation technique to help bigger groups to focus their energy and attention on solving complex problems.

World Cafe enables this approach by creating a relaxed atmosphere where participants are able to self-organize and explore topics relevant and important to them which are themed around a central problem-solving purpose. Create the right atmosphere by modeling your space after a cafe and after guiding the group through the method, let them take the lead!

Making problem-solving a part of your organization’s culture in the long term can be a difficult undertaking. More approachable formats like World Cafe can be especially effective in bringing people unfamiliar with workshops into the fold. 

World Cafe   #hyperisland   #innovation   #issue analysis   World Café is a simple yet powerful method, originated by Juanita Brown, for enabling meaningful conversations driven completely by participants and the topics that are relevant and important to them. Facilitators create a cafe-style space and provide simple guidelines. Participants then self-organize and explore a set of relevant topics or questions for conversation.

6. Discovery & Action Dialogue (DAD)

One of the best approaches is to create a safe space for a group to share and discover practices and behaviors that can help them find their own solutions.

With DAD, you can help a group choose which problems they wish to solve and which approaches they will take to do so. It’s great at helping remove resistance to change and can help get buy-in at every level too!

This process of enabling frontline ownership is great in ensuring follow-through and is one of the methods you will want in your toolbox as a facilitator.

Discovery & Action Dialogue (DAD)   #idea generation   #liberating structures   #action   #issue analysis   #remote-friendly   DADs make it easy for a group or community to discover practices and behaviors that enable some individuals (without access to special resources and facing the same constraints) to find better solutions than their peers to common problems. These are called positive deviant (PD) behaviors and practices. DADs make it possible for people in the group, unit, or community to discover by themselves these PD practices. DADs also create favorable conditions for stimulating participants’ creativity in spaces where they can feel safe to invent new and more effective practices. Resistance to change evaporates as participants are unleashed to choose freely which practices they will adopt or try and which problems they will tackle. DADs make it possible to achieve frontline ownership of solutions.

7. Design Sprint 2.0

Want to see how a team can solve big problems and move forward with prototyping and testing solutions in a few days? The Design Sprint 2.0 template from Jake Knapp, author of Sprint, is a complete agenda for a with proven results.

Developing the right agenda can involve difficult but necessary planning. Ensuring all the correct steps are followed can also be stressful or time-consuming depending on your level of experience.

Use this complete 4-day workshop template if you are finding there is no obvious solution to your challenge and want to focus your team around a specific problem that might require a shortcut to launching a minimum viable product or waiting for the organization-wide implementation of a solution.

8. Open space technology

Open space technology- developed by Harrison Owen – creates a space where large groups are invited to take ownership of their problem solving and lead individual sessions. Open space technology is a great format when you have a great deal of expertise and insight in the room and want to allow for different takes and approaches on a particular theme or problem you need to be solved.

Start by bringing your participants together to align around a central theme and focus their efforts. Explain the ground rules to help guide the problem-solving process and then invite members to identify any issue connecting to the central theme that they are interested in and are prepared to take responsibility for.

Once participants have decided on their approach to the core theme, they write their issue on a piece of paper, announce it to the group, pick a session time and place, and post the paper on the wall. As the wall fills up with sessions, the group is then invited to join the sessions that interest them the most and which they can contribute to, then you’re ready to begin!

Everyone joins the problem-solving group they’ve signed up to, record the discussion and if appropriate, findings can then be shared with the rest of the group afterward.

Open Space Technology   #action plan   #idea generation   #problem solving   #issue analysis   #large group   #online   #remote-friendly   Open Space is a methodology for large groups to create their agenda discerning important topics for discussion, suitable for conferences, community gatherings and whole system facilitation

Techniques to identify and analyze problems

Using a problem-solving method to help a team identify and analyze a problem can be a quick and effective addition to any workshop or meeting.

While further actions are always necessary, you can generate momentum and alignment easily, and these activities are a great place to get started.

We’ve put together this list of techniques to help you and your team with problem identification, analysis, and discussion that sets the foundation for developing effective solutions.

Let’s take a look!

  • The Creativity Dice
  • Fishbone Analysis
  • Problem Tree
  • SWOT Analysis
  • Agreement-Certainty Matrix
  • The Journalistic Six
  • LEGO Challenge
  • What, So What, Now What?
  • Journalists

Individual and group perspectives are incredibly important, but what happens if people are set in their minds and need a change of perspective in order to approach a problem more effectively?

Flip It is a method we love because it is both simple to understand and run, and allows groups to understand how their perspectives and biases are formed. 

Participants in Flip It are first invited to consider concerns, issues, or problems from a perspective of fear and write them on a flip chart. Then, the group is asked to consider those same issues from a perspective of hope and flip their understanding.  

No problem and solution is free from existing bias and by changing perspectives with Flip It, you can then develop a problem solving model quickly and effectively.

Flip It!   #gamestorming   #problem solving   #action   Often, a change in a problem or situation comes simply from a change in our perspectives. Flip It! is a quick game designed to show players that perspectives are made, not born.

10. The Creativity Dice

One of the most useful problem solving skills you can teach your team is of approaching challenges with creativity, flexibility, and openness. Games like The Creativity Dice allow teams to overcome the potential hurdle of too much linear thinking and approach the process with a sense of fun and speed. 

In The Creativity Dice, participants are organized around a topic and roll a dice to determine what they will work on for a period of 3 minutes at a time. They might roll a 3 and work on investigating factual information on the chosen topic. They might roll a 1 and work on identifying the specific goals, standards, or criteria for the session.

Encouraging rapid work and iteration while asking participants to be flexible are great skills to cultivate. Having a stage for idea incubation in this game is also important. Moments of pause can help ensure the ideas that are put forward are the most suitable. 

The Creativity Dice   #creativity   #problem solving   #thiagi   #issue analysis   Too much linear thinking is hazardous to creative problem solving. To be creative, you should approach the problem (or the opportunity) from different points of view. You should leave a thought hanging in mid-air and move to another. This skipping around prevents premature closure and lets your brain incubate one line of thought while you consciously pursue another.

11. Fishbone Analysis

Organizational or team challenges are rarely simple, and it’s important to remember that one problem can be an indication of something that goes deeper and may require further consideration to be solved.

Fishbone Analysis helps groups to dig deeper and understand the origins of a problem. It’s a great example of a root cause analysis method that is simple for everyone on a team to get their head around. 

Participants in this activity are asked to annotate a diagram of a fish, first adding the problem or issue to be worked on at the head of a fish before then brainstorming the root causes of the problem and adding them as bones on the fish. 

Using abstractions such as a diagram of a fish can really help a team break out of their regular thinking and develop a creative approach.

Fishbone Analysis   #problem solving   ##root cause analysis   #decision making   #online facilitation   A process to help identify and understand the origins of problems, issues or observations.

12. Problem Tree 

Encouraging visual thinking can be an essential part of many strategies. By simply reframing and clarifying problems, a group can move towards developing a problem solving model that works for them. 

In Problem Tree, groups are asked to first brainstorm a list of problems – these can be design problems, team problems or larger business problems – and then organize them into a hierarchy. The hierarchy could be from most important to least important or abstract to practical, though the key thing with problem solving games that involve this aspect is that your group has some way of managing and sorting all the issues that are raised.

Once you have a list of problems that need to be solved and have organized them accordingly, you’re then well-positioned for the next problem solving steps.

Problem tree   #define intentions   #create   #design   #issue analysis   A problem tree is a tool to clarify the hierarchy of problems addressed by the team within a design project; it represents high level problems or related sublevel problems.

13. SWOT Analysis

Chances are you’ve heard of the SWOT Analysis before. This problem-solving method focuses on identifying strengths, weaknesses, opportunities, and threats is a tried and tested method for both individuals and teams.

Start by creating a desired end state or outcome and bare this in mind – any process solving model is made more effective by knowing what you are moving towards. Create a quadrant made up of the four categories of a SWOT analysis and ask participants to generate ideas based on each of those quadrants.

Once you have those ideas assembled in their quadrants, cluster them together based on their affinity with other ideas. These clusters are then used to facilitate group conversations and move things forward. 

SWOT analysis   #gamestorming   #problem solving   #action   #meeting facilitation   The SWOT Analysis is a long-standing technique of looking at what we have, with respect to the desired end state, as well as what we could improve on. It gives us an opportunity to gauge approaching opportunities and dangers, and assess the seriousness of the conditions that affect our future. When we understand those conditions, we can influence what comes next.

14. Agreement-Certainty Matrix

Not every problem-solving approach is right for every challenge, and deciding on the right method for the challenge at hand is a key part of being an effective team.

The Agreement Certainty matrix helps teams align on the nature of the challenges facing them. By sorting problems from simple to chaotic, your team can understand what methods are suitable for each problem and what they can do to ensure effective results. 

If you are already using Liberating Structures techniques as part of your problem-solving strategy, the Agreement-Certainty Matrix can be an invaluable addition to your process. We’ve found it particularly if you are having issues with recurring problems in your organization and want to go deeper in understanding the root cause. 

Agreement-Certainty Matrix   #issue analysis   #liberating structures   #problem solving   You can help individuals or groups avoid the frequent mistake of trying to solve a problem with methods that are not adapted to the nature of their challenge. The combination of two questions makes it possible to easily sort challenges into four categories: simple, complicated, complex , and chaotic .  A problem is simple when it can be solved reliably with practices that are easy to duplicate.  It is complicated when experts are required to devise a sophisticated solution that will yield the desired results predictably.  A problem is complex when there are several valid ways to proceed but outcomes are not predictable in detail.  Chaotic is when the context is too turbulent to identify a path forward.  A loose analogy may be used to describe these differences: simple is like following a recipe, complicated like sending a rocket to the moon, complex like raising a child, and chaotic is like the game “Pin the Tail on the Donkey.”  The Liberating Structures Matching Matrix in Chapter 5 can be used as the first step to clarify the nature of a challenge and avoid the mismatches between problems and solutions that are frequently at the root of chronic, recurring problems.

Organizing and charting a team’s progress can be important in ensuring its success. SQUID (Sequential Question and Insight Diagram) is a great model that allows a team to effectively switch between giving questions and answers and develop the skills they need to stay on track throughout the process. 

Begin with two different colored sticky notes – one for questions and one for answers – and with your central topic (the head of the squid) on the board. Ask the group to first come up with a series of questions connected to their best guess of how to approach the topic. Ask the group to come up with answers to those questions, fix them to the board and connect them with a line. After some discussion, go back to question mode by responding to the generated answers or other points on the board.

It’s rewarding to see a diagram grow throughout the exercise, and a completed SQUID can provide a visual resource for future effort and as an example for other teams.

