Mathematics
A spaceship moving at an incredible rate.
CORE DECISIONS:
What: This mathematics lesson will build off of a science lesson in which students will have created a model solar system. We will use this science content as a context within which to explore the concept of rate. Chapin & Johnson (2006) define rate as the ratio we have “[w]hen two different types of measures are being compared.” In this lesson, the students will be comparing distance and time (a rate that we call velocity, or speed). I will introduce tables and number lines as useful tools for working with rate problems; these tools emphasize rates as ratios by directly comparing the two measurements visually. The ultimate goal will be for the students to realize that working with rates is multiplicative. Given a velocity and an amount of time, for instance, one can multiply to find the distance traveled. On the other hand, when one has a velocity and a distance, or a distance and amount of time, one can use division to find the missing information.
How: Along with the content, I want my students to have an opportunity to work collaboratively on meaningful and challenging math problems. However, discussion-based math learning is not a norm in my classroom, and thus could not be implemented without clear expectations for how it will look. Therefore, my lesson will follow two parallel progressions: as the students tackle more challenging material, they will also be expected to work more and more collaboratively. I will begin with a short introduction, during which I will tie-in the science lesson from the previous day to hook the students in. From there, I will lead some guided practice with a couple of word problems. It is during the guided practice that I will try to lay some of the groundwork for the collaborative work to follow, by employing some of the talk moves cited by Shindelar (2009). As the students move into collaborative group work, I will also attempt to do some “social coaching” as the need arises (Shindelar, 2009). The lesson will conclude with a wrap-up discussion centered on the final word problem. This will also serve as a final, formative assessment of the students’ understanding of the content.
Why: The students in my classroom have very little opportunity to work collaboratively, especially during math time. I do not expect to overturn established classroom practices and for a discussion-based, collaborative math class to suddenly flourish. Nonetheless, I do want to give the students a chance to experience it, and I also want to take the opportunity to practice this particular type of teaching. Furthermore, it pertains to my inquiry question about how to re-engage academically marginalized students (two such students will be in my small group). I would like my students to begin to see each other as assets within the classroom. In the context of math, I would like them to see that their peers may have different but effective ways of solving problems, and that solving math problems can be done collaboratively, not just individually.
I chose this particular content – rate as an expression of a ratio – because it simultaneously reviews a concept at which they have already looked while laying a foundation for concepts to be covered in-depth later on. These students have done some work with rate, but the results on their homework have been inconsistent, suggesting that they could benefit from further review. This area of focus, then, seems particularly relevant for my group of students.
PEDAGOGICAL FOCUS:
In line with my guiding question, a primary goal of this lesson will be to enact a more cooperative model of student interaction within this small group setting. I have designed tasks that I believe will foster such cooperation, as well as provide an opportunity to engage with challenging math in a interesting context. As they work on these tasks, I will seek to incorporate discussion into the lesson, and this will be my pedagogical focus.
LESSON PLAN:
Objectives: Students will be able to solve a word problem that compares different rates; collaborate with peers to solve problems; verbalize strategies for solving word problems. The Common Core State Standards for sixth grade mathematics addressed in this lesson:
- CCSS-M 6.RP 2 (“Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.”)
- CCSS-M 6.RP 3a (“Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables…”)
- CCSS-M 6.RP 3b (“Solve unit rate problems including those involving…constant speed…”)
Hook: We will begin by recalling the science lesson from the previous day, in which the students constructed a model solar system. I will ask one student to verbally summarize the lesson, as I draw a diagram of the planets on the chalkboard. We will recall the vast distances of the outer planets, remembering that, even in our tiny scale, we did not have enough space to fit the whole model; in reality, these distances are impossible to really comprehend. To transition into the math, I will tell the students that they will get the opportunity to act as space explorers later on, but first they have to do a little “training.” (8 minutes)
Guided instruction: I will begin by posing some easier problems to the group. These easy problems will not be related to the science lesson, but will be easily relatable for the students: “If you can bike 10 miles in an hour, how many miles can you bike in 2 hours? 5 hours?” Before they set to work, I will provide some strategies that may be useful for them in solving rate problems: a rate table, and a double number line. On the chalkboard, I will model how they look and how they may be used. The students will then have time to solve the problem independently, putting up their thumb to show they are finished. When all students are finished, I will ask one to come to the chalkboard and explain how they solved the problem. If students used different strategies, I may have one or two others do the same. Next, I will provide a rate problem with different information: “If you are biking somewhere 15 miles away, and it takes you three hours, how fast are you biking?” In both cases, I will allow time for all students to come to an answer. If students appear to need extra practice, I can pose a few more problems, following the same pattern.
