TEACHING STRATEGIES: USING MISCONCEPTIONS TO REPAIR UNDERSTANDING AND LOOKING FOR EFFICIENCY
An interesting theme that emerged from the collective debrief was the analysis of sophistication of strategies. This multi-grade makeup of grades 3-8 teachers pushed the teams to think across the learning progression and the algebraic vertical connections. In the planning of the lesson, teachers anticipated that students would most likely generate concrete, pictorial, numeric, and tabular approaches which were considered to be more accessible for early grades and graphical, verbal, and algebraic approaches to be more sophisticated. The team also considered presenting a graphical approach if it was not generated by the students, so that students would see the rate of change and be encouraged to interpret the graph based on the context of the problem.
In the sixth-grade lesson called Aleah’s Spending, the teachers wanted to focus on the multi-representational aspect of algebra. The research lesson was taught in a low- performing mathematics class and the teacher who led the lesson thought that many of her students needed concrete manipulatives. They decided that they would have $10 and $5 bills available for manipulation. The sixth-grade teacher did not anticipate any of the students creating a graph but did decide to have graph paper available just in case. During the observation of the lesson, the other teachers-observers noted how students negotiated meaning among team members and how they worked together to make sense of the problem. As indicated by the host teacher’s reflection, which she shared during the debrief, many responses and strategies surprised her. She stated,
Timeia originally started working with a group that used concrete models to solve the problem, but she quickly moved off to work on her own and began recording repeated subtraction from both Aleah’s bank account and her brother’s. This was one of the strategies we thought that students may use in our lesson plan; unfortunately Timeia did not get to finish showing her work for this strategy, but it is clear that she sees a repeated pattern occurring in the problem. (see Figure 6.4)
The essential understanding also included how the repeated subtraction that was used is related to the constant slope that was illustrated using a graphical approach.
Figure 6.4 Repeated subtraction method and the graphical approach with error but showing initial ideas about the pattern of change.
While there is an error in the calculation, the notion of “decreasing” or “negative slope” starts to evolve as they start to formalize their observation using algebraic reasoning.
Supporting productive struggle in learning mathematics. One exciting mathematical happening was that one of the students that the host teacher anticipated having the most trouble with the problem actually gravitated toward graphing, which the team of teachers perceived as a more sophisticated and unfamiliar approach. She stated her surprise in her reflection:
Lastly, one of my students who I anticipated having trouble with the task surprised me when he immediately went to graphing the two situations (see Figure 6.4)—Aleah’s money and her brother’s money. While it took most of his time to set up the scale and the data points did not match up to the days, Thomas could quickly see a pattern of the decreasing values of money in their bank accounts. Seeing Thomas gravitate toward creating a graph was a topic of our conversation during our debriefing session. One of my colleagues even suggested having him graph his multiplication facts; something that he normally struggles with. May be a graph is just more sophisticated to him and something new that he will grasp on to.
Elicit and use evidence of student thinking to build procedural fluency from conceptual understanding. Examine the following students’ artifacts and look for efficiency in their problem-solving approach and what the different approaches offers meaning to the solution. One of the important ways to bring meaningful mathematics discourse in the classroom is to ask students to look for the connection from different solution strategies. The ability to analyze among different representations deepens one’s understanding.
The following examples were offered for students to compare the tabular approach, algebraic expressions, and the graphical representations created by his students (see Figure 6.5). Andrew approached the problem using a double-sided table, which showed the decrease in money for both of the brothers. He kept subtracting until he found that David had more money and then counted how many days it had been. Andrew found an efficient way to create the table, and instead of listing all the amount decreased each day, Andrew decided to use a number sentence to skip down 10 days using (18 - 10) x 10 = 80 and (22 - 10 ) x 5 = 60 and used the notation ... to show that the pattern continued.
Brian used the days and created 180 - 10y for Brian and 105 - 5y for David, which eventually led to the discussion of solving for 180 - 10d = 110 - 5d. When Andrew finished quickly, the teacher encouraged him to solve the problem in a different way to verify his work. Andrew decided that he would graph the values on the table and found the point of intersection was the day at which they had the same amount of money and that the 15th day David had more. Andrew was able to formally show his algebraic thinking through both his table and his graph, however, had some difficulty verbalizing his findings. But overall, he shows that he is developing skills to think in abstract ways.
Brian’s strategy below (see Figure 6.5) also shows how he made sense of the problem by starting with a tabular approach but instead of counting down to the amount each day, he decided to abstract from computation and make a general rule using symbols. This is an important skill that Driscoll (1999) identifies as one of the important algebraic habits of mind. Abstracting from computation allows for one to make a generalized rule for a pattern of change. Brian realized that the amount left in the wallet of the brothers was going to be the 180 - 10 times the number of days for Dan and 110 - 5 times the number of days for Nick. The students at this level were not ready to set up an inequality statement from what they had but were at the cusp of this new understanding.
This teacher shared the evidence of the depth in which he examined his student’s solution strategies, use of representations, sophistication of ideas, and justification. The teacher had solved this rich task during the summer institute and had collaborated with other teachers to anticipate multiple strategies and sophistication of methods, which provided the specialized mathematical knowledge to perform this analysis. As a result, the teacher was prepared to analyze students’ work and marked the learning progression displayed in his classroom and determined where he might go with his students to push their mathematical thinking forward.
In particular, the benchmark problem used across grade levels not only developed the discussion among teachers about the vertical progression of algebraic thinking but also the need for multiple representations and how such representations evolves across grade levels. At the final research lesson presentation, the lesson study team presented their collective learning. The third-grade teacher explained how her students observed patterns in the tables the students had created (such as in Figure 6.4), the sixth- and eighth-grade teachers shared their students’ work with the table, equations, and graphs (see Figures 6.5, 6.6) and started discussing the notion of rate of change which led to the discussion of “slope.”
Following this, they talked about “steepness” which helped the third-grade teachers connect the tabular approach to the graphical approach. The eighth-grade teacher also
Figure 6.5 Andrew's efficient strategies and graph.
Figure 6.6 Brian's strategy, short cut.
shared how she could potentially use this activity in her classroom for a stock market game the students were involved in and connect the activity to the topic on linear equations and system of equations that she will teach later in the year.