Math 506A: Research on the Learning of Mathematics: Fall 2006

Tuesdays 5:15 - 7:45 pm                                                      Math East 143/141

 

               Cynthia Anhalt                            Marta Civil                                     Javier Díez Palomar

               canhalt@math.arizona.edu     civil@math.arizona.edu               jadiez@math.arizona.edu

               Math East 244                                             Math 203                                        Math East 240B

               626-1522                                        621-6873                                        621-8306

                                            

This course has two main objectives (breadth and depth):

1)      To familiarize students with some of the key pieces of research in mathematics education (with particular attention to research on student learning). So, in this sense, it is a survey course into main areas of research on learning.

2)     To allow for students to go in depth in an area of their interest through a series of assignments during the semester, culminating in a poster presentation.

 

Required Text

Carpenter, T. P., Dossey, J. A., & Koehler, J. L. (2004) (Eds.). Classics in Mathematics Education Research. Reston, VA: NCTM

 

There will several other readings handed out during the semester either via e-mail as PDF attachments or as photocopies.

 

About the course

This course will be run as a seminar with a pair of students each week facilitating the discussion of the articles. To ensure the flavor of a seminar, students’ participation is crucial.  A major part of this course is reading and discussing ideas/issues related to teaching and learning mathematics as they relate to the readings; therefore adequate time spent preparing for each class period is expected of every student – read assigned material thoroughly and take some notes as you read.

 

There will be three main “activities” in this course

 

In-class work: as discussion facilitators you will be expected to prepare some questions and / or activities related to the readings assigned for your session. As a participant, your responsibility will be to have carefully read the articles to help your colleagues as they facilitate the discussion.

 

PROJECT I: Each student will become knowledgeable with some of the literature in a particular area of student learning of mathematics (e.g., children’s understanding of the concept of area). This knowledge will be built as students collect, read, interpret, compare, synthesize, and write about a set of research articles/chapters related to a topic of their choosing (to be agreed upon by the student and instructors). This project will be broken down into two smaller pieces as follows:

Assignment 1 – Project I: A two to three page write up on the topic of your choice that includes: rationale (why did you choose this topic) and a review of one research article related to your topic. The review should 1) briefly describe the research subjects and methods, (2) summarize key findings and recommendations, and (3) offer your personal commentary / critique. [Due September 26]

Assignment 2 – Project I: A 6-8 page review of three research articles related to your topic (the article used for assignment 1 can be one of them). These articles must be OK’d by the instructors in advance. Students will bring two copies of their paper to class on Tuesday October 17 to exchange with other students as part of a peer discussion of each other’s papers that will take place the following Tuesday (October 24).

Final Review of the Literature Paper – Project 1: The paper for assignment 2 will be revised (according to instructors’ and peers’ feedback) and expanded to a 10-14 page review of 6 to 7 articles, due Tuesday December 5. There should be one section on recommendations for further research (based on both your readings and your own study (see Project 2)).

 

PROJECT II (can be done in pairs): This will be a mini research project in which you explore further the topic of your choice for project 1 by carrying out a small research study. The main goal is for you to develop some experience doing a research project in an area of students’ understanding of a topic. Although there are many approaches possible, in this course we will emphasize the use of task-based interviews as the research approach. So, for example, if you topic of interest is “children’s understanding of area,” your research project may consist of a task-based interview to explore 2 or 3 children’s understanding of area. Replication of methods used by other researchers is acceptable. On November 7th we will have an in class session where you exchange your research ideas / instruments for peer and instructor feedback. Bring 2 to 3 copies of a draft to this meeting to be shared with a small working group.

You will present this research project as a poster the day of the final exam (Tuesday December 12). In addition to the poster, you will turn in a final write-up that describes methods and results and situate findings within the literature on the topic. Expected length for this write-up is approximately 5 to 7 pages.

 

Format for Papers

APA style (fifth edition) is required for all write-ups. All write-ups should be double-spaced with 1 inch margins on all sides. Block quotes should be single-spaced and indented. Use a 12-point font and a standard font style (i.e., CG Times, Arial, or Times New Roman).

