Part II: Impact on Math Planning and Instruction
In the previous section, I wrote of how my experience in the field this year took place within a context of high-stakes testing. The pressure for students to perform well on the PSSA undoubtedly impacted my Classroom Mentor’s pedagogical choices, just as it impacts those of many other teachers today. These choices often stood in contrast to the methods about which I was reading in my courses, and which I would like to incorporate into my future teaching practice. However, I also acknowledge that in the future, like this year, I will almost certainly be operating under the same pressure. In this section I will attempt to describe how I attempted, in this year of student teaching, to reconcile the ideal with the real, so to speak.
Let me return to where I began: the dailies. As I wrote in the Introduction, we were introduced on the first day of Math Methods to the idea that there are two possible student outcomes in math instruction, what Skemp (1976) calls “instrumental understanding” and “relational understanding.” The latter may bring “more immediate” and “more apparent” results, making it attractive for test preparation (Skemp, 1976, p. 27). However, Boaler (1999) showed that students who received instruction aimed at instrumental understanding gained a type of knowledge that was “nontransferable” and “useful for little more than taking tests” (p. 1). On the other hand, relational understanding could be described as “a capability and intellectual power that will transcend the boundaries of the classroom” (Boaler, 1999, p.2); surely that is preferable.
Let me return to where I began: the dailies. As I wrote in the Introduction, we were introduced on the first day of Math Methods to the idea that there are two possible student outcomes in math instruction, what Skemp (1976) calls “instrumental understanding” and “relational understanding.” The latter may bring “more immediate” and “more apparent” results, making it attractive for test preparation (Skemp, 1976, p. 27). However, Boaler (1999) showed that students who received instruction aimed at instrumental understanding gained a type of knowledge that was “nontransferable” and “useful for little more than taking tests” (p. 1). On the other hand, relational understanding could be described as “a capability and intellectual power that will transcend the boundaries of the classroom” (Boaler, 1999, p.2); surely that is preferable.
Artifact C
After establishing, then, that relational understanding was the better goal to pursue for our students, we began to look at instruction. My notes from our September 18th class show a rough outline of two types of math instruction (see Artifact C, right). The first, indicated by the red arrow, highlights some characteristics of what was referred to as “modal practice.” As you can see from the notes, instrumental understanding is listed as a typical outcome of this type of instruction. The second model, indicated by the green arrow, and fostered by the National Council for Teachers of Mathematics (NCTM), is characterized by an “increased emphasis on problem solving [and] communication” (Artifact C).
All of this is to explain why I wanted to change the way the dailies -- in this case the Daily Math Practice -- were taught. As we began to look at specific instructional strategies that have been shown to build relational understanding, I was inspired by two in particular. First, I knew I wanted to incorporate some semblance of a "number talk" into these lessons (Parrish, 2010). Specifically, I wanted to end the practice of calling on a student for an answer, saying whether he was right or wrong, and moving on. I wanted to talk through problems, having the students articulate their thinking even if they did not arrive at the right answer. Along the same lines, I wanted to incorporate more discourse into the lessons, discourse of the type described by Chapin, O'Conner, and Anderson (2003), with my students "able to make mathematical conjectures, present evidence, voice agreement and disagreement with the claims of others, and support their own positions" (p.5). In other words, I wanted to reroute the lines of communication that had already been established as the way the dailies were managed.
I faced some immediate barriers in orienting the Daily Math Practice lessons toward building relational understanding, however. For example, my Classroom Mentor allows the students to use calculators. As a result, most of the students use the calculators even to solve addition and subtraction problems. Although I knew that this would inhibit their opportunities to build their numerical reasoning skills, I sensed that removing access to the calculators would be too drastic a shift from the norm, at least at first. What I did instead was encourage the students to try a problem without the calculator, saying something like, "You can use the calculator if you'd like, but see if you can challenge yourself to solve it in your head." The first time I tried this, a few of the students (the higher-performing ones) did set their calculators down (field notes, 10/2013; see Appendix, Artifact D). However, I soon realized that unless I issued this challenge every time, the students' first instinct was to pick up their calculator. This was a battle that I did not want to fight, so I decided instead to keep my focus on promoting greater discourse.
