Appendix A
Final Project Report 2003
Connecting Mathematics for Elementary Teachers (CMET)
I. The Project and Project Goals
The CMET project attempts to connect prospective elementary
teachers’ learning of mathematics in mathematics content courses with how
children understand and learn mathematics.
In this proof-of-concept NSF CCLI-EMD DUE 0126882 grant supplementary
materials were developed for a typical first mathematical content course for
elementary teachers. An expanded supplement of teaching notes was developed for
the instructors. More specifically, the
materials explored how children learn and understand mathematics, the
mathematics actually taught in the elementary school, and the important
connections at the elementary level to mathematics and the technological skills
children will learn in middle and secondary school. The project’s primary emphasis was on
mathematics from children’s perspectives and how this is directly related to
the mathematics they are learning and will eventually be teaching children.
The goals of the project were:
1.
Improve pre-service elementary teachers’
understanding of how children learn and understand mathematics.
2.
Enhance pre-service elementary teachers’
understanding of mathematics.
3.
Help
prospective elementary teachers’ connect the mathematics they are learning with
the mathematical concepts they will be teaching children.
4.
Facilitate prospective elementary teachers
understanding of the connection between the mathematics they will be teaching
in elementary school to the mathematics children will learn in middle and high
school.
5.
Improve pre-service elementary teachers,
teaching of mathematics to children.
II. Description of Implementation and Assessment Activities
This section will describe who used the CMET materials and the extent to which the materials were used. The CMET materials were piloted with three groups. Data from one control group was also collected.
Group
1: The
principal investigator piloted CMET in two sections at Purdue University North
Central, a regional campus, Fall 2002.
The mathematical content textbook used was: Mathematics for
Elementary Teachers: An Interactive
Approach. Second Edition, by Sonnabend, T., 1993.
Group
2: The second group consisted of two
sections at
Group
3: The control group consisted of two
sections at
Group 4: The last group was comprised of two
sections at
Data
Sources
· An end of the semester Questionnaire was given to Groups 1 & 2. The questionnaire is included in Appendix A.
· At mid-semester Group 2 completed a quiz on which they were asked what they thought of the CMET supplement at that time.
· Student solutions to selected test and final questions for Groups 2, 3 & 4.
· Interviews:
Group 1 1 student
Group 2 14 students
Group 3 10 students
Group 4 4 students
The interview questions are included in Appendix B.
· The instructors for Group 2 and 4 were interviewed.
· The coordinator’s notes given to the teaching assistants in Groups 2, 3, & 4.
· Tests and Finals given to all groups.
· Written feedback solicited from the coordinator
All the interviews were transcribed.
III. Results and Analyses
The ongoing assessment conducted by the project team used varying methods of data analysis. On students’ test responses and on selected answers on the questionnaires quantitative analyses were conducted. A qualitative analysis of data from the one quiz, questionnaires, and interviews involved looking for themes and patterns across the data—an adaptation of the constant comparative method (Glazer and Strauss, 1967). The data analysis was triangulated by having three project staff members independently look over the data and draw their own conclusions. The results were then compared and compiled into this document. A fourth member of the project team then analyzed the compiled results and made further revisions.
The evaluation focused on answering the following four interrelated questions.
- What were students’ general perception and reaction to the CMET Supplement?
- What were the instructors’ and the coordinator’s perception and reaction to the CMET Supplement?
- How well did the project achieve its five goals and what evidence is available that supports these claims?
- How can the CMET Supplement be improved or refined?
Multiple data sources were used for each question. The data sources were frequently applicable to more than one question.
A. What
were students’ general perception and reaction to the CMET Supplement?
One question on the questionnaire, given to Groups 1 & 2,
addressed whether students viewed the CMET supplement as beneficial.
Group 1,
Group 2,
Was the CMET supplemental beneficial? Group 1
Was the CMET supplemental beneficial? Group 2
The vast majority of students, (93% currently) of Groups
1&2, found the supplement Very
to Somewhat Beneficial. Eighty-one percent of the students in Group 1
found the supplement Very Beneficial
as opposed to only 29% in Group 2. This difference is most likely explained
by the fact that Group 1 was taught by the principal investigator of the
project who thoroughly understood the materials and how they might be
integrated into the course. He also had a vested interest in their success. While the results from Group 1 cannot be used
to claim that the CMET supplement is worthwhile, the results suggest that if
the materials are used as intended, then students are more likely to perceive
them as very beneficial.
