Lesson Analysis
In this section, I will analyze the lesson according to the four dimensions of analysis from Staub et al.: Tasks, Discourse, Tools, and Norms. This analysis will be done both from the perspective of my own decision-making – much of which was improvised during the lesson itself – and from a more reflective perspective, looking at how the lesson as implemented compared to my mathematical, pedagogical, and philosophical goals (many of which are outlined in the Core Decisions analysis). I will conclude with an analysis of student learning as observed in the lesson.
Tasks
During my initial planning, the target mathematical objectives for the lesson began with basic understandings related to the measurement of length, including the use of a ruler and concepts of whole- and half-inch units. It built upon this to extend into more complex concepts of measurement, involving combination and iteration of measurements and the measurement of objects that exceed the length of measurement tools. In retrospect, although I had reviewed the chapter in Chapin and Johnson (2006) which elaborates on concepts of measurement prior to writing the lesson plan, I should have placed more of a focus on decomposing the understandings described above into the concepts laid out by Chapin and Johnson (e.g. conservation, transitivity, etc.). Upon beginning the lesson, it quickly became clear that the students were struggling with the broader concepts of the lesson, and therefore I was forced to deconstruct them in a fairly ad hoc manner, and find ways to teach students various skills and concepts related to basic measurement. Initially, I emphasized techniques for using a ruler and measurement of half-inches; each of these, I quickly discovered, needed to be considered in terms of even more fundamental concepts as well. For use of a ruler, I focused in on the principle of conservation (in order to show that placement relevant to the ruler doesn’t change the length of the object, and thus, if you line it up improperly, the label it lines up with does not accurately reflect the object’s length). For measuring half inches, I focused on the idea of partitioning whole units to create fractions.
In the lesson as implemented, there were essentially five “tasks” (defined loosely because, since I was mostly improvising and reacting to the students, some of the tasks evolved and incorporated multiple parts):
Prior to these, I had done my introduction, as described in the Revised Lesson Plan. While I think some of my attempts to connect with their personal experiences were a bit clumsy, and while I clearly made a management mistake by handing out the rulers before this section, I think this served as a fairly effective introduction.
Tasks
During my initial planning, the target mathematical objectives for the lesson began with basic understandings related to the measurement of length, including the use of a ruler and concepts of whole- and half-inch units. It built upon this to extend into more complex concepts of measurement, involving combination and iteration of measurements and the measurement of objects that exceed the length of measurement tools. In retrospect, although I had reviewed the chapter in Chapin and Johnson (2006) which elaborates on concepts of measurement prior to writing the lesson plan, I should have placed more of a focus on decomposing the understandings described above into the concepts laid out by Chapin and Johnson (e.g. conservation, transitivity, etc.). Upon beginning the lesson, it quickly became clear that the students were struggling with the broader concepts of the lesson, and therefore I was forced to deconstruct them in a fairly ad hoc manner, and find ways to teach students various skills and concepts related to basic measurement. Initially, I emphasized techniques for using a ruler and measurement of half-inches; each of these, I quickly discovered, needed to be considered in terms of even more fundamental concepts as well. For use of a ruler, I focused in on the principle of conservation (in order to show that placement relevant to the ruler doesn’t change the length of the object, and thus, if you line it up improperly, the label it lines up with does not accurately reflect the object’s length). For measuring half inches, I focused on the idea of partitioning whole units to create fractions.
In the lesson as implemented, there were essentially five “tasks” (defined loosely because, since I was mostly improvising and reacting to the students, some of the tasks evolved and incorporated multiple parts):
- Using the ruler
- Inches and half inches
- Conservation
- Independent measurement
- Discussion of measurement strategies
Prior to these, I had done my introduction, as described in the Revised Lesson Plan. While I think some of my attempts to connect with their personal experiences were a bit clumsy, and while I clearly made a management mistake by handing out the rulers before this section, I think this served as a fairly effective introduction.
Math - lead-in.avi from Jesse Gottschalk on Vimeo.
1: Using the ruler
My goals for this section were 1) to assess whether the group understood how to use a ruler properly, and 2) to provide a mini-lesson as needed to support them if that understanding was lacking.
For this, I used two different materials from my tool kit – a pencil and a fake dollar bill – and asked different students to measure each. When the first student struggled with his measurement, I led a brief mini-lesson on the use of a ruler. (Note: this clip follows one of the students making the same mistake that I am modeling here.)
