MULTIPLE REPRESENTATIONS AND STRATEGIES AS TOOLS TO CULTIVATE VISIBLE THINKING IN MATHEMATICS
One of the major benefits of orchestrating a lesson with an inquiry approach is that it provides opportunities for students’ visible thinking to take center stage in our math classrooms. Visible thinking (Ritchhart & Perkins, 2011) is a systematic, research- based approach to integrating the development of students’ thinking with content learning across subject matters. Researchers and authors, Ron Ritchhart and David Perkins, stated that visible thinking provides opportunities to cultivate students’ thinking skills and dispositions, and to deepen content learning while stimulating greater motivation for learning and positive disposition toward learning. This shift in classroom culture toward a community of enthusiastically engaged thinkers and learners is further explored at their website (http://www.visiblethinkingpz.org) and see Appendix MMI Toolkit 5—Visible Thinking Strategies in mathematics.
In our research lessons, we promote visible thinking through Poster Proofs, which encourages students to publish their mathematics strategies for others to evaluate just as mathematicians justify their thinking. Having students display their thinking through different representations makes thinking visible and allows for teachers to better assess whether a student or a group of students conceptually understands a problem.
To distinguish representations used by teachers and students, Lamon (2001) stated that representations can be “both presentational models (used by adults in instruction) and representational models (produced by students in learning), which can play significant roles in instruction and its outcomes” (p. 146). A teacher who has strategic competence can use these representations as “pedagogical content tools, devices such as graphs, diagrams, equations, or verbal statements that teachers intentionally use to connect students thinking while moving the mathematical agenda forward” (Rasmussen & Marrongelle, 2006, p. 388).
Another way to integrate visible thinking routine in problem solving is to use Same and Different Venn Diagrams where students use a think-pair-share strategy to actively explain, reason, and agree to disagree their thinking while looking for similarities and differences in their approaches to a problem. We encourage students to create Poster Proofs with their solutions as another visible thinking activity that they display through a Math Gallery Walk, to encourage students to explore diverse perspectives and multiple strategies. Finally, one of our favorite routines is to ask students to write an exit pass using the prompt, I used to think... but now I think... which is a visible thinking routine for reflecting on how and why our thinking has changed (see Appendix MMI Toolkit 5—Visible Thinking Strategies).
The modeling activity can begin by asking the students sitting at the various tables to introduce themselves by shaking hands and at the same time asking them how many different handshakes happened at their table. It is very common to hear about
Figure 1.4 Pictorial approach. Source: Authors.
misconceptions that may involve the total number of handshakes at the table being twice of what it should be or the total number being the square of the number of people.
For instance, a table of three students may instantly say, since there are three people and since it takes two people to shake hands, the answer is three times two, which is six. Another response might be that the answer is three squared, which is nine. Such misconceptions introduce great teaching moments. For example, one can ask the table of three students to stand up and start shaking their hands, while the rest of the class keeps a count. They will quickly realize that they are engaged in an active-learning exercise that helps to identify that they had double counted the number of handshakes.
As reinforcement, another geometric connection can be made at this point by drawing three dots that would refer to the students at the table. If an edge between the dots would represent a handshake, then it is easy to connect to the numerical answer obtained through the active-learning process to the number of edges in the picture (see Figure 1.4).
Another useful approach to help the students think about the problem would be to create an organized list to record the handshakes, which illustrates the fine difference between a permutation and a combination. The active-learning exercise where the students physically shook hands, the geometric approach of drawing dots with edges indicating handshakes, as well as creating the organized list brings out another important approach, which is verbalizing the process. For instance, saying that the first person shook hands with two people and then the second person shakes only one more mathematically translates to the total number of handshakes being 2 + 1.
This verbal reasoning not only helps the students to make a connection to the active-learning, geometry, and organized list approaches, but also helps them to think ahead and start seeing a pattern when there are more people at the table. For instance, a table with four students may immediately notice that this pattern leads to 3 + 2 + 1. This then could lead to a tabular approach where one could record the relationship between the number of students at a table and the corresponding handshake count as a sum of natural numbers (see Figure 1.5).
Although we want the students to ultimately converge upon the fact that there is some hidden formula that this investigation process is going to lead them to, we want to help them discover this through their mathematical work. Naturally, we want them
Figure 1.5 Tabular approach. Source: Authors.
Figure 1.6 Algebraic approach. Source: Authors.
to next abstract from the computation that they have worked out. For instance, we next ask how many different handshakes happen if the room had 25 people in it.
The students at this point recognize the importance of being able to learn how to abstract. It is common to see them verbalize or argue that the answer is the sum of all the natural numbers 1 + 2 + 3 + ... + 24, but not knows how to go about computing this. This is a good place to help them learn and discover triangular numbers and their connection to the Gauss formula through efficient algebraic approaches (see Figure 1.6).
Even after, it is evident by applying this formula to the original problem with 25 people, resulting in a solution of (24 x 25) / 2, teachers have the opportunity to talk about taking advantage of important arithmetic operations that can help compute such answers without having to rely on a calculator. For instance, talking about associative property of multiplication helps us to think of one half times (24 x 25) as (/ x 24) times 25, which yields the final answer of 12 x 25.