 

 

JOURNAL LIST

 

Educational Studies in Mathematics (ESM) - http://www.springerlink.com/content/1573-0816/

Focus on Learning Problems in Mathematics (FLPM)

For the learning of Mathematics (FLM) - http://flm.educ.ualberta.ca/

Journal for Research in Mathematics Education – Monograph - http://www.jstor.org/journals/08839530.html

Journal for Research in Mathematics Education (JRME) –

http://my.nctm.org/eresources/journal_home.asp?journal_id=1

Journal of Mathematical Behavior [JMB] –

http://www.elsevier.com/wps/find/journaldescription.cws_home/620200/description#description

Journal of Mathematics Teacher Education [JMTE] - http://springerlink.metapress.com/content/1573-1820/

Mathematical Thinking and Learning (MTL) (It’s published by Lawrence Erlbaum; http://www.erlbaum.com

Mathematics Educator [ME] - http://math.coe.uga.edu/tme/tmeonline.html

 


 

 

Day:

Topic:

Readings (due on date indicated on far left column):

Assignments:

Facilitators:

Aug 22nd

Course Overview

“Combining Cognitive and Sociocultural approaches in mathematics education research”

 

 

 

Aug 29th

Intro to research (interviewing)

The role of cognition

Š         Brenner, M.E. (2006), (In Green, Camilli & Elmore)

Š         Brenner, M. E. (1998), (AEQ)

Š         Erlwanger, S. H. (1973), (ch. 5, Classics...)

 

 

Sept 5th

Student Thinking /Arithmetic

Š         Lampert, M. (1990), (ch. 12, Classics...)

Š         Carpenter el al. (1997), (JRME)

 

 

Sept 12th

Geometry

Measurement

Š         van Hiele, P. M. (1959/1985), (ch. 6, Classics...)

Š         Barrett, Clements, Klanderman et al. (2006), (JRME)

Š         Battista, M.T. (2004), (MTL)

 

 

Sept 19th

Fractions

Š         Cramer et al (2002), (JRME)

Š         Empson, S.B. (2003), (JRME)

Š         Mack, N.K. (2001), (JRME)

 

 

Sept 26th

Algebraic thinking

 

Š         Chazan, D. & Yerushalmy, M. (2003), (ch.9, Research companion...)

Š         Carraher et al. (2006), (JRME)

Š         Knuth et al. (2006), (JRME)

Project I-

Assignment 1 due

(1 article)

 

Oct 3rd

 

Reasoning & proof

Š         Yackel, E., Hanna. G. (2003), (ch. 15, Research companion...)

Š         Reid, D.A. (2002), (JRME)

Š         Healy, L. & Hoyles, C. (2000), (JRME)

 

 

Oct 10th

Constructivism / Emergent/

Sociocultural/

Ethnomathematics

Š         Steffe, L. & Kieren, T. (1994), (ch.7, Classics...)

Š         Cobb, P. & Yackel, E. (1996), (ch. 17, Classics...)

Š         D’Ambrosio, U. (1985), (ch. 15, Classics...)

 

 

Oct 17th

Research methods

Š         Doyle, W. (1988). (Educational Psychologist)

Š         Pirie, S. (1997). (JRME Monograph, 9)

Project I-

Assignment 2 due

(3 articles) (papers to exchange with peers)

 

Oct 24th

Paper discussion / Peer feedback

Working Groups

 

 

Oct 31st

Representation

Š         Smith, S. P. (2003), (ch. 18, Research companion…)

Š         Goldin, G. & Shteingold, N. (2001), (Cuoco & Curcio)

 

 

Nov 7th

 

Research proposal discussion / peer feedback

 

Working Groups

Project II-

Research proposals due--bring drafts to class

 

Nov 14th

Multiplicative Structures / Proportional Reasoning

Š         Vergnaud, G. (1988), (ch.8, Classics...)

Š         Lesh, R., Behr, M., Post, T. (1987), (In Janvier).

 

 

Nov 21st

Mathematics & cultural context

Š         Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985), (ch. 14, Classics...)

Š         Bishop, A. J. (1988), (ch. 16, Classics...)

Project II-

Informal report on progress of research project

 

Nov 28th

Mathematics & language

Š         O’Connor, M. (2001), (ESM)

Š         Lampert, M. & Cobb, P. (2003), (ch. 16, Research companion...)

Š         Moschkovich, J. (2002), MTL

 

Project II-

Informal report on progress of research project

 

Dec 5th

 

 

TBA

Project I-

Final review of literature due (6-7 articles)

 

Dec 12th

Final Exam

Poster Presentations

Final Research Project II write-up due

 



 

Reference / Resource List [Partial; more references will be added during the semester]

 

Barret, J. E., Clements, D.H., Klanderman, D., Pennisi, S-J., & Polaki, M.V. (2006). Students’ coordination of geometric reasoning and measuring strategies on a fixed perimeter task: developing mathematical understanding of linear measurement. Journal for Research in Mathematics Education, 37(3), 187-221.