The next two questions address if students feel the supplement should be required for other students in the same course and if they would like a supplement like this for their next mathematical content course for elementary teachers.
Do you think the CMET
supplement should be required for MA 137? Group 1
Do you think the CMET
supplement should be required for MA 137?
Group 2
Would you like a supplement like this in your other
mathematical content courses (MA 138 & 139)? Group 1
Would you like a supplement like this in your other
mathematical content courses, (MA 138 & 139)? Group 2
Slightly more students indicated that they would like the supplement for their own use in future classes.
The majority of students found the CMET materials beneficial and believed that they should be used in future courses. Students’ responses on the questionnaire and the interviews also support this conclusion.
Several students indicated, in both interviews and the questionnaires, that this supplement would be useful when they become teachers.
“I will keep this supplement and use it when I teach.”
Overall, the majority of students found the supplement beneficial. There were four specific reasons why students found the material valuable (presented in order of significance). The supplemental materials…
- Helped them understand how children learn math.
- Presented a different approach or a new perspective for learning math.
- Provided practical examples.
- Reinforced what had been covered in class.
Other minor reasons stated for finding the supplement valuable included that it broke information down into more understandable pieces and provided a more personal approach.
B. What were the instructors’ and the
coordinator’s perception and reaction to the CMET Supplement?
First Instructor
The first instructor of The West Lafayette sections, group
2 was interviewed after the completion of the semester. She had a Bachelor’s in Mathematics and a
Master’s degree in education. She had
taught one year of high school and one year of middle school. She had never
taught a mathematical content course for elementary teachers, but had taught
college algebra at Purdue West Lafayette (PUWL). She had more education experience than the
typical Teaching Assistant at PUWL, but she had no elementary school teaching experience.
Her overall perception of her students’ view of the CMET supplement was that they enjoyed it, especially “the ones that … really want to teach.” She indicated that there were some students who you could tell that did not want to be teachers.
The students were only assigned to read and do problems in Chapters 1 and 3. However, she indicated that she did use the supplement in her teaching.
Sometimes I would put
those up, children’s ideas for solving problems, and ask how did this work….
I’d say… these are children’s ideas of how they’re thinking… See how their
ideas are so different than theirs and kind of get into a discussion about
that.
She wanted her students to figure out why the example of children’s thinking worked. One point she made was that students did not always know when examples she was using came out of the CMET supplement. However, she did not make up the test and she felt compelled to cover all the traditional material on the test. The students in this group were not tested over the CMET materials. Given these constraints, she would have liked to incorporate more of the CMET materials into her teaching but did not feel she could do so.
Significantly, she indicated that she learned about how children learn math from this supplement. Her main point in addressing this was that adults thought processes are so different from children.
She believed this supplement would influence their future teaching. “I think this book is filled with so much information… a lot of the students said they were really going to use this and keep it…” Two of her students did actually use the supplement in working with children.
Just recently one
of my students told me that she went home and worked with some special needs children. … They could not remember their
multiplication [facts] and she said she remembered to review it as repeated
addition…. She said, you know, he [the
child] felt so much better because he actually knew what was going on.
She believed that the students with more educational experience, juniors and seniors, saw more value in the supplement than some of the freshmen. She believed that some freshmen just looked at this class as another math class. “They haven’t been in the classroom and they don’t know how teaching is. They just know I’m taking a math class…” She described a story about how a Junior had used the supplement in her work to help a child understand why you carry.
Another group of students whom she believed that the supplement was particularly beneficial for were her weaker mathematical students, “especially the students who have struggled in mathematics, who say I hate math or I will never do well in it. They really have, I think, benefited from it [the supplement].” Another time she said, “…if they struggle, they love the supplement.”
