Multiple Representation Justifications
“As an introduction to mathematical justifications, I gave my Algebra 2 students a persuasive essay to read. Then they discussed what made it a good argument with their teams. As a class we came together to get a list of characteristics for a good argument on the board. At least that was the plan. It was first hour Monday morning, so it didn’t go quite as well as I had hoped. Students were sleepy and not very willing to talk. There was little discussion with their teams and few volunteered ideas when we listed them on the board. I think students were overlooking the basic yet essential elements. I asked them if the author just listed facts and then told you what to believe. They responded, “No,” and then finally told me that they explained each fact. In math, this is like giving a reason for each statement. They knew these things! They had just finished Geometry last year, which was full of flowcharts, proofs, and other justifications. Unfortunately, they weren’t discussing these ideas. I think students could have gotten more out of the activity. Hopefully they will keep these characteristics in mind for Thursday’s lesson.”
This frustrated rant of a journal entry was written right after Tuesday’s attempt to introduce today’s lesson on justifications with multiple representations. (More information on introducing justifications with non-math examples can be found in my last blog post.) As you can imagine, I was worried going into today’s lesson since it requires students to transition between a function’s graph, table, and equation. Students should build off their ability to justify a statement using one mode (as they did in Algebra and Geometry) to now interconnect three modes. This is not easy, but it is extremely beneficial.
I wanted to prepare students for these justifications by referencing the elements of a good persuasive essay, such as explaining each statistic and using multiple supporting arguments from a variety of sources. Since the original discussion did not go well, we took a bit of time to re-examine the list before jumping in. These same elements are the key to a well-developed justification. By using multiple representations to support a statement, students learn to write quality justifications and (perhaps more importantly) interact the content at a much deeper level.
As I was researching how to help student form these many justifications, I came across an article by Jodie Hunter and Glenda Anthony titled Developing Students’ Use of Justification Strategies. This article gave specific examples of students’ multiple modes of solving an algebraic pattern problem, and even included pictures and student quotes. The overwhelming result from the study was that by engaging in numerical, verbal, and visual tasks, students better comprehended the pattern and were able to generalize their findings. The article also highlighted what I saw in my classroom and in the activity I did with my GMWP colleagues, which was that students had an increased chance of catching their own mistakes and students were able to understand how the problem could be solved/explained using different methods.
Now, all of this sounds wonderful, but again it was difficult for students to get to this point. I started by going through an example with the three justifications I wanted students to use, a graph, table, and equation. To facilitate the kinds of discussions and writing I was looking for, I continually reminded students that they must use all three representations and that the justification for each should say very specifically how the representation is used to prove their statement. I was very picky! Students even started to get annoyed, which I took as a complement to my persistence. To justify a function’s x-intercepts, some students said, “It says it in the table,” or “You just look at the graph.” With more prompting, I was able to get them to say things like, “You look for x-values in the table that have a y-value of 0,” or “You look for where the graph crosses the x-axis.” Although students were getting annoying with my badgering, it was amazing to see how many students hadn’t made the connection to the equation (plugging 0 in for y and then solving for x) until they were forced to write that they looked for y=0 in the table to find the x-values. In a few cases, I didn’t need to answer or ask many more questions about the equation justification once I made them re-write their table and graph justifications. Victory!
To give students a bit more freedom with these justifications, I then had them try justifying a statement that they came up with using the function investigation questions we developed earlier in the chapter. (There is a separate journal entry that turned into a rant about getting students to come up with their own function investigation questions, but that’s a topic for another time.) The questions they had were weak, but after doing an intense multiple representation justification, they were able to come up with more specific questions to address. The answer to these questions, in relation to the function we were studying, became the statement to justify. Some of these were more basic, or were related to the justification they had already done. It was interesting to see that students hadn’t made connections between these topics, such as domain/range and asymptotes (a newer topic), until they realized the justifications were very similar.
For those of you wanting to try this in your own classroom, my first suggestion is to reserve a large chunk of time! My students spend 40 minutes on each of their first two statement justifications, though they were fairly difficult topics. By the third, and possibly easiest statement, it took them about 20 minutes. Consequently, I would recommend starting with an easier statement first. Perhaps choose one with familiar vocabulary, equations, and representations. This will allow students to focus on the justification process without overwhelming them with too much new content. I would also open the process up to interpretation. Let students come up with their own statements to justify. Encourage students to use the representations in different ways, and let them make/find mistakes. Lastly, hold students accountable for correct work. Remember, the goal is to make connections and eventually generalize the content knowledge based on these specific justification examples. Maybe students could even write a justification for the generalization! In order to get to this point, students need accurate details, so hopefully this process irons out any misconceptions.
If you have tried this in your class and have any questions or suggestions, please share! I will continue to work on these types of justifications with my classes, and I hope to share more math-specific tips soon.