After this, I will transition into the science tie-in. I will ask the students if speed has to be measured in miles per hour. If students say yes, then I will ask them why they think so. In the spirit of number talks, I will withhold judgment on the students’ thinking. Either way, I will begin to talk about how scientists often measure distances in space, because the distances are so vast. I will introduce the astronomical unit (AU, defined as 149,597,871 km), asking students why it might be beneficial to create a unit like this. A question like, “Would it be useful to measure the distance from here to New York City in inches?” might help them see why different units are needed for different distances. Then, to get them more comfortable using AUs as a unit, I will hand out a worksheet with a table that needs to be completed:
The students may choose to work independently or with peers. When completed, we will review the answers and ask how they came to them. (25 minutes)
Group work: Finally, the class will work on a difficult problem involving rate:
To encourage collaboration, I will only make two copies of this word problem. They will be placed in manila folders in the back of the room, on which “TOP SECRET” is written. I will have some extra copies with me, however, in case students decide they want their own copy to read off of. However, having only two copies available initially will necessarily lead the students to work in one or two groups. The wording of the problem also encourages this. I will provide the rest of the lesson for them to work on this problem (with the exception of five minutes at the end, which I will save for a discussion). As they are working, I will bounce between the groups, listening in and offering to clear up points of confusion. (15 minutes)
Closing: The students will reveal their answer and explain how they reached it. If there is time, I would also like to ask students how they felt about working as a group. I will collect their work, which will also serve as an assessment. (10 minutes)
Total time: 55-60 minutes
FINAL ANALYSIS:
This lesson was held on a Thursday afternoon, while the majority of the class was participating in a weekly dance class. My small group consisted of the five boys who choose not to do dance. Fortunately, we were able to use the regular classroom as a space. We pushed five desks together to create small ‘table,’ so that each student could see each other as well as the blackboard. In the group, I had two students with IEPs, one who has been recommended for an IEP meeting, and two relatively high-performing students. The first three students tend to need considerable support during math, while the other two do not.
A couple of unfortunate developments occurred in the lead-up to this lesson, which I think are worth mentioning. First, while I had intended for all five of these students to participate in my science lesson the previous day, only two of them were ultimately able to. As a result, we spent more time reviewing the science content from that lesson than I had planned. Seeking to turn the responsibility for discourse over to the students right away, I asked one of the students who was there to summarize the lesson; he did so very thoroughly, and I overheard one student say, “Aw man, I wish I could’ve been there.” By reviewing the model solar system we had constructed, all of the students were able to share at least some understanding of the vast planetary distances that would be incorporated into the math. The second development, briefly, is that the students ended up taking their Benchmark Assessment in Mathematics that morning. I worried that they would be burnt out on math for the day. Fortunately, that proved not to be the case (more on that later).
I designed this lesson to encourage collaboration, as well as reflection and communication. I did this for two reasons. First, I agree that “[r]eflecting and communicating are the processes through which understanding develops (Hiebert et al., 1997). Second, I felt that these emphases would increase the confidence and motivation of the students, especially the strugglers (Gilles, 2007). I developed tasks that I felt could allow room for these emphases in the lesson. Thus, I attempted to create word problems that the students would find either relevant (the biking problems) or exciting (the space problems). I specifically worded the final word problem to emphasis communication, referring to the students as an “elite team” of NASA engineers [emphasis added]. I especially liked the final word problem because I felt it represented something closer to the way that math is done in the real world: collaborating with fellow professionals to solve challenging problems.