My goals for this section were 1) to assess whether the group understood how to use a ruler properly, and 2) to provide a mini-lesson as needed to support them if that understanding was lacking.
For this, I used two different materials from my tool kit – a pencil and a fake dollar bill – and asked different students to measure each. When the first student struggled with his measurement, I led a brief mini-lesson on the use of a ruler. (Note: this clip follows one of the students making the same mistake that I am modeling here.)
Math - Ruler use.avi from Jesse Gottschalk on Vimeo.
While this was not a particularly engaging lesson (students were quite distracted in parts, although they also seemed to tune in at some critical points), I wanted this part of the lesson to be both hands-on and also visible for all students. In reflection, there was a much better way that I could have accomplished these goals: I could have given each pair one pencil and one ruler, and asked them to decide together how long the pencil was. If both pairs had answered it correctly, I could have challenged the individual students who I suspected were struggling more with the concepts to explain how they and their partners had gotten their correct answers; if one pair had answered it correctly but the other pair had gotten it wrong, then that could have been the opportunity for them to engage in trying to explain the principles they applied to measuring, which could have helped me assess their understandings more quickly and deeply than I accomplished in the actual lesson.
2: Inches and half-inches
Because this was a relatively new, important, and complex concept for some of the students (only one out the four of them seemed to have a strong degree of familiarity with half-units), it seems pretty clear that this was deserving of its own lesson with much more attention and a variety of tasks to explore it. Students could have practiced measuring divisible objects to become more comfortable with the idea of halves, before being introduced to the idea that inches can be divided into halves. However, this also carries embedded in it the concept of inches as a continuous unit of measurement which can be partitioned into subunits (Chapin & Johnson 101), which I did not sufficiently explain. If nothing else, I should have been better equipped to explain during the discussion of ruler use that inches are representative of the space between marks rather than the marks themselves, and should have made sure that all students were comfortable with that concept before moving on to partitioning.
2: Inches and half-inches
Because this was a relatively new, important, and complex concept for some of the students (only one out the four of them seemed to have a strong degree of familiarity with half-units), it seems pretty clear that this was deserving of its own lesson with much more attention and a variety of tasks to explore it. Students could have practiced measuring divisible objects to become more comfortable with the idea of halves, before being introduced to the idea that inches can be divided into halves. However, this also carries embedded in it the concept of inches as a continuous unit of measurement which can be partitioned into subunits (Chapin & Johnson 101), which I did not sufficiently explain. If nothing else, I should have been better equipped to explain during the discussion of ruler use that inches are representative of the space between marks rather than the marks themselves, and should have made sure that all students were comfortable with that concept before moving on to partitioning.
Math - Halves from Jesse Science on Vimeo.
3: Conservation
Another centralized, guided discussion, although this one was a bit more teacher-centric. The need for this came out of my informal assessment of the student – watch the clip for a sense of how I transitioned into it (pay close attention to the final seconds).
Another centralized, guided discussion, although this one was a bit more teacher-centric. The need for this came out of my informal assessment of the student – watch the clip for a sense of how I transitioned into it (pay close attention to the final seconds).
Math - conservation 1 from Jesse Gottschalk on Vimeo.
“It’s because your head is not made of a ruler, so you don’t really know.” What an amazing encapsulation of this student’s current understanding! To me, this not only shows a lack of a sense of conservation - it also shows a lack of understanding of the connection between measurable features (e.g., length), measurement tools (e.g., ruler), and the actual act and purpose of measurement. To explore and push further on this area, I decided to attempt to activate a piece of knowledge that I assumed (perhaps wrongly) most of the students would have at least an idea about – their own heights. I chose this because it seemed like the most likely measurement that they would have memorized, which therefore allowed me to ask “are you still that height when you’re not being measured?”
Math - Conservation 2 from Jesse Science on Vimeo.
I thought at the time that this was a fairly effective framing, although some of the students still seemed to struggle with conservation afterwards. In reflection, however, this didn’t seem to be a particularly engaging discussion – perhaps it would have worked better with a hands-on component, maybe even involving having the students actually measure each other.
