Knowledge of Children’s Learning of Mathematics: A Common Denominator in Preservice, Teacher, and Parent Education David Feikes David Pratt Sarah Hough Purdue North Central Purdue North Central University California Santa Barbara

Copyright © 2007 Purdue University North Central

Connecting Mathematics for Elementary Teachers (CMET) Connecting Mathematics for Elementary Teachers (CMET) NSF CCLI Grants DUE & DUE The views expressed in this paper are those of the authors and do not necessary reflect those of NSF.

Chapter 1Problem Solving Chapter 2Sets Chapter 3Whole Numbers Chapter 4Number Theory Chapter 5Integers Chapter 6Rational Numbers - Fractions Chapter 7Decimals, Percents, and Real Numbers Chapter 8Geometry Chapter 9More Geometry Chapter 10Measurement Chapter 11Statistics/Data Analysis Chapter 12Probability Chapter 13Algebraic Reasoning

Mathematical Content Courses for Elementary Teachers Focus on How Children Learn Mathematics! Methods Courses Methods Courses Graduate Courses Graduate Courses

CMET Materials: Descriptions, written for prospective elementary teachers, on how children think about, misunderstand, and come to understand mathematics. These descriptions are based on current research and include:  how children come to know number  addition as a counting activity  how manipulatives may embody (Tall, 2004) mathematical activity  concept image (Tall & Vinner, 1981) in understanding geometry In addition to these descriptions the CMET materials contain:  problems and performance data from the National Assessment of Educational Progress (NAEP)  Problems and performance data from the Third International Mathematics and Science Study (TIMSS)  our own data from problems given to elementary school children  questions for discussion.

Preservice Teachers Preservice elementary teachers’ mathematical knowledge, beliefs, and efficacy about teaching and the learning of mathematics can be developed by focusing on how children learn and think about mathematics in content courses.

Teachers and Parents Methods – Journaling, Interviews, Reviews Participants were asked to describe: what they learned by using the materials times when they considered or used information from the project materials in their teaching or with their children how they used this information, when they did something differently or tried something new based on an idea from the text any instance where a child’s mathematical thinking was like or unlike that described.

Indicators of: Verification of CMET Learning Influence Practice

Teachers - Verification [On children’s understanding of graphs] I can hear my class talk about “being the winner” if their choice is picked by most. It’s kind of like being the first one to arrive in the classroom. [Probability] At the third grade level, personal experiences will even influence how a child answers reading comprehension questions. They will choose the answer they would like it to be, rather than the correct answer.

Teachers - Learning 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 was 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.

Teachers - Learning Children focus on “filling in the blanks” when using pre-partitioned shapes instead of looking at the meaning of the fraction. Students need the mental images in order to understand geometry. I was helpful to see how the children progress through the stages of measurement. I also liked the different concepts of measurement; iteration was a new term for me. I learned about things not to do, like stringing problems together, which misrepresents the facts.

Teachers - Learning I found it interesting that one of the problems children have with decimals is that they try to understand them by relating to their prior understanding of whole numbers. I also enjoyed reading about invariance of shape. Every year I draw a “downward” triangle on the board and ask my students what “this” is. Each year they say, “An upside down triangle”.

Teachers - Learning I liked the part where it includes the exact verbiage that a child said in regards to the decimals. It gives teachers an opportunity to be inside the head of a child. This chapter refreshed why we count the number of decimal factors and why we move the number of decimal places in the divisor to make a whole number. I had forgotten why. It was just how it is suppose to be. It is just the rule. It reminded me so that I can explain it to the children. When there is an explanation to give to the children, they believe it and trust it.

Teachers – Influence Practice Many of my kindergarten students counted one, two, three, four, five, seven and I didn’t understand why. [Later she wrote] One-to- one correspondence is how young children count. I also forgot how important it is to let children count on their fingers. In my fifth grade classroom, some children are still …[counting on their fingers] and the try to hide their hands.

