Someone asked today on math-teach@mathforum.org why it is that students fail at math. His answer?

This struck me as a rather useless tautology, a non-explanation purporting to reveal all. I could not see what he subsequently offered as any sort of realistic answer. I thought instead about Davydov's curriculum.

The short answer: They don't get it.

The long answer...

What is it that they don't get? They don't get the chain of reasoning involved in mathematics.

This struck me as a rather useless tautology, a non-explanation purporting to reveal all. I could not see what he subsequently offered as any sort of realistic answer. I thought instead about Davydov's curriculum.

I first became aware of the mathematical curriculum of V. V. Davydov and his colleagues early in 2009 when several people independently wrote to ask me my opinion of his "measurement-based" approach to elementary mathematics. I had no idea who Davydov was or what his approach entailed.

After reading some papers that these folks sent me, I knew that Davydov followed and grounded his work in some key ideas of Vygotsky, and that at least one attempt had been made in the US to develop and implement a measurement-based elementary curriculum: MEASURE UP! at the University of Hawaii's Curriculum Research and Development Group. One of those developers was someone I knew, Barbara Dougherty, now at Iowa State University.

Further, I learned that Jean Schmittau at SUNY@Binghamton had written several papers about the Davydov curriculum and had, in fact, translated Davydov's elementary books (which comprise an early-grade three-year program) into English in the late 1990s and used these translations to work with elementary students in a local school district.

Perhaps coincidentally, at the time I became aware of and began looking into Davydov's work, I became interested and embroiled in the subject of my previous post (and many others): the controversy Keith Devlin generated beginning with his June 2008 column on multiplication, "It Ain't No Repeated Addition." By the time I jumped in, first to oppose Devlin's views and later to agree with them, the arguments about whether there was any reason NOT to teach young students that multiplication is repeated addition (MIRA) and whether in fact it was ever true to claim that MIRA is correct was raging hot and heavy in the blogosphere. That didn't stop me from putting both feet in my mouth (I took umbrage at Devlin, a mathematician I'd once met briefly and whom I respected as a solid thinker, trying to tell elementary teachers how to teach basic mathematics to kids.) And then I actually started to think, both about the nature of multiplicative reasoning and problems I saw in typical elementary school mathematics education (and our way of "schooling" so many students into intellectual passivity) that Davydov's approach seemed well-positioned to address.

As I began to post about both Davydov and the multiplication/MIRA issues both on my blog and on a couple of on-line discussion lists, I noticed how quickly the educational conservatives lined up against Devlin, against Davydov, against anything that wasn't precisely how they were taught or taught others themselves in the cases where they instructed on elementary mathematics (usually in remedial situations).

One might expect that this approach would have great appeal to anyone concerned with how little is asked of US students for the most part in elementary mathematics teaching and curricula. And given the fact that for many disadvantaged students, reading is a huge impediment to using many of the often-maligned progressive reform elementary books such as EVERYDAY MATHEMATICS and INVESTIGATIONS IN NUMBER, DATA & SPACE, the following should be enormously heuristic:

I find it interesting, however, that in speaking with several US mathematicians and mathematics educators who either have developed or are in the process of developing curricular materials (or both) based on Davydov's ideas and work state that they don't feel that Davydov's original materials would fly in the US (in translation, of course).

I strongly suspect that the concern is not so much for the ability of kids to succeed with Davydov's materials (after all, if they work for kids in Russia, and they are still successful there (at least up to about 10 years ago; I have no more updated information) as evidenced by the 1999 testing at the end of third grade of 2300 Rusian students who used the El'Konin-Davydov curriculum. According to Catherine Sophian's latest book, Vorontsov concluded based on the results of the testing that

In fact, I suspect the problem, such as it is, has little to do with kids, but rather with the low ability and willingness of so many US elementary teachers (and beyond) to teach mathematics without step-by-step guidebooks and their relatively weak knowledge of the relevant mathematics and content-based pedagogy. Considering that any non-traditional mathematics program introduced in the US that does not have the imprimatur of such educationally conservative groups as Mathematically Correct, NYC-HOLD, and a host of think tanks that condemn each and every "progressive math" program being used in this country, but praise uncritically particular programs from high-scoring Asian countries (particularly Singapore) for reasons that may not be completely non-political or grounded in reality for American kids and teachers, are subject to criticism for being TOO verbal, TOO hard, TOO sparse AND TOO watered-down (I am not kidding; the willingness of these self-proclaimed curriculum experts to offer utterly contradictory criticism in order to destroy programs they don't like is remarkable) it takes a daring teacher, school or district indeed to risk using something that is so starkly non-verbal as are the original Davydov materials.