SQUID   #gamestorming   #project planning   #issue analysis   #problem solving   When exploring an information space, it’s important for a group to know where they are at any given time. By using SQUID, a group charts out the territory as they go and can navigate accordingly. SQUID stands for Sequential Question and Insight Diagram.

16. Speed Boat

To continue with our nautical theme, Speed Boat is a short and sweet activity that can help a team quickly identify what employees, clients or service users might have a problem with and analyze what might be standing in the way of achieving a solution.

Methods that allow for a group to make observations, have insights and obtain those eureka moments quickly are invaluable when trying to solve complex problems.

In Speed Boat, the approach is to first consider what anchors and challenges might be holding an organization (or boat) back. Bonus points if you are able to identify any sharks in the water and develop ideas that can also deal with competitors!   

Speed Boat   #gamestorming   #problem solving   #action   Speedboat is a short and sweet way to identify what your employees or clients don’t like about your product/service or what’s standing in the way of a desired goal.

17. The Journalistic Six

Some of the most effective ways of solving problems is by encouraging teams to be more inclusive and diverse in their thinking.

Based on the six key questions journalism students are taught to answer in articles and news stories, The Journalistic Six helps create teams to see the whole picture. By using who, what, when, where, why, and how to facilitate the conversation and encourage creative thinking, your team can make sure that the problem identification and problem analysis stages of the are covered exhaustively and thoughtfully. Reporter’s notebook and dictaphone optional.

The Journalistic Six – Who What When Where Why How   #idea generation   #issue analysis   #problem solving   #online   #creative thinking   #remote-friendly   A questioning method for generating, explaining, investigating ideas.

18. LEGO Challenge

Now for an activity that is a little out of the (toy) box. LEGO Serious Play is a facilitation methodology that can be used to improve creative thinking and problem-solving skills. 

The LEGO Challenge includes giving each member of the team an assignment that is hidden from the rest of the group while they create a structure without speaking.

What the LEGO challenge brings to the table is a fun working example of working with stakeholders who might not be on the same page to solve problems. Also, it’s LEGO! Who doesn’t love LEGO! 

LEGO Challenge   #hyperisland   #team   A team-building activity in which groups must work together to build a structure out of LEGO, but each individual has a secret “assignment” which makes the collaborative process more challenging. It emphasizes group communication, leadership dynamics, conflict, cooperation, patience and problem solving strategy.

19. What, So What, Now What?

If not carefully managed, the problem identification and problem analysis stages of the problem-solving process can actually create more problems and misunderstandings.

The What, So What, Now What? problem-solving activity is designed to help collect insights and move forward while also eliminating the possibility of disagreement when it comes to identifying, clarifying, and analyzing organizational or work problems. 

Facilitation is all about bringing groups together so that might work on a shared goal and the best problem-solving strategies ensure that teams are aligned in purpose, if not initially in opinion or insight.

Throughout the three steps of this game, you give everyone on a team to reflect on a problem by asking what happened, why it is important, and what actions should then be taken. 

This can be a great activity for bringing our individual perceptions about a problem or challenge and contextualizing it in a larger group setting. This is one of the most important problem-solving skills you can bring to your organization.

W³ – What, So What, Now What?   #issue analysis   #innovation   #liberating structures   You can help groups reflect on a shared experience in a way that builds understanding and spurs coordinated action while avoiding unproductive conflict. It is possible for every voice to be heard while simultaneously sifting for insights and shaping new direction. Progressing in stages makes this practical—from collecting facts about What Happened to making sense of these facts with So What and finally to what actions logically follow with Now What . The shared progression eliminates most of the misunderstandings that otherwise fuel disagreements about what to do. Voila!

20. Journalists  

Problem analysis can be one of the most important and decisive stages of all problem-solving tools. Sometimes, a team can become bogged down in the details and are unable to move forward.

Journalists is an activity that can avoid a group from getting stuck in the problem identification or problem analysis stages of the process.

In Journalists, the group is invited to draft the front page of a fictional newspaper and figure out what stories deserve to be on the cover and what headlines those stories will have. By reframing how your problems and challenges are approached, you can help a team move productively through the process and be better prepared for the steps to follow.

Journalists   #vision   #big picture   #issue analysis   #remote-friendly   This is an exercise to use when the group gets stuck in details and struggles to see the big picture. Also good for defining a vision.

Problem-solving techniques for developing solutions 

The success of any problem-solving process can be measured by the solutions it produces. After you’ve defined the issue, explored existing ideas, and ideated, it’s time to narrow down to the correct solution.

Use these problem-solving techniques when you want to help your team find consensus, compare possible solutions, and move towards taking action on a particular problem.

  • Improved Solutions
  • Four-Step Sketch
  • 15% Solutions
  • How-Now-Wow matrix
  • Impact Effort Matrix

21. Mindspin  

Brainstorming is part of the bread and butter of the problem-solving process and all problem-solving strategies benefit from getting ideas out and challenging a team to generate solutions quickly. 

With Mindspin, participants are encouraged not only to generate ideas but to do so under time constraints and by slamming down cards and passing them on. By doing multiple rounds, your team can begin with a free generation of possible solutions before moving on to developing those solutions and encouraging further ideation. 

This is one of our favorite problem-solving activities and can be great for keeping the energy up throughout the workshop. Remember the importance of helping people become engaged in the process – energizing problem-solving techniques like Mindspin can help ensure your team stays engaged and happy, even when the problems they’re coming together to solve are complex. 

MindSpin   #teampedia   #idea generation   #problem solving   #action   A fast and loud method to enhance brainstorming within a team. Since this activity has more than round ideas that are repetitive can be ruled out leaving more creative and innovative answers to the challenge.

22. Improved Solutions

After a team has successfully identified a problem and come up with a few solutions, it can be tempting to call the work of the problem-solving process complete. That said, the first solution is not necessarily the best, and by including a further review and reflection activity into your problem-solving model, you can ensure your group reaches the best possible result. 

One of a number of problem-solving games from Thiagi Group, Improved Solutions helps you go the extra mile and develop suggested solutions with close consideration and peer review. By supporting the discussion of several problems at once and by shifting team roles throughout, this problem-solving technique is a dynamic way of finding the best solution. 

Improved Solutions   #creativity   #thiagi   #problem solving   #action   #team   You can improve any solution by objectively reviewing its strengths and weaknesses and making suitable adjustments. In this creativity framegame, you improve the solutions to several problems. To maintain objective detachment, you deal with a different problem during each of six rounds and assume different roles (problem owner, consultant, basher, booster, enhancer, and evaluator) during each round. At the conclusion of the activity, each player ends up with two solutions to her problem.

23. Four Step Sketch

Creative thinking and visual ideation does not need to be confined to the opening stages of your problem-solving strategies. Exercises that include sketching and prototyping on paper can be effective at the solution finding and development stage of the process, and can be great for keeping a team engaged. 

By going from simple notes to a crazy 8s round that involves rapidly sketching 8 variations on their ideas before then producing a final solution sketch, the group is able to iterate quickly and visually. Problem-solving techniques like Four-Step Sketch are great if you have a group of different thinkers and want to change things up from a more textual or discussion-based approach.

Four-Step Sketch   #design sprint   #innovation   #idea generation   #remote-friendly   The four-step sketch is an exercise that helps people to create well-formed concepts through a structured process that includes: Review key information Start design work on paper,  Consider multiple variations , Create a detailed solution . This exercise is preceded by a set of other activities allowing the group to clarify the challenge they want to solve. See how the Four Step Sketch exercise fits into a Design Sprint

24. 15% Solutions

Some problems are simpler than others and with the right problem-solving activities, you can empower people to take immediate actions that can help create organizational change. 

Part of the liberating structures toolkit, 15% solutions is a problem-solving technique that focuses on finding and implementing solutions quickly. A process of iterating and making small changes quickly can help generate momentum and an appetite for solving complex problems.

Problem-solving strategies can live and die on whether people are onboard. Getting some quick wins is a great way of getting people behind the process.   

It can be extremely empowering for a team to realize that problem-solving techniques can be deployed quickly and easily and delineate between things they can positively impact and those things they cannot change. 

15% Solutions   #action   #liberating structures   #remote-friendly   You can reveal the actions, however small, that everyone can do immediately. At a minimum, these will create momentum, and that may make a BIG difference.  15% Solutions show that there is no reason to wait around, feel powerless, or fearful. They help people pick it up a level. They get individuals and the group to focus on what is within their discretion instead of what they cannot change.  With a very simple question, you can flip the conversation to what can be done and find solutions to big problems that are often distributed widely in places not known in advance. Shifting a few grains of sand may trigger a landslide and change the whole landscape.

25. How-Now-Wow Matrix

The problem-solving process is often creative, as complex problems usually require a change of thinking and creative response in order to find the best solutions. While it’s common for the first stages to encourage creative thinking, groups can often gravitate to familiar solutions when it comes to the end of the process. 

When selecting solutions, you don’t want to lose your creative energy! The How-Now-Wow Matrix from Gamestorming is a great problem-solving activity that enables a group to stay creative and think out of the box when it comes to selecting the right solution for a given problem.

Problem-solving techniques that encourage creative thinking and the ideation and selection of new solutions can be the most effective in organisational change. Give the How-Now-Wow Matrix a go, and not just for how pleasant it is to say out loud. 

How-Now-Wow Matrix   #gamestorming   #idea generation   #remote-friendly   When people want to develop new ideas, they most often think out of the box in the brainstorming or divergent phase. However, when it comes to convergence, people often end up picking ideas that are most familiar to them. This is called a ‘creative paradox’ or a ‘creadox’. The How-Now-Wow matrix is an idea selection tool that breaks the creadox by forcing people to weigh each idea on 2 parameters.

26. Impact and Effort Matrix

All problem-solving techniques hope to not only find solutions to a given problem or challenge but to find the best solution. When it comes to finding a solution, groups are invited to put on their decision-making hats and really think about how a proposed idea would work in practice. 

The Impact and Effort Matrix is one of the problem-solving techniques that fall into this camp, empowering participants to first generate ideas and then categorize them into a 2×2 matrix based on impact and effort.

Activities that invite critical thinking while remaining simple are invaluable. Use the Impact and Effort Matrix to move from ideation and towards evaluating potential solutions before then committing to them. 

Impact and Effort Matrix   #gamestorming   #decision making   #action   #remote-friendly   In this decision-making exercise, possible actions are mapped based on two factors: effort required to implement and potential impact. Categorizing ideas along these lines is a useful technique in decision making, as it obliges contributors to balance and evaluate suggested actions before committing to them.

27. Dotmocracy

If you’ve followed each of the problem-solving steps with your group successfully, you should move towards the end of your process with heaps of possible solutions developed with a specific problem in mind. But how do you help a group go from ideation to putting a solution into action? 