The most beneficial aspect, from her point of view, was that the supplement showed how children think about mathematics and that connected with the mathematics she was trying to teach in class. “I think it is the child’s perspective. Every time you put that in there, I think that opened the students’ mind to what I was trying to teach in class.” She reiterated this point on several occasions, “I think that was the best learning tool for them… when they actually saw this came from a child, and this is their perspective.” She wanted her students to know more than just how children thought about mathematics. She wanted them to value that knowledge. “It was … just trying to get them to understand the value of actually understanding why having a child work it their own way is so much more important than just giving them it and saying do it this way all the time.”
Her emphasis was probably not that of a typical teaching assistant given her educational background. She wanted her students to understand mathematics and she believed her students should teach for understanding. “My big push is a child should understand where material is coming from. It shouldn’t just be coming out of the sky.” She believed that her students should understand mathematics so they could teach for understanding. However, not all her students thoroughly understood mathematics. “I am trying to teach them how and why, but they don’t even know how yet themselves.”
She believed that the supplement helped her students gain a better understanding of mathematics and helped to create a dialogue to discuss her students’ mathematical misconceptions.
The instructor believed that the CMET supplement encouraged students to explore their own misconceptions about mathematics. Children’s understandings of mathematics provided a context for discussion about their own mathematical misunderstandings. One example in the supplement describes how a college student related her misunderstanding of addition in elementary school (p. 27). “We talked about it…this brought out a couple
of stories from students [of] some things they always thought were wrong.” In conjunction with this other students mentioned to her that they performed mathematical calculations correctly without understanding what they were doing.
The instructor indicated that she would like to teach this class again using the supplement. “I would like to be able to use it again because I know the ideas behind it.” She felt by using the supplement again, she could do a better job. “I don’t think I did a good job of tying everything into the big picture.” She now understood the supplement better and could fit it with the big ideas of the course. This is another indication of her learning and the influence of the supplement on her.
This teaching assistant suggested minor changes such as: moving material between the instructor’s supplement and the students supplement, clarifying some of the text, and providing more examples of children’s mathematical thinking. Overall, she enjoyed using the supplement and believed that it gave her and her students a better understanding of how children learn and think about mathematics.
Second Instructor
The second instructor of the Spring West Lafayette sections, group 4, was also interviewed after the completion of the semester. He was an international graduate student working on a master’s degree in Industrial Engineering. He had only been a teaching assistant in large lecture, calculus courses. He had no public school experience. This was the first time he had ever taught a class on his own. He viewed the CMET supplement as a tool for “understanding of the level of understanding of elementary school children in this country”. For him, the CMET supplement offered a different approach or techniques to teaching the mathematical concepts he was trying to teach his students. He indicated that if students did not understand a concept presented in the textbook they could read about a different approach in the CMET supplement. He did not see it as being different from other approaches to teaching mathematical content to prospective elementary teachers. Further, when an alternate approach was presented in the CMET supplement to a concept that students already understood, such as adding positive and negative numbers, the students did not like learning about these approaches.
They didn’t appreciate
some of the methods. … They knew
probably the rules …, how it works, and this was an additional set of rules and
they didn’t like it.
He indicted that he did learn about the different ways in which children think about mathematics. Specific examples he mentioned were, “counting or different kinds of counters and things like how they look at multiplication, like they look at repeated addition.”
It was clear that this instructor did not understand the intent of the CMET supplement. He viewed mathematics as rules and procedures and his goal was to help his students understand these procedures. He did suggest that the beginning of the instructor supplement contain an overview which describes how the supplement is intended to be used and which provides a “big picture” of its purpose.
Coordinator
The coordinator of the instructors for the mathematical
content courses for elementary education majors on the PUWL campus created the
daily teaching notes for the instructors and all tests. He did not teach the
control sections or use the CMET supplement in his sections of the first
content course. The second semester he
found it “easy to integrate the materials into the course” as long as the
project staff provided him with suggestions for each lesson. He suggested that the CMET instructor
supplemental materials “give specific suggestions on pacing, how to weave the
text into lessons, etc.”
During the first semester, without suggestions, a large
portion of the supplement was not assigned to the students. The coordinator had a master’s degree,
experience as an editor for a mathematics education (NCTM) journal for teachers,
and middle school teaching experience. He
was hired for his educational background to coordinate all the mathematical
content courses for elementary teachers.
