I observed a fifth grade girl doing her mathematics homework. The content area

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Coral McLaughlin
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1 Teacher s Inquiry into Children s Knowledge and Learning EDUC 334 and 335 Diane Beiswanger I observed a fifth grade girl doing her mathematics homework. The content area being covered in class was geometry. My first observation was on March 24, when the weather had been very nice and spring break was just around the corner for the students. This young girl, Ann (pseudonym), was called in to do her homework after having had time outside to be physically active in the nice weather. I assume she was nourished and hydrated because she had already eaten dinner for the evening. As she began her work, she was enthused and attentive. She spent some time getting her work area set up at the kitchen table. She began to work on her math assignment, 24 problems working with triangles. Her posture was good and I sensed her energy was high. She completed the first eight problems in about 10 minutes. These were problems classifying triangles as scalene, isosceles or equilateral. As she reached the next section of the homework she seemed to get disinterested. She gave a big sigh and laid her head on her arm on the table. She continued working, rolling her head from one arm to the other. Sometimes she picked it up and set her chin in her hand. She took about 16 minutes to complete the next eight problems in the lesson. These were problems classifying triangles as acute, right or obtuse. By this time she had been sitting still for nearly 30 minutes. Her demeanor was one of exhaustion. She seemed to have the weight of a long Mathematics lesson on her shoulders. She was allowed to get up and take a break. Her break consisted of 50 jumping jacks, running around the house three times and about seven push-ups. She

2 told me the exercise would get her brain thinking again. Her break lasted about 7 minutes. After the break, she sat down to the last eight problems in the lesson. These problems were the most difficult in that they required her to calculate the degree measures of angles. The first four of these problems were done with enthusiasm and interest. As she progressed through the assignment, she became slower in her calculations. Additionally, what she had previously done in her head now needed to be written on paper and calculated. She knew she had to come up with 180 total for each triangle. At times she couldn t seem to figure out the number of degrees she needed to reach the 180 total. This was especially true when she was at a sub-total of the degrees of the angles that ended in something other than 10 degrees. She eventually finished the assignment after having spent about 45 minutes on 24 problems. After the first third of the assignment, she was no longer enjoying herself. The break helped rekindle enthusiasm and interest, but it was short lived as she wrangled with the tougher parts of the assignment. My second observation of Ann was right after spring break, on April 5. The weather was again sunny and warm. This assignment was on circumference of circles and was 14 problems long. This time Ann had not had time to play outside after school, and she was more fidgety. I used the minute-by-minute recording method. As she began the lesson, she was focused on her work with her head in her hand. Within a minute, she began talking to herself as she did her work. For the next few minutes, she worked earnestly. Then she began erasing and rewriting her work.

3 Her concentration seemed broken. She began using her fingers to calculate the problems. She laid her head on her arm on the table while working. This is when I noticed all calculations for finding the circumference were to be done with manual calculations. Seven of the problems were to find the circumference using pi as Seven of them were to use pi as 22/7. The calculations were slowing her down and seemed to cause a lack of focus. Ann spent the next 15 minutes working with her head down on the table rolling from one arm to the other. Through most of this time she talked to herself as she worked. While she was going through the motions of working, I sensed a distraction, as if she were pretending to concentrate and do her work, but was really not focused on the task. During this time, she had three bouts of heavy erasures where she seemed to be starting over with problems. At this point, Ann sought help with a problem. Getting up from the table to find help seemed to be rejuvenating for her. After getting up from the table and some attention from an adult, she went back to work. The next five minutes seemed more focused. Her head was on her hand while she concentrated and calculated. Her pencil was in her mouth and on her head. While she was nearly as fidgety as during the earlier part of the assignment, she seemed more focused and directed. After 30 minutes, Ann had completed the 14 problem assignment on circumference of circles using manual calculations. When I asked her why she was erasing some of her work, she said it was because she didn t want the teacher to see all of her work (calculations), only the stuff they were supposed to show. Again, a break seemed to trigger a renewed focus on her work.