Dotmocracy – or Dot Voting -is a tried and tested method of helping a team in the problem-solving process make decisions and put actions in place with a degree of oversight and consensus. 

One of the problem-solving techniques that should be in every facilitator’s toolbox, Dot Voting is fast and effective and can help identify the most popular and best solutions and help bring a group to a decision effectively. 

Dotmocracy   #action   #decision making   #group prioritization   #hyperisland   #remote-friendly   Dotmocracy is a simple method for group prioritization or decision-making. It is not an activity on its own, but a method to use in processes where prioritization or decision-making is the aim. The method supports a group to quickly see which options are most popular or relevant. The options or ideas are written on post-its and stuck up on a wall for the whole group to see. Each person votes for the options they think are the strongest, and that information is used to inform a decision.

All facilitators know that warm-ups and icebreakers are useful for any workshop or group process. Problem-solving workshops are no different.

Use these problem-solving techniques to warm up a group and prepare them for the rest of the process. Activating your group by tapping into some of the top problem-solving skills can be one of the best ways to see great outcomes from your session.

  • Check-in/Check-out
  • Doodling Together
  • Show and Tell
  • Constellations
  • Draw a Tree

28. Check-in / Check-out

Solid processes are planned from beginning to end, and the best facilitators know that setting the tone and establishing a safe, open environment can be integral to a successful problem-solving process.

Check-in / Check-out is a great way to begin and/or bookend a problem-solving workshop. Checking in to a session emphasizes that everyone will be seen, heard, and expected to contribute. 

If you are running a series of meetings, setting a consistent pattern of checking in and checking out can really help your team get into a groove. We recommend this opening-closing activity for small to medium-sized groups though it can work with large groups if they’re disciplined!

Check-in / Check-out   #team   #opening   #closing   #hyperisland   #remote-friendly   Either checking-in or checking-out is a simple way for a team to open or close a process, symbolically and in a collaborative way. Checking-in/out invites each member in a group to be present, seen and heard, and to express a reflection or a feeling. Checking-in emphasizes presence, focus and group commitment; checking-out emphasizes reflection and symbolic closure.

29. Doodling Together  

Thinking creatively and not being afraid to make suggestions are important problem-solving skills for any group or team, and warming up by encouraging these behaviors is a great way to start. 

Doodling Together is one of our favorite creative ice breaker games – it’s quick, effective, and fun and can make all following problem-solving steps easier by encouraging a group to collaborate visually. By passing cards and adding additional items as they go, the workshop group gets into a groove of co-creation and idea development that is crucial to finding solutions to problems. 

Doodling Together   #collaboration   #creativity   #teamwork   #fun   #team   #visual methods   #energiser   #icebreaker   #remote-friendly   Create wild, weird and often funny postcards together & establish a group’s creative confidence.

30. Show and Tell

You might remember some version of Show and Tell from being a kid in school and it’s a great problem-solving activity to kick off a session.

Asking participants to prepare a little something before a workshop by bringing an object for show and tell can help them warm up before the session has even begun! Games that include a physical object can also help encourage early engagement before moving onto more big-picture thinking.

By asking your participants to tell stories about why they chose to bring a particular item to the group, you can help teams see things from new perspectives and see both differences and similarities in the way they approach a topic. Great groundwork for approaching a problem-solving process as a team! 

Show and Tell   #gamestorming   #action   #opening   #meeting facilitation   Show and Tell taps into the power of metaphors to reveal players’ underlying assumptions and associations around a topic The aim of the game is to get a deeper understanding of stakeholders’ perspectives on anything—a new project, an organizational restructuring, a shift in the company’s vision or team dynamic.

31. Constellations

Who doesn’t love stars? Constellations is a great warm-up activity for any workshop as it gets people up off their feet, energized, and ready to engage in new ways with established topics. It’s also great for showing existing beliefs, biases, and patterns that can come into play as part of your session.

Using warm-up games that help build trust and connection while also allowing for non-verbal responses can be great for easing people into the problem-solving process and encouraging engagement from everyone in the group. Constellations is great in large spaces that allow for movement and is definitely a practical exercise to allow the group to see patterns that are otherwise invisible. 

Constellations   #trust   #connection   #opening   #coaching   #patterns   #system   Individuals express their response to a statement or idea by standing closer or further from a central object. Used with teams to reveal system, hidden patterns, perspectives.

32. Draw a Tree

Problem-solving games that help raise group awareness through a central, unifying metaphor can be effective ways to warm-up a group in any problem-solving model.

Draw a Tree is a simple warm-up activity you can use in any group and which can provide a quick jolt of energy. Start by asking your participants to draw a tree in just 45 seconds – they can choose whether it will be abstract or realistic. 

Once the timer is up, ask the group how many people included the roots of the tree and use this as a means to discuss how we can ignore important parts of any system simply because they are not visible.

All problem-solving strategies are made more effective by thinking of problems critically and by exposing things that may not normally come to light. Warm-up games like Draw a Tree are great in that they quickly demonstrate some key problem-solving skills in an accessible and effective way.

Draw a Tree   #thiagi   #opening   #perspectives   #remote-friendly   With this game you can raise awarness about being more mindful, and aware of the environment we live in.

Each step of the problem-solving workshop benefits from an intelligent deployment of activities, games, and techniques. Bringing your session to an effective close helps ensure that solutions are followed through on and that you also celebrate what has been achieved.

Here are some problem-solving activities you can use to effectively close a workshop or meeting and ensure the great work you’ve done can continue afterward.

  • One Breath Feedback
  • Who What When Matrix
  • Response Cards

How do I conclude a problem-solving process?

All good things must come to an end. With the bulk of the work done, it can be tempting to conclude your workshop swiftly and without a moment to debrief and align. This can be problematic in that it doesn’t allow your team to fully process the results or reflect on the process.

At the end of an effective session, your team will have gone through a process that, while productive, can be exhausting. It’s important to give your group a moment to take a breath, ensure that they are clear on future actions, and provide short feedback before leaving the space. 

The primary purpose of any problem-solving method is to generate solutions and then implement them. Be sure to take the opportunity to ensure everyone is aligned and ready to effectively implement the solutions you produced in the workshop.

Remember that every process can be improved and by giving a short moment to collect feedback in the session, you can further refine your problem-solving methods and see further success in the future too.

33. One Breath Feedback

Maintaining attention and focus during the closing stages of a problem-solving workshop can be tricky and so being concise when giving feedback can be important. It’s easy to incur “death by feedback” should some team members go on for too long sharing their perspectives in a quick feedback round. 

One Breath Feedback is a great closing activity for workshops. You give everyone an opportunity to provide feedback on what they’ve done but only in the space of a single breath. This keeps feedback short and to the point and means that everyone is encouraged to provide the most important piece of feedback to them. 

One breath feedback   #closing   #feedback   #action   This is a feedback round in just one breath that excels in maintaining attention: each participants is able to speak during just one breath … for most people that’s around 20 to 25 seconds … unless of course you’ve been a deep sea diver in which case you’ll be able to do it for longer.

34. Who What When Matrix 

Matrices feature as part of many effective problem-solving strategies and with good reason. They are easily recognizable, simple to use, and generate results.

The Who What When Matrix is a great tool to use when closing your problem-solving session by attributing a who, what and when to the actions and solutions you have decided upon. The resulting matrix is a simple, easy-to-follow way of ensuring your team can move forward. 

Great solutions can’t be enacted without action and ownership. Your problem-solving process should include a stage for allocating tasks to individuals or teams and creating a realistic timeframe for those solutions to be implemented or checked out. Use this method to keep the solution implementation process clear and simple for all involved. 

Who/What/When Matrix   #gamestorming   #action   #project planning   With Who/What/When matrix, you can connect people with clear actions they have defined and have committed to.

35. Response cards

Group discussion can comprise the bulk of most problem-solving activities and by the end of the process, you might find that your team is talked out! 

Providing a means for your team to give feedback with short written notes can ensure everyone is head and can contribute without the need to stand up and talk. Depending on the needs of the group, giving an alternative can help ensure everyone can contribute to your problem-solving model in the way that makes the most sense for them.

Response Cards is a great way to close a workshop if you are looking for a gentle warm-down and want to get some swift discussion around some of the feedback that is raised. 

Response Cards   #debriefing   #closing   #structured sharing   #questions and answers   #thiagi   #action   It can be hard to involve everyone during a closing of a session. Some might stay in the background or get unheard because of louder participants. However, with the use of Response Cards, everyone will be involved in providing feedback or clarify questions at the end of a session.

Save time and effort discovering the right solutions

A structured problem solving process is a surefire way of solving tough problems, discovering creative solutions and driving organizational change. But how can you design for successful outcomes?

With SessionLab, it’s easy to design engaging workshops that deliver results. Drag, drop and reorder blocks  to build your agenda. When you make changes or update your agenda, your session  timing   adjusts automatically , saving you time on manual adjustments.

Collaborating with stakeholders or clients? Share your agenda with a single click and collaborate in real-time. No more sending documents back and forth over email.

Explore  how to use SessionLab  to design effective problem solving workshops or  watch this five minute video  to see the planner in action!

visual model problem solving

Over to you

The problem-solving process can often be as complicated and multifaceted as the problems they are set-up to solve. With the right problem-solving techniques and a mix of creative exercises designed to guide discussion and generate purposeful ideas, we hope we’ve given you the tools to find the best solutions as simply and easily as possible.

Is there a problem-solving technique that you are missing here? Do you have a favorite activity or method you use when facilitating? Let us know in the comments below, we’d love to hear from you! 

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thank you very much for these excellent techniques

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visual model problem solving

Facilitation skills can be applied in a variety of contexts, such as meetings, events, or in the classroom. Arguably, the setting in which facilitation skills shine the most is the design and running of workshops.  Workshops are dedicated spaces for interaction and learning. They are generally very hands-on, including activities such as simulations or games designed to practice specific skills. Leading workshops is an exciting, rewarding experience! In this piece we will go through some of the essential elements of workshop facilitation: What are workshops? Workshops are a time set aside for a group of people to learn new skills, come up with the best ideas, and solve problems together.…

A notebook and a computer

So, you’ve decided to convene a workshop, a special time set aside to work with a team on a certain topic or project. You are looking for brilliant ideas, new solutions and, of course, great participation. To begin the process that will get you to workshop success, you’ll need three ingredients: participants willing to join, someone to facilitate and guide them through the process (aka, you) and a detailed agenda or schedule of the activities you’ve planned. In this article we will focus on that last point: what makes a good agenda design? Having a good agenda is essential to ensure your workshops are well prepared and you can lead…

visual model problem solving

What are facilitation skills and how to improve them?