Consequently, he believed that the courses as he had been previously
coordinating were meeting the needs of the students. “I think part of that difficulty arises
because I have a set of lessons that I’m comfortable with, so integrating
something new into those lessons is a challenge.” Initially he did not see a need to integrate
the materials into the course.
He described one aspect of teaching children mathematics that he learned by using the supplement.
One specific thing that
I learned had to do with the concept of ten.
Until reading about this in your materials, I had assumed that the
concept of ten as a unit way pretty easy for students. I’d seen many suggestions for using
manipulatives (base-ten blocks or an equivalent), but I’d never seen anything
suggesting that even with the manipulatives it’s a big step for children to
think of ten as a unit.
His views are reflective of one of the problems in the
mathematical education of elementary teachers and that is, very few people teaching
these content courses have actual elementary school experience. Although instructors of these courses may be
highly qualified mathematically and educationally, the CMET supplement can
serve as a valuable resource to address a pressing need.
One other key point the coordinator made was that at many institutions, “there is a history of wanting to keep these courses as teaching mathematics, not teaching methods of teaching mathematics.” Consequently, he believed that we should focus the supplement and the dissemination of the supplement on its intent to enhance the mathematical knowledge of pre-service teachers.
C. How
well did the project achieve its five goals and what evidence is available that
supports these claims?
Goal 1: Improve pre-service
elementary teachers’ understanding of how children learn and understand
mathematics.
The overriding comment from students about CMET was that it showed them how children think and how children view and solve problems in a variety of ways.
The supplement gave
fantastic examples of how children in real situations dealt with problems. I learned that each child is different in how
they think mathematically.
Students commented that they learned about children’s problem-solving methods and felt that they were better able to think like a child or see math from a child’s perspective.
It gave great examples
how children learn and think about math.
I learned that is a process of building upon what they know and learn,
and I was able to see math from a child’s perspective.
They realized that a child may be “wrong,” but is still trying to make sense out of mathematics. They came to see that children may view math differently than the teacher and that different children think differently and learn at different rates.
It helped me understand
their way of thinking and helped me to realize they learn in different ways
than I do.
They appreciated that CMET pointed out where children may have difficulty and that it pointed out the different levels children progress through. They learned that children need opportunities to use manipulatives and to draw pictures to represent their thinking. Several students remarked that they had learned the importance of counting activities in the early grades.
It gave me a whole
perspective of their level of thinking.
I didn’t realize how important counting was in everything.
On the questionnaires, one clear distinction between groups was the inclusion of specific details by the regional respondents about how children learn. Regional campus surveys included specifically that children learn how to add by counting (mentioned by 7 respondents). Also mentioned was how children specifically understood the concept of zero, communicative property, and the concept of 10.
Goal 2: Enhance
pre-service elementary teachers’ understanding of mathematics.
The summary of students’ questionnaire and interview comments suggests that CMET’s emphasis on different ways children solve problems had helped the MA 137 students develop more ways to solve problems.
It improved my
understanding of math by showing me different ways to do one problem.
Many students felt that the course’s and CMET’s emphasis on understanding the “why?” of mathematics, on solving problems in a way that makes sense, and on explaining one’s reasoning had helped to improve their own understanding of mathematics. They often indicated that they had memorized mathematics in the past and had not understood it. The supplement helped them figure the “why”.
It
helped me to better understand why certain rules work, rather than just
memorizing steps.
I
think that I learned a lot of procedures through my years of math and really
didn’t know what they meant. I just
memorized and used them.
I think it helped me
understand math better. I could do many
types of problems but never really understood how I got there.
Several students explained that CMET “breaks problems down” in a way they could understand and also said that they had come to enjoy problem solving.
[The supplement] broke
it down more than the book and made it easier to understand.
While some students indicated that using the supplement did not improve their understanding of mathematics, it did help improve their understanding of the mathematics that children will do.
Analyses of specific test and final questions revealed no significant differences between the mathematical achievement of the students in the control group, Group 3, and Group2, the students using the same textbook the same semester. Basically their mathematical achievement was identical.
