Statement for the National Mathematics Panel
November 6th, 2006
Presented by: Dr. Judit Moschkovich
Associate Professor, Education Department, University of California, Santa Cruz
Representing TODOS: Mathematics for All
Thank you for this opportunity to address this distinguished panel. Your task is extraordinarily important to the mathematics and mathematics education community, and to the millions of students in our country for whom your recommendations will have significant consequences. I write as a representative of TODOS: Mathematics for All, an organization that advocates for an equitable and high quality mathematics education for all students, in particular Hispanic/Latino students as well as students from minority populations such as Native Americans, African Americans, and others.
My remarks represent TODOS as well as my own professional experience as a mathematics instructor at the university level and as a researcher in mathematics education for over 15 years. My career in mathematics education began when, after receiving a B.S. in Physics, I taught Algebra courses as a lecturer in the Mathematics Department at San Francisco State University. I received my Ph.D. in mathematics education in 1992 and have been conducting research in classrooms since then. I have been involved in mathematics education at many levels: I have served as a member of the editorial panel for the Journal for Research in Mathematics Education and as the Chair of the AERA Special Interest Group for Research in mathematics Education (2004-2006). I am the author of many research articles and chapters in edited books. I teach a course that introduces future secondary mathematics teachers to evidence-based research in mathematics education and courses for Ph.D. students in mathematics education. My research for the past 10 years has focused on the study of the relationship between language and learning mathematics, especially for Latino English learners.
I will limit my observations and recommendations to address one of the points listed on the Executive Order with regards to the Report to the President on Strengthening Mathematics Education. I include a list of research references at the end. Point for discussion, (c): The processes by which students of various abilities and backgrounds learn mathematics, with implications for instruction, teacher education, assessment, and materials development.
The main question I will address is “What are the needs of Latino English learners in mathematics classrooms?” Before I can address that question, I would like to lay the groundwork by first considering the following questions:
1. What is mathematical proficiency?
2. What is “conceptual understanding” in mathematics? Why is it important?
3. Why is communication important for learning mathematics?
1. What is mathematical proficiency?
A current description of mathematical proficiency comes from the book “Adding it up: helping children learn mathematics” published in 2001 by the National Research Council (edited by Kilpatrick, Swafford, and Findell). The NRC volume defines the intertwined strands of mathematical proficiency as: a) Conceptual understanding (comprehension of mathematical concepts, operations, and relations); b) procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently, and appropriately); c) strategic competence (formulating, representing, and solving mathematical problems (novel problems, not routine exercises); d) adaptive reasoning (logical thought, reflection, explanation, and justification); and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).
I will address only the first two components, procedural fluency and conceptual understanding. Fluency in performing mathematical procedures is what most people imagine we mean when we say “learning mathematics.” Conceptual understanding is more difficult to define and less well understood by parents, administrators, and beginning teachers. So, what is conceptual understanding? It involves the connections, reasoning, and meaning that learners (not teachers) construct. Conceptual understanding is more than performing a procedure accurately and quickly. It involves understanding why a particular result is the right answer and what the meaning of that result is, i.e. what the number, solution, or result represents.
Another aspect of conceptual understanding involves connecting representations (such as words, drawings, symbols, diagrams, tables, graphs, equation, etc.), procedures, and concepts (Hiebert & Carpenter, 1992). For example, if a student understands addition and multiplication, we would say they have learned to make connections between these two procedures, and expect that they would be able to explain how multiplication and addition are related (for example, that multiplication can be described as repeated addition). If they understand the procedures for multiplying and dividing negative numbers, we would say they have learned to make connections between these two procedures, and expect that they would be able to explain how the procedures for multiplication and division are similar, different, and explain why.
2. Why is conceptual understanding important?
One might think, “OK fine, so some researchers think that students need to be able to draw pictures and explain what they are doing when they perform a procedure, but what is the big deal about conceptual understanding? Why can’t students just learn their multiplication facts or learn that the right procedure to divide fractions is to “invert and multiply” and be done with it? I certainly don’t think I understand most of the arithmetic I learned and yet I have made it through school. Why does my child’s learning need to include conceptual understanding?”
One answer to these questions is that conceptual understanding and procedural fluency are closely related, even if we, as adults, do not now remember understanding a particular procedure when we learned that procedure. Research in cognitive science (Bransford, Brown, & Cocking, 2000) has shown that people remember better, longer, and in more detail if they understand, actively organize what they are learning, connect new knowledge to prior knowledge, and elaborate. Children will remember procedures better, longer, and in more detail if they actively make sense of procedures, connect procedures to other procedures, and connect procedures to concepts and representations. Rehearsal may work for memorizing a grocery list (and even then organizing the list will improve memorization). Rehearsal, however, is not the most efficient strategy for remembering how to perform demanding cognitive tasks. The research evidence is clear. The best way to remember is to understand, elaborate, and organize what you know (Bransford, Brown, & Cocking, 2000).
3. Why is communication important for learning mathematics?
One might think, “OK, I can see why my child needs to develop conceptual understanding, but what is all the fuss about communication in the mathematics classroom? I always did math by myself at my desk.” Communication is important because it supports conceptual understanding. The more opportunities a learner has to make connections among multiple representations, the more opportunities that learner has to develop conceptual understanding. But not all kinds of communication will support conceptual understanding in mathematics. Communication needs to be focused on important mathematical ideas. Classroom communication that engages students in evidence based arguments by focusing on explanations, arguments, and justifications builds conceptual understanding. Communication should include multiple modes (talking, listening, writing, drawing, etc.), because making connections among multiple ways of representing mathematical concepts is central to developing conceptual understanding.
