Using Concrete Tools for Improving Problem Solving Strategies
Years ago, a childhood friend and I found pictures from a birthday party. We were about 7 years old, and we both laughed at her outfit. The top had a variety of colors in a pattern, and the pants had a variety of different colors in a different pattern. When we asked her mother about this, the response was simple. “That was what you wanted to wear, and you didn’t want to go if you couldn’t wear your favorite outfit.”
As a teacher, I applaud my friend’s mother for allowing her to make decisions! In today’s world, most decisions are made for children. Clothes are chosen in advance; play dates are made. At school, many decisions are made for practical reasons. How, then, can we encourage decision-making in other contexts effectively?
As teachers, we know the importance of incorporating the Standards of Mathematical Practice in all tasks. One standard is “Use appropriate tools strategically.” In the definition of this standard, the description of tools lists pencil and paper, concrete models, rulers, calculators, and software, among others. Concrete models are a broad category. How can we encourage students to use the appropriate tools within this category?
I suggest giving students access to multiple tools! Also, give each group a red plastic cup to collect smaller objects such as Color Tiles. In my room, these manipulatives are easily available for all students: Color Tiles (squares), Two-Color Counters, toothpicks, beans, Snap Cubes®, Fraction Towers, Base Ten Blocks, Pattern Blocks, stacking counters, string, rulers, dice, paper, and colored pencils. Other materials are available upon request too.
Give groups a problem-solving task. Students think independently, then discuss the task as a group. The first decision the group makes is how to represent the problem most efficiently. One person from the group gets the manipulatives needed, then students begin to work on the task. If the group decides things are not working, they may return those tools and get something different. Eventually, students take a gallery walk to view the work of all groups. Finally, lead a classroom discussion about which tools were most effective and why they were effective.
The 2020–2021 school year was challenging for everyone. Manipulatives had to be cleaned between each use, and students were distanced. A colleague and friend, new to a math specialist position, wanted to give all third-grade students a choice of tools. This spring, she asked all in-person third graders to choose their tools to represent a problem-solving task involving area.
What is your prediction? In many classes, students are given square Color Tiles to use for these types of tasks. All of these students had used these tiles when learning the concept of area. You might predict that all students would, therefore, choose this tool.
In fact, students chose a variety of tools, including Color Tiles, Two-Color Counters, Snap Cubes®, and Base Ten Blocks (rods only). A little more than half of the groups selected Color Tiles. The teacher, from observing, knew that not all students who selected the square Color Tiles had a mathematical reason. These are the two types of models she observed from groups. Which of the following models represents an understanding of area? What misconceptions might the other groups have?
Some groups used square Color Tiles because other groups were using them. Some groups used square Color Tiles because they had used them in class. Only about half of the groups who selected square tiles were able to verbally explain they were using them because they were square, and area is measured in square units. Those groups also represented area appropriately. The students were polite, but from their expressions, you could tell they thought it was an obvious reason!
The online students were given square tiles in a digital format. In hindsight, the teacher wished she had given them an option of tools, too. Students can use websites that offer digital manipulatives, such as Brainingcamp.com, then share their work. By just sharing access, students choose the manipulative from many tools. (They also have the option to build at home and take a picture.)
Even with online tools, teachers can identify misconceptions. Consider the following examples:
Using square tiles to represent area is obvious to teachers, but it is surprising that the reason these squares effectively represented the concept of area was only obvious to about 25% of the students in third grade. What can we learn from this? Students need to learn to make good decisions in all areas of life and that includes choosing appropriate mathematical tools. For that to happen, students must have a choice!
As you begin the 2021–2022 school year, consider having students choose their own manipulatives as one way to use appropriate tools strategically!
Standards of Mathematical Practice