Goal 3: Help prospective elementary teachers’ connect the mathematics they are learning with the mathematical concepts they will be teaching children
Several students noted that some of the CMET content (e.g., counting, whole number concepts, and fractions) was exactly what they would be teaching.
What I learned is
exactly what I will be teaching, even the “why” we do what we do part.
Many students saw the connection between how children learn mathematic and the mathematics they will be teaching children. Their focus became more child centered.
It
helped me to switch my thinking to the way a child would think so I can teach
in a way that children will understand.
There was a strong tie
in between what I learned and what I’d be teaching because the supplement
offered specific examples of children’s math.
The documentation of this goal is evidenced in the many comments of students that the supplement first influenced them to think about how children understand mathematics and second, how this will influence their future teaching of children mathematics.
When
we learn math we had to think about how children would see the problem and come
up with an answer. This will help us
understand how to teach them, if we know what they’re thinking.
Goal 4: Facilitate prospective
elementary teachers understanding of the connection between the mathematics
they will be teaching in elementary school to the mathematics children will
learn in middle and high school.
Student responses to an interview question related to this goal suggests that CMET fell short in achieving it. Students struggled to think of any connection. Although not able to mention anything specific, one student did quite reasonably point out that learning different strategies and developing an understanding of procedures and concepts in elementary school math will help prepare children for middle school and high school math.
Goal 5: Improve pre-service elementary teachers, teaching of mathematics to children.
Although this goal cannot be directly assessed, students’ comments frequently touched on how what they learned from the CMET supplement might affect their future teaching. The three most significant responses among from the survey data included:
- Learned new methods including use of hands on materials, and ways to present problems.
- Will use the supplement directly as an aid to teaching in the future.
- Able to see learning math from a child’s perspective and be more understanding about how children learn.
A common response from students was that they were going to keep the supplement and use it when they taught.
I like the CMET
book. It describes in detail the steps
of how to teach kids. I want to keep it
and use it when am a teacher. I have
highlighted things. I just like it how
it teaches different ways and how children will do things differently.
Many of their comments were general comments about how CMET had helped them learn how to teach children or learn how to teach math.
It helped me to switch
my thinking to the way a child would think so I can teach in a way that
children will understand.
They felt that CMET had suggested different teaching styles and teaching strategies that they would be able to employ. As a consequence of learning about the variety in children’s mathematical thinking, many students developed the belief that a teacher would need a variety of ways to teach a topic and would need to know and understand many different ways of solving and explaining problems. Many students stated that they would encourage children to use different methods, to solve problems in ways that make sense to them, and to explain their thinking.
I will be more aware of
math from a child’s perspective taking into account how they learn different
principles in math and tailor my teaching methods to suit their learning
styles.
They felt that as teachers they would need to focus on the “why?” and not just on having children memorize procedures (and one student suggested that emphasizing the “why?” could make math more fun).
I want to be sure my
students know the why, how, who, and where, so it is not just memorization; and
so it will be more fun for them.
In this regard, they felt CMET was helpful in preparing them to be able to explain math and “break things down” for children. (Note that this suggests that the students do not really understand the idea of a problem-solving approach to instruction or the reason for encouraging children to solve problems in ways that make sense to them, i.e., the students still see themselves as the conveyors of mathematical knowledge.) Finally, a few students garnered ideas for lessons or activities from CMET and one student simply said that CMET encourages “teacher thinking.”
D. How can the CMET Supplement be improved
or refined?
These suggestions represent a synthesis of student and instructor comments, interviews, and observations made by the coordinator and the principal investigators.
1. As noted above, students made several specific suggestions regarding CMET content. These included expanding and clarifying the fractions section, including more K-1 examples, expanding the patterns section and more fully discussing the importance of patterns, clarifying the concept of 10, and shortening the properties section.
2. Some “cosmetic” changes students suggested were to include blank pages or spaces for notes, include more illustrations, provide references for further reading, and put the citations for references at the end of each chapter rather than in the text.
3. Many students asked that CMET include more suggestions about how to teach. However, doing so would not seem to fit with either the nature of the MA 137 course or the intended goals of CMET. Explicit attention to teaching methods, beyond the “what is a problem-solving approach to instruction?”, is probably best left to a methods course or to the modeling provided by the MA 137 instructor.