At this point, we once again help the students to make a real-world connection to what they are computing. For example, 12 x 25 could be thought of as 12 quarters, which by proportional reasoning up and down can be seen to be $3.00 or 300 cents (since four quarters is a dollar). Such simplicity in solving problems helps teachers
Figure 1.7 Picture proof. Source: Authors.
to recognize the power of using real-world examples to reinforce computation. Along with simplicity, it is also important for students to learn to appreciate and use mathematical formulas to help provide greater insight into solving complex problems.
Revisiting this newly discovered formula for the original example of three people yields 1 + 2 + 3 = (3 x 4) / 2. While this equation may seem obvious, it provides a natural challenge to prove that an additive representation on the left side of the equation equals a multiplicative representation on the right-hand side. One may think of proving this using mathematical induction for the general case, but how does one convince students that this equation makes sense using a simple mathematical model?
Consider the illustration shown that denotes a pictorial proof. If each addend on the left can be thought of as a square, the 1 + 2 + 3 is denoted by the group of squares on the left that have boundary denoted by solid lines. Adding an exact copy to this makes a rectangular grid whose area is 3 x 4, and since we only wanted the sum of one of those copies, we can divide this area by 2. Such simple pictorial illustrations not only help to provide a simple proof of the mathematical statement, but also provide opportunity for students to test hypotheses by doing the mathematics using manipulatives (see Figure 1.7).
While the approaches that we have taken so far to solve the handshake problem may seem elementary, one can also take advantage of advanced concepts students learn in middle school and beyond involving combinations to solve this problem. For example, the handshake problem can be simply solved with some basic knowledge of combinations, as it only takes two people to shake hands, and since there are n people to choose these two people from, the answer is the formula, which may also be referred to as the number of combinations of n people selected two at a time. Note that in order to increase the cognitive demand of the task, the students may be asked to make predictions on the function that represents the number of handshakes from the values obtained from the tabular approach.
One approach to accomplish this may be to plot the set of ordered pairs obtained from the tabular approach on a graph to determine the nature of the function (see Figure 1.8). This will later lead to connections to high-school standards, where one may also combine this with observations of the change in the у-values, which can then help to determine the quadratic function for H(n) that we are looking for via finite differences as shown below (see Figure 1.9). Having such flexibility in problem solving not only helps the students to engage in multiple representations in problem solving, but also helps the teachers to scaffold the task to accommodate cognitive demand at multiple levels.
Once the students are taken through the journey of doing the problem and abstracting from computation, it is important to also teach them the skill of undoing a problem. For example, we go on to ask the following question: “If you were at a party
Figure 1.8 Graphical approach. Source: Authors.
Figure 1.9 Finite difference approach that appears later in high-school topics. Source: Authors.
where 190 different handshakes happen, how many people were there?” At this point, the students recognize that the answer to the handshake problem is given and they must now work backward to find out the number of people in the room. Going through this, they recognize the power of being able to work backward to solve problems, which is a very powerful problem-solving strategy.
The work in Figure 1.10 is based on reflection from a teacher from a professional development that we offered along with a poster their team created that summarizes the discussion on the multiple solution strategies to the handshake problem where the star was used as a way to assess their representational fluency.
As mentioned earlier, the choice of the problem that we want the students to grapple with should not only be mystifying but also be interesting. This means that besides learning multiple approaches to solving the handshake problem, it is important to discuss related problems where such triangular number patterns naturally emerge. For example, as a follow-up to our discussion, we introduce the following Christmas song,
Figure 1.10 Assessing multiple representations in problem solving. Source: Authors.
The 12 days of Christmas. The first few lines are as follows: On the first day of Christmas, my true love sent to me, A partridge in a pear tree. On the second day of Christmas, my true love sent to me, Two turtle doves, and a partridge in a pear tree. The song continues with three French hens on the 3rd day, four calling birds on the 4th day, five golden rings on the 5th, and so on, up to the 12th day. As the teachers read through this problem, they seem to immediately notice a connection to the triangular number patterns. These connections can also be seen in the Pascal’s triangle (see Figure 1.11).
Figure 1.11 Pascal's triangle. Source: Authors.
As shown here, the handshake problem is a great example of a task that not only engages the participants whether they are students or teachers in rich problem solving, but also helps them to expand their algebraic habits of minds, namely doing and undoing, building patterns from representations, and abstracting from computation (Driscoll, 1999). For educators, it also provides the opportunity to learn about five practices (Smith & Stein, 2011) including:
- • Anticipating: The teacher can anticipate the type of strategies that students might employ, along with misconceptions that can be turned into opportunities.
- • Monitoring: It is important for teachers to provide their students with the opportunity to engage in problem solving, to monitor students to identify the various strategies that are discussed, and to understand how students think as they solve the problem.
- • Selecting: This step requires teachers to select the order in which the strategies will be shared so everyone benefits from learning all strategies.
- • Sequencing: This allows the teacher to sequence the approaches being presented, whether that be from simple to complex or in a sequence that supports the lesson agenda. This might mean starting with the acting out strategy, moving to looking for patterns, and concluding with finding an algebraic formula, or it might mean starting with a common misconception so that students can repair their understanding.
- • Connecting: As a last step, the teacher has the opportunity to facilitate a discussion that helps to connect all of the various strategies displayed and shared during the discussion. This helps students to see connections not only within and across the strategies, but also to related problems. For example, this problem could be changed to accommodate the distribution of valentine cards in an elementary classroom or to find the number of diagonals of n-sided regular polygon.