Updated: Oct 31, 2022
When you read the words, “Math problem solving” what do you think of? If you were anything like me, it included a word problem, we completed after I taught the math lesson. It was a time that my students and even I dreaded.
No matter what I tried, my students had difficulty taking what we learned in the math lesson and transferring it into "problem-solving".
Where was the disconnect? What should I change as a teacher?
After extensive research, I realized I was going about problem-solving all wrong. I limited problem-solving to just word problems. However, problem-solving occurs in every part of our lives. Every time we face a problem, we go through a problem-solving process to find a solution.
I also realized the reason my students were struggling with the problem-solving plans I had taught in class was because it didn't promote critical thinking. The "problem-solving" plan I taught them was just a checklist problem-solving plan.
In a checklist problem-solving plan, the students are to follow each step in order. The students in the problem below circle the numbers, underline the question, box keywords, evaluate, and solve. Seems pretty simple, right? But take a closer look at the problem. I blacked out all the information that is not circled, boxed, or underlined because students only pay attention to this information.
This doesn't give students much left to work with once they circle, underline, and box it all up. So what do students do? ADD. They just add it all up because they don't know what else to do.
Can you blame them though? What else are they supposed to do? According to CUBES, they did everything we asked them to. So where's the disconnect? What's missing?
The THINKING piece!
It is essential to remember that problem solving isn't a checklist or a linear process and it certainly doesn't only occur when solving a word problem. Problem-solving is all about flexible THINKING.
How do we go about teaching students to rely less on checklists and more on authentic and flexible thinking?
DEFINE, EXPLORE, SOLVE
After many years of research and trial and error, I developed the problem-solving plan DEFINE, EXPLORE, & SOLVE.
“Define, Explore, and Solve” serves as a guide for students as they work through problems while allowing for the thinking process to happen authentically.
It is a great tool for students to work through word problems, but it is not exclusive to just word problems. “Define, Explore, Solve” can be used to teach math content during STEM, in other content areas, or even in everyday life.
So let's get to it. What exactly is Define, Explore, Solve?
The define piece should be pretty basic, right? "What is the question asking?" or "What problem am I trying to solve?" One thing I have found while developing DES (define, explore, solve) is that students struggle with this piece the most. They want to jump straight into solving the problem whether it be a word problem, computation problem, STEM activity, etc. They don't take the time to truly understand what the actual problem is before solving it.
The late CEO of Apple, Steve Jobs, once said: "If you define the problem correctly you almost have the solution."
Let that sink in for a minute. How many times when we ask students what problem are they trying to solve they respond with "I don't know?". This response always baffled me. What do you mean you don't know? How are you solving a problem when you don't have a clear understanding of what that problem actually is?
Intentionally defining the problem keeps us from just going through the motions. It keeps us focused on what and how we are going to solve the problem.
The graphic above shows how this process is completed with a word problem and the same process applies when using this in other problem-solving situations. When we DEFINE the problem we need to find the "bottom line". What specifically are we trying to solve?
Explore is where the bulk of the thinking takes place. After we define the problem we have to sort through all of the information that is in front of us and decide how we will use it to solve the problem.
The 4 main thinking strategies in the explore section include Self- Questioning, Visualizing, Important facts (Determine Importance), and Making Connections (Schema). We are going to take a look at each of these skills in isolation today and you will also teach these skills in isolation. However, when it comes time to actually put these skills into practice they will not be used in isolation. You will find that once students are familiar with this process these skills will be used fluidly. They can be used in any order as needed.
In the next 4 sections, we will explore in-depth the 4 different thinking skills. Let’s get started!
Explore: SELF- QUESTIONING
Self- Questioning is critical. Thinking can not happen without self-questioning. In fact, the entire problem-solving process relies on self-questioning. It's what builds our understanding of what the problem is and how we can solve it. Self-questioning helps students to interact with the problem and sparks curiosity and creativity when solving the problem. We address it during the Explore piece but in reality, it is used throughout the entire process.
In the graphic above you will find some examples of questions students can use to help prompt their thinking. However, students are not limited to just these questions. There may be other questions students find that are helpful to help clarify their thinking. I encourage you to explore these different questions as a class and include them in the list.
‘If a picture speaks a thousand words then in Mathematics a picture can spawn a thousand ideas’ - Shehnaz Saronwal
Visualizing is the foundation of comprehension. In order to understand and solve a problem, we have to make our thinking visible.
In the book, “Building Mathematical Comprehension” Laney Sammons writes “Educators must teach students to draw upon their own inner resources to generate mental images of what they read and the mathematics with which they work to strengthen their abilities to construct meaning.”
In order for students to draw upon their own inner resources, teachers must give students learning opportunities to CREATE these inner resources. This happens in a variety of ways.
Let’s look at an example.
“Mrs. Smith wants her students to be fluent in addition facts. She gives the students timed tests and notices that her students are not performing well. What skills are students missing that are hindering them from being fluent in basic facts?”