In fact, this could have been a better way to introduce the concept of conservation; assuming this came after students had had practice measuring lengths, the activity could have gone something like this: I could have had a measuring tape posted on the wall, and had students measure each other (perhaps I should have them measure me first, in order to make sure everyone sees the proper technique). When a student told me they were four feet tall, I could have said, “nonsense, you’re only one foot tall.” When they argued with me, I could have shown them another measuring tape, starting two feet off the ground; according to this measurement, they would have only been two feet tall. Then they could have talked to each other and finally had to explain to me whether and why they were still four feet tall. To reinforce this prior to their independent measurements, I could have shown them images of an object aligned and misaligned with a ruler, and asked them to choose the ones that were properly aligned and determine the proper lengths (or done this in the form of a worksheet).
4. Independent measurement
By the time I finally returned to my originally prepared measurement task, I was sufficiently distracted that I unfortunately forgot about the framing scenario I had cooked up to accompany it (described in the revised lesson plan) – although by this point, we were distanced enough from my carpentry example that it probably would have come off more as a gimmick than an authentic task. Still, the task certainly came off as fairly dry and more frustrating than engaging for some of the students – at least at first. With additional direction and suggestion from the teacher, some of the students eventually overcame frustration and were successful; others did a lot of measuring but wrote little or nothing on their graphic organizers. When one student finished early, I suggested that she find other objects that could match the given lengths on her paper; when she finished that, I suggested that she help her partner by finding objects in the
It was only during this section that I remembered another concept which I had identified as valuable in my Core Decisions, but which I had neglected to prepare to teach explicitly: estimation. I tried to present this as a skill for students who were struggling to fill in their work-sheets, and this new strategy was used with success by the students I shared it with (the student in this video had been struggling for several minutes to find a four-inch object).
In fact, this could have been a better way to introduce the concept of conservation; assuming this came after students had had practice measuring lengths, the activity could have gone something like this: I could have had a measuring tape posted on the wall, and had students measure each other (perhaps I should have them measure me first, in order to make sure everyone sees the proper technique). When a student told me they were four feet tall, I could have said, “nonsense, you’re only one foot tall.” When they argued with me, I could have shown them another measuring tape, starting two feet off the ground; according to this measurement, they would have only been two feet tall. Then they could have talked to each other and finally had to explain to me whether and why they were still four feet tall. To reinforce this prior to their independent measurements, I could have shown them images of an object aligned and misaligned with a ruler, and asked them to choose the ones that were properly aligned and determine the proper lengths (or done this in the form of a worksheet).
4. Independent measurement
By the time I finally returned to my originally prepared measurement task, I was sufficiently distracted that I unfortunately forgot about the framing scenario I had cooked up to accompany it (described in the revised lesson plan) – although by this point, we were distanced enough from my carpentry example that it probably would have come off more as a gimmick than an authentic task. Still, the task certainly came off as fairly dry and more frustrating than engaging for some of the students – at least at first. With additional direction and suggestion from the teacher, some of the students eventually overcame frustration and were successful; others did a lot of measuring but wrote little or nothing on their graphic organizers. When one student finished early, I suggested that she find other objects that could match the given lengths on her paper; when she finished that, I suggested that she help her partner by finding objects in the
It was only during this section that I remembered another concept which I had identified as valuable in my Core Decisions, but which I had neglected to prepare to teach explicitly: estimation. I tried to present this as a skill for students who were struggling to fill in their work-sheets, and this new strategy was used with success by the students I shared it with (the student in this video had been struggling for several minutes to find a four-inch object).
Math - Four inches from Jesse Science on Vimeo.
In reflecting on this lesson, there are quite a few ways the task could have been better-constructed to help students meet the goal of practicing measurement and building familiarity with various lengths. First, I should have used many fewer objects. I had provided a large number, with the thinking that it would be useful for the later activities, and because I thought this task otherwise might be too easy for the students; in retrospect, I should have limited the “distractor” objects, and given each student a much more narrow selection. Second, I should have explicitly presented estimation as a skill to use in this task – perhaps I should have had each student use the ruler and show me how to demonstrate one inch with their finger, then asked the students to tell me which of several objects (without using a ruler) was one inch long. On the one hand, this is in essence teaching an entirely separate concept from the others focused on in this lesson – transitivity – so perhaps this would have been too complicated to introduce here. On the other hand, without this skill the activity becomes much harder, so presenting it as an option to help students seems like it would be worthwhile (and provide a new dimension for me to assess). Third, this task did not require any partner work, which I would like to change. It was designed to be like the tasks I used in my Science lesson, where students did brief independent work and then were challenged to build on each others’ work while working in partners; however, the independent work ended up taking a long time and the partner work didn’t end up happening.