Battista, M. T. (2004). Applying cognition-based assessment to elementary schools students’ development of understanding of area and volume measurement. Mathematical thinking and learning, 6(2), 185-204.

Bishop, A. J. (1988/2004). Mathematics education in its cultural context. In T.P. Carpenter, J.A. Dossey, & J.L. Koehler (Eds.), Classics in mathematics education research, (pp. 200-2007). Reston, VA: NCTM.

Bihop, A. J., Clements, K., Keitel, C., Kilpatrick, J., & Laborde, C. (Eds.) (1996). International handbook of mathematics education. Boston, MA: Kluwer.

Brenner, M. E. (1998). Adding cognition to the formula for culturally relevant instruction in mathematics. Anthropology & Education Quarterly, 29(2), 214-244.

Brenner, M.E. (2006). Interviewing in educational research. In J. l. Green, G. Camilli, & P. B. Elmore (Eds.), Handbook of complementary methods in education research (pp. 357-370). Mahwah, NJ: Lawrence Erlbaum.

Carpenter, T. P., Franke, M.L., Jacobs, V.R., Fenema, E., & Empson, S.B. (1997). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29(1), 3-20.

Carpenter, T., Fennema, E., Peterson, P., Chiang, C., & Loef, M. (1989/2004). Using knowledge of children’s mathematics thinking in classroom teaching: an experimental study. In T.P. Carpenter, J.A. Dossey, & J.L. Koehler (Eds.), Classics in mathematics education research, (pp. 134-151). Reston, VA: NCTM.

Carpenter, T.P., Dossey, J.A., & Koehler, J.L. (Eds.) (2004). Classics in mathematics education research. Reston, VA: NCTM.

Carraher, D.W., Schliemann, A.D., Brizuela, B.M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87-115.

Carraher, T.N., Carraher, D.W., & Schliemann, A.D. (1985/2004). Mathematics in the streets and in schools. In T.P. Carpenter, J.A. Dossey, J.L. Koehler (Eds.), Classics in mathematics education research, (pp. 186-193). Reston, VA: NCTM.

Chambers, D.L. (2001) (Ed.). Putting Research into Practice in the elementary Grades: Readings from Journals of the NCTM. Reston, VA: NCTM.

Chazan, D., & Yerushalmy, M. (2003). On appreciating the cognitive complexity of school algebra: research on algebra learning and directions of curriculum change. In J. Kilpatrick, G. Martin, D. Schifter (Eds.), A research companion to principles and standards for school mathematics, (pp. 123-135). Reston, VA: NCTM.

Cobb, P. & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31(3/4), 175-190.

Cobb, P. & Yackel, E. (1996/2004). Constructivism, emergent, and sociocultural perspectives in the context of developmental research. In T.P. Carpenter, J.A. Dossey, & J.L. Koehler (Eds.), Classics in mathematics education research, (pp. 208-226). Reston, VA: NCTM.

Cramer, K.A., Post, T.R., & delMas, R.C. (2002). Initial fraction learning by fourth- and fifth- grade students: a comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33(2), 111-144.

D’Ambrosio, U. (1985/2004). Ethnomathematics and its place in the history and pedagogy of mathematics. In T.P. Carpenter, J.A. Dossey, & J.L. Koehler (Eds.), Classics in mathematics education research, (pp. 194-199). Reston, VA: NCTM.

Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction. Educational Psychologist, 23, 167-180.

Empson, S.B. (2003). Low-performing students and teaching fractions for understanding: an interactional analysis. Journal for Research in Mathematics Education, 34(4), 305-343.

English, L.D. (Ed.) 2002. Handbook of International research in Mathematics Education. Mahwah, New Jersey, London: Lawrence Erlbaum Associates Publishers.

Erlwanger, S.H. (1973/2004) Benny’s conception of rules and answers in IPI mathematics. In T.P. Carpenter, J.A. Dossey, J.L. Koehler (Eds.), Classics in mathematics education research, (pp. 48-59). Reston, VA: NCTM.

Goldin, G. & Shteingold, N. (2001). Systems of representations and the development of mathematical
concepts. In A. Cuoco & F. Curcio (Eds.) The roles of representation in school mathematics 2001 yearbook (pp. 1-23). Reston, VA: National Council of Teachers of Mathematics.

Grouws, D. A. (Ed.) (1992). Handbook of research on mathematics teaching and learning. New York, NY: Macmillan.

