ENTRY 6: MODIFYING THE CONNECTED MATHEMATICS PROJECT
Description of Entry:
At the end of each CMP investigation, students are asked a set of Mathematical Reflection questions, in which they explain the major concepts of the investigation in their own words. However, in one of the investigations, I did not think that the CMP's four questions were sufficient in covering the vast amount of information we had discussed. Therefore, I modified the reflection questions and created a worksheet. I have included the original questions and several student-completed versions of my worksheet.
In my sixth grade honors class, three students shared their conjectures for finding the lowest common multiple of two numbers. These were methods created by those students and not mentioned in the text. After discussing each of the conjectures, I assigned the class a worksheet that I created that asked them to explain why the conjectures work. I have included the handout that restated the conjectures, the assignment that asked for an explanation for the conjectures, and three student responses.
Program Goals and Targets:
I demonstrate my subject matter knowledge (1A) by designing and creating material that supplements the CMP units. I demonstrate my understanding of subject matter assessment (1D) by observing student progress of material and basing instruction on that evaluation. Finally, I demonstrate my ability to adapt for differences (2B) in my assessment of the student responses, by understanding how learners differ in their approaches to learning.
Reflection:
The Mathematical Reflections worksheet I created for Investigation 5 of the Prime Time unit was a lot different than the book's original version. The reason that I changed and added questions was because the content that my classes discussed was more than the original scope of the investigation. To be sure that all of my students understood these concepts, I tried to develop a more comprehensive set of reflection questions.
The three completed worksheets I selected to include were from students who are very different in ability levels. Student A is highly skilled in the computational aspect of mathematics as well as in expressing her understanding in words. Student B is a student with good mathematical ability, but whose writing skills are at a third grade level. Student C is an average math student, who has a little difficulty explaining his answers. I found it interesting to compare how these students responded to the questions that I posed.
For example, question two asked the students why we do not include one when we write the factorization of a number. Student A clearly stated that "1 is a factor of every number" and the factor tree for the number would continuously add ones. She made it clear that she understood that continuously adding one to a factorization is insignificant, by mentioning that "you would just be making it longer and harder." She showed an example to illustrate her point.
Student B stated that it was because all numbers can be divided by one. He implied that since all numbers are divisible by one, it is insignificant to include it in the factorization. However, he did not explicitly state this, which leaves one to wonder whether this is a result of an incomplete understanding or because he simply did not write it down. Because of his writing disability, I believe it is most likely the latter.
Student C stated that it was because the prime factorization, would go on and on. This explanation is also a little vague. He implied that including one in the factorization would give an infinitely long factorization string, however he did not explicitly state why that was significant. However it is clear that he understood that such a factorization was unnecessary.
Question three also brought up some interesting responses. The students were asked to explain how to use the prime factorization of a number to find all of the factor pairs. Student A's written explanation was a little unclear here. She wrote that you should first make a factor tree, but the rest of her statement is not completely coherent. The example she wrote on the right side helps. She took different combinations of the prime factorization and multiplied them together to make the factors.
Student B's explanation was even more difficult to understand. He stated you should make a table and fill in all of the prime factorizations like his example. He also mentioned that you should continue until you cannot do anymore. The response was missing a lot of details for clarification, such as the actual number and its prime factorization that his example was based on. It would have also been more clear had he described the process of filling in the table. Again, I believe that his lack of clarification was due more to his writing ability than his mathematical understanding.
Student C, on the other hand, showed that he did not understand. He included even less written explanation than Student B. He did, however, include a factor tree for 100 as an example. But he circled the prime factorization and labeled it as a factor pair. This makes me wonder if he even has a complete understanding of what a factor pair is, let alone how to find one.
Textbooks are used in nearly every math classroom across the country. "Given the wide availability of textbooks, plus research that shows that they dominate middle-grades instruction, it might be reasonable to assume that teachers rely too much on the textbook to teach the content of the curriculum" (Muth & Alvermann, 1999, p. 96). However, I believe that I have an ethical responsibility to avoid such dependency.
None of the questions I discussed in this entry were in the original set of reflection questions. If I had not modified the assignment, I would not have been able to assess my students' understanding of these important concepts. Thus, I believe that it is important that teachers be willing to modify textbook curriculum. And since each class of students is different, it should be the students that dictate the content and the pacing of that content, not the text book.
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