Here’s one possibility for how I could change the activity going forward, to better meet the goal of practicing measuring while also encouraging collaboration and discussion, and hopefully being more engaging: Each student gets an identical list, which includes the lengths of all the assembled objects – but each student only gets half of the objects. Students cannot measure their partner’s objects (although they can observe and provide advice); however, both students must record all the objects on their sheets, including their partners’ objects.
5. Discussion
I think this discussion – in which I asked students to name strategies that had been useful to them – could have been more valuable with two circumstances changed. First, if the previous activity had been done in partners, then they would presumably have observed more strategies and had a broader selection to choose from in sharing, as well as perhaps a deeper understanding of the strategies they shared (whether from learning from their partners or from the experience of sharing their reasoning with their partners). Second, the students – who were fairly distracted at many points during the lesson, as you have probably noticed in the videos – were not particularly engaged, especially after the excessively long and in some cases frustrating independent work period, which I’m sure made them less interested in focusing on sharing or listening during this discussion. With a more tight, engaging lesson, this hopefully would have been different as well.
Here’s one possibility for how I could change the activity going forward, to better meet the goal of practicing measuring while also encouraging collaboration and discussion, and hopefully being more engaging: Each student gets an identical list, which includes the lengths of all the assembled objects – but each student only gets half of the objects. Students cannot measure their partner’s objects (although they can observe and provide advice); however, both students must record all the objects on their sheets, including their partners’ objects.
5. Discussion
I think this discussion – in which I asked students to name strategies that had been useful to them – could have been more valuable with two circumstances changed. First, if the previous activity had been done in partners, then they would presumably have observed more strategies and had a broader selection to choose from in sharing, as well as perhaps a deeper understanding of the strategies they shared (whether from learning from their partners or from the experience of sharing their reasoning with their partners). Second, the students – who were fairly distracted at many points during the lesson, as you have probably noticed in the videos – were not particularly engaged, especially after the excessively long and in some cases frustrating independent work period, which I’m sure made them less interested in focusing on sharing or listening during this discussion. With a more tight, engaging lesson, this hopefully would have been different as well.
Math - Discussion from Jesse Science on Vimeo.
Discourse
First of all, as my analysis of Tasks as well as my Core Decisions make clear, I am fairly sold on the value of making student discussion, collaboration, and argumentation central to the lesson – not only because I believe that students learn more effectively with these, but because it is more engaging and encourages all students to be active learners. To some extent, all of the changes I consider in the Tasks section would serve to promote student participation in active discourse that is student-centric. As I believe is evident in both the videos from the Tasks section and my analysis, there was tremendous room for improvement in this regard in the lesson as it was implemented.
Perhaps most importantly, the discussions were predominantly teacher-centric. Although I asked a lot of questions, most of them were “fill-in-the-blanks” variety; this was symptomatic of the whole lesson, which I fear ended up deficient in “worthwhile tasks” (although I believe that some of my alterations in the Tasks section could make the lessons better fit the sort of tasks recommended by Hiebert et al. (1997), in terms of “encouraging reflection and communication” and hopefully “leaving behind important residue”[i]), and consequently was likely quite discouraging of more active participation in thoughtful discussion. Meanwhile, I spent a lot of time helping students answer my questions; while I believe that my use of revoicing was valuable, I took it too far in terms of adding my own contributions to my restatements of student thoughts. I believe that one of the reasons I did this was because I kept subconsciously trying to get us back on track for my prepared lesson, and thus wanted to find ways to get the students to the necessary concepts faster – which, needless to say, is essentially against the principles of learning that have been central to our Math Methods class. The other reason that I suspect I made my voice dominant was because students did not seem to be paying attention to each other, and to some extent they were paying closer attention to me; on some occasions, I tried to reinforce the classroom norms of paying attention to each other, but in others I think I chose to speak to ensure that I would get some higher degree of students hearing the concepts. This was likely self-reinforcing, as it presumably set up the expectation that if something stated was truly important, I would repeat it. Rather than dominating the conversations, I believe I could have addressed these two factors mentioned above – the first, by having better prepared for the lessons I ended up teaching (although improvisation will, I’m sure, always be a skill I need to draw on, so encouraging student ownership of discussion while improvising is a skill that I need to attentively practice); the second, by finding other ways to get the students more engaged (primarily by having more engaging tasks!).