Closing: The students will reveal their answer and explain how they reached it. If there is time, I would also like to ask students how they felt about working as a group. I will collect their work, which will also serve as an assessment. (10 minutes)
Total time: 55-60 minutes
FINAL ANALYSIS:
This lesson was held on a Thursday afternoon, while the majority of the class was participating in a weekly dance class. My small group consisted of the five boys who choose not to do dance. Fortunately, we were able to use the regular classroom as a space. We pushed five desks together to create small ‘table,’ so that each student could see each other as well as the blackboard. In the group, I had two students with IEPs, one who has been recommended for an IEP meeting, and two relatively high-performing students. The first three students tend to need considerable support during math, while the other two do not.
A couple of unfortunate developments occurred in the lead-up to this lesson, which I think are worth mentioning. First, while I had intended for all five of these students to participate in my science lesson the previous day, only two of them were ultimately able to. As a result, we spent more time reviewing the science content from that lesson than I had planned. Seeking to turn the responsibility for discourse over to the students right away, I asked one of the students who was there to summarize the lesson; he did so very thoroughly, and I overheard one student say, “Aw man, I wish I could’ve been there.” By reviewing the model solar system we had constructed, all of the students were able to share at least some understanding of the vast planetary distances that would be incorporated into the math. The second development, briefly, is that the students ended up taking their Benchmark Assessment in Mathematics that morning. I worried that they would be burnt out on math for the day. Fortunately, that proved not to be the case (more on that later).
I designed this lesson to encourage collaboration, as well as reflection and communication. I did this for two reasons. First, I agree that “[r]eflecting and communicating are the processes through which understanding develops (Hiebert et al., 1997). Second, I felt that these emphases would increase the confidence and motivation of the students, especially the strugglers (Gilles, 2007). I developed tasks that I felt could allow room for these emphases in the lesson. Thus, I attempted to create word problems that the students would find either relevant (the biking problems) or exciting (the space problems). I specifically worded the final word problem to emphasis communication, referring to the students as an “elite team” of NASA engineers [emphasis added]. I especially liked the final word problem because I felt it represented something closer to the way that math is done in the real world: collaborating with fellow professionals to solve challenging problems.
A student explains his work at the board while his peers look on.
I did sense that pre-existing classroom norms would be an obstacle in my pursuit of a more collaborative, communication-oriented math lesson, however. These students are used to a type of math instruction that is heavily oriented toward instrumental understanding and test preparation (Skemp, 1976). Therefore, I came into this lesson prepared to speak explicitly about (or directly model) what I meant when, for instance, I asked students to explain their work. This turned out to be only partially the case, however. As we worked on the bike problems, the students did an admirable job of standing at the blackboard in front of their peers and explaining their work, including both students with IEPs, whose confidence levels are quite low after years of academic struggles. In her comments, my Penn mentor noted that this was quite impressive, and was a testament to how comfortable the students must have felt to take that risk. The students’ tended only to verbalize their process, however, and not their reasoning. For example, one student explained that he “multiplied ten times two, and I got twenty.” He drew the numbers in the standard algorithm form and wrote the answer. When I probed for more, often by asking why they did this or that, they seemed to grow more uncomfortable. Because confidence and motivation were important foci of my lesson, I made sure to avoid keeping any particular student “on the spot” for too long.
As we moved into more difficult work with the rate table, however, the struggling students became a little more lost. As they worked independently on the rate table, I helped one student who was starting to lose some steam, and I found myself taking a very process-oriented approach with him. My instruction became focused on getting this student to the correct answer rather than building his mathematical thinking. Again, my guiding question for this assignment involved boosting the confidence of struggling learners — and this happens to be a particularly fragile student. What is interesting, though, is how quickly and easily I decided that focusing on process alone would keep his spirits up. I think this goes back to norms, and the understanding that both he and I have an internalized notion that the ‘point’ of doing math is to get the right answer. Along the same line, I maintained a commitment to ‘finishing’ the lesson despite the fact that the students were generally finding the content too hard. I did extend the time spent on guided practice to provide the students more scaffolding, but I ultimately pushed through to the final word problem. In a sense, the lesson superseded the actual mathematical understanding. I recognize both of these issues as potentially problematic in my math instruction.