4. As noted, the students in the questionnaires and the interviews saw very little connection the mathematics they will be teaching in elementary school to the mathematics and technological skills children will learn in middle school and high school. These connections are explicitly made in the CMET supplement. However, these connections need to be emphasized more and more of them made.
5. A few students suggested including more examples from elementary textbooks (i.e., more of the sample text worksheets). However, one the student indicated these worksheets should either be removed or emphasized more. The worksheets need some brief commentary accompanying them that highlights their strengths and weaknesses, or some questions that ask students to think about how children might solve the problems on the worksheets and what they might learn from them.
6. A large number of students indicated that the chapter review questions were not helpful. They suggested replacing them with problems or activities that would focus more on children’s methods of solving problems. For example, they suggested problems that would ask them to use children’s methods to solve a problem. One student also suggested having actual discussion questions rather than questions whose answers could be looked up in the CMET text.
7. The students almost uniformly liked the examples of classroom situations and the examples of children’s mathematical thinking. Still, many asked for even more examples of how children think and solve problems. Several students suggested including examples from “actual children” in the form of excerpts from interviews or transcripts of classroom episodes.
8. Quite a few students indicated that, in places, CMET was too wordy or too lengthy. Although they did not mention specifics (except for the properties section), the most likely offenders are the counting types and concept of 10 sections.
9. The Chapters and Sections need to be divided into subsection such as 3.11, etc. This will make it easier to align the supplement with different mathematics textbooks for elementary teachers. Often students were instructed to read to the middle of a page so that the material corresponded to what they were doing in their textbooks.
10. Move references, other than quotes, to the Instructor Supplement and use paraphrases of research based findings concerning how children learn and understand mathematics.
11. Reconfigure the problems and Questions for Discussion. The revised supplements will have four separate components. First, mathematical problems for prospective elementary teachers to solve. Second, children’s solutions to these problems with further comments about how the children solved the problems. Third, new and revised Questions for Discussion moved to the end of each section. Fourth, a limited number of class activities interspersed in the text. Again these activities are designed to help pre-service teachers understand how children understand mathematics.
12. Possible test and quiz questions added to the Instructor’s Supplement.
IV. Dissemination Activities
The dissemination activities of the CMET Project have just begun and are ongoing.
Presentations
January 2004 Integrating Knowledge of How Children Learn and Understand Mathematics into Mathematical Content Courses for Elementary Teachers”, the Association of
Mathematics
Teacher Educators (AMTE) Annual Conference, San Diego ,
CA
April 2003 Research into Practice: Connecting
Mathematics for Elementary School Teachers. NCTM National Meeting, San
Antonio , TX
January 2003 Connecting Mathematics for Elementary
Teachers in Pre-service Mathematics Content Courses. NCTM Regional Meeting,
Indianapolis , IN
Book Chapters
2004 (Submitted) Connecting Preservice Teachers’ Learning of
Problem Solving with Children’s Problem Solving. In The Work of Mathematics Teacher
Educators: Exchanging Ideas for
Effective Practice. Eds. T. Watanabe
& D. Thompson. AMTE Monograph.
V. Conclusions
Overall, the CMET Supplement
appeared to help students construct knowledge of how children learn and
understand mathematics. We believe that
by using these materials prospective elementary teachers did connect the
mathematics they were learning with the mathematics they will be teaching
children. Further, many students’
indicated that the supplement will influence their future teaching of
mathematics to children. The primary
influence was on their new knowledge of how children think mathematically.
After taking this class
a few semesters ago, I didn’t have the child’s viewpoint and what and how they
think. Now I do and with more strategies
to teach them.
Some students did indicate that it improved their
understanding of mathematics; however, most indicated that it did not. More significantly though, many students
indicated that using the supplement did improve their understanding of how
children see and understand mathematics.
From the perspective of the project team, understanding how children
view mathematics requires a reconstruction of a pre-service elementary
education major’s own mathematical knowledge.
References
Glaser, B. G.
& Strauss, A.L. (1967). The
discovery of grounded theory: Strategies
for Qualitative research.