Below is a video clip that I think is worth watching; it shows several of the challenges to discourse (student distraction, my being dominant, me specifically calling a student’s answer wrong), but it also shows a bit of teacherly self-awareness and adaptation (note how I take advantage of the disciplinary pause at 0:32 to self-correct, shifting from explaining to asking for explanation).
First of all, as my analysis of Tasks as well as my Core Decisions make clear, I am fairly sold on the value of making student discussion, collaboration, and argumentation central to the lesson – not only because I believe that students learn more effectively with these, but because it is more engaging and encourages all students to be active learners. To some extent, all of the changes I consider in the Tasks section would serve to promote student participation in active discourse that is student-centric. As I believe is evident in both the videos from the Tasks section and my analysis, there was tremendous room for improvement in this regard in the lesson as it was implemented.
Perhaps most importantly, the discussions were predominantly teacher-centric. Although I asked a lot of questions, most of them were “fill-in-the-blanks” variety; this was symptomatic of the whole lesson, which I fear ended up deficient in “worthwhile tasks” (although I believe that some of my alterations in the Tasks section could make the lessons better fit the sort of tasks recommended by Hiebert et al. (1997), in terms of “encouraging reflection and communication” and hopefully “leaving behind important residue”[i]), and consequently was likely quite discouraging of more active participation in thoughtful discussion. Meanwhile, I spent a lot of time helping students answer my questions; while I believe that my use of revoicing was valuable, I took it too far in terms of adding my own contributions to my restatements of student thoughts. I believe that one of the reasons I did this was because I kept subconsciously trying to get us back on track for my prepared lesson, and thus wanted to find ways to get the students to the necessary concepts faster – which, needless to say, is essentially against the principles of learning that have been central to our Math Methods class. The other reason that I suspect I made my voice dominant was because students did not seem to be paying attention to each other, and to some extent they were paying closer attention to me; on some occasions, I tried to reinforce the classroom norms of paying attention to each other, but in others I think I chose to speak to ensure that I would get some higher degree of students hearing the concepts. This was likely self-reinforcing, as it presumably set up the expectation that if something stated was truly important, I would repeat it. Rather than dominating the conversations, I believe I could have addressed these two factors mentioned above – the first, by having better prepared for the lessons I ended up teaching (although improvisation will, I’m sure, always be a skill I need to draw on, so encouraging student ownership of discussion while improvising is a skill that I need to attentively practice); the second, by finding other ways to get the students more engaged (primarily by having more engaging tasks!).
Below is a video clip that I think is worth watching; it shows several of the challenges to discourse (student distraction, my being dominant, me specifically calling a student’s answer wrong), but it also shows a bit of teacherly self-awareness and adaptation (note how I take advantage of the disciplinary pause at 0:32 to self-correct, shifting from explaining to asking for explanation).
Tools
In some ways, this lesson was tool-centric. Taken in its entirety, the lesson wasn’t just intended to teach the use of a measurement tool (rulers) – it meant to use measurement tools as a means for teaching a deeper understanding of measurement concepts. The use of the ruler was meant to promote concepts of continuity and unit iteration; the more advanced measuring tasks were meant to ultimately expand upon the concept of iteration and promote a more abstract understanding of measurement. Students would eventually learn to use a ruler to measure objects that seemingly extend beyond the capacity of the ruler to measure.
As implemented, however, I didn’t get to any of those more advanced concepts. I did, however, try to let the concepts and the tool use inform each other; thus, I focused explicitly on teaching how to use a ruler (asking students how to line up an object properly), but then circled back to the relevant concepts and used them to explain more theoretically why the ruler should be used in this way (showing that an object doesn’t change length when measured improperly), thus hopefully providing an applied context for their understanding that would deepen comprehension and simultaneously strengthen their understanding of proper tool use.
Norms
Because I had originally seen this as a predominantly partnership-driven lesson, the norms that I had spent the most time considering were those regarding respectful and productive collaboration – norms such as active participation, attentive listening, thoughtful reflection, and honest but respectful argumentation. I also wrote in my Lesson Plan that “These same conversational norms will apply in full-group discussions.”