To work toward this goal, I used Shindelar (2009) as a resource for some specific examples of "talk moves" aimed at encouraging student talk. The first and perhaps easiest technique to try was not settling for just an answer on problems that required more than just punching numbers into a calculator. Instead, I asked students to raise their hands if they had an answer and if they were willing to explain how they got it. This was a break from the typical question/answer/correct or incorrect paradigm previously at work during the Daily Math Practice lessons. Artifact E is a transcript from an early attempt at this promotion of discourse, an interaction between myself (RC) and a student (M):
All of this is to explain why I wanted to change the way the dailies -- in this case the Daily Math Practice -- were taught. As we began to look at specific instructional strategies that have been shown to build relational understanding, I was inspired by two in particular. First, I knew I wanted to incorporate some semblance of a "number talk" into these lessons (Parrish, 2010). Specifically, I wanted to end the practice of calling on a student for an answer, saying whether he was right or wrong, and moving on. I wanted to talk through problems, having the students articulate their thinking even if they did not arrive at the right answer. Along the same lines, I wanted to incorporate more discourse into the lessons, discourse of the type described by Chapin, O'Conner, and Anderson (2003), with my students "able to make mathematical conjectures, present evidence, voice agreement and disagreement with the claims of others, and support their own positions" (p.5). In other words, I wanted to reroute the lines of communication that had already been established as the way the dailies were managed.
I faced some immediate barriers in orienting the Daily Math Practice lessons toward building relational understanding, however. For example, my Classroom Mentor allows the students to use calculators. As a result, most of the students use the calculators even to solve addition and subtraction problems. Although I knew that this would inhibit their opportunities to build their numerical reasoning skills, I sensed that removing access to the calculators would be too drastic a shift from the norm, at least at first. What I did instead was encourage the students to try a problem without the calculator, saying something like, "You can use the calculator if you'd like, but see if you can challenge yourself to solve it in your head." The first time I tried this, a few of the students (the higher-performing ones) did set their calculators down (field notes, 10/2013; see Appendix, Artifact D). However, I soon realized that unless I issued this challenge every time, the students' first instinct was to pick up their calculator. This was a battle that I did not want to fight, so I decided instead to keep my focus on promoting greater discourse.
To work toward this goal, I used Shindelar (2009) as a resource for some specific examples of "talk moves" aimed at encouraging student talk. The first and perhaps easiest technique to try was not settling for just an answer on problems that required more than just punching numbers into a calculator. Instead, I asked students to raise their hands if they had an answer and if they were willing to explain how they got it. This was a break from the typical question/answer/correct or incorrect paradigm previously at work during the Daily Math Practice lessons. Artifact E is a transcript from an early attempt at this promotion of discourse, an interaction between myself (RC) and a student (M):
Artifact E
This brief exchange serves as a useful example of some of the difficulties I ran into early on. Reading the transcript shows that the student actually provided the right answer and even began to explain how he arrived at his answer in a way that indicated solid mathematical reasoning, having broken down 135 first into 100. I was careful to avoid saying whether he was correct or incorrect, because I wanted to shift the emphasis away from right and wrong, and toward talking through problems. Almost all of a sudden, though, he seemed to lose confidence in his thinking. He asked me what I meant by groups of ten, as though he was suddenly unsure that he had even understood the question. I tried to encourage him, and "revoice" his thinking (Shindelar, 2009, p. 175). Rather than return to his original answer, however, he seemed to take a guess. I again tried to guide him back to his original thinking, but he continued to express confusion. I tried to keep the discussion centered on the students by asking if anyone would like to help (but nobody raised their hand).