As we moved into more difficult work with the rate table, however, the struggling students became a little more lost. As they worked independently on the rate table, I helped one student who was starting to lose some steam, and I found myself taking a very process-oriented approach with him. My instruction became focused on getting this student to the correct answer rather than building his mathematical thinking. Again, my guiding question for this assignment involved boosting the confidence of struggling learners — and this happens to be a particularly fragile student. What is interesting, though, is how quickly and easily I decided that focusing on process alone would keep his spirits up. I think this goes back to norms, and the understanding that both he and I have an internalized notion that the ‘point’ of doing math is to get the right answer. Along the same line, I maintained a commitment to ‘finishing’ the lesson despite the fact that the students were generally finding the content too hard. I did extend the time spent on guided practice to provide the students more scaffolding, but I ultimately pushed through to the final word problem. In a sense, the lesson superseded the actual mathematical understanding. I recognize both of these issues as potentially problematic in my math instruction.
The seating could have been structured more effectively.
I think another reason that this particular student (cited in the previous paragraph), as opposed to the other struggling learners, needed extra help from me is due to the seating arrangement. My pedagogical focus was to foster class discussion and collaboration, and I set up the desks so that the students could easily talk amongst each other. However, this student ended up next to a quiet student who prefers to work independently. Meanwhile, the other three students had already formed their own little group. That left this one young man all alone, so to speak. In hindsight, it may have benefited me (and him) to think more carefully about seating.
Yet overall, the students were admirable in their willingness to engage in discussion and interdependence with their peers. There were moments that reminded me how new it was for them (for instance, they would inevitably look at me when they explained their strategies at the chalkboard, rather than at their peers) but there were impressive moments, too. For example: as I turned the students loose to work on the final word problem, one of the high-performing students turned to one of the strugglers and, rather than ask the struggler if he needed help, asked the struggler to help him. This is indicative of the extent to which the students demonstrated compassion and a willingness to help each other learn, which can contribute largely toward promoting the confidence of the lower students (Gilles, 2007). Infusing discussion and teamwork into the lesson was essential in reaching this goal, yet it is hard to say whether it helped them with the content.
As I wrote before, the content proved to be too difficult. The ratio table ended up serving as a useful tool for the students, and they relied on it throughout the lesson. However, none of the students came close to the answer in the final problem, as can be seen from their work:
Yet overall, the students were admirable in their willingness to engage in discussion and interdependence with their peers. There were moments that reminded me how new it was for them (for instance, they would inevitably look at me when they explained their strategies at the chalkboard, rather than at their peers) but there were impressive moments, too. For example: as I turned the students loose to work on the final word problem, one of the high-performing students turned to one of the strugglers and, rather than ask the struggler if he needed help, asked the struggler to help him. This is indicative of the extent to which the students demonstrated compassion and a willingness to help each other learn, which can contribute largely toward promoting the confidence of the lower students (Gilles, 2007). Infusing discussion and teamwork into the lesson was essential in reaching this goal, yet it is hard to say whether it helped them with the content.
As I wrote before, the content proved to be too difficult. The ratio table ended up serving as a useful tool for the students, and they relied on it throughout the lesson. However, none of the students came close to the answer in the final problem, as can be seen from their work:
I had to walk them through it as we neared the end of our time. I believe this was the weakest component of the lesson. Yet, its difficulty may have served a purpose, too. As I wrote before, this was the end of the day after the students had already taken a long math exam. Yet, as my Penn mentor noted, the students worked hard for a full hour and I even had a student come up after we finished to ask follow up questions. Perhaps it was the difficulty of the problem that motivated them? I do believe that the students sensed a difficulty that they wanted to “resolve and discuss,” which Hiebert et al. (1997) note as a component of an important mathematical task.
I believe this raises an interesting question, which also serves as a useful transition into my reflection: Can a math lesson be successful if none of the students get the right answer? My Penn mentor thought the lesson was overwhelmingly a success, and yet a look at the students work reveals that they probably lacked the proper tools and conceptual understandings to tackle much of the content.