Facilitation skills are the abilities you need in order to master working with a group. In essence, facilitation is about being aware of what happens when people get together to achieve a common goal, and directing their focus and attention in ways that serve the group itself.  When we work together at our best, we can achieve a lot more than anything we might attempt alone. Working with others is not always easy: teamwork is fraught with risks and pitfalls, but skilled facilitation can help navigate them with confidence. With the right approach, facilitation can be a workplace superpower.  Whatever your position, career path, or life story, you probably have…

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Visual problem solving with flowcharts and mind maps

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What’s life without problems? Probably a little boring, if we’re being honest. If everything were perfect all the time, there would be no challenges, and things would get pretty monotonous. This is a rather optimistic view on what many believe to be an aggravating part of life. No matter how you feel about problems, one thing is true: problems are inevitable . You can’t always control how many problems you encounter in your life, but you can learn better ways to solve them. So, what can we do for those really complex issues that aren’t easily solved? Visual problem solving is the perfect way to see solutions and break down complex issues.

Make your own flowchart with Gleek .

What is visual problem solving?

Visual problem solving is the process of using aids like charts or diagrams to display all the aspects of a problem in order to find viable solutions. When problem solving, sometimes it’s hard to see what’s causing the problem, or other relationships and correlations that are affecting whatever it is you’re working on. Two common methods for problem solving include mind maps and flowcharts . A mind map is a non-linear diagram, used for making new ideas or breaking down complex issues. A flowchart is a linear diagram, used for making action plans and describing processes.

5 steps to solve problems

Identify the true problem

Maybe you know what the issue is in clear terms, or perhaps it’s still a little confusing. A good way to get a concrete vision of the problem you need to solve is to pose it as a question, or a short statement. You might come up with something like ‘our sales have dropped’, or, as a question ‘what can we do to increase sales?’.

Get information

Now that you have a clear objective to solve, the next step is to gather all the relevant information that pertains to the issue. This can look like statistics, comments from customers, employee feedback, and more. Once you’ve collected the data, you’ll need to analyze it from all angles to get a clear view on the topic.

Brainstorming session

Get any and all potential solution ideas out on the table. Doesn’t matter how silly an idea seems, just put anything that comes to mind on the drawing board. This is where your visual aids will really come in handy, especially mind maps. You might need more than one chart, depending on how complicated the issue is.

Choose the best idea(s)

Whether on your own or with a team, you’ll have to eliminate the potential solutions that just won’t work. To find the solution that’ll work best, it’s good to analyze it in the same way you did the problem – by looking at potential outcomes, and all facets involved.

Make an action plan

So you think you’ve found the perfect solution! Now what? If your problem is complicated, usually the solution will be too. Here is where another visual aid, like a flowchart, will be helpful. Map out the specific steps you need in order to implement your solution. Then, it’s time to put your plan into action.

These are just the basic steps you can use to start problem solving. You may find that other actions are needed during your own journey.

Common mistakes when problem solving

Mistakes? We all make them from time to time. Here are some common mistakes we are prone to when trying to fix problems.

Undefined problem – When identifying the problem, it’s possible that the problem is too big, multi-faceted, or too complex to tackle all at once. A way to avoid this is to break the problem down into chunks, following common themes.

More problems arise – This isn’t always a direct result of anything we do, but it can happen nonetheless. The best way to deal with more problems that arise when you’re trying to solve the original one is to think of the possible things that could go wrong during the solution stage. When you’re prepared for any situation, you’ll rarely have any setbacks.

No action plan – Finding a way to solve your problem doesn’t mean that the planning is over. On the contrary, you need to create a strategy to properly execute your solution so you won’t end up with a half-solved problem and even more issues than you started with.

When to use flowcharts

One way to chart your problems and progress is through flowcharts. For those who like to think in a step-by-step or linear fashion, flowcharts are the best way to visualize things. Let’s have a look at some situations that are best suited to flowcharts.

Big problems – Flowcharts can help break down a large problem or solution into specific steps or stages from start to finish.

Decision trees – This type of flowchart is helpful when diagramming actions that will happen as a result of other actions, whether they be in a software system or actions taken by people.

Cause and effect – Similar to a decision tree, a cause and effect flowchart is where you can analyze the potential results of various actions, past or present.

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Check out our 20 flowchart templates that you can also easily edit !

When to use mind maps

Mind maps are great for brainstorming sessions, and non-linear problem solving. Here are some situations that are best visualized through a mind map.

Finding the problem – So, what is the problem exactly? Sometimes it’s hard to see. Making a mind map offers you the opportunity to see all the moving parts involved with a situation, and how they relate to one another, and can help you suss out the true problem.

Core and branching ideas – You start with a core idea, such as ‘online sales’, then add related ideas or issues branching off from that, like maybe ‘ad revenue’, or ‘social media campaigns’. Then those ideas can have their own branches. This is an easy way to analyze all aspects of a problem.

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Source: Problem Solving with Mind Maps (Tutorial)

Looking to create your own flowchart? Gleek has the solution for you. With Gleek, you can create your own flowcharts using a text-based command center, without ever using your mouse. Not only can you create flowcharts, you can create many other UML-based diagrams that will wow your colleagues and bring new life to your presentations. Get started for free today .

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Get a Copy of Cardsmith's Ultimate Guide to Using Brainstorming!

6 simple steps to visual problem solving.

Oct 30, 2018 | Productivity , Project management , Remote Work , Strategy , Thinking tools

brainstorming cartoon

Chances are, you’d describe yourself as a visual thinker. Scientific studies reveal between 65% and 80% of people fit this category. Visual thinking likely played a big role in our evolution; the ability to see a threat and react to it ensured our survival. Communicating ideas visually significantly improves effectiveness. A 1986 study from the University of Michigan showed that presentations using visuals were 43% more persuasive.  [source: http://misrc.umn.edu/workingpapers/fullpapers/1986/8611.pdf ] Visual tools help us think more creatively and efficiently, especially when grappling with complex problems or pondering big decisions. Involving your team members or colleagues will create a larger pool of ideas to help solve your challenge. The more ideas you incorporate from people who understand the problem and its complexity, the more options you will have to solve it. Here at Cardsmith, we focus on visual problem solving every day. We’ve found that six simple steps can take you from struggle to success.

Here’s how to amp up your visual problem solving:

1: start with a clear objective..

Before brainstorming, you’ll want to set up the problem you’re aiming to solve. Try posing it as a question. For instance, “How can we increase our sales?” Or you might ask, “80% of our sales come from one large customer. How can we reduce the risk this poses to our business?”

2: Dump out all the Legos ®.

Get all the ideas out on the table. This is where you express as many ideas as possible in a rapid flow without judging, structuring or organizing them. You don’t want to disrupt the free flow of ideas. Even the crazy ones count, and the ones that don’t seem to fit into any particular category. These ideas may be symptoms of the problem or related thoughts, fears and hopes. Don’t worry if your thoughts are long or complex, and don’t try to edit them. Whatever is in your mind, dump it out! Once you write all these things down, your mind can relax. Move on when you feel like all participants’ brains are empty. Read more about “Dumping out the Legos” .

3: Create right-sized information chunks.

When working with cards at this point, make each one discrete and stick to single sentences. Avoid using the word “and” because that will introduce too many ideas at once. A card in Cardsmith or 2” x 2” sticky notes have the perfect amount of space to enforce this rule. These constraints also help you be concise and clear about each thought. If you have longer ideas from the previous step, this is the time to edit those down to make them shorter and more focused. Here are some examples of information chunks, right-sized by using Cardsmith cards.

Cards with information chunks

Figure 1. Cardsmith Cards with information “chunks”.

4: Remember that problem solving is a creative act.

Be intentional about whether you are in the divergent or convergent phase of the creative process. These are phases of Design Thinking, the cognitive process from which design concepts emerge. A divergent phase is when you are looking for more: lots of thoughts, many ideas, etc. “Dump out the Legos” is the perfect example of a divergent activity. At some point, you’ll want to transition from divergent to convergent thinking. Convergent means using criteria to sort, group or organize the ideas. It’s making decisions that focus on fewer, rather than many. When you transition to the divergent phase, create a new framing question or objective, then repeat the process.

divergent and convergent phases

Figure 2. Example of divergent and convergent phases in Design Thinking.

5: Get clear on the problem before proceeding to solutions.

This concept is related to #4 but worth calling out as its own step. Often we think we understand the problem at hand and jump into problem-solving mode prematurely. First, spend a moment getting clear about the exact problem, and consider if it is the most important problem to solve. You will be more effective in the long run. Ask yourself questions like, “Are we jumping into solution space too quickly?” “Do we truly understand the problem?” “Is this the most important problem to solve now?” If you are not 100% certain, use step #1 to brainstorm all the possible problems. Try constraining the brainstorm to one area. For example, if you know you are having problems with sales, brainstorm to understand all the symptoms of the sales problem before moving on to seek a solution. Here’s a hypothetical brainstorm around this question:

problem brainstorming example

Figure 3. Problem Brainstorm in Cardsmith.

6: Select the right visual tool

Think about the type of question being asked. Would a tree, map or list best suit your problem-solving process? There are many options for mapping diagrams such as mind maps, fishbone diagrams, and affinity diagrams. We will discuss these options more in another post, but for now let’s focus on the affinity diagram. This is a great visual tool to use immediately after dumping out the Legos. Affinity diagramming is simply a way to group like with like. Cardsmith makes it easy to drag your ideas into clusters. If you are working in a team, doing this together yields benefits beyond the creation of the diagram itself, as you will discuss in detail why certain cards belong together with others. This will reveal hidden assumptions useful to downstream decision processes.   Here is an affinity diagram example, based on the above problem brainstorm:  

Affinity diagram

Figure 4. Affinity Diagram in Cardsmith.

Try out visual problem solving to improve your own life!

Now that you know the 6 steps, try this challenge to become more experienced in creative visual problem solving:

  • Create a Cardsmith board called “Life Problems”. Find a quiet time and place to reflect on all your dissatisfactions with your life. Add the first card to the board and title it something like, “What things in my life are less than ideal?”
  • Take a moment to thank your mind for all the problems and complaints it tracks on your behalf. Tell your mind it’s okay to think these things, and now is a chance to get free from such a heavy burden, by putting everything on the board.
  • Then just start dumping out all of the Legos. Write down any thought that comes, whether negative or positive. You have just practiced steps #1 through #4!
  • Now create an affinity diagram by grouping related cards together. You’ll likely start to see themes or areas of your life that you’d like to improve. I did this myself and while I came up with 27 things that I’m not happy about, when I created the affinity diagram I realized there were only three unique areas that I’d like to improve. Most of the cards are under health and fitness, so wellness activities are clearly worth pursuing!