4 Ann and I met for an interview to assess her understanding of the geometry skills she had been discovering over the past two school weeks. Using appropriate activities from the Chapter 15 handout on Geometry, we talked about two dimensional shapes because it was known this content was covered in class. We began with a discussion on polygons. Ann was able to define a polygon as a closed shape made with straight lines. I asked her to tell me what closed meant. She responded that it meant, The lines are all connected. From there she identified a number of polygons and non-polygons correctly. The polygons we discussed included parallelograms such as the square, rectangle and rhombus, as well as the trapezoid, triangle and irregular polygons. Ann was able to identify all shapes correctly as a polygon or non-polygon. We expended our assessment to talk about quadrilaterals and parallelograms. We discussed quadrilaterals and parallelogram, including the differences between the various kinds of parallelograms. Through a series of exploratory questions using some of our H.O.T. tools, Ann was able to correctly explain the specific features of the various shapes. While it was clear she knew the specifications for most of the polygons, she had some difficulty articulating what she knew. The following are some of her responses: Quadrilateral: polygon with four sides Parallelogram: quadrilateral with two sets of equal sides Square: quadrilateral, parallelogram, four equal sides, all 90 (angles) Rectangle: quadrilateral, parallelogram, two sets of equal sides, all 90 (angles)

5 Rhombus: quadrilateral, parallelogram, four equal sides, different angles. She was not able to articulate two pair of equal angles. She used a comparison to the trapezoid saying, it can t be like a trapezoid, pointing to the angles, but it doesn t have 90 angles like a square. Trapezoid: quadrilateral, two parallel sides and two equal sides, but not the same sides. The parallel ones aren t the equal ones. Triangle: three sides, angles add up to 180 Hexagon: six sides Pentagon: five sides It was clear she understood the basic specifications of the polygons we discussed. She seemed to be missing a little of the vocabulary. Also, when questioned, she knew the angles of a square total 360 and, with a little pointed questioning, was able to deduce the angles of a rectangle total 360. She did not know the angles of all quadrilaterals total 360. Our discussion then turned to triangles. Ann easily identified scalene, isosceles and right triangles. She enthusiastically volunteered all she knew about the specifications of each type of triangle, obviously getting the hang of our encounter. She also understood the properties of equilateral triangles, noting they are a special kind of isosceles triangle. She was able to correctly categorize a variety of triangles as equilateral, scalene, isosceles and right triangles. Our final element of assessment was to look at lines of symmetry. Ann understood lines of symmetry for a rhombus, trapezoid, hexagon, square and rectangle. However, parallelograms, triangles, pentagons and irregular polygons gave her some

6 difficulty. Her tendency was to draw the lines of symmetry between vertices. This worked well for polygons with an even number of vertices, but caused confusion when there were not two vertices opposite each other, through which a line of symmetry could be draw. Based on my observations and interview with Ann, I recommend spending time exploring lines of symmetry on polygons with an odd number of vertices and irregular polygons. The use of a mirror held across the various polygons would allow her to quickly and easily try a variety of lines in an attempt to discover the true, if any, lines of symmetry for these polygons. As she views the image created by the polygon on the paper and the mirror combined, she will find a line of symmetry when the polygon shared with the mirror matches the complete polygon on the paper. This is the interventional experience I shared with Ann. We began by setting the mirror on lines of symmetry on polygons for which she knew the lines of symmetry. In this way she understood how the process worked and learned to trust the strategy. As we progressed through some of the more challenging polygons, her inclination was to connect the vertices. When this technique did not reveal an image created by the polygon on the paper and the mirror combined that matched the complete polygon on the paper, Ann became frustrated. It took some directed questioning for her to try strategies that did not involve connecting vertices. Once she discovered this truth, she enthusiastically began manipulating the mirror to find the true lines of symmetry of the polygons. She even went back to the polygons whose lines of symmetry were familiar to her to see if she could find other lines of symmetry.