REFLECTION:
I ultimately agree with my Penn mentor that this lesson was successful, although it is likely true that I learned more than the students did. It is clear to me that these students are willing, or maybe even yearning, to engage in a more discussion-based and collaborative type of math lesson. Though it went against the established norms, this group of boys verbalized their strategies and asked for and offered help to one another. Moving forward, I would like to incorporate more of this into the whole group setting. I am fortunate also that this happens to be a particularly nice group of sixth graders. Though much of the usual middle school teasing and drama still plays out, I have seen tremendous moments of compassion from each of student in this class at different times. Hopefully, that will make future attempts to incorporate this type of pedagogy equally smooth.
Having written that, I must also acknowledge the fact that I reverted to a more traditional, teacher-centered and process-oriented mode of instruction when the students encountered difficulty. Going back toward easier content may have solved this, but I must also gain experience with, and trust in, maintaining my pedagogical focus in the face of difficulty. What do expert teachers do? How much do you let students struggle, and is there ever a time to just walk them through the problem to get to the right answer? These are questions that linger with me following the lesson.
Whatever the answers, it is evident to me that I must start at an ever more basic level when introducing more difficult concepts, such as rate. This is doubly true if I am enacting pedagogical techniques and tasks that shift the cognitive demand onto the students. Since this was the case, the students and I would likely have been better served with less demanding content like the fourth and fifth grade teacher cited by Oakes & Lipton (1999). I probably could have stuck with just the bicycle problems and the students would have been able to build a more solid understanding of the content.
PENN MENTOR NOTES:
I believe this raises an interesting question, which also serves as a useful transition into my reflection: Can a math lesson be successful if none of the students get the right answer? My Penn mentor thought the lesson was overwhelmingly a success, and yet a look at the students work reveals that they probably lacked the proper tools and conceptual understandings to tackle much of the content.
REFLECTION:
I ultimately agree with my Penn mentor that this lesson was successful, although it is likely true that I learned more than the students did. It is clear to me that these students are willing, or maybe even yearning, to engage in a more discussion-based and collaborative type of math lesson. Though it went against the established norms, this group of boys verbalized their strategies and asked for and offered help to one another. Moving forward, I would like to incorporate more of this into the whole group setting. I am fortunate also that this happens to be a particularly nice group of sixth graders. Though much of the usual middle school teasing and drama still plays out, I have seen tremendous moments of compassion from each of student in this class at different times. Hopefully, that will make future attempts to incorporate this type of pedagogy equally smooth.
Having written that, I must also acknowledge the fact that I reverted to a more traditional, teacher-centered and process-oriented mode of instruction when the students encountered difficulty. Going back toward easier content may have solved this, but I must also gain experience with, and trust in, maintaining my pedagogical focus in the face of difficulty. What do expert teachers do? How much do you let students struggle, and is there ever a time to just walk them through the problem to get to the right answer? These are questions that linger with me following the lesson.
Whatever the answers, it is evident to me that I must start at an ever more basic level when introducing more difficult concepts, such as rate. This is doubly true if I am enacting pedagogical techniques and tasks that shift the cognitive demand onto the students. Since this was the case, the students and I would likely have been better served with less demanding content like the fourth and fifth grade teacher cited by Oakes & Lipton (1999). I probably could have stuck with just the bicycle problems and the students would have been able to build a more solid understanding of the content.
PENN MENTOR NOTES:
WORKS CITED:
Chapin, S. H., & Johnson, A. (2006). Math Matters: Understanding the Math You Teach, Second Edition. Sausalito, CA: Math Solutions.
Gilles, R.M. (2007). Cooperative Learning: Integrating Theory and Practice. Thousand Oaks, CA: Sage Publishing, Inc.
Hiebert, J. et al. (1997). Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NH: Heinemann.
Oakes, J. & Lipton, M. (1999). Teaching to Change the World, (pp. 215-26). Boston: McGraw Hill.
Shindelar, A. (2009). Maintaining Mathematical Momentum through “Talk Moves.” In B. Herbel-Eisenmann & M. Cirillo (Eds.) Promoting purposeful discourse, (pp. 165-178). Reston, VA: NCTM.
Skemp, R. (1976). Relational Understanding and Instrumental Understanding.Mathematics Teaching, 77, pp. 22-26.