What will you tackle next?

Now that you know the six simple steps, what problem shall we solve next? Tell us how you’ve used this process—and Cardsmith cards—to solve complex problems in your own work or life.

More Good Stuff

  • Feature Update: Custom Card Sizes Are In The House
  • How To Write Rapidly With Cardsmith
  • Why Your Remote Team Needs Online Sticky Notes For Brainstorming
  • Visual Tools for Coaching #1 – The Have/Want Matrix
  • Sticky Notes on Steroids: A Simple Tool for Any Facilitator’s Unique Process

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The Bar Model is one of the most frequently used tool in the Concrete-Pictorial-Abstract approach. Sometimes called Model Method, it is a very powerful visual problem-solving heuristic that serves as a foundation to algebraic thinking, and is used as early as first grade.

visual model problem solving

About Bar Models

Bar Modeling uses rectangular bars to represent relative quantitative values, and was first developed by the Ministry of Education in Singapore in the 1980s to help students solve word problems. It is a very powerful visual problem-solving heuristic that serves as a foundation to algebraic thinking.

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Using Bar Models

Bar Modeling, or sometimes called Model Method, uses rectangular bars to represent relative quantitative values, and was first developed by the Ministry of Education in Singapore in the 1980s to help students solve word problems. It is a very powerful visual problem-solving heuristic that serves as a foundation to algebraic thinking, and is used as early as first grade.

At lower elementary levels,

bar models help them visualize relationships between quantities that may belong to two different entities. For example, the problem below is usually taught by bar modeling in second grade.

There are 824 girls in the auditorium. There are 125 more girls than boys. How many boys are there?

A simple bar model

Bar models are also used when solving multiplication, division or fraction of a set problems in upper elementary. Here, students are introduced to the concept of defining a “unit” in the bar models, which is a basic place-holder for some unknown quantity. The following example illustrates how the bar model can help students visualize a word problem.

Amy has some flowers. Bob has 3 times as many flowers as Amy. Together, they have 120 flowers. How many flowers does Amy have?

Common units in bar model

Bar models are also helpful when introducing algebra in middle school. Since they learn to visualize unknown “units” in bar models early on, the students are more comfortable dealing with symbols as temporary place-holders and can also more easily visualize word problems. For example, the same problem above may also appear as an algebra question in middle school.

Using bar model to visualize algebraic problems

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SIS For Teachers

Visual Models: Your Secret Weapon for Word Problem Comprehension

Oct 1, 2020

Tape diagrams, model drawings, visual models, bar, model, unit bar – different names for the same thing: strategies to help with the comprehension of word problems for students. 

Sometimes in math, each book or teacher takes a name for a really great idea and twists it around to call it something else. Because this is often the case, and because we work with so many different math series, I’ll be calling this concept visual models. 

visual model problem solving

Visual models are a comprehension strategy that is amazing for helping students solve word problems. I think back to how I taught word problems to my students. I remember thinking it made perfect sense – circle the numbers, underline the words, box important information, right? For the most part, that procedure really worked out well when students were doing part-whole addition, part-whole subtraction and part-whole missing addend problems. But I vividly remember thinking of problems that were a lot more complex, which might be what I know now as multiplicative comparison problems or additive multiplicative comparison and realizing that my fancy procedure didn’t hold up. 

When we started to get more of our state testing, and the standards started to become more firm than they were when I first started teaching 20 some years ago, we had a few strategies – guess and check, make a table. There were lots of different icons at the top of the problems and I remember thinking, Gosh, how am I helping kids to know which strategy to use for which problem? 

Fear of word problems is also a very real thing, for students and even adults. In fact, when I took the GRE to get into my master’s program at Oakland University in Michigan, there was a question on the test that said something like, Fran drove eight more miles than Sam to work but three miles less than and twice the amount … my eyes glazed over and I thought, “Oh no! Not one of these problems!” I tried to draw diagrams and find the best (longest!) way possible to solve the problem.

visual model problem solving

To be honest with you, most of the time, I felt like it was just kind of a guessing game and the result was that a lot of kids began to subscribe to the “guess and check” method. We also tried the chart method, where you have a T-chart with key vocabulary that indicates the type of operation the problem requires. If it says sum it will be addition, if it says difference” it will be subtraction. Kids just aren’t learning to think and analyze the problems. Usually, when the going gets tough, because they don’t understand, students resort to an appeal to their teacher for help.

When most kids read a story problem, they want to get right to the nitty gritty. They look at the problem and decide – add or subtract? Typically, these kids are really more interested in solving it and figuring out what operation they’re going to use, than really understanding what the problem is asking.

For the majority of my teaching career, up until about eight years ago, I felt like we needed a strategy to help all students with all the different types of problems. I felt like we needed a common language around problem solving, which didn’t really exist at the time. 

Is there a solution? Is there a way that we could tailor problem solving for students as early as kindergarten all the way through eighth grade? There is: using visual models to solve word problems. 

Much of the information for visual models comes from research that’s been done in Singapore. The process works for everything, all the different types of problems that students are going to encounter: part-whole problems, additive comparison, multiplicative comparison, additive multiplicative, fractions, and as you go up even higher, ratios and proportions. 

Sometimes, people in Michigan or the US think, wait a minute, why are we doing something that is from another country? It’s just good math!! Using visual models to give students a comprehension strategy that helps them solve word problems is golden! 

I really wanted a process that can be consistent from teacher to teacher, grade level to grade level, that could follow a student all the way up. In most of our M 3 Building Math Mindsets project schools, we proudly display this step-by-step poster that walks students through being able to break down and answer problems, both the simple and the more complex.

Why do we need a process? 

This tutorial video, Word Problems with Visual Models: Basics, will explain the need for a process for students to be able to follow with visual models, from as early as first grade, students can be doing visual models with proportional units to help them understand part-whole addition, part-whole subtraction, part-whole missing addend, and even additive comparison. As students start to get a little bit older, they start to no longer be able to use a proportional model because we’re not just talking about five jelly beans any more. We might be talking about 29 jelly beans, and we’re certainly not going to proportionally draw out 29 boxes. 

Many students that are in first grade feel very frustrated, maybe thinking that there’s no purpose in being able to do a visual model because they know, from reading the problem, that we add or subtract. It’s “easy peasy”! But, what they don’t realize is that, the second a more complex problem comes up, they will stop in their tracks. 

visual model problem solving

For example, if we were to say that Karen brought 48 ice cream cups to the soccer game. One third of those cups were chocolate chip ice cream, three fourths of the remainder were strawberry, and the rest were vanilla. How many vanilla cups did Karen bring to the soccer game?

Students’ reactions to a problem like this are probably pretty similar to yours: Huh? Three fourths of the remainder – what’s the remainder? Should I find a common denominator? Okay, I circled all the numbers and I underlined the words, but I don’t really know what this is actually asking!

Quite quickly, problems can go from very simplistic adding problems to really more complex problems like this one. If we teach this step-by-step process of using visual models while the problems are simple, the students will be able to use it most effectively when the problems do get more complex. 

Moving towards Non-Proportional Thinking

visual model problem solving

This spring, we started talking about the journey to help kids connect proportional and non-proportional thinking with our Math4Littles series (catch up here!). Beginning with real objects in the physical world, moving to quantitative pictures, math work mats, and finally to journals beginning in Kindergarten and then 1st grade, where we really start to make the transition official. 

So what comes after we have this understanding? Part-whole problems with a non-proportional representation!

For your instructional convenience, we have created some really amazing videos that you can use during face-to-face or virtual instruction that will help students gain understanding about part-whole problems! The videos come with a PowerPoint tutorial that you can use in the classroom, a reference poster, a student journal that mirrors the presentation, and a blank journal template that you can customize based on the types of problems that you’re working on with your students. Check out our sneak peeks to see what you can expect!

visual model problem solving

It’s all at your fingertips! Our M 3 Members have access to download each of the six bundles at no cost , or you can buy the bundles individually in our store for less than a cup of coffee.

These Word Problems with Visual Models bundles are for you if…

PWMA

  • …you hate coming up with problems. Me too! And usually, the sample problems in our math books don’t flow the way we want our kids to learn. First, we want kids to learn part-whole addition, then part-whole subtraction, and then maybe part-whole missing addend. Next, I want to give them a mixed review before I add the next type of problem, which might be part-whole multi-step. You can completely customize the flow of problems by mixing and matching the bundles.
  • …you love a good anchor chart/poster. Each bundle has a unique poster, featuring the studious Professor Barble, that can be blown up for a classroom wall, or printed small on a bookmark that the kids could have in their journals. The posters will help students learn to recognize, and develop familiarity with, the types of drawings and problems they’ll encounter in elementary school.
  • …you don’t love creating PowerPoints. That part is done for you! Of course, we include the original PowerPoint file in the bundle, so you could always edit the presentation if you wanted to add your own problems, but if you don’t, no problem! It’s ready to go into your classroom – virtual or face-to-face – tomorrow! 

Check out our new Word Problems with Visual Models series and let us know how it goes! 

Check out our Math Word Problem Journals – available for 1st through 5th grade! Each bundle includes a year’s worth of problems, detailed answer key, and more! Available for purchase in our store !

Next week – we’ll look at visual models that go with comparison word problems! See you then!

*Addition , *Subtraction , *Word Problems , Audience - K-5 , Audience - Lower Elementary (K-2) , Audience - Upper Elementary (3-5) | 0 comments

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Light Bulb Moments in Math

Enhance Students’ Understanding of Solving Equations with Visual Models

Try to think back to when you first learned about how to solve equations. You were probably in 6th or 7th grade and most likely learned that you solve equations using procedures with inverse operations. Honestly, most of us probably teach our students how to solve equations exactly the same way. Students tend to master the skill of solving equations because they can follow the procedures of performing inverse operations on both sides of the equation. The reality, though, is that procedures are not enough for students to develop a deep, long-lasting understanding of what equations truly represent. 

Solving equations is a foundational topic for so many other concepts and skills in math, and students must have a solid conceptual understanding of what equations are, what they do, and what they mean before they learn the procedures of how to solve equations algebraically. In the real world, students won’t write and solve equations to figure out problems. Instead, they’ll use number sense, logic, and arithmetic. We can better support our students and help them develop a natural understanding of solving equations with visual models.

activities for solving equations with visual models on a desk with notebooks, writing utensils, and a calculator

What are visual models?