Healy, L. & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396-428.

Kelly, A. E. & Lesh, R. A (Eds.) (2000). Handbook of research design in mathematics and science education. Mahwah, NJ: Lawrence Erlbaum.

Kilpatrick, J., Martin, G., & Schifter, D. (Eds.) (2003). A research companion to principles and standards for school mathematics. Reston, VA: NCTM.

Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) (2001). Adding it up. Helping children learn mathematics. Washington, D.C.: National Academic Press.

Knuth, E.J., McNeil, N.M. & Alibali, M.W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 292-312.

Lampert, M. & Cobb, P. (2003). Communication and language. In J. Kilpatrick, G. Martin, D. Schifter (Eds.), A research companion to principles and standards for school mathematics, (pp. 237-249). Reston, VA: NCTM.

Lampert, M. (1990/2004). When de problem is not the question and the solution in not the answer: mathematical knowing and teaching. In T.P. Carpenter, J.A. Dossey, J.L. Koehler (Eds.), Classics in mathematics education research, (pp. 152-171). Reston, VA: NCTM.

Lannin, J.K. (2005). Generalization and justification: the challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231-258.

Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, G. Martin, D., & Schifter (Eds.), A research companion to principles and standards for school mathematics, (pp. 179-192). Reston, VA: NCTM.

Lesh, R., Behr, M., & Post, T. (1987). Rational Number Relations and Proportions. In C. Janvier (Ed.), Problems of Representations in the Teaching and Learning of Mathematics (pp. 41-58). Hillsdale, NJ: Lawrence Erlbaum.

Mack, N.K. (2001). Building on informal knowledge through instruction in a complex content domain: partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267-295.

Moschkovich, J. (in press). Using two languages when learning mathematics. Educational Studies in Mathematics.

Moschkovich, J. (2002). A situated and sociocultural perspective on bilingual mathematics learners. Mathematical Thinking and Learning, 4, 189-212.

NCTM (1993) (Jensen, R. J., Ed.). Research ideas for the classroom: Early childhood mathematics. Reston, VA: NCTM.

NCTM (1993) (Owens, D. T., Ed.) Research ideas for the classroom: Middle grades mathematics. Reston, VA: NCTM.

NCTM (1993) (Wilson, P. S., Ed.) Research ideas for the classroom: High school mathematics. Reston, VA: NCTM

NCTM (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

O’Connor, M. (2001). Can any fraction be turned into a decimal? A case study of a mathematical group discussion. Educational Studies in Mathematics, 46, 143-185.

Pirie, S. (1997). Working toward a design for qualitative research. In A. Teppo (Ed.), Qualitative research methods in mathematics education. Journal for Research in Mathematics Education, Monograph Vol. 9, (pp. 79-97). Reston, VA: NCTM.

Reid, D.A. (2002). Conjectures and refutations in grade 5 mathematics. Journal for Research in Mathematics Education, 33(1), 5-29.

Saxe, G. B., Taylor, E. V., McIntosh, C. & Gearhart, M. (2005). Representing fractions with standard notation: A developmental analysis. Journal for Research in Mathematics Education, 36, 137-157.

Schoenfeld, A. (2002). Research methods in (mathematics) education. In L. D. English (Ed.), Handbook of international research in mathematics education, (pp. 435-487). Mahwah, New Jersey: Lawrence Erlbaum Associates.

Smith, S. P. (2003). Representation in school mathematics: children’s representations of problems. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics, (pp. 263-274). Reston, VA: NCTM.

Sowder, J. & Schappelle, B. (2002) (Eds.). Lessons Learned from Research. Reston, VA: NCTM.

Steffe, L. & Kieren, T. (1994/2004). Radical constructivism and mathematics education. In T. P. Carpenter, J. A. Dossey, & J. L. Koehler (Eds.), Classics in mathematics education research, (pp. 68-83). Reston, VA: NCTM.

van Hiele, P. M. (1959/1985). The child’s thought and geometry. In T.P. Carpenter, J.A. Dossey, & J.L. Koehler (Eds.), Classics in mathematics education research, (pp. 60-67). Reston, VA: NCTM.

Vergnaud, G. (1988/2004). Multiplicative structures. In T.P. Carpenter, J.A. Dossey, & J.L. Koehler (Eds.), Classics in mathematics education research, (pp. 84-97). Reston, VA: NCTM.

Yackel, E. & Hanna, G. (2003). Reasoning and proof. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics, (pp. 227-236). Reston, VA: NCTM.