In order for students to be fluent in basic addition facts, they need to visually decompose numbers in their head. If students aren’t able to do this, then they are most likely struggling with subitizing, doubling, skip counting, and/or making 10. All of which require visualization skills to complete. Mrs. Smith will need to go all the way back and identify which of these skills students are struggling with and reteach with concrete models, then scaffold it to the abstract. We call this CRA (Concrete, Representations, Abstract). Once the students have a strong understanding of the concrete models, then they move to the picture models, and then they can move to the abstract which in this case is the addition facts test.
It’s tempting as teachers for us to jump straight to the abstract. However, if we want students to make visual connections, which is again the foundation of comprehension, then we need to take the time to make math visible.
Below are some ways you can make math visible to your students.
Explore: IMPORTANT FACTS
Important facts vs. Interesting detail.
When we are trying to solve a problem, we are often faced with the task of sorting through which information we need and which information is interesting but unnecessary to solve the problem.
Students will go through the problem and determine which information is an important fact or interesting detail. I used to call interesting details, “trash”. However, I realized that information wasn’t trash. It was there to help us understand what was going on in the problem but wasn’t necessarily something we needed to solve the problem. Now instead of calling it trash we just call it an interesting detail. Meaning it's not something we need to solve the problem.
An important skill we use during this thinking piece is Chunk and Think. This is where we go through all of the information, chunk, and think "Will this information help me solve my problem or answer my question?"
Below you will find an example of how you would go through this process in a word problem.
Students will go through the problem and determine which information is an important fact or interesting detail. I used to call interesting details, “trash”. However, I realized that information wasn’t trash. It was there to help us understand what was going on in the problem but wasn’t something we necessarily needed to solve the problem.
As we work through this problem, we would go through it and first make sure we understood what the question was asking. In the problem above, we wanted to know how many pieces of mint gum we had.
Now it's time to Think and Chunk. We go back and reread the problem and ask “Will this help me find how many pieces of mint gum there were?” If the answer was yes we would circle it or highlight it to show we are going to use that information to solve the problem. We would then move on and read our next chunk.
Word problems are just one way we use the skill "Determine Importance." This skill is used in all problem-solving scenarios, which we will explore in greater detail in the month of March.
Explore: MAKING CONNECTION
When I was a 4th-grade math teacher, I had to teach my students how to do 2 digit by 2 digit multiplication. It filled me with dread when the time came to teach this skill. I knew from experience that students were going to struggle with what to multiply and what to do when they had to drop to the second line and add a zero. There were tricks and sayings students could use to work through the process, but I also knew from experience that even with these tricks my students would still struggle. Not because they weren’t capable, but because the complexity of the algorithm would intimidate them.
I decided to try something I hadn’t done before. I put the problem 24 x 17 on the board and told them to solve it with their table partner. There were some groans and a lot of “but we haven’t been taught this!” I told them to think about what they knew from 3rd grade about multiplication and to use that information to find a way to solve it.
After a few minutes, the students settled in and found ways to solve the problem. After about 10 minutes I had students solving the problems in a variety of ways and they were doing it correctly! The pride they had in themselves is something I will never forget.
Soon other students started to come up with ways to solve the problem. I had the students share with the class how they solved the problem and why they solved it this way. The students had made connections from 3rd grade and could apply what they already knew to a more challenging problem.
In our everyday lives, we all have unique experiences which allow us to make connections in different ways. The same applies when we are tackling math problems. Because of these unique experiences, it is important that as teachers we encourage flexible and creative thinking. There’s more than one way to make a connection, therefore, there’s more than one way to solve a problem.
However, too often students are looking for the “right” way to solve a math problem. A lot of this stems from the procedural strategies like problem-solving checklists and algorithms that we teach our students. Checklists and algorithms have their place in math but they can not solely stand on their own.
Creative and flexible thinking must also be a part of the equation.
A bonus is students may see a different way to solve a problem that you may never have thought of. This can strengthen and deepen not only your student's understanding but your own personal understanding of the problem as well. Some of my best teaching strategies came from students that were allowed to share the strategies they created based on their own personal understanding and connections.
Below is a graphic that goes into even more detail about Making Connections.
This graphic gives tips and question stems to help guide students as they make connections while solving problems. We will go into this in greater detail in future blog posts/
Solving the problem seems pretty straightforward? Just take the numbers and do the math. But there’s so much more to it than that. During the explore section students will continue to go through the 4 different thinking skills to identify what the problem is and what information is needed to solve the problem. As students go through the process to solve the problem you will find that the 4 thinking strategies will come into play but now in just a different capacity.
Being consistent with these skills will help students work through the problem-solving process. In the next month, we are going to look at Define, Explore, Solve in greater detail. I will share how this looks in word problems, stem activities, computation problems, etc. I will also provide activities you can use in your classroom to begin implementing in your classroom.