Visual models are concrete pictures, images, or diagrams that enable students to “see” the math they are doing. They are an essential component of learning mathematics because they allow students to use critical thinking, problem solving skills, and sensemaking to develop a deep understanding of key mathematical concepts. Popular visual models and representations used in math include:

visual model problem solving

  • Fraction bars and circles
  • Base 10 blocks
  • Area models
  • Ratio tables and bars
  • Number lines
  • Integer discs/counters
  • Algebra tiles
  • Tape diagrams
  • Hanger diagrams

Using visual models and representations to introduce a new topic can empower students by providing them the opportunity to discover problem solving strategies and procedures on their own. Classes that spend time developing a conceptual understanding of mathematics topics often don’t need to spend as much time teaching procedures because the mathematical procedures and methods follow naturally from the use of visual models and other means of discovery.

Let’s check out two ways that you can enhance students’ understanding of solving equations with visual models: tape and hanger diagrams.

Tape Diagrams as Visual Models for Solving Equations

Tape diagrams can be used to demonstrate and solve a variety of one-step, two-step, and multi-step equations. They show that values on both sides of the equal sign are equivalent and make it more obvious what operations are needed to solve for the unknown values. Consider this example for a one-step equation.

solving equations with visual models; solving the equation 3x=15 visually with a tape diagram and algebraically

Each of the three boxes ( x ) is the same size and, therefore, represents the same amount (15 divided by 3 = 5). Here is another example of a one-step equation using a tape diagram.

solving x+14=40 visually with a tape diagram and algebraically

To find the value of x , students subtract 14 from 40 to get 26. Using simple arithmetic to find the unknown values in the tape diagram then leads directly and naturally to solving the same equation algebraically with inverse operations. 

Do tape diagrams work with two-step equations and harder multi-step equations? They sure do!

solving 2x+11=27 visually with a tape diagram and algebraically

Subtracting 11 from 27 gives the total amount for the two unknown values. Because the difference of 16 is split evenly in half, the value of x is 8.

solving 2x+54=5x+18 visually with a tape diagram and algebraically

The key to using tape diagrams for equations with the variable on both sides is for students to figure out what amounts overlap and what amounts are left over. In the case of 2 x + 54 = 5 x + 18, the top bar has 36 more to make 54, and the bottom bar has an additional 3 x to make 5 x . Splitting the extra 36 in thirds equals 12.

Those are some examples of how you can introduce solving equations with tape diagrams, but there’s another fantastic visual model that you can also try with your students. I happen to like this next model even better.

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Hanger Diagrams as Visual Models for Solving Equations

Tape and hanger diagrams both do a good job of modeling equivalence with solving equations, but I prefer hanger diagrams because they show how both sides of the equations balance more obviously. Hanger diagrams are also called mobiles. Let’s start with a one-step equation example. 

solving 2x=28 visually with a tape diagram and algebraically

The hanger/mobile is balanced, which means that both sides have the same weight of 28. Students can divide 28 in half to find that each square has a weight of 14. Using simple arithmetic directly and naturally translates to solving the same equation algebraically with inverse operations.

Here’s another one-step equation modeled by a hanger diagram.

solving x+10=31 visually with a tape diagram and algebraically

Both sides are balanced with the same weight of 31. In order for the left side to have a total weight of 31, the square must weigh 21 (31 – 10 = 21).

Guess what? We can use hanger diagrams to model two-step and multi-step equations, too.

solving 3x+7=13 visually with a tape diagram and algebraically

Both sides have a total weight of 13. The left side includes a known weight of 7, so the remaining weight of the three squares must add up to 6, which makes each box have a weight of 2.

solving 6x+13=4x+45 visually with a tape diagram and algebraically

Of the rectangles, the left side has an extra 2 x . Of the circles, the right side has an extra 32. Since the weight on each side is even, x must be 16. The total weight on each side of the hanger checks out at 109.

Resources for Using Tape and Hanger Diagrams with Your Students

Allowing your students to explore solving equations with visual models can make a powerful difference in improving and deepening their level of understanding. When students work with visual models, they can discover how their strategies of using simple arithmetic and logic translate to solving equations algebraically. The mathematical procedures for solving equations will come more naturally to students once they’ve developed a conceptual understanding of how equations work. As my examples in this post demonstrate, it is useful to show the visual/conceptual and procedural methods side-by-side so students can make connections between the two representations.

If you enjoyed the strategies and examples I shared in this blog post and want to try using visual models for solving equations with your students, I encourage you to take a look at these helpful resources. Get ready for your students to experience light bulb moments and make connections like never before!

1) Solving Equations Resources from Light Bulb Moments in Math (That’s me!)

  • One- and Two-Step Equations with Visual Models
  • Multi-Step Equations with Visual Models
  • Solving Equations with Visual Models Bundle

cover image for One- and Two-Step Equations with Visual Models

These resources are some of my bestsellers because teachers and students love this unique visual approach to solving equations. Each of these activities is editable, scaffolded, and can be easily differentiated to meet your students’ needs. If you’re ready to jump in all the way with tape and hanger diagrams, the Solving Equations with Visual Models Bundle contains each of the first two resources but is discounted at 20% off ! Such a deal!

2) SolveMe Mobiles

If you’d like to try an interactive digital version of visual models for solving equations, these mobiles are perfect! I’ve used these puzzles with my own classes, and they are always a hit. 

The different shapes have different weights. The goal is to find the weight of each shape so that both sides of the mobile are balanced. As students test out different weights, the mobile will either stay level (if balanced correctly) or teeter to one side (if unbalanced). Students can begin with the Explorer level of puzzles and work their way through the Puzzler and Master levels.

Click HERE to learn more about how these awesome mobile puzzles work and how you can use them with your students.

3) Khan Academy and Delta Math

If you’re already familiar with Khan Academy and Delta Math and use these programs with your students, get ready to love them even more. They have assignments about writing and solving equations with visual models! I totally geeked out when I found these practice problems. I compiled a list of the assignments to help make life a bit easier for you.

On Khan Academy:

  • Identify Equations from Visual Models (Tape Diagrams)
  • Identify Equations from Visual Models (Hanger Diagrams)
  • Solve Equations from Visual Models

These practice sets only include one-step equations from the 6th grade curriculum, but they are a good introduction and confidence-builder for students before moving on to two-step and multi-step equations.

On Delta Math:

  • Tape Diagram Model – Linear Equations
  • Tape Diagram Model – Linear Equations (Multiple Choice)
  • Tape Diagram Model – Equations with the Distributive Property
  • Tape Diagram Model – Equations with the Distributive Property (Multiple Choice)

If you have a Delta Math account, look for these topics under “Understand Equations/Context” in 7th Grade (green headings). If you don’t have a Delta Math account, stop right now and sign up for one . IT’S FREE and one of the best online math programs available.

4) Sign up for my email list!

Once your students have mastered how to solve equations with tape and hanger diagrams, they’ll need to practice solving equations algebraically. Want a quick and fun resource that you can use with your students immediately? Sign up for my email list , and you’ll get a FREE EXCLUSIVE COPY of Solving Equations Placemat Activities . These engaging activities are editable, scaffolded, and include fully-worked answer keys. Get your copy now!

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Visual Problem Solving with Mind Maps and Flowcharts

Updated on: 25 July 2023

Everyone has problems, and we spend most of our working lives solving them. For those who find this quite negative, problems can be also termed as Issues, Challenges or Opportunities.

Some people are especially gifted at problem-solving while others struggle. Some are only good at solving some types of problems, while some other are simply great at finding viable solutions for any problem. Society generally calls the latter, smart.

What if I was to tell you that there’s a simple way to solve any problem you may encounter. In fact, it can be regarded as the smart way to solve problems.

Before we get into it, let’s see how people really fail at solving problems.

Problem-Solving   Fails

You Solve the Wrong Problem

Well, if you don’t know what the problem area is and don’t understand it very well, you’ll probably solve a problem that actually doesn’t exist while the actual problem remains as it is.

You Solve It Half Way

Again, this happens if you don’t know what the full problem is. Identifying and understanding the problem is so important before you start.

You Solve it but New Problems Show Up

This is typical when you don’t know much about the background about the problem area. If you know nothing about computers and you try to fix a broken computer, you probably won’t get very far and will likely make it worse.

You Don’t Know How

Well, obviously if you are trying to solve a problem that you have no clue about, this is going to be hard. When that’s the case, get the help of an expert in the domain the problem you are trying to solve belong to.

How to Solve Any Problem

As it’s quite clear the first step to solving any problem is understanding it thoroughly. Apart from getting a domain expert involved, the best trick I can bring you in is to draw it out. If you are a visual person this is the first thing you should do.

Different kinds of problems require different diagrams, but mind maps and flowcharts are common solutions to most problems.

Thinking Around the Problem

To get a background idea on what the problem and problem area is, mind maps can help greatly. Start with the core idea and branch out as you think about various aspects of the problem.

Mind map for visual problem solving

A mind map is a good place to start visual problem solving ( click on image to create your own mind map )

After thinking about wide aspects of the problem, it’s best to document what the immediate context of the issue is.

To do this, a concept map helps. A concept map is a diagram where you use various shapes to show areas of the problem and how they are connected.

Breaking It Down

Any big problem can be broken into a series of smaller problems. These are usually connected so a flowchart helps . Break the problem into smaller steps with a flowchart.

If you are analyzing an existing solution and trying to optimize it, a flowchart makes perfect sense as it also does the ‘defining’ part of the problem as well.

Flowcharts are also great for visual problem solving

Analyze your problem further with a flowchart

Once you have broken down the problem into smaller easily solvable problems in a flow chart, you can start creating another chart for the solution as well.

Getting Help

You should always get help if it’s available when you are solving any problem. A second opinion or a second pair of eyes can help a lot in getting the optimal solution.

Tools to Aid Visual Problem Solving

While there is a myriad of tools to help you draw things, Creately is definitely one of the easiest ways to visualize your problem.

We support mind maps, flowcharts, concept maps and 50+ other diagram types which you can use for visual problem-solving.

Our professionally designed templates and productivity features  help you just focus on the drawing as it’s really easy to draw a beautiful diagram in it.

It also comes with built-in real-time collaboration so it helps when you want to get someone else to collaborate on your problem.

Other choices for drawing diagrams to solve problems include Dia, Google Draw or even Microsoft office packages.

Join over thousands of organizations that use Creately to brainstorm, plan, analyze, and execute their projects successfully.

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Visual problem-solving: The secret sauce of successful teams

Reading time: about 7 min

  • Strategic planning

Our brains process visual cues faster and more easily than written or verbal cues. That’s why many of us rely on landmarks to help us navigate around town and why outdoor advertising is heavy on images and light on text.

Because images can often convey meaning more effectively than words, implementing visuals into your problem-solving will help your team easily find solutions faster.

What is visual problem-solving?

Visual problem-solving includes using visual aids like flowcharts, maps, diagrams, and sticky notes in your problem-solving process. 