Nonetheless, I think that teachers in inner-city, poverty-stricken, and other challenging environments would, if properly helped to understand how to teach and implement a Davydov-like program, be interested in mathematics books that don't have the serious disadvantage of being inaccessible to non-native and native English speakers alike who struggle with reading English.

Obviously, to fully evaluate the potential of a program like that described by Schmittau and Morris, it would be necessary both to examine the problems and see the instructional guides that come with them, and/or to view classrooms in which children are taught by competent teachers who have been trained effectively in its use. Thus, it is impossible to draw definitive conclusions about exactly what students are doing and how they are led to do it. While the reported results are naturally intriguing, it remains an open question, to my mind at least, as to whether American teachers could be brought to use willingly (and ultimately successfully for their students) a translation, with or without adjustments, of the original Davydov problems and guides. Absent DIRECT access to the problems or teacher materials, and knowing that replication of Schmittau's and her colleagues' research has not been attempted elsewhere. However, neither is it fair to draw any definitive conclusions from that. One would also want to look at the research emerging from various Americanized programs that purport to take an approach similar to Davydov's (though clearly they do not follow the entire Vygotskian theoretical approach as Davydov and his colleagues implemented it).

All the above said, I find it curious that some critics feel well-positioned to draw any conclusions either about Davydov's work or what is actually behind much of American student failure in mathematics. No expansion upon the tautology that students fail because they "don't get it" is likely to be adequate to the task unless one has looked adequately at approaches that purport to be able to take students to much higher degrees of success in mathematics, on average, by instead of giving students pre-algebraic experiences that are numerical, giving them pre-numerical experiences that are algebraic. I'd say that 2300 Russian students succeeding with such an approach is worthy of serious consideration as at least one piece of the puzzle as to how to remedy student failure. Offering up tautologies and then speculating about the causes based on one's own limited imagination? Probably not so much.

I wondered why they would not find Vygotsky's theories and Davydov's implementation of them in early mathematics curricula appealing or at least interesting. After all, that viewpoint is all about the idea that kids, left to their own devices or to typical educational approaches to teaching them mathematics, will have little reason to grapple with abstraction. Schmittau and Morris make this clear when they write regarding Vygotsky and Luria that they

"found in their studies of the development of primates, 'primitive' peoples, and children, that cognitive development occurs when one is confronted with a problem for which previous methods of solution are inadequate [Comment: not unlike how much mathematics has developed, by the way]. Hence, Davydov's curriculum is a series of very deliberately sequenced problems that require children to go beyond prior methods, or challenge them to look at prior methods in altogether new ways, in order to attain a complete theoretical understanding of concepts. More importantly, their consistent engagement with this process develops the ability to analyze problem situations at a theoretical rather than an empirical level, and thus to form THEORETICAL rather than EMPIRICAL generalizations, which is the distinguishing feature of Davydov's work." ("The Development of Algebra in the Elementary Mathematics Curriculum of Davydov," THE MATHEMATICS EDUCATOR, 2004, p. 62)

One might expect that this approach would have great appeal to anyone concerned with how little is asked of US students for the most part in elementary mathematics teaching and curricula. And given the fact that for many disadvantaged students, reading is a huge impediment to using many of the often-maligned progressive reform elementary books such as EVERYDAY MATHEMATICS and INVESTIGATIONS IN NUMBER, DATA & SPACE, the following should be enormously heuristic:

"The curriculum itself. . . consists of nothing but a carefully developed sequence of problems, which children are expected to solve. The problems are not broken down into steps for the children, they are not given hints, and there is no didactic presentation of the material. There is nothing to read but one problem after another. The third grade curriculum, for example, consists of more than 900 problems. Teachers, in turn, present the children with these problems, and they do not affirm the correctness of solutions; rather, the children must come to these conclusions FROM THE MATHEMATICS ITSELF [emphasis added]. The children learn to argue their points of view without, however, becoming argumentative." (loc. cit.)

