Monday, September 20, 2010

11 Interesting Articles on Math Education

Last year I was on sabbatical, working on a book. Playing With Math: Stories from Math Circles, Homeschoolers, and the Internet is nearing completion, and will most likely be published in 2011. It's full of great stories, puzzles, and ideas from over 20 authors.

To get that sabbatical, I had to make a proposal, and I was told that just editing a book wouldn't get me a whole year sabbatical. So I added a few more bits and pieces to my proposal, and now my 'sabbatical evidence' is due in 2 days. One of the agreements I made was to read 15 books and 15 articles from a list I provided. Below are my annotations for 11 of the articles (I've left out the ones I didn't care for). I've already posted about most of the books.

These articles are all good. If you haven't read the Treisman article, and you have any interest in social justice issues in relation to math education, do read it. The Hoyles article is also a classic. It differentiates between school-math and the math people create on their own, that they don't even think of as math. You may want to skim that one, but you'll find some gems in it.

Andreescu, T., and Mertz, J, Cross-Cultural Analysis of Students with Exceptional Talent in Mathematical Problem Solving. In Notices of the AMS V55, Num 10, 2008.
Janet Mertz spoke on this at the 2009 Great Circles conference hosted at the Mathematical Sciences Research Institute, in Berkeley. This talk is available as a Quicktime movie, and includes more discussion of the data than what is in the article.

Larry Summers, while president of Harvard, made the claim that the likeliest reason for the paucity of tenured women math professors was brain differences between men and women. (He hypothesizes that there is more variability in intelligence among men than among women, so there would be more extremely smart and more extremely dumb men than women.) One woman professor walked out, and then encouraged colleagues to do some statistical analysis. The percentage of women among those who rank most highly in math competitions varies widely from one country to another, from below 5% to over 20%. This is also reflected in the percentage of women in tenured math faculty in different countries (below 5% in some western European counties and over 20% in Portugal and a few other countries). This research shows that culture is a big component in girls’ and women’s achievement in math, making clear our inability to disentangle the effects of biology and culture.

Ball, D. L. , Working on the inside: Using one's own practice as a site for studying mathematics teaching and learning. In Kelly, A. & Lesh, R. (Eds.). Handbook of research design in mathematics and science education, (pp. 365- 402). [Link is to pdf.]
Ball analyzes how 3 different teacher-researchers (herself, M. Lampert, and R. Heaton) use their own teaching as a way of researching how teachers teach and students learn. In her introduction she discussed pre-services teachers’ misconceptions about math:
…what they believed was often at odds with what the teacher educators wanted them to think or know. For example, many believed that mathematical ability is innate and that many people simply cannot be good at mathematics. Most thought of mathematics as a cut-and-dried area of truths to be memorized and procedures to be practiced.
She discusses the necessity of using oneself as research subject because of the rarity of teachers doing the kind of teaching one might want to analyze.

Ball, D. L., & Bass, H., Interweaving Content and Pedagogy in Teaching and Learning to Teach: Knowing and Using Mathematics. In J. Boaler (Ed.), Multiple Perspectives on the Teaching and Learning of Mathematics (pp. 83-104). [Link is to pdf.]
The most valuable idea in this article for me is related to the notion of ‘compression’ – once we learn something well in math, it gets compacted, and seems simpler. To teach, we need to reverse that process:
…Mathematics is a discipline in which compression is central. Indeed, its polished, compressed form can obscure one’s ability to discern how learners are thinking at the roots of that knowledge. … Because teachers must be able to work with content for students in its growing, not finished, state, they must be able to do something perverse: work backward from mature and compressed understanding of the content to unpack its constituent elements.
The complex skills needed to teach math well are illustrated through classroom examples. [More Ball chapters and articles available online.]

Ball, D. L., Hill, H., and Bass, H., Knowing Mathematics for Teaching: Who Knows Mathematics Well Enough To Teach Third Grade, and How Can We Decide? In American Educator Fall 2005. [Link is to pdf.]
The authors looked at what special knowledge of math is needed by teachers, in order to effectively teach it. The article gives background on why they structured their research the way they did:
…there are legitimate competing definitions of mathematical knowledge for teaching....
Our aim is to identify the content knowledge needed for effective practice and to build measures of that knowledge that can be used by other researchers. The claim that we can measure knowledge that is related to high-quality teaching requires solid evidence.

The article also gives some preliminary results which may be promising regarding social justice:
… the size of the effect of teachers’ mathematical knowledge for teaching was comparable to the size of the effect of socioeconomic status on student gain scores. This … suggests that improving teacher’s knowledge may be one way to stall the widening of the achievement gap as poor children move through school.

Benson, S. & Findell, B., A Modified Discovery Approach to Teaching and Learning Abstract Algebra. In Innovations in Teaching Abstract Algebra, MAA Notes #60, eds. Allen Hibbard & Ellen Maycock, pp. 11-17, 2002.
The author taught almost entirely through worksheets he designed to get groups of students working on the material. His ability to observe the groups allowed him to get more clarity about what they understood than when he’d lectured. When they didn’t understand the congruence relationship, he got them to discuss what they had shown so far, and would not ‘give them the answer’. In spite of letting go of a ‘calendar’ and taking the time needed for students to work out their own understandings, he found that this class actually covered more material than the usual.