Visual problem-solving helps you to:

  • Understand the problem : Visuals reveal the way that different elements relate to each other, offering enhanced clarity.
  • Simplify the problem : Visuals help identify past patterns in your work, allowing you to apply previous solutions to current issues. Visuals can also break down complex elements into smaller, more manageable pieces.

Your exact visual problem-solving process might vary, but it will likely include the following basic steps:

  • Identify and understand the problem
  • Gather information that pertains to the problem
  • Brainstorm and identify missing pieces
  • Select the best solution
  • Develop a plan and implement the solution
  • Review the results and revise as necessary

How can you make problem-solving more visual?

Let’s break down the steps mentioned above to further explore how visuals can take your team to the next level of problem-solving.

Step 1: Identify and understand the problem

There are several different visual tools you can use that will help you identify and better understand the problem at hand. Here are just a couple of suggestions.

A mind map is a visual tool that closely aligns with how our brains work. You start with a central idea and then bounce from one thought to the next in a non-linear fashion while identifying relationships as you go.

visual problem solving

Steps for identifying a problem with a mind map might include:

  • In the center of a page, digital canvas, or whiteboard, write the problem you’re dealing with.
  • Surrounding your central idea, add what you think might be possible causes of the problem. Connect these ideas back to your initial problem with lines or arrows.
  • Continue to branch out from each of the ideas circling your central problem. Add examples, details, and any information that will help you to further identify the root problem and its causes. Be sure to show connections between ideas while keeping the most important ideas closest to the center.
  • Use different colors, diagrams, and shapes to organize the different levels of thought—anything that makes sense to you and helps you identify the most pertinent information.

Mind maps have an endless number of uses. Students can utilize mind maps to brainstorm essay topics, creatives can implement mind maps into their workflow to collect strategy ideas, and management teams can use mind maps to illustrate potential effects of  company-wide initiatives.

Cause-and-effect diagrams

visual problem solving

To draw a cause-and-effect diagram :

  • In a box at the right side of a canvas or whiteboard, write your main issue or problem.
  • Draw a horizontal line from the left edge of the box. This line is known as the spine.
  • From the spine, draw angled branches that represent a potential cause of the main problem. Each branch can also have sub-branches that contain information that relate to the stated cause.

By analyzing the relationships between your potential causes, you can more easily pinpoint (and solve) the core problem.

Step 2: Gather information that pertains to the problem

Gather information from surveys, website statistics, and so on. Then examine this feedback through pie charts, graphs, sticky notes, and more. By presenting the information visually, you’ll be able to analyze it more efficiently.

Step 3: Brainstorm and identify missing pieces

When brainstorming, encourage quantity over quality. As they say, no ideas are bad ideas when brainstorming. Reinforcing this attitude will empower your team to think quickly, creatively, and collaboratively. It will also provide a platform for everyone to feel comfortable sharing ideas. 

Unsurprisingly, this step also generally becomes more effective when visual diagrams are implemented. As previously mentioned, mind maps in particular are great tools to use in a brainstorming session . 

Step 4: Select the best solution

You should have a board or canvas full of ideas now. Next, you’ll need to eliminate any proposed solutions that seem less feasible or effective than the others. Once you’re left with plausible ideas, analyze and discuss the potential outcomes of each to determine which one might work best to solve your problem.

You’ll often find that there isn’t just one obvious fix to your problem. You might need to combine several ideas into a solution that will best suit your needs.

One simple exercise developed by Tom Wujec, a pioneer in business visualization, asks you to take two to three minutes to draw your process for making toast . This helps emphasize both how different people think, as well as how many approaches there are to solving a problem. 

An exercise like this encourages collaboration and provides your team with applied practice in brainstorming and identifying the best ideas.

Step 5: Develop a plan and implement the solution

To implement a solution, you need to develop a plan—even just a simple one. A plan ensures that your team understands the path to success and knows what actionable steps to take next.

Flowcharts are effective visual aids in plan creation because they typically represent a more linear set of sequences. Using specific shapes and connectors to represent steps and decision paths, it’s easy for people to understand a process flow from start to finish. For example, use a flowchart template to help new hires understand the organization of a company or to illustrate the steps of a work task, like using the copy machine or replacing the printer ink. Or, use flowcharts on a marketing team to visually prepare for a product rollout or an upcoming campaign.

visual problem solving

After implementing the solution, share the flowchart so everyone has access to the step-by-step plan. This helps your team members understand their role in the overall solution process.

Step 6: Review the results and revise as necessary

You may have fixed your problem, but that doesn’t mean you’re completely finished with the problem-solving process. Continue to monitor the implementation of your solution and analyze its results. That way, you can revise the process if necessary to increase effectiveness.

Flowcharts can be useful in this step, as well, to preemptively identify issues and process bottlenecks, anticipating and mitigating potential problems before they even occur.

Things to avoid with visual problem-solving

Cut down on inefficiency and wasted time by avoiding these common problem-solving mistakes:

  • Not having a well-defined problem : Avoid trying to tackle too much at once. Break your problem down into smaller pieces and work your way up to the bigger problems.
  • Giving up if your solution doesn’t work : Sometimes a clear answer is difficult to find, but exhaust all potential resources before throwing in the towel.
  • Experiencing new problems when an initial solution is implemented : Don’t expect all issues to be solved right away. Ensure you’ve analyzed every facet of a possible solution in order to avoid future problems down the road. 
  • Failing to create an action plan : Without a strategic action plan, your team will struggle to align and act. Make sure to include a testing period within your plan. 

As you get more comfortable using charts, diagrams, and other visuals, you’ll find that it’s easier for team members to quickly align and process important information. Potential problems will be easier to spot and data will make more sense.

Supercharge your problem-solving with a virtual whiteboard like Lucidspark. Lucidspark’s  visual collaboration tools bring the whole team together no matter where they are located, keep ideas organized, and empower you to take action on the best ideas.

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Take your problem-solving to the next level with Lucidspark.

About Lucidspark

Lucidspark, a cloud-based virtual whiteboard, is a core component of Lucid Software's Visual Collaboration Suite. This cutting-edge digital canvas brings teams together to brainstorm, collaborate, and consolidate collective thinking into actionable next steps—all in real time. Lucid is proud to serve top businesses around the world, including customers such as Google, GE, and NBC Universal, and 99% of the Fortune 500. Lucid partners with industry leaders, including Google, Atlassian, and Microsoft. Since its founding, Lucid has received numerous awards for its products, business, and workplace culture. For more information, visit lucidspark.com.

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A complete guide to understand mind mapping, how mind maps are often used, and steps to make a mind map of your own.

visual model problem solving

Let's explore how visual collaboration is empowering people to work in new and more productive ways and how you can begin to implement this practice on your own team.

Bring your bright ideas to life.

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Visual Model of the Problem

Standardizing the visualization of problem statements is a complex task due to several reasons:

Diversity of Problems : There is a vast number of problems in computer science and mathematics, each with its own nuances and specificities. It’s challenging to create a standard visualization method that effectively covers all problems.

Subjectivity : Visualization often depends on individual interpretation and understanding. What works for one person might not work for another. Some people might find a tree-based visualization more intuitive for a particular problem, while others might prefer a matrix-based approach.

Complexity : Some problems are simple to visualize, while others might involve multi-dimensional data or complex transformations, making a standard form of visualization difficult.

Evolution of Methods : As research progresses, new methods of problem-solving and therefore visualization are developed. Standardization could potentially hinder this innovation and progress.

Flexibility : Having non-standardized visualizations allows for flexibility in teaching and understanding. Different teachers and learners can adapt their visualizations to best suit their needs.

Certain types of problems have standardized visualizations. For example, tree-based problems are usually visualized as trees, and problems involving two-dimensional data are often visualized as matrices or grids. However, these visualizations are more like conventions that have emerged due to their effectiveness, rather than rigidly imposed standards.

Translating Representations

A problem statement can be represented in a textual format for reading, as a equation or using mathematical notation, as a diagram. These different representation can be translated from one form to another. A person who is dominant in one form, can translate the representation in another form, so that they can solve the problem.

The ability to translate problem statements between different representations (textual, mathematical, visual, etc.) is a powerful skill that can greatly enhance problem-solving abilities. Each representation has its strengths and is better suited for certain types of problems or stages of problem-solving:

Textual : This is typically how problems are initially presented. A well-written problem statement should clearly define the problem space, inputs, outputs, and any constraints. This form is accessible to everyone and doesn’t require any specialized knowledge beyond language comprehension.

Mathematical : Mathematical notation can often succinctly express relationships and constraints that would be verbose in text. For certain problems, such as those in the domains of algorithms, data structures, and optimization, translating the problem into a mathematical form can make it significantly easier to solve.

Visual : Visual diagrams can be particularly helpful for understanding spatial relationships, data flow, complex structures, etc. They provide an intuitive understanding that’s often harder to achieve with text or mathematical notation alone.

Being comfortable in multiple representations and being able to translate between them allows one to leverage the strengths of each. For example, one might first understand the problem through its textual representation, then translate it to a mathematical model to devise a solution, and finally use a visual diagram to communicate the solution to others.

In fact, in fields like data science and machine learning, this ability is highly valued. Problems are often initially presented in text, and professionals have to translate them into mathematical models to apply machine learning techniques. Then, they have to communicate their results to non-technical stakeholders, which often involves visual representations.

Categories of Diagrams

Visual models for problem-solving and representation do have classifications, but these are often more context-dependent or domain-specific than mathematical models. A few general categories include:

Flowcharts : Flowcharts are often used to visually depict processes or algorithms. They use standardized symbols to represent different steps or decision points, and arrows to show the flow of control.

Graphs and Networks : These are used to represent relationships or connections between entities. They are fundamental to a wide range of fields, including computer science (graph theory), physics (network science), and social science (social network analysis).

Spatial Diagrams : This category includes all diagrams that represent spatial relationships. This might include geographic maps, floor plans, or molecular structures in chemistry.

Data Visualizations : This includes bar charts, scatter plots, line graphs, and so on. These visualizations represent data in a graphical format to aid understanding and interpretation.

Concept Maps and Mind Maps : These are used to represent relationships between concepts or ideas. They are often used in educational contexts to aid learning.

Systems Diagrams : These diagrams are used to represent complex systems. They might be used, for instance, in engineering to represent a mechanical system, or in business to represent a supply chain.

Each of these categories has its own conventions and “language,” and some also have further sub-classifications. However, the boundaries between categories are often blurry, and a single problem may be represented with multiple types of visual models.

In the end, the goal of any model - whether mathematical or visual - is to aid understanding and problem-solving. The choice of model depends on the problem, the intended audience, and the specifics of the situation.

Benefits of Drawing

The process of drawing diagrams, working through examples, and exploring those diagrams can be incredibly beneficial when solving coding problems.

