Implementing Lesson Study Summary

Monday - Friday, July 10 - 14, 2006

Monday July 10, 2006

Today Jo taught the lesson to the high school geometry class while the rest of the group members and facilitators observed. Nicole and Megan from Learning from Teaching cases also observed the lesson.

We sat down to debrief and discuss how the lesson went. The question was asked, "What was our goal?" The goal was to emphasize that perpendicular segments are the measures that determines the area, and not necessarily the side measures. After reflecting on today's lesson, everyone agreed we needed to address the issue of time. The class started at 9:07, and most of the class time was spent working in groups. It was only at 9:50 that the students completed the task and group discussion begun. We were not able to get as far in class discussion as we would have liked, and we did not get to the parallelogram worksheet we were hoping to do. Towards the end of the lesson, the class was finally discussing and addressing the key concepts and making connections we were hoping to make as a group. If we had more time at this stage in the lesson, we all agreed the lesson would have definitely been more effective.

We discussed what happened during the group work to cause this time strain. The original rancher problem now had much clearer constraints and visuals(in comparison to Friday's test lesson), however, much time was still spent veering off from the topic we wanted to focus on. Students were spending a great deal of time creating other quadrilaterals and triangles and took a long time to form a parallelogram. Many of them ruled out the parallelogram right away because they assumed it formed the same area (side times side) as the rectangle.

This was a mistake we saw a lot in all groups. Students remembered the formula bXh, but then immediately said, oh it's the same area as the rectangle. They simply multiplied 5X8. They clearly did not know the definition of what was a base and what was a height. Their initial reaction was that the task was impossible. Some groups drew in the height as 5, but then the side measure was clearly not 5 and so they were not following the constraints of the problem. It was mentioned that we could spend more time asking questions like, "what do you mean by base, or what is height?"

Other students were drawing a height of 5 and base of 8, but the other side length had a different length(other than 5). These students were not sticking to the constraints.

One solution that was suggested was hinging the sides of fence together in the manipulatives so that the students would not waste time forming other quadrilaterals like kite, trapezoid, etc. This would help guide the students to a parallelogram much earlier in the process. We also agreed would should provide a hands-on manipulative for each student. Also, it would remove the issue of the students creating a parallelogram with a ht of 5 and base of 8, but a different side length.

Another improvement that could be made is changing the questions slightly so that the students first are asked what other shapes could be made. (hinges included) Then we can agree as a class that the shape is a parallelogram. Since many groups were making the same misconceptions, it would have been valuable to bring the class together to clear up some common errors and then set them off in groups again. After the class agrees it is a parallelogram, we could then ask the students to find the area. This eliminates much of the wasted time questioning what type of shape is created. Some group members also suggested that we tell the students to make a parallelogram. Universally, the consensus was that we did not want to waste a lot of class time on the discovery of the parallelogram itself.

Some other issues that were raised were group dynamics. In one group, one student immediately came up with the answer, however, the group did not seem to discuss the reasoning as to how they arrived to their answer. It became very clear that not everyone in the group understood where their answer came from. Jo called on a group member and the student was not able to explain the answer. Furthermore, the student who did get the answer did no further work and when Jo suggested to find another area, he said there was no other area he could get. We discussed alternate questions to ask, such as "how many possible areas could you make?", or "is this the only parallelogram you can make with these side measures?" "Why or why not?"

Megan mentioned that group A had elements of a great discussion, however, the discussion was not pursued further. They were starting to discuss that the maximum area would be the rectangle and anything other than that would create a parallelogram with a height less than 5, and therefore a smaller area. We had hoped this group would have developed this discussion further. We also talked about how to facilitate or allow this discussion to occur in all groups. One related observation was that the students did very little to justify their work. One solution to these problems is working in pairs. This would alleviate some of the group dynamic issues and would allow students to become more active participants in their group. Another suggestion was to ask questions that would help facilitate the discussion. One question that needed to be written on the original problem is "justify" or "prove how you found your area."