7 My experience with Ann confirmed a number of principles I have learned in my teaching education, so far. I think the most interesting was the significant impact of physical activity on brain function and concentration. When Ann had been able to get physical activity before doing her homework, her attention span was significantly longer (10 minutes vs. two minute) than when she had not had time to be physically active before doing her homework. Additionally, the 7 minutes break for physical exertion had a powerful affect on her engagement in the remainder of the lesson. This is consistent with what I learned in my PESS 200 class. I also learned that Ann, as a learner, likes direction rather than exploration. I believe she was shown lines of symmetry that connected vertices. I think this is why she continued to use that strategy even when it was impossible to make it work. My questioning had to become quite directed, leading her to the idea of trying a line of symmetry that didn t use two vertices as its end points. Once directed to this new strategy, she was very excited about trying it on new polygons and familiar polygons. Lastly, I learned Ann struggles with calculations. While I would certainly encourage math games that facilitate the memorization of facts, I would not hinder the progress and joy of learning other mathematical concepts because of this difficulty. I would allow Ann to lay out the formula required for her calculations and then use a calculator to complete the work. No one needs to know how to multiply anything by 22/7. In future encounters with Ann, I would use what I have learned to facilitate the learning experience. I would be sure she had plenty of physical activity before sitting down to instruction. I would also take breaks for physical activity about every 10

8 minutes to avoid fatigue rather than have to recuperate from it. To promote her own exploration rather than directed learning, I would encourage her to try new ideas that came from her own head. I would push her to explore and discover rather than try someone else s suggestion. If I were able to spend another week with Ann, I would like to explore the following activities with her. Based on our experience with lines of symmetry, I would ask her to draw an open shape. Using the mirror, I would ask her to draw the remainder of the shape to turn it into a polygon. This would strengthen her understanding of lines of symmetry which do not have vertices as their end points. I would ask her to think about what specifications her open shape had to have in order for her mirrored shape to be a polygon. Another concept I would explore is the sum of the degrees in the angles of polygons. She already understands that the number of degrees in a triangles totals 180 and a square and rectangle totals 360. On one day I would begin with hands-on activities that allow her to confirm what she believes to be true. I would ask her to anticipate what she thinks she will find when she puts the points of a triangle or the corners of a square all together on a fixed point. The following day we would expand this idea to include the polygons for which she did not know the sum of the angles. Hopefully this would allow her to uncover an understanding of why the rote learning she has been taught is true. I hope that she may also discover the pattern found in the sum of the degrees of the angles of a polygon as she explores polygons with more angles. We would spend yet another day exploring the angles of polygons using the covering strategy or the number of angles around a fixed point strategy. This would

9 empower her to discover the exact measure of angles through direct manipulation of objects. A question to consider with this activity might be, Why do all of the polygons with four or more angles completely cover a fixed point? The last area I would like to explore with Ann is circles. She worked so hard to calculate the circumference of circles in her math assignment. I would like for her to have some fun with that. Using a can full of tennis balls, I would ask her to speculate as to which is larger, the height (three times the diameter of the balls) of the can or the circumference of the can. Then using the tennis balls and string to measure, she will hopefully discover the mathematical truth about circumference which her eyes may not have revealed. Using string to measure the circumference and diameter of other cylinders, I hope she will be able to extrapolate the consistency of the concept of pi. Working with Ann was very inspiring. While I certainly hope to one day have a class of my own, I know I could find great satisfaction working with children on a one-toone basis. When my grandmother retired from teaching, she spent 20 years working with two brothers confined to wheel chairs and unable (in those days) to go to school. I remember how much joy she found in that work. Now I better understand what she had found.

An angle that has a measure less than a right angle.

An angle that has a measure less than a right angle. Unit 1 Study Strategies: Two-Dimensional Figures Lesson Vocab Word Definition Example Formed by two rays or line segments that have the same 1 Angle endpoint. The shared endpoint is called the vertex.