Problem Understanding : Visual representations like diagrams can help us better understand the problem. They can clarify relationships between different elements, show the flow of data or control, and provide a big-picture view of what we’re trying to achieve.

Solution Development : Diagrams can help in generating solutions by providing a canvas to map out our thoughts. We can use them to try out different approaches, spot issues, and see how different components of our solution relate to each other.

Debugging : When a solution isn’t working as expected, diagrams can help us understand where things are going wrong. By mapping out the current state and tracing through the steps our code is taking, we can often spot the issue more easily.

Communication : Diagrams can be invaluable in explaining our thought process to others. This can be especially important in a collaborative setting, such as pair programming or a technical interview.

Formulating Equations : By drawing diagrams and exploring them, we can often observe patterns or relationships that might not be immediately obvious. These insights can then be formulated as equations or inequalities, which can guide our solution.

Remember that diagrams are just one tool in a problem solver’s toolkit. Different problems might call for different approaches, but visualizing problems is a powerful technique that can often lead to breakthroughs.

Brain · Continuous Improvement · Creativity · Cultural Enablers · Engagement · Problem Solving · Visual Management

Visual problem solving.

Tom Wujec is a Fellow at Autodesk, the world’s leader in 2D & 3D design software.  He has brought several software applications to market, including SketchBook Pro, PortfolioWall, and Maya which won an Academy Award for its contribution to the film industry.  Given Tom’s expertise with technology and software, some might find it surprising to learn that he is also a pioneer in the use of simple, interactive visuals to help teams solve problems.  Using images, sketches, and animations, Tom and his team make complex ideas understandable by making them visible and tangible.

So why should Tom’s work at Autodesk be of interest to you?  As I have written before, neuroscientists have discovered that we don’t actually “see the world as it is.”  Rather, our brain filters the information it receives based on past experiences to create the view of the world we have around us.  As Tom describes it, our brains conduct a “visual interrogation” of everything we see by asking a series of questions and creating a mental model based upon the answers.  The depth and variety of questions our brains ask (where, how, location, number, why, color, when, shape, size, what) is dictated by the richness of the images it encounters.  The richer and more diverse the images, the more of the brain’s three primary regions are utilized in processing the image to create meaning.

Since one of our goals as leaders is to improve the effectiveness of our organizational problem solving, Tom’s work provides us with a couple of interesting lessons:

First, make problem solving more visual.  Rather than merely using data points and words to describe, analyze and solve problems, use images.  Images help the brain clarify ideas, identify underlying patterns of logic, and create meaning.  As opposed to numbers and words, a good visual invites the eyes to dart around and engage the entire brain to create a visual logic and make sense of the information to which it is being exposed.  The more fully the brain is engaged in the act of analyzing and creating meaning, the richer the outcome of the problem solving activities will be.

Second, make your problem solving more interactive.  The act of creating a visual narrative of the problem solving process is critical to the team’s ownership of the problem as well as their engagement in finding a solution.  The more the team creates the visual logic used to tell the story of the problem and what caused it, the more vested they will be in the outcome.

Creating a culture of continuous improvement requires both engaging people in the process of identifying and solving problems as well as providing them with the tools to do so.  Most traditional approaches to problem solving fail to inspire people and generate creative solutions.  They lack both a visual component to kick the entire brain into action as well as a sufficient level of interactivity to create ownership between the team and the problem.  By challenging teams to use images to identify underlying patterns and create meaning, you just might be surprised at the improvement in both the quality of thought as well as improvement ideas.

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Made for Math

Word Problem-Solving: Easy Visual Models for Dyscalculia

Word Problems Watermelon

Makes sense considering many people can relate to the pain and suffering of solving word problems but especially students with a math learning disability. In this article, we’ll talk about why word problems are so difficult, explore what the research suggests, and share an effective model to help students! Join in the conversation with researcher Dr. Xin of Purdue and learn all about visual word problem-solving.

Table of Contents

Watch video: dyscalculia and word problem solving (ep 16), why is word problem-solving so difficult for students but especially for those with a math learning disability, why keyword strategies for solving word problems is not a reliable approach, what word problem method does the research suggest, how do model-based strategies help with word problems.

​Why is word problem-solving so difficult for students but especially for those with a math learning disability?

The semantics and language of math can be super strange and completely separate from real-life (Boaler, 2015). There are many instances of dual use for language such as being third in a race or eating a third of a pie. To make it even more complex, students with math learning disabilities typically have weaker working memory which makes working with numbers while reading a word problem difficult.

dual language math

Many teachers are confused about how to guide students in solving word problems. Generally, teachers find themselves coaching students to use keywords and match them to an operation. This causes students to mostly guess and check. However, this method is ineffective as students are making surface-level analyses instead of looking for deeper structures through relationships between the context and numbers (Parmar et al., 1996).

Another common method teachers use is the four-step strategy: read, plan, solve, and check. This is a general heuristic procedure. Or basically, this is another fancy way of saying, “Let’s guess and check”.

In the video below, the What Works Clearinghouse shared from Dr. Beckmann why keyword strategy for solving word problems is an oversimplified tool.

What Works Clearinghouse shows there is strong evidence for instruction on solving word problems that is based on common underlying structures. What does that even mean?

Word problems generally follow a pattern that can be translated into algebraic models which means we can solve for an unknown quantity. The sooner we teach students this skill set, the better!

Representing problems in mathematical model equations (e.g., part + part = whole, or unit rate x number of units = product), students do not have to memorize numerous rules to make decisions on the choice of operation for finding the solution; rather, the mathematical models, which depict mathematical relations involved in the problem, provide students with a defined algebraic equation for the solution.

What word problem-solving approach do we recommend?

Dr. Xin created a program called Conceptual Model-Based Problem Solving (COMPS) which simplified other programs and moved it to a model-based algebraic approach. We love this program because it allows students to learn skills that will work for the long haul of mathematics.

Xin borrowed from the simplicity of Story Grammar to create simple models that students can use. You may have been taught this model at some point in your teaching career. Every story has a clear beginning, middle, and end.

Likewise, simple models in mathematics help students organize their thinking.

The Made for Math team tested the COMPS program with our students and found that this method helps students connect their math manipulatives to the algebraic models. Let’s look at the two models!

Part-Part-Whole

visual word problem solving

These types of word problems are additive, meaning that we’re taking two parts and creating a whole. Here students are explicitly taught how to identify which parts of the word problem are the parts and which are the whole. A variable or question mark is used as a placeholder for the unknown value in this model.

Equal Group

visual word problem solving

These types of word problems are multiplicative, meaning that we are identifying the product, unit rates, and the number of units. Again, we can use a variable or question mark as a placeholder for the unknown value in this model.

We’ve found that students at Made for Math loved this approach. It demystified the word problem-solving process and made it easier to organize thinking without the painful process of guess n’ check which was fraught with errors!

With the COMPS program, students are engaged and excited to learn how to tackle word problems. Let’s make word problems fun. 😁 

🎲 Is multisensory math right for your child? 🎲

There’s no need to wait for a diagnosis to receive help for dyscalculia. We have a team of experts ready to help. To assist you in making sure the multisensory approach would be helpful, take this handy quiz.

MFM Authors

Jennie Miller

Jennie Miller

Marketing Assistant

is our Marketing Assistant and content creator here at Made for Math. Jennie loves being part of a company that is working to make mathematics accessible to children with dyscalculia.

visual model problem solving

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Course: 6th grade   >   Unit 7

  • Same thing to both sides of equations
  • Representing a relationship with an equation
  • Dividing both sides of an equation
  • One-step equations intuition
  • Identify equations from visual models (tape diagrams)

Identify equations from visual models (hanger diagrams)

  • Solve equations from visual models
  • (Choice A)   2 + k = 18 ‍   A 2 + k = 18 ‍  
  • (Choice B)   2 k = 18 ‍   B 2 k = 18 ‍  
  • Your answer should be
  • an integer, like 6 ‍  
  • a simplified proper fraction, like 3 / 5 ‍  
  • a simplified improper fraction, like 7 / 4 ‍  
  • a mixed number, like 1   3 / 4 ‍  
  • an exact decimal, like 0.75 ‍  
  • a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

A focus fusion attention mechanism integrated with image captions for knowledge graph-based visual question answering

  • Original Paper
  • Published: 13 February 2024

Cite this article

  • Mingyang Ma 1 , 2 ,
  • Turdi Tohti 1 , 2 ,
  • Yi Liang 1 , 2 ,
  • Zicheng Zuo 1 , 2 &
  • Askar Hamdulla 1 , 2  

Visual question answering tasks based on the knowledge graph are dedicated to integrating rich information in the knowledge graph to deal with complex questions that cannot be solved by image features alone while focusing on improving the performance of fundamental visual question answering tasks. The core of this task is to achieve effective cross-modal information fusion and resolve the semantic gap between images and text, thereby predicting answers more accurately. However, current visual question answering methods face challenges such as sparse information, single fusion features, and excessive computational burden. Given the sparsity of image regions related to questions in visual question answering tasks, traditional fusion methods such as linear pooling and cross-attention, while capable of effectively handling interactions between different modalities, engage the question with the entire image globally. It introduces unnecessary noise and increases computational complexity. To solve these problems, we propose a focus fusion attention mechanism (FFAM) integrated with image captions, effectively reducing noise and computational burden by focusing on the topk high-relevance areas. In addition, we adopt the advanced BLIP-2 model to generate image captions and introduce it as a new modality into the fusion process, breaking through the limitation of relying solely on features generated by the image encoder. Although introducing the knowledge graph increases the possibility of model processing complexity and noise, our method still shows powerful effects. On the F-VQA dataset, our model improved by 2.57% compared to the baseline model without the knowledge graph and achieved an accuracy of 86.35% with the knowledge graph.

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visual model problem solving

Data availability

The dataset supporting the results of this study can be publicly available, with the VQAv2.0 dataset available in https://visualqa.org/ and the F-VQA dataset available in https://github.com/wangpengnorman/FVQA .

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This work has been supported by the National Natural Science Foundation of China (62166042, U2003207), Natural Science Foundation of Xinjiang, China (2021D01C076), and Strengthening Plan of National Defense Science and Technology Foundation of China (2021-JCJQ-JJ-0059).

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Mingyang Ma, Turdi Tohti, Yi Liang, Zicheng Zuo & Askar Hamdulla

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Ma, M., Tohti, T., Liang, Y. et al. A focus fusion attention mechanism integrated with image captions for knowledge graph-based visual question answering. SIViP (2024). https://doi.org/10.1007/s11760-024-03013-7

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This CEO built his company for $60 in one weekend—it brought in $80 million last year: 'You can copy my model'

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I was fired from Facebook in my 20s—now I make $3.3 million running my own tech company

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