Teachers – Influence Practice The paragraph about multiplication thinking strategies was very helpful to me. I began using this in class just recently. My students are eager to share their thinking strategies. I believe once they can express their thought process in words, it helps solidify the sense making aspect of math. It is hard for teachers to try to think like students. Many times, you are teaching the children and they are not getting it. And, you keep persisting and cannot figure out why they do not get it. And, cannot see what or why they are thinking what they do. This part [of CMET] explains that. I worked with a student one night that was having problems with integers. We were working on the rules for multiplying and dividing negative numbers. I was able to use this chapter to help explain.

Teachers – Influence Practice I liked the section called Key Concepts of Measurement. That section was very detailed. I liked the detailed pictures. The explanations of actual children doing the activities were great. I have tried several of the activities with the students that I work with at Sylvan. I also found the area problem where you cover up or erase part of the area to be beneficial. I tried this with my students to see who really could visualize and understand the concept of area. The time section was very helpful. I now understand why time is so confusing. They can’t see it!! They can’t touch it!! Conservation is a hard skill, so it makes sense why elapsed time can be so hard and confusing for students.

Teachers – Influence Practice This chapter has allowed me to accept a student’s self-generated algorithm for solving a computation problem. If the child is able to explain their process, and it correctly leads to the answer, I will encourage this child to continue to use this algorithm. One of my students recently created his own algorithm for subtracting two-digit numbers that required “trading.” In the problem 64 – 29, Austin came up with a “partial differences” algorithm. He subtracted 60 – 20 = 40, and then subtracted 9 – 4 = 5, he then took 40 – 5 = 35. This algorithm worked for him, and he taught it to a friend who was struggling with “trade- first” subtraction.

Teachers – Influence Practice I didn’t feel that my students really understood the concept of congruence, so I took a suggestion from CMET, and we physically cut out shapes to compare congruency. Students are now able to determine congruency without having to physically manipulate the shapes. I have been making a conscious effort to incorporate geometry into all areas of my teaching. I had an occasion to discuss the problem: = ___ + 5 with my second grade students. We have since been working several problems like this during calendar, and they really seem to get the concept of equality.

Parent - Verification As a parent of an almost 4 year old, I tried the measure activity with his footsteps. It is so neat to see he did exactly what you illustrated was typical at that developmental level.

Parent - Verification I thought you’re example of time and speed conservation was a nice way to present the problem that children face when trying to understand time. I did a similar activity with my pre-schooler. He too thought that the sand moved faster when he moved faster. It is very interesting to put these examples into action.

Parent - Verification I found myself actually having my 4 year old collect data with a mini clipboard. I had him go around our house when we had company lately and see who was wearing jeans, pants, or skirts. He drew pictures and reported his “data” to us at dinnertime. It was so neat to see him do this. I have since used this with my friends, mothers as well, and they too have been pleasantly surprised by their little statisticians!

Parent - Learning [A parent talking on her daughter’s understanding of place value] I think she understands place value to a certain degree, but I know she doesn’t totally get it. She knows that she must trade or borrow, but I don’t think she truly understands why. I would like to get a set of unifix cubes for home for when she does her homework to help her better understand this concept.

Parent – Influence Practice A parent of a sixth grader In her journaling, she bulleted the following changes:  restraining myself to not "jump in" and allow her to experience problem solving for herself  becoming more open to using other resources/tools to assist her, as "my way" of thinking/teaching, may be more harmful than helpful to her

Parent – Influence Practice CMET changed how she helped her daughter with her mathematics homework. She found herself more willing to let her daughter experience mathematics rather than just telling her how to do it She began to realize that her way of helping her daughter may not be the best approach

Concluding Comments by a Teacher …you’re not teaching in the CMET how or when to teach concepts, but rather giving some insight as to how children think about and learn these. This chapter made me realize that this is the first text I have ever read that aims to help the reader understand what children have been and will be going through when attempting to learn math. This is such a neat idea because not only will pre-service teachers and parents understand their children’s mathematical thinking better, but they will also have more empathy for them.

Summary  Focusing on knowledge of children’s mathematical thinking raises prospective teachers’ efficacy to understand and teach mathematics as well as having an impact on their beliefs about mathematics and its teaching.  Initial qualitative evidence suggests that using this approach also influences teachers and parents in their teaching and work with children.

Looking to the Future We need letters of commitments from faculty to attend a faculty workshop or use CMET in either your content or methods courses for our next grant! We are seeking a funding source to support the development of parent resources and parent research.