Dweck, C., Caution: Praise Can Be Dangerous. In American Educator, Spring 1999.
Carol Dweck has written a book, Mindset (2006), which says about the same things this article does, at much more length. The article describes her thesis and her research much more concisely and (in my opinion) effectively. Her claim (proved by her research) is that praising a student’s intelligence makes them wary of harder tasks and of looking dumb, but praising their effort encourages them to tackle harder tasks and enjoy it. She also looks at people who think intelligence is fixed and compares them to people who think effort can change one’s intelligence. People with the second mindset are able to develop their potential much more effectively than those with a ‘fixed intelligence’ mindset.

Hoyles, C., Noss, R., & Pozzi, S., Proportional Reasoning in Nursing Practice. In Journal for Research in Mathematics Education, Vol. 32, No. 1 (Jan., 2001), pp. 4-27
Research has repeatedly shown people doing mathematics in their work situations more effectively than they can as students. Most people say they do no math in their work, because they don’t recognize that what they’re doing is mathematical.

These studies suggest that adults are adept at solving proportional problems in everyday or work situations but often employ informal strategies that are tailored to the particular situation and are not easily identified with formal school-taught methods. (From page 6 of pdf.)

The authors of this article look at nurses’ calculations of drug dosages and found these calculations to be more flexible and fluent than the 'nurses Rule’ they’d been taught.

Kato, Y., Honda, M., & Kamii, C., Kindergartners Play Lining Up the 5s: A Card Game to Encourage Logico-Mathematical Thinking. In Mathematical Behavior, 13(1), 55-80, 2006.
The authors describe their research studying video of children playing a very simple card game. They are looking for ways to describe progress in logical (‘logico-mathematical’) thinking skills. I am impressed with how much thinking kids need to develop to do things that seem utterly simple to adults.
The … categories that the players created are much more abstract than those children can create in sorting activities involving squares, rectangles, “red ones,” “blue ones,” and so on. The seriation involved in “cards to be used first, second, and last” is likewise much more abstract than what can be done with Montessori sticks and cylinders. If we had to set standards for mathematics in kindergarten, we would never think of including the high-level logic that we saw in Lining Up the 5s.

Play has long been valued in early childhood education, and we will do well to analyze it with depth and precision not only in card games but also in other kinds of play that naturally appeal to young children.
My belief, which continues to be affirmed by all the research I’ve done this year, is that we would do well to continue to let children learn through play, for as long as they wish. Perhaps then they’d choose to study hard later, for the sheer pleasure of the learning.

Schoenfeld, A., A Highly Interactive Discourse Structure. In Social Constructivist Teaching, Volume 9, pages 131–169. 2002
A quarter of this article is devoted to transcripts of two very different classes, a high school physics class, and a third grade math class. The two classes turn out to share a structure in the interactions between the teacher and students. Schoenfeld has created a flowchart of this structure, but I find a summary more useful:
Teacher starts by giving context and background for topic.
• Asks class: “What (else) can you say about [this topic]?”
• Calls on a student.
• Does their response raise other issues? (If so, deal.)
• Is clarification, expansion or reframing useful? (If so, deal.)
• Would more discussion be useful? (If so, deal.)
[I have created a sheet to remind myself of this summary, to help me get out of lecture mode.] Deborah Ball taught the third grade class, which was videotaped, and is cited in many researchers' work.

Schoenfeld, A. H., What do we know about mathematics curricula? Journal of Mathematical Behavior, 13(1), 55-80, 1994.
Schoenfeld discusses what we do and don’t know about the math curricula we need. “The ‘constructivist perspective’ is better grounded in empirical and experimental evidence than the theory of evolution; we should just assume it and get on with our business (while working … hard, of course, to flesh it out and understand it more fully).” But, on the other hand, the best balance between traditional ‘content’ and the development of problem-solving skills is unclear, and “If, for example, what we now call ‘algebra’ is distributed through the curriculum in bits and pieces and learned in specific problem solving or applied contexts, how do we know when and to what degree students will have the relevant algebraic skills to deal with problems they will encounter?”

The article includes an excellent section on the value and uses of proof, which starts with:
There are, I think, three roles of proof that need to be explored and understood: the unique character of certainty provided by air-tight mathematical arguments, which differs from that in any other discipline and is part of what makes mathematics what it is; the fact that proof need not be conceived as an arcane formal ritual, but can be seen as the mere codification of clear thinking and a way of communicating ideas with others; and the fact that for mathematicians, proving is a way of thinking, exploring, of coming to understand – and that students can and should experience mathematical proving in the same ways.

Treisman, U., Studying Students Studying Calculus: A Look at the Lives of Minority Mathematics Students in College. In The College Mathematics Journal, V 23, #5, Nov 1992.
Uri Treisman’s work with first-year calculus students at UC Berkeley is famous. Black students were uniformly failing this course. These students were not underprepared; they were the cream of the crop in many ways. No one really understood the problem. Treisman decided to follow the students home, to get real information. He looked at the Black students and the Asian students. What were they each doing when they studied? Both the Black and Asian students started out studying and completing homework for about 8 hours a week alone. But then the Asian students got together in groups of friends and discussed the homework. If one person had a different answer, they could learn from the others. If they all had different answers, they knew they were lost on a particular problem.

Treisman knew they needed to find a way to encourage group discussions among the Black students. They put together a workshop program in which students were asked to work on especially challenging problems in groups. “Our idea was to construct an anti-remedial program for students who saw themselves as well prepared.” The students who went through this program did significantly better than the average for all students. Treisman concludes with thoughts about how to change all calculus courses to include this sort of engaging work.


  1. I was planning to get something done this week but now..... :) Thanks for these great articles.

  2. This is awesome! Thanks for sharing this.

  3. Glad it's useful, and not just an empty exercise to prove I was working last year. ;^)

    Maybe I'll post a list some day of all the other articles I read, that weren't on my list.


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