I find it interesting, however, that in speaking with several US mathematicians and mathematics educators who either have developed or are in the process of developing curricular materials (or both) based on Davydov's ideas and work state that they don't feel that Davydov's original materials would fly in the US (in translation, of course).

I strongly suspect that the concern is not so much for the ability of kids to succeed with Davydov's materials (after all, if they work for kids in Russia, and they are still successful there (at least up to about 10 years ago; I have no more updated information) as evidenced by the 1999 testing at the end of third grade of 2300 Rusian students who used the El'Konin-Davydov curriculum. According to Catherine Sophian's latest book, Vorontsov concluded based on the results of the testing that

"'pupils learning in [this] educational system completely fulfill the requirements of the existing state standard.' The percentages of students who succeeded on 'tasks corresponding to the "standard" level of elementary school' ranged from 86 to 96% with the exception of a task that involved dividing multi-digit numbers, on which 76% of the pupils were successful. In a comment reminiscent of the debate surrounding reform mathematics in the United States, Vorontsov goes on to state, 'The results obtained dispel the myth, current in pedagogical circles, of the poor results of mastery of subject matter among children in [these] programs.'" (Sophian, Catherine: THE ORIGINS OF MATHEMATICAL KNOWLEDGE IN CHILDREN, 2007, p. 168).

In fact, I suspect the problem, such as it is, has little to do with kids, but rather with the low ability and willingness of so many US elementary teachers (and beyond) to teach mathematics without step-by-step guidebooks and their relatively weak knowledge of the relevant mathematics and content-based pedagogy. Considering that any non-traditional mathematics program introduced in the US that does not have the imprimatur of such educationally conservative groups as Mathematically Correct, NYC-HOLD, and a host of think tanks that condemn each and every "progressive math" program being used in this country, but praise uncritically particular programs from high-scoring Asian countries (particularly Singapore) for reasons that may not be completely non-political or grounded in reality for American kids and teachers, are subject to criticism for being TOO verbal, TOO hard, TOO sparse AND TOO watered-down (I am not kidding; the willingness of these self-proclaimed curriculum experts to offer utterly contradictory criticism in order to destroy programs they don't like is remarkable) it takes a daring teacher, school or district indeed to risk using something that is so starkly non-verbal as are the original Davydov materials.

Nonetheless, I think that teachers in inner-city, poverty-stricken, and other challenging environments would, if properly helped to understand how to teach and implement a Davydov-like program, be interested in mathematics books that don't have the serious disadvantage of being inaccessible to non-native and native English speakers alike who struggle with reading English.

Obviously, to fully evaluate the potential of a program like that described by Schmittau and Morris, it would be necessary both to examine the problems and see the instructional guides that come with them, and/or to view classrooms in which children are taught by competent teachers who have been trained effectively in its use. Thus, it is impossible to draw definitive conclusions about exactly what students are doing and how they are led to do it. While the reported results are naturally intriguing, it remains an open question, to my mind at least, as to whether American teachers could be brought to use willingly (and ultimately successfully for their students) a translation, with or without adjustments, of the original Davydov problems and guides. Absent DIRECT access to the problems or teacher materials, and knowing that replication of Schmittau's and her colleagues' research has not been attempted elsewhere. However, neither is it fair to draw any definitive conclusions from that. One would also want to look at the research emerging from various Americanized programs that purport to take an approach similar to Davydov's (though clearly they do not follow the entire Vygotskian theoretical approach as Davydov and his colleagues implemented it).

All the above said, I find it curious that some critics feel well-positioned to draw any conclusions either about Davydov's work or what is actually behind much of American student failure in mathematics. No expansion upon the tautology that students fail because they "don't get it" is likely to be adequate to the task unless one has looked adequately at approaches that purport to be able to take students to much higher degrees of success in mathematics, on average, by instead of giving students pre-algebraic experiences that are numerical, giving them pre-numerical experiences that are algebraic. I'd say that 2300 Russian students succeeding with such an approach is worthy of serious consideration as at least one piece of the puzzle as to how to remedy student failure. Offering up tautologies and then speculating about the causes based on one's own limited imagination? Probably not so much.

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