Again, this pattern of exchange was not uncommon in the first few weeks of my attempts to change the nature of the dailies. Students would often provide the right answer, or something close to it, but then falter when they were asked to explain how they arrived at it. In Artifact E, and in other similar instances, I began to believe that it was the students' not knowing whether they were right or wrong that made them hesitant and uneasy about explaining their thinking. After all, these students had previously had their answers greeted with a quick "good" or "no, not that" or something along those lines; in other words, they always knew where they stood. I believe that, when I did not tell them that they were correct (even when they were), they assumed that they were wrong and thus began to backtrack.
I decided to modify my teaching a bit in response to this phenomenon. I wanted to see if students felt more comfortable sharing their reasoning if they knew they had given a correct answer, trying to find more of a balance between what they were used to and what I had envisioned. Here is another transcript from a Daily Math Practice lesson a few weeks later, after making the adjustment:
Again, this pattern of exchange was not uncommon in the first few weeks of my attempts to change the nature of the dailies. Students would often provide the right answer, or something close to it, but then falter when they were asked to explain how they arrived at it. In Artifact E, and in other similar instances, I began to believe that it was the students' not knowing whether they were right or wrong that made them hesitant and uneasy about explaining their thinking. After all, these students had previously had their answers greeted with a quick "good" or "no, not that" or something along those lines; in other words, they always knew where they stood. I believe that, when I did not tell them that they were correct (even when they were), they assumed that they were wrong and thus began to backtrack.
I decided to modify my teaching a bit in response to this phenomenon. I wanted to see if students felt more comfortable sharing their reasoning if they knew they had given a correct answer, trying to find more of a balance between what they were used to and what I had envisioned. Here is another transcript from a Daily Math Practice lesson a few weeks later, after making the adjustment:
Artifact F
While this is a single incident and involves a different student than in Artifact E, it is indicative of the increased comfort that the students seemed to have in sharing how they got their answers. Whether this was due to their increasing familiarity with my expectations or with the change I had made in telling them whether or not their answer was correct I do not know for sure. However, I noticed that when I told the students their answer was correct, they were much more confident -- and patient -- when explaining their reasoning.
But what about the students who provided an incorrect answer? In many cases, I still asked them to explain how they got their answer so that they could find out where they went wrong. Interestingly (and surprising to me), the students in this case also seemed more willing to talk through their thinking about the problem without reverting to the uncertainty or backtracking that was typical before.
However, the transcript in Artifact F points to another issue I was facing in trying to make the Daily Math Practice more discourse-based, more like a number talk. I noticed that, once somebody had given a correct answer, stopped paying attention. The extra talk in the back of the classroom as student 'O' explained his reasoning was not unique to that particular instance. The focus on 'just getting the right answer on the paper' proved to be a sticky habit. So while there seemed to be progress, I still hoped for more. I wanted students who also got the right answer to ask themselves if they solved it a different way; on the other hand, I wanted those students who had an incorrect answer to not just replace it with the correct one, but to listen closely for an explanation that might clear up their confusion.
But what about the students who provided an incorrect answer? In many cases, I still asked them to explain how they got their answer so that they could find out where they went wrong. Interestingly (and surprising to me), the students in this case also seemed more willing to talk through their thinking about the problem without reverting to the uncertainty or backtracking that was typical before.
However, the transcript in Artifact F points to another issue I was facing in trying to make the Daily Math Practice more discourse-based, more like a number talk. I noticed that, once somebody had given a correct answer, stopped paying attention. The extra talk in the back of the classroom as student 'O' explained his reasoning was not unique to that particular instance. The focus on 'just getting the right answer on the paper' proved to be a sticky habit. So while there seemed to be progress, I still hoped for more. I wanted students who also got the right answer to ask themselves if they solved it a different way; on the other hand, I wanted those students who had an incorrect answer to not just replace it with the correct one, but to listen closely for an explanation that might clear up their confusion.