However, I had originally anticipated introducing these norms explicitly while introducing the first partner activity (since, prior to that, I hadn’t been accounting for much discussion); because that activity never happened, I failed to find an appropriate time to explicitly lay out the norms I was expecting. I think that such a direct outline of norms would have been extremely valuable, particularly because our classroom has otherwise allowed for very little collaboration, and has had little recent discussion of classroom norms (other than in the form of teacher discipline); therefore, I feel it would have been important to make explicit that the expectations on them were different from their normal classroom expectations, and that their success would rely on their taking responsibility within the activity, and on their active and respectful collaboration with one another.
Instead of giving a general outline of normative expectations, I was left using verbal reinforcement to address significant breaches of my own expectations – singling out distracted students to ask them to pay attention to one another and to focus on the necessary tasks. While this usually worked with individual cases, the fact that it was repeatedly needed emphasizes for me that I should have found more ways to address this on a whole-class basis. Further, I am concerned that the extent to which I tended to dominate conversation, revoicing and building upon much of what students said, may have undermined my attempts to get students to speak and listen to one another.
Student Learning
Unfortunately, my assessment checklist proved to be of little value – nearly all components of it related to parts of the lesson that I didn’t end up teaching. The only items on the checklist which were relevant were the first three: “Uses ruler properly”; “Success measuring integer units, 1-12”; and “Success measuring fractional units, ½-12.” However, as I discussed in the Tasks section, I ultimately ended up focusing on a variety of categories of understanding subordinate to these items; therefore, I didn’t feel that any of these items could really capture student understanding. I also didn’t have any assessments well-crafted to carefully assess any of the subordinate concepts. However, there were ways that I can assess the students’ graphic organizers and the class discussion to draw some conclusions about student understanding.
In some ways, this lesson was tool-centric. Taken in its entirety, the lesson wasn’t just intended to teach the use of a measurement tool (rulers) – it meant to use measurement tools as a means for teaching a deeper understanding of measurement concepts. The use of the ruler was meant to promote concepts of continuity and unit iteration; the more advanced measuring tasks were meant to ultimately expand upon the concept of iteration and promote a more abstract understanding of measurement. Students would eventually learn to use a ruler to measure objects that seemingly extend beyond the capacity of the ruler to measure.
As implemented, however, I didn’t get to any of those more advanced concepts. I did, however, try to let the concepts and the tool use inform each other; thus, I focused explicitly on teaching how to use a ruler (asking students how to line up an object properly), but then circled back to the relevant concepts and used them to explain more theoretically why the ruler should be used in this way (showing that an object doesn’t change length when measured improperly), thus hopefully providing an applied context for their understanding that would deepen comprehension and simultaneously strengthen their understanding of proper tool use.
Norms
Because I had originally seen this as a predominantly partnership-driven lesson, the norms that I had spent the most time considering were those regarding respectful and productive collaboration – norms such as active participation, attentive listening, thoughtful reflection, and honest but respectful argumentation. I also wrote in my Lesson Plan that “These same conversational norms will apply in full-group discussions.”
However, I had originally anticipated introducing these norms explicitly while introducing the first partner activity (since, prior to that, I hadn’t been accounting for much discussion); because that activity never happened, I failed to find an appropriate time to explicitly lay out the norms I was expecting. I think that such a direct outline of norms would have been extremely valuable, particularly because our classroom has otherwise allowed for very little collaboration, and has had little recent discussion of classroom norms (other than in the form of teacher discipline); therefore, I feel it would have been important to make explicit that the expectations on them were different from their normal classroom expectations, and that their success would rely on their taking responsibility within the activity, and on their active and respectful collaboration with one another.
Instead of giving a general outline of normative expectations, I was left using verbal reinforcement to address significant breaches of my own expectations – singling out distracted students to ask them to pay attention to one another and to focus on the necessary tasks. While this usually worked with individual cases, the fact that it was repeatedly needed emphasizes for me that I should have found more ways to address this on a whole-class basis. Further, I am concerned that the extent to which I tended to dominate conversation, revoicing and building upon much of what students said, may have undermined my attempts to get students to speak and listen to one another.
