The following is the reflections from Sean Gilmour (’16) after reading and implementing Five Practice for Orchestrating Productive Mathematics Discussions by Margaret S. Smith and Mary Kay Stein in his classroom.
The Importance of Discussion
From this text, I hoped to glean ways to engage my students in activities with multiple entry points thus I can meet students where they are. I teach 7th and 8th grade students at many different ability levels in each class. As one might assume, it is common and convenient for me to rely on the most prolific and confident young mathematicians to drive a typical classroom discussion. These students enable me to cover the most material and get straight to correct answers.
Still, the most ‘productive’ discussion should reach and be digestible for all students. On a base level, state-testing scores, successful schools are awarded through growth, which means students with the lowest scores can make the largest impact. Moreover, in math, it is especially important to allow students to struggle with rigorous problems so that they can develop their critical thinking skills, as well as to weigh multiple approaches to solving problems – both the correct ones and those that demonstrate common mistakes.
Two words in the title, “orchestrate,” and “productive,” offer two daunting challenges for a first year teacher. Any teacher knows that it isn’t difficult for students to lead a discussion on just about anything. The teacher’s challenge is subtly guiding the discussion that you can meet designated goals by the end of a class. When students arrive at their own conclusions, it cognitively resonates much more than when a teacher blatantly states them.
Summary of the Five Practices
In order, Smith and Stein’s 5 Practices are:
It is crucial for the teacher to prepare, well beyond the typical amount, when planning a class-wide discussion. Anticipating first necessitates a teacher to set a clear, achievable goal for the task. The direction of class’ discussion reacts to students’ responses, but the end goal must be certain. With this in mind, the teacher must select a task with ‘High Demands.’ These have connections to the underlying concepts at the core of mathematics – and thereby at the core of a person’s understanding of mathematics – rather than something that relies on rote memorization or repetition of a procedure. On the other hand, ‘low demand’ tasks rely on memorization and repetition.
Here is an example: In the first couple of exercises, students have a visual representation of the problem – as seems compulsory with Geometry. However, a clearly predetermined protocol and formula can solve the problem. Even the ‘real world’ bicycle example gives another image like the earlier problems, making it effectively identical.
Their counter allows students to actually interact with the shapes, measuring their sides, hands-on. Students are then allowed to make observations and “conjectures” about the mathematical underpinnings of the exercise. It becomes more of a ‘science’ experiment than the typical mathematical drill.
The next steps occur during the class itself. First, during monitoring, the teacher observes as students work with partners or in groups to solve the problem and prepare to share their methods with the class. The teacher may use questioning to preemptively draw out students’ mathematical reasoning.
All the while, the teacher reflects on their anticipatory preparation, and then uses these notes to select important information for students to share. Often, by actively selecting certain groups or students to contribute, the teacher purposefully creates opportunities for students across all levels to contribute to the dialogue. Typically, teachers (including myself, admittedly) will ask for volunteers to share responses. While this may streamline discussion, it excludes students who may be most confused and in need of assistance as well as narrowing the rigor and breadth of discussion. The end goal of selecting is all about specifying ‘what’ needs to be said, not ‘who’ will or wants to say something.
In the penultimate step, sequencing, the teacher imagines the best order for selected students to share. Ideally, the optimal sequence is like a story ; the class meanders through different strategies before arriving at the climax of the core math concept. As I said before, the teacher includes incorrect strategies since these allow the teacher to address and engage all levels of understandin g, and then get closer and closer to the solution. Reaching the final solution is not possible without acknowledging all strategies for approaching a solution. Mistakes are allowed to be made, since students need to learn from them.
Then, at the end, the teacher connects all of the students’ contributions. This is the stage where the teacher truly orchestrates everything, but again, everything is led by students’ responses to the problems. The teacher allows students to consider theirs and their peers’ strategies in their own words before ultimately guiding them to the core mathematical concept. At two recent Professional Development sessions, I gleaned a lesson that is very fitting here. Students are mathematicians instinctually. Whether information is presented through a graph, an algebraic equation, or something else, all students canpick up the most important details of a problem. However, it is up to the teacher to connect students’ realizations and ground these in mathematics.