We also talked about students are dependent on being told or simply "knowing" the answer or formula. Many students came up with the parallelogram, but could not remember the formula for the area. Some of these students simply asked their teacher who just told them the formula. Other students used their old notes and resources and one group used the area of a triangle formula A = 1/2a b sinc in order to find the area of the parallelogram. While the use of this formula was very interesting and a great method of solving the problem, we questioned whether or not these students really understood where the formula came from and why it works. Another concern that was addressed is that when students make their posters, make sure they are drawn large enough so everyone can read. Another concern was that students perhaps felt they had to use EVERY material provided for them in their bag of materials, and they were possibly wasting time getting hung up on that.

We suggested that the teacher could have asked a few more leading questions in the rectangle problem:

  • How did you find the area of the rectangle?
  • Why does multiplying 5 by 8 give you the area?
  • What is the relationship between the sides?
  • What is base and what is height? (This question could really be asked several times, even during the parallelogram problem)
  • Have the students draw the rectangle on grid paper and use that as a tool to explain the method of solving for the area

Another issue that was raised was that on the posters the student's put up, we only saw the final result. We were not able to view their thought process or work. It is also valuable to view student's incorrect answers as a method of learning. One possible solution that was mentioned is to use "placemat", as we had thought of earlier in the lesson planning process. This way each student's individual work AND the collaborative work would all be on the poster.

Another note on the student work was that when one group posted their solution, it came up "side-ways" b/c of the nature of the adhesive on the paper. We adjusted the orientation of the paper so that the base would be on the bottom. However, one point was made that perhaps we could have left the paper oriented like that so we could have a discussion of base and height. We could use this opportunity to emphasize that the base does not have to be on the bottom. A suggestion was made that we could at first turn the paper like the students originally wanted, but then switched it around during the discussion in order to bring up this point. We also mentioned that we could have worked for 7-10 extra minutes since we got a late start. Towards the end of the lesson, the concept was really getting addressed and there was a very nice discussion. However, we ran out of time because we started the discussion piece only in the last 10 minutes of the lesson. After much discussion we asked ourselves, did we accomplish our goal? We agreed that we got much closer today than we did on Friday. Some of the suggested solutions, including hinging the fence pieces, more guided discussion on the discovery of a parallelogram, grouping the students in pairs, and including more questions of discovery and justification would all contribute to creating a much more effective lesson achieving our goal.

Tuesday July 11, 2006

Some more discussion on the lesson is occurring on this day. One point was made that two different orientations were shown for the rectangle. One was horizontal, one was vertical,(either base of 8 or 5). However, what if the rectangle was displayed diagonally? Another observation of yesterday's lesson was that 3 groups drew a base of 8 in their parallelogram and only one group used 5.

An observation was made yesterday that students calculated the height and got 4.6, but then rounded to 5. They concluded that the area was then 40 again. Jo jumped in to question them and they were able to resolve this issue. The comment was made that perhaps Jo could have discussed this issue with the class.(time permitting) Another comment that was made about our lesson was that we overestimated the students prior knowledge. We assumed they would know and remember the formula for the area of a parallelogram, but most of them did not remember. Making this assumption really created some of the time strain in the lesson. Also, had we not made this assumption, perhaps our lesson would have been designed differently. Today there were some comments on collecting and studying the student work. We noticed that some of the students were not using graph paper effectively as a tool. Jo said that perhaps her suggestion to have students draw the rectangle problem on graph paper would help alleviate this problem. A comment was made that the act of collecting student work allowed us to discover much more about what was going on then we even saw when observing the lesson itself.

Now we are all working on making modifications to the lesson plan and its structure, as well as modifications to the lesson rationale. Today we will also work on our creating a presentation for Thursday. We must present our lesson study to the entire SSTP group. Since Jo and James have already taught lessons, we agreed that they should not have to present to the group. Irma, Eileen and Anita will split up the presentation equally. James will work on the power point presentation and e-mail it to everyone.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.