VISIBLE THINKING IN MATH: USING A MODELING MATH MAT

To encourage the connections among multiple representations, the teachers asked students to map out their thinking using the Modeling Math Mat (see Figure 5.3).

The Modeling Math Mat allowed them to construct their understanding of the picture model in four ways:

1. visually with a drawing,

2. verbally with words,

3. using a function table, and

4. expressing as a rule and an equation

all of which promote algebraic thinking. After making connections among representations, we had the students transfer the pattern on to a coordinate graph to see the relationship to previous patterns.

In the first growing tower, students could quickly identify that the towers grew by 1 more block each time. The verbal explanation came very naturally for students. The difference between a novice and an advanced level (see Figure 5.5) according to the teacher was how the advanced student was able to describe in words the pattern using examples and specific vocabulary, “Stage 1 has six blocks and to get from 1 to 6 you add 5. So each stage is the stage number plus 5, for example, stage 100 equals 105.”

Figure 5.3 Modeling Math Mat to connect the five representations for math thinking. Source: Authors.

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One can tell in this response how the students are looking for the function rule by looking horizontally in the table from the input number to the output number instead of relying only on the recursive pattern as the novice does. For example, the novice response was, “You add a block to the top of the block to go step by step.” In addition, the novice uses numbers as a way to show the computation, while the advanced student is able to abstract from computation arriving at the general rule for the pattern in multiple versions, “x plus 5=y; add 5 to the stage #; x plus 5 equals y; the sum of x and five equals y.”

The next two patterns had multiplication in the equation. Since the students were so confused by the move from addition to multiplication in equation, one teacher noted that may be at the beginning of such lessons the host teacher should show students that there are four different possibilities for writing equations—four different operations.

Figure 5.4 Growing pattern 1 for the tower.

Figure 5.5 Growing pattern 2 for the tower.

It is not always addition, but it can also be subtraction, multiplication, or division so students would not get stuck on the patterns like pattern #2 (see Figure 5.5) and pattern #3 (see Figure 5.6) as some of them did when they did not see a pattern by considering only addition and got frustrated because of that. This helps teachers to also realize that additive thinking is a linear process, but multiplicative thinking is far more complex for students that can involve multiple processes.

Pattern #3 not only expects the students recognize that there is a fixed number that does not change (y-intercept) and then added to this fixed part is a multiplicative pattern which is two times the height of the chair (where the 2 refers to the slope). As illustrated in figure 5.6, creating a table to understand the pattern can help in the students’ understanding as well. While the table presented does not show this, it may be noted that there is a constant difference of 2 in the number of cubes from one stage to the next. Also note that the table starts at a height of the chair being 2 which is a reasonable point to start but if the pattern was continued backward then the number of cubes corresponding to height of the chair being zero would be -1 which is the у-intercept.

The teacher’s reflection below noted how she would use these pattern cards throughout the unit.

Thinking about how I may implement this lesson differently next year, I would want to try one pattern once a week throughout the unit. We would start the first pattern at the beginning of the unit so I would be able to identify students’ background knowledge, and as the weeks progress, I would relate each part of the unit to the task. After the first pattern, we’d go into our review of graphing and I would connect the need for graphing to representing the pattern on a coordinate plane. I would also use the task to identify that graphing points doesn’t always make a picture (like many seem to think from graphing activities in elementary school). Each week, we’d explore a new pattern and make connections that are the most relevant at the time. Perhaps one week the focus will be on how the function tables relate to the graph, another week on the difference between arithmetic and geometric sequences, another week may be identifying the coefficient vs the constant. As the unit progresses, the patterns will get more difficult and the connections will (hopefully) become more deeply ingrained in student thinking and understanding. Perhaps students will even get to the point where I can give them a table and they can create a pattern

Figure 5.6 Pattern 3 for the tower with a coefficient.

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in addition to the other representations. This could be a wonderful example of unique student work where each student is given the same table and they create a poster of the model including their own patterns.

Thinking Prompt: How docs the Modeling Math Mat allow a teacher to assess students’ representational fluency?