Tiling with the Seventh Grade
Smith and Stein use several case studies to show the Five Practices at work. This format is obviously much appreciated by a teacher hoping to implement them, but each case study was in the context of introducing a concept. At this stage of the year, I would be using this system to review material, standards, and concepts I’ve already taught. I knew some of my students would be able to remember and fall back on things they’ve already memorized, which would keep me from living up to the true intention of the Five Practices. However, I hoped this process would help me to flesh out lingering places of confusion and misconceptions in other students.
The first task I took through the 5 Practices model was based on one of the case studies in the book, Tiling a Patio. In it, students have a set of patterns, where the amount of ‘tiles’ changes at a constant rate from one pattern to another. Originally, the task was devised to depict slope-intercept form, an 8th Grade standard, but I amended it so that it could address 7th Grade standards surrounding Ratios and Proportional Relationships (7.RP). Given the time constraints I have at this point of the year, I had to tackle 3 goals in this task :
- Recognize Proportional Relationships change at a constant rate.
- Model Proportional Relationships through multiplication, specifically the formula y=kx.
- Determine graphs of proportional relationships cross the origin and are straight lines.
Firstly, these is probably far too many disparate big ideas to cover evenin an ideal, student-led math discussion. However, with Easter Break a clear and present danger at the time, I felt it necessary to multitask. Still, these concepts are all intertwined as sub-sections of the same standard (7.RP.2.a-d) .
Before the discussion itself, I anticipated two main sticking points. My task had two examples. In one set of patterns, the tiles would change by three every time, and students would have to come to the conclusion that ‘Pattern 0’ would have three less tiles than Pattern 1 – that is they would have 0 tiles.
The second set of patterns would change by two each time, meaning that ‘Pattern 0’ would have 1 tile.
Many students did indeed fall into a trap when approaching this part. It was easy for them to continue increasing at a constant rate by simply adding. But firstly, some gave pause to what to do when they would have to go down a pattern from Pattern 1 to Pattern 0 (i.e. subtract).
Secondly, many students incorrectly assumed that both patterns must start with 0 tiles. The second set changed at a constant rate of two tiles, but, that meant it should have started with 1 tile, which would mean it wasn’t proportional. However, many students easily recognized this rate of change, then defied it to make it proportional.
There was a table where students recorded the amount of tiles in a given pattern, and when they picked up the rate of change, they extended the pattern. Thus, at the beginning students could continually add by three or two. However, for the last two ordered pairs, I broke the pattern, forcing students to use a rule based on the formula, y=kx, with x being the pattern, and y being the number of tiles.
This would necessitate students realizing my second intended goal and using a multiplication equation to represent the changes from pattern to pattern. Indeed, many students & subtracting at the predetermined rate.
There were exceptions, as students were willing to ‘break’ the prior rule.
I suppose it’s always important to teach students to read carefully. But, those that did correctly multiply in order to find an output, were able to make the connection between the formula, y=kx, and proportional relationships perhaps better so than they did the first time I covered this topic several months ago.
All in all, students liked going through this exercise and were willing to complete it on their own. While I have often heard complaints about the difficulty of work I give out, my students found this assignment accessible, rigorous, and rewarding. As I was monitoring students, they were quick to make connections on their own. When I questioned them, they were comfortable reasoning in their own words, rather than falling back on what a friend said. If and when students did express doubt, they simply didn’t expect the answer to be so intuitive and so easy to find.
As I wrapped up my lesson during the final stages it was seamless for me to bring up my anticipated mistakes, as students didn’t mind sharing when selected. All in all, this task seemed to build student’s confidence since the task was accessible even when their answers were wrong. The tactile nature of adding tiles was easy to grasp, as opposed to earlier in the year when I used concepts that I deemed important and realistic, but maybe too abstract for a middle schooler, such as calculating miles per gallon. Ultimately, when I connected the change to multiplication and a straight line on the graph, which is caused by constant addition by the same amount, students were clamoring to make the realization themselves.
Tiling a Patio with Eighth Graders
A few weeks later, I revisited the themes activity with my 8th graders. As I said before, this task was initially designed to depict Slope-Intercept Form. This concept is central to several standards across two domains in the Mississippi CCSS (8.F.2-4, 8.EE.5).