Student Learning
Unfortunately, my assessment checklist proved to be of little value – nearly all components of it related to parts of the lesson that I didn’t end up teaching. The only items on the checklist which were relevant were the first three: “Uses ruler properly”; “Success measuring integer units, 1-12”; and “Success measuring fractional units, ½-12.” However, as I discussed in the Tasks section, I ultimately ended up focusing on a variety of categories of understanding subordinate to these items; therefore, I didn’t feel that any of these items could really capture student understanding. I also didn’t have any assessments well-crafted to carefully assess any of the subordinate concepts. However, there were ways that I can assess the students’ graphic organizers and the class discussion to draw some conclusions about student understanding.
The graphic organizers reinforce my informal assessments of student understandings: two of the students, by the end of the lesson at least, were fairly comfortable using the ruler to measure objects, while two had much greater challenges. Interestingly, the boy who only wrote one item on his worksheet did in fact find more items than this – his partner helped him find a 2-inch object, which he showed to me. The fact that he didn’t write it seems to speak to his comfort with the writing task (he is often reluctant to take part in writing tasks in the classroom), not just to his success with measurement (although clearly he struggled with this as well). This case also testifies to the potential value of partner work: if the student had been collaborating with someone more comfortable with writing, it would have taken some of the pressure off of the student and hopefully freed him up to focus more on the mathematics. Overall, the graphic organizers seem to give me a more general impression of student facility with basic measurement, rather than any more specific determination of student facility with subordinate concepts.
When presented with questions related to conservation, three of the four students seemed to grasp the concept (although, lacking a task to directly assess their understanding of this concept, it is possible that they were simply agreeing with their peers); interestingly, the one who continued to vocally disagree when asked if object lengths remained constant when measured differently was the same student who was most successful in the measurement task. I believe that, in her case, her comfort with measurement and relevant mathematical tasks were in conflict with this (and possibly other) fundamental misapprehensions; therefore, providing her with authentic tasks aimed to help her confront these misconceptions could be valuable at helping her move on to significantly more complex measurement concepts. At the same time, such tasks would also benefit the other students, even though they did not evidence struggle with conservation specifically to the same extent – I hope that the sorts of alternative tasks that I outlined in the Tasks section would focus in more on specific skills in a way that would be more assessable.
Meanwhile, at various points at least three of the students demonstrated a decent understanding of the concept of a “half,” though only one of them did so initially. Though the students disagreed to some extent on how to measure one half on a ruler (two of them pointed to the ½ mark, one pointed to the 1½ mark), these three all seemed to have a basic understanding of the meaning of ½, though clearly one that could be strengthened and reinforced (by the end of the discussion on halves, all three had come into agreement regarding the ½ mark).
References:
Chapin, S.H. & Johnson, A. (2006). Math Matters: Grades K-8: Understanding the Math You Teach (2nd edition). Sausalito, CA: Math Solutions.
Hiebert, J. et al. (2007). Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NJ: Heinemann.
When presented with questions related to conservation, three of the four students seemed to grasp the concept (although, lacking a task to directly assess their understanding of this concept, it is possible that they were simply agreeing with their peers); interestingly, the one who continued to vocally disagree when asked if object lengths remained constant when measured differently was the same student who was most successful in the measurement task. I believe that, in her case, her comfort with measurement and relevant mathematical tasks were in conflict with this (and possibly other) fundamental misapprehensions; therefore, providing her with authentic tasks aimed to help her confront these misconceptions could be valuable at helping her move on to significantly more complex measurement concepts. At the same time, such tasks would also benefit the other students, even though they did not evidence struggle with conservation specifically to the same extent – I hope that the sorts of alternative tasks that I outlined in the Tasks section would focus in more on specific skills in a way that would be more assessable.
Meanwhile, at various points at least three of the students demonstrated a decent understanding of the concept of a “half,” though only one of them did so initially. Though the students disagreed to some extent on how to measure one half on a ruler (two of them pointed to the ½ mark, one pointed to the 1½ mark), these three all seemed to have a basic understanding of the meaning of ½, though clearly one that could be strengthened and reinforced (by the end of the discussion on halves, all three had come into agreement regarding the ½ mark).
References:
Chapin, S.H. & Johnson, A. (2006). Math Matters: Grades K-8: Understanding the Math You Teach (2nd edition). Sausalito, CA: Math Solutions.
Hiebert, J. et al. (2007). Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NJ: Heinemann.