A student explains his work during my Term III math lesson
I will return to my work with the Daily Math Practice lessons at the end. Around this same time, however, I was planning and implementing my Term III lessons. Though my inquiry question was not the same then as it is now, I still took the opportunity to incorporate a discourse-based approach into my math lesson. In the rationale for my lesson plan I wrote that, while I did not "expect to overturn established classroom practices and for a discussion-based, collaborative math class to suddenly flourish," I did want the students "to see that their peers may have different but effective ways of solving problems" (Term III math lesson plan, 2012; see Appendix, Artifact G). In this lesson, I had five students who worked collaboratively to solve a difficult problem involving rate. To warm them up, however, they did some independent work and then shared how they got their answers, displaying their work on the board (see photo, right). The small group setting allowed me to implement some of the practices I envisioned for the Daily Math Practice with greater ease. The students were in fact eager to go up to the board and show their work, including one student with significant academic struggles who therefore rarely chooses to share. Although the content proved too difficult, I felt the lesson was a success in terms of implementing the pedagogical strategies I envisioned (a more in-depth analysis of the lesson can be found here, along with my Penn Mentor's observations). After this lesson, I felt an invigorated sense that I could in fact work away from the teacher-centered "modal practice" toward a math practice that would build relational understanding.
By the end of the fall semester, then, I had seen some progress in my Daily Math Practice and felt my Term III lesson had been a success. Further, some of my Penn Mentor's notes in the Descriptive Profile bolstered my confidence even more in terms of implementing a new approach to math in the classroom. She used words like "creative," "student centered," and "authentic" to describe my lessons. Parts of my Classroom Mentor's Descriptive Profile, however, offered a reminder of what was to come:
By the end of the fall semester, then, I had seen some progress in my Daily Math Practice and felt my Term III lesson had been a success. Further, some of my Penn Mentor's notes in the Descriptive Profile bolstered my confidence even more in terms of implementing a new approach to math in the classroom. She used words like "creative," "student centered," and "authentic" to describe my lessons. Parts of my Classroom Mentor's Descriptive Profile, however, offered a reminder of what was to come:
As you can see, my Classroom Mentor was delineating between the math work that I had been doing and the work that was to come: preparing the students for the PSSA. Beginning a few weeks later, we embarked on a race to cover all of the eligible content before the test began in April. Whereas before we had been using Math in Context as our math curriculum, we began to use Measuring Up, a curriculum specifically designed to prepare students for the PSSA. The structure of the lessons themselves made it difficult to implement any of the pedagogical strategies I had been trying before, since their very layout emphasized teacher-led direct instruction followed by assessment pages in which students regurgitate the information, often in true/false or multiple-choice format (see Appendix, Artifact I). Additionally, I was sensitive to the pressure that my Classroom Mentor was feeling and did not feel it was appropriate but to help her in a push to prepare the students as best as possible for the test.
That said, I continued to use the Daily Math Practice lessons to pursue my pedagogical goals of a math instruction geared toward relational understanding by continuing to expect increased communication. As the year went on, the students grew more and more accustomed to explaining their path to an answer. They began to give their explanations without my provocation. Furthermore, sometime around early March, students began raising their hand to say that they had done it a different way, and wanting to share their own reasoning (field notes, 2013; see Appendix, Artifact J). Ultimately, I believe I had seen enough progress to know that achieving new pedagogical math goals was possible, but I felt keenly aware of how the PSSA had limited my opportunities for pursuing them. Yet, much of my acquiescence to the test prep reality was due to the fact that this was, in the end, not my classroom. I was a student teacher. What, then, are the implications for my future practice?
(References are cited here.)
That said, I continued to use the Daily Math Practice lessons to pursue my pedagogical goals of a math instruction geared toward relational understanding by continuing to expect increased communication. As the year went on, the students grew more and more accustomed to explaining their path to an answer. They began to give their explanations without my provocation. Furthermore, sometime around early March, students began raising their hand to say that they had done it a different way, and wanting to share their own reasoning (field notes, 2013; see Appendix, Artifact J). Ultimately, I believe I had seen enough progress to know that achieving new pedagogical math goals was possible, but I felt keenly aware of how the PSSA had limited my opportunities for pursuing them. Yet, much of my acquiescence to the test prep reality was due to the fact that this was, in the end, not my classroom. I was a student teacher. What, then, are the implications for my future practice?
(References are cited here.)