Rather than adapt it to connect with language and techniques I had taught, like I did with 7th graders, I left the task in the original form I found online. Students had a more complex tile pattern and would have to assign the pattern to an algebraic rule, which would have to be in Slope-Intercept Form. So, my learning goals were for students to be able to:
- Recognize the rate of change, or slope
- Recognize the starting rate of the pattern, or y-intercept
- Use these to create a rule in Slope-Intercept Form (y = mx+b)
Just like the 7th Graders, I went into this task as a review activity, with a lot of indirect anticipation of how students would address this problem and what challenges they would confront. Similarly, I knew my students would easily be able to recognize the rate of change, i.e. the ‘slope.’ This made sense given the standards they presumably mastered in the year prior in exercises like the Tile Diagrams.
This prior assumption was met. Most students quickly picked up on the pattern and (with some guidance) began to make an input-output table recording the numbers of tiles in each pattern.
Most students realized that the rate of change was two white tiles. But there was a large degree of variation as some students realized that the pattern didn’t line up with a simple rule of multiplication, as in a proportional relationships. I could see that some students had second guessed their earlier, correct assumptions regarding the pattern, then tried to make new, incorrect rules to explain the data.
In its unaltered form, students didn’t have to find a ‘Patio 0,’ which 7th graders did. This meant that the assignment’s instructions wouldn’t give them the impetus to find a y-intercept. Therefore, many students used only the rate of change when asked to assign a rule to the pattern.
A couple of clever, creative students didn’t assign a specific rule, but still were able to extend the pattern. They may not have realized it, but they bypassed their 8th Grade standards and went directly into Algebra I standards. Obviously, this was exciting for me to see.
Another point of confusion came up when students compared the patterns of the black tiles and white tiles. The task’s initial instructions described how each kind of tile progressed but didn’t indicate which pattern would be the focal point. Given my fixation on the ultimate mathematical goals, I admittedly didn’t consider that this would become an issue. However, many students missed my target because they focused on the pattern of the black tiles. Unlike the 7th Grade task, I was interested to see how students confronted the task in its original form without my guidance. This way, it required the higher level reasoning of a High-Demand Task. Looking back, it should have been easy for me to see how the introduction could have confused students about which pattern, or patterns, to analyze. I caught this about midway through, and stressed my students to only focus on the white tiles. Still, it was interesting for me to see how students seemed to grapple with comparing and linking the patterns between the two sets of tiles.
In hindsight, the pattern of the black tiles was interchangeable with the patio number. If students analyzed the ratios of black tiles to white tiles, they would have reached the same conclusions. However, when instructions said to analyze the tiles, students decided that they only had to analyze one set of tiles, and naturally, they looked at the simpler pattern. Unfortunately, they unknowingly didn’t have a chance to meet my goals for the exercise.
Nonetheless, the breadth of strategies used to confront this exercise intriguedigued and pleased me. Only a handful of students were truly close to the ideal rules for the pattern and the true solution as a result. Nonetheless, there seemed to be a pretty small correlation between students’ grades and their ability to confront the task. Perhaps this had a lot to do with my lack of direction, as students were free to understand the problem subject only to their own imagination. Then, as as I provided a little guidance, many students had the “Ah, ha!” moment as they now had to compromise their findings with an algebraic rule.
As I went about selecting and sequencing class discussion, I was able to draw from numerous perspectives, strategies, and personalities. All the while, students never seemed to see themselves as part of a hierarchy. At this point in the year, it obviously would have been ideal for students to understand how this related to specific contents and standards, but the variety of entry points and strategies for solutions led to a lesson and discussion that I’m quite proud of.
After reading Five Practices, and testing them with these exercises, I feel much more prepared for delivering and teaching with high level tasks. In our Math content group, our mandate has always been to give students rigorous content and let students rise to the occasion. Accordingly, I, from the start, laid out rigorous tasks for students even though they often complained and yearned for an easily replicated procedure.
After reading Five Practices, I feel much more ready and excited for how I, as a teacher, can manage and orchestrate these tasks. My experiences, especially the Tiling a Patio, have proven that inspiration can come from all levels of mathematical achievement. With the pragmatic rules I have learned from this book, I am now eager to recreate and capitalize on my successes so that whole classes can recognize their innate connection with that most daunting and elusive of subjects: Math.