4. What are the needs of Latino English Learners in mathematics classrooms[1]?
Latino students need to develop both procedural fluency and conceptual understanding, because these are two central aspects of mathematical proficiency. They also need to have opportunities to engage in mathematical communication, because this is a central way to develop both procedural fluency and conceptual understanding. Now, one might think “Sure, classroom discussions may support conceptual understanding. But English learners can’t participate in mathematical discussions because they are just learning English” or “English learners just need to learn math vocabulary. Then they can participate in mathematical discussions.”
These may seem like common sense claims. However, research shows that English language learners, even as they are learning English, can participate in discussions where they grapple with important mathematical content (for examples of lessons where English Learners participate in a mathematical discussion see Moschkovich, 1999a and Khisty, 1995). Instruction for this population should not emphasize low-level language skills over opportunities to actively communicate about mathematical ideas. One of the goals of mathematics instruction for students who are learning English should be to support all students, regardless of their proficiency in English, in participating in discussions that focus on important mathematical ideas, rather than on pronunciation, vocabulary, or low-level linguistic skills. By learning to recognize how English learners express their mathematical ideas as they are learning English, teachers can maintain a focus on the mathematical concepts as well as on language development.
Research also describes how mathematical communication is more than vocabulary. While vocabulary is necessary, it is not sufficient. Learning to communicate mathematically is not merely or primarily a matter of learning vocabulary. During discussions in mathematics classrooms students are also learning to describe patterns, make generalizations, and use representations to support their claims. The question is not whether students who are English learners should learn vocabulary but rather how instruction can best support students as they learn both vocabulary and mathematics. Vocabulary drill and practice is not the most effective instructional practice for learning either vocabulary or mathematics. Instead, vocabulary and second language acquisition experts describe vocabulary acquisition in a first or second language as occurring most successfully in instructional contexts that are language rich, actively involve students in using language, require both receptive and expressive understanding, and require students to use words in multiple ways over extended periods of time (Blachowicz & Fisher, 2000; Pressley, 2000). To develop written and oral communication skills students need to participate in negotiating meaning (Savignon, 1991) and in tasks that require output from students (Swain, 2001). In sum, instruction should provide opportunities for students to actively use mathematical language to communicate about and negotiate meaning for mathematical situations.
In conclusion, I would like to thank the National Mathematics Panel for this opportunity to address these concerns and represent TODOS.
Dr. Judit Moschkovich
Associate Professor
Education Department
University of California, Santa Cruz
1156 High Street
Santa Cruz, CA 95064
References
Blachowicz, C. and Fisher, P. (2000). Vocabulary instruction. In M. Kamil, P. Mosenthal, P. D. Pearson R. Barr, (Eds.), Handbook of Reading Research, Volume III. Mahwah, NJ: Lawrence Erlbaum Associates, 503-523.
Bransford, J., Brown, A., Cocking, R. (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.
Hiebert, J. and Carpenter, T. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of Research in Mathematics Teaching and Learning. New York: Macmillan, pp. 65-97.
Khisty, L. L. (1995). Making inequality: Issues of language and meanings in mathematics teaching with Hispanic students. In W. G. Secada, E. Fennema, & L. B. Adajian (Eds.), New directions for equity in mathematics education (pp. 279-297). New York: Cambridge University Press.
Khisty, L. L., McLeod, D., & Bertilson, K. (1990). Speaking mathematically in bilingual classrooms: An exploratory study of teacher discourse. Proceedings of the Fourteenth International Conference for the Psychology of Mathematics Educator, 3, 105-112. Mexico City: CONACYT.
Kilpatrick, Swafford, & Findell (2001). Adding it up: helping children learn mathematics. National Research Council. Washington, DC: National Academy Press.
Moschkovich, J.N. (in press) Bilingual Mathematics Learners: How views of language, bilingual learners, and mathematical communication impact instruction. To appear in N. Nasir and P. Cobb (Eds.), Diversity, Equity, and Access to Mathematical Ideas. New York: Teachers College Press.
Moschkovich, J. N. (in press). Using two languages while learning mathematics. To appear in Educational Studies in Mathematics.
Moschkovich, J. N. (2002). A Situated and Sociocultural Perspective on Bilingual Mathematics Learners. Mathematical Thinking and Learning, 4(2&3), 189-212.
Moschkovich, J. N. (1999a). Supporting the participation of English language learners in mathematical discussions. For the Learning of Mathematics, 19(1), 11-19.
Moschkovich, J.N. (1999b) Understanding the needs of Latino students in reform-oriented mathematics classrooms. In L. Ortiz-Franco, N. Hernandez, and Y. De La Cruz (Eds.), Changing the Faces of Mathematics (Vol. 4): Perspectives on Latinos. Reston, VA: NCTM, 5-12.
Pressley, M. (2000). What should comprehension instruction be the instruction of? In M. Kamil, P. Mosenthal, P. D. Pearson R. Barr, (Eds.), Handbook of Reading Research, Volume III. Mahwah, NJ: Lawrence Erlbaum Associates, 545-561.
Savignon, S. (1991). Communicative language teaching: State of the art. TESOL Quarterly, Vol. 25, No. 2, 261-277.
Swain, M. (2001). Integrating language and content teaching through collaborative tasks. Canadian Modern Language Review, Volume 58, No. 1, 44-63.
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[1] I address this question in more detail in a chapter titled “Understanding the needs of Latino students in reform-oriented mathematics classrooms” (Moschkovich, 1999b).