Building Problem-Solvers Through Visual Aids

BACK TO THE BASICS – Building Problem- Solvers through Visual Aids

Suppose a 4^th grader is given the following problem: The difference between two numbers is 17. The sum of the numbers is 107. Find the two numbers.

Many older students cannot solve this problem so how is a 4^th grader supposed to be able to solve this problem??? With proper training, this problem can be solved quite easily by a 4^th grader. We will revisit this problem and see a straight-forward solution later in this paper.

As teachers, we want to develop the student’s understanding when we teach math, not just teach how to do math.  Knowing how to do calculations is important, but understanding mathematics is the goal. If we develop a student’s understanding of math concepts, then we are laying a strong foundation for their future understanding of more difficult concepts. Our goal is to help students develop the tools necessary so that they will be good problem solvers and critical thinkers in the future.

 In the early grades we usually develop understanding using objects (manipulatives) and then move to drawings .  For example:  Strip diagram showing 8 + ? = 14.      

As students get older, student learning transitions from using objects to using drawings and eventually moves to writing equations.  Many times lessons drop the use of drawings to focus on calculating answers.  This may result in students not connecting previous knowledge to current concepts and may lead to them not understanding current concepts.  Most of us are familiar with Polya’s Problem-solving process which should be used when solving problems. Polya mentions many problem-solving strategies including drawings but in this paper we will focus on the importance of drawings.

I see that there is a disconnect in many students’ understanding as they move away from concrete and visual models. I think it is good to eventually transition away from physical objects but I think we should not transition away from drawings. We should continue to model concepts using math drawings to develop problem-solvers, but the use of physical objects can be reduced.

The Common Core mentions problem-solving and Algebraic Thinking beginning in Kindergarten. I have seen first -hand how using drawings to model word problems and concepts in math from first grade on is vital to developing understanding of concepts, problem-solving skills, and Algebraic thinking.  Yes! Believe it or not, first graders use Algebraic thinking! Children can understand basic Algebra concepts even in first grade.  For example, first graders can comprehend that B + R = R + B. They could think of it as “blue + red” equals “red + blue”. Consider using linking cubes (cubes that attach together) to build a tower of red and blue cubes. Let’s say we build a tower with 4 red cubes on bottom and 3 blue cubes on top. A child can see that we now have a tower of 7 cubes and notice that the “amount of red cubes” + “amount of blue cubes” = 7 cubes. (Red + Blue = 7 or 4 + 3= 7). Now if we flip the tower upside down, we still have 7 cubes in our tower but we can think of the total number of cubes as “amount of blue cubes” + “amount of red cubes” = 7cubes. (Blue + Red = 7 or 3 + 4 = 7). They will notice that we did not gain or lose any cubes by turning the tower upside down. They can also understand that no matter how many red cubes or blue cubes we have, we can think of the total number of cubes in the tower as either “amount of red cubes” + “amount of blue cubes” (Red + Blue) or as “amount of blue cubes” + “amount of red cubes”(Blue + Red).

First graders can also comprehend a basic example of the Distributive Property of Multiplication over Addition. For example, they can comprehend that 2 · (3 + 4) = 2 · 3 + 2 · 4. The mathematical equation is a bit too complicated for them at this stage, BUT they are definitely capable of understanding the thinking process! We can model 2 · (3 + 4) = 2 · 3 + 2 · 4 by using linking cubes. Consider the following: Use linking cubes to build two towers. Each tower has three red linking cubes on bottom and 4 blue linking cubes on top. Now we can show we have two groups of (3 + 4) blocks. Notice I can separate the red blocks from the blue blocks in each tower without losing or gaining any blocks. If I do this, I now have two groups of red plus two groups of blue. So I have two groups of three red blocks and two groups of four blue blocks. (In essence, I have 2 · 3 + 2 · 4 blocks. Notice I started with 2 · (3 + 4) blocks). We could extend the students’ thinking and mention that 2 groups of “red + blue” blocks is the same amount of blocks as 2 groups of red + 2 groups of blue. (We are hinting at the fact that 2 · (R + B) = 2 · R + 2 · B.)

I believe that often a continuation of pictorial modeling is missing in the lower grades. Students should regularly encounter concrete and visual models for as many different concepts as possible, beginning in Kindergarten, and see these models on a continual basis up through 8^th grade.  They should also see visual models of concepts in high school as the need arises. This will get the students used to trying to visualize the concepts so they can get their heads wrapped around different aspects of the concept and better understand the concept. Having students first create a pictorial model of the concept will also train them to be good problem-solvers and have courage to attempt new problems they have never seen before.   If the students develop a habit of visualizing concepts or situations given in word problems, then when they get to more difficult problems, it will be easy for them to try to create a visual model and not just say to themselves, “Well, I have never seen a problem like this before. I do not even  know where to start.”

We are training them to be critical thinkers and problem- solvers if we help them develop a habit of first either building a concrete model or visual model then moving on to a more abstract model. I have my master’s degree in mathematics and I still use this problem - solving strategy when I encounter a problem I am trying to solve and I am having difficulty understanding how to solve the problem.

Even though we want the students to transition from only using concrete and visual models, to eventually include using equations to model problems, they need to develop a habit of starting with concrete and visual models when they are starting any problem. They might not always need the manipulatives or the visual models once they develop understanding of a difficult concept or problem but we still want them to be in the habit of thinking about using concrete or visual models if they get stuck or do not know where to begin.

 Let’s revisit the 4^th grade problem that I mentioned at the beginning of this paper:

The difference between two numbers is 17. The sum of the numbers is 107. Find the two numbers.

Now, most adults know how to solve that problem using Algebra, but how is a 4^th grader supposed to be able to solve that problem? Those were my thoughts exactly until I realized the student could use visual aids to model the situation.  The given problem is not difficult for a 4^th grader to solve if they have made a habit of using concrete models and visual models to solve problems since Kindergarten or 1^st grade. The 4^th grader would naturally gravitate to using concrete or visual models and solve the problem quite quickly. The student might draw something like this:

From the drawing, it is easy to see that if we just remove the extra 17 from the total of 107, then we are left with two parts of the same size that total 90. (since 107 – 17 = 90). To find the value of one of those parts we just need to divide 90 by 2. So we can see that the value of one of those parts is 45. So my two numbers must be 45 and 45 +17 = 62.

As teachers we should lay a foundation for future grades. We should aim to stretch (expand) our students thinking (stretch them a little past the current grade level material.) We should teach our students to think, reason, and analyze concepts and given problems. We should build understanding so that when our students see an unfamiliar problem, they will have the tools they need to help them be successful in solving that problem.

Years ago, I had the privilege of using these techniques with my children.  My children developed a strong understanding of math concepts and also developed great critical thinking skills, problem-solving skills, and algebraic thinking skills.  So much so that recently one of my children modeled a real - world problem (they were having difficulty solving) with a system of equations and solved that system of equations rather quickly. The interesting thing is that this child had not been in an Algebra class in 8 years but could visualize the problem and knew how to write equations to solve the problem.

In conclusion, going back to the basics of regularly using visual aids to help students understand situations presented in problems helps students become better problem-solvers.