Why is work by teachers, math ed profs good but math foundations bad?

Throughout most of the math education debate is the idea that hard work by teachers and math education researchers is good, but work by math foundations people is bad.  Math foundations gets no respect in math ed.   Why is that?

Part of it is fear.  Math ed people for the most part and most teachers do not know math foundations.  They were not taught them in college.

In the US, the Peano Axioms are not typically taught to pre-service teachers.  As we know, at least once they were taught to them at PH-Heidelberg.

Part of it is the failure of 1960s era New Math reforms.  That gave math foundations a bad name.  The main problem with 1960s New Math was it did not push the Peano Axioms and the definition of addition and multiplication of natural numbers.

New Math pushed set theory and maybe a few other things, but it failed to push the road from counting in the Peano Axioms using the successor function through defining addition and multiplication of natural numbers and then proving the properties of these functions.

Part of the reason for this was the Edmund Landau Foundations of Analysis book used the Kalmar definition of addition, and this definition is hard to understand.  That was the main book in the 1960s on Peano Axioms.  That book also covers up the important sequence in the Dedekind book through order of natural numbers, the recursion theorem and then the definitions of addition and multiplication recursively.

The meaning of elementary math is in math foundations and there alone.  The meaning is not applications.  Telling students implicitly that the meaning of natural numbers or fractions is the applications is misleading because it is wrong.

The short road through elementary math is the road of abstraction. That road is laid out in the Dedekind 1888 book.  The Dedekind book lightens the cognitive load because it contains the path through the very beginning of elementary math.  However, it takes hard work itself to learn.  It lacks examples and exercises so it is not suitable for self study by most people.



About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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5 Responses to Why is work by teachers, math ed profs good but math foundations bad?

  1. Michael Paul Goldenberg says:

    Upon what are you basing your claim that discussing mathematical foundations is viewed as bad by mathematics educators? And which math educators do you stipulate feel that way?

    Second, are you sufficiently well-versed in the actual work of the various New Math projects from the ’60s to make the gross generalization that the New Math (there was no THE New Math) was all about set theory? How much of your claim is predicated on nothing more than seeing the Dolciani texts, which are the ones most Americans who even remember New Math saw and associate as being THE New Math?

    Finally, on what grounds (research, personal opinion, or what?) do you claim that studying the Peano Postulates/Axioms are the “The short road through arithmetic”? For whom, exactly? Teachers? Kids? Mathematicians? Joe Sixpack?

    • Asmuth and Rips discuss the Peano Axioms and mathematical induction in the math ed context. So do a few others. At PH-Heidelberg they teach Peano Axioms at least one time to pre-service teachers. See link at below.


      Borovik talks about Peano Axioms for education as an impossible dream but still uses them a little in his books. Milgram also claims they are used in Russia at lower grades in an appropriate form.

      I have given the Peano Axioms in a form for the number line.


      I have also given a way to define addition using two number lines.


      My book goes through some of this as well. I can give you a complimentary copy for Nook or Kindle version if you like. Let me know which platform you prefer.

      The next volume on geometry of addition is in preparation and has a wealth of tools for teaching to use.

      Peano Axioms are the shortest way through arithmetic if we have to start at the Peano Axioms. I believe the number line version can be taught to roughly somewhere around first graders.

      Difficulties with fractions show a confusion both on the part of educators and on students. the path to fractions is hard to think through, but I believe the best is natural numbers, then fractions as operators on natural numbers with limited domains and then their extension to a type of number.

      In my book I show that place value notation is recursive and that addition using place value is recursive. This is somewhat obvious, but many seem to overlook it.

      The Asmuth and Rips papers are supportive of using Peano Axioms and mathematical induction. I also do comparisons with standards of learning partly in the book and possibly in earlier posts to show that the addition complexity of mathematical induction over current common core standards is not that great.

      I think the entire math ed project for everyone can and should be restructured around the Peano Axioms. What is not the explanation is not helpful. Math foundations clears that out.

      The Dedekind book, “What are numbers and what should they be?” is vastly underestimated by all except a few like David Joyce.

      • Michael Paul Goldenberg says:

        No Nook or Kindle, but appreciate the offer.

        Not sure that I trust Dr. Milgram on matters of math education. While it may be true that some Russian or former Soviet schools do what he says, it’s also true that V.V. Davydov’s elementary math books and those influenced by them (I have a set of the latter in the original Russian, but my Russian is far too weak to translate them fluently and I have not had the time to invest to do it as painfully slowly as it goes given my limitations) teach a measurement approach to elementary mathematics that has had some influence in the US (specifically at U of Hawaii, then later in work being done at one of the branches of Cal State, and at Iowa State). Not sure I’ve heard from Milgram on that, unless maybe there’s an underlying connection. I will check with a friend who grew up in Russia and has often had a rather different take on claims about Soviet education in math than I hear from somewhat more conservative American sources (among whom I number Milgram).

        But let’s give him the benefit of the doubt and say, “Sure, Jim.” So what? I’m not sure that we can claim with any degree of certainty that if something was done or is done in some Russian schools that it follows either that it’s good or that it would be sensible to follow suit here. Lots of things are done differently in lots of places, but it takes more than knowing that to decide to model US math ed on any one of them. And I say that fully believing that we DO need to do a lot of things differently and can learn from other countries.

        I’ve downloaded the article by Asmuth, et al. and will give it a look.

        Germany isn’t close to my ideal model for the teaching of mathematics, particularly based on what I’ve seen myself from the TIMSS videos, as well as the analysis by Stigler and Heibert in THE TEACHING GAP. So what’s done at PH-Heidelberg, unless it’s paired with a better instructional model, isn’t likely to get me too excited.

        I just ordered the Dedekind book via interlibrary loan. Should be of interest.

        In any event, I’m very cautious about your project. And I am curious as to what you’ll have to say to the questions I’ve raised previously that you’re failed to answer thus far, particularly on New Math and your choice to (apparently) rely on the work of someone like David Klein as a trustworthy source for historical, factual information. He’s around my age, so he didn’t exactly have involvement in that development, but he definitely has axes to grind. The article you cite is very flawed, on my view, and I find the accounts of people who were involved in the projects much more plausible, particularly given the original documents I’ve looked at, than Klein’s politically-influenced version.

        You may just not have gotten around to some of what I’ve been raising. If that’s the case, I look forward to your responses.

    • Who do you suggest as an alternative source for history of 20th century in math education?

      Nook and Kindle have aps for reading on a computer that don’t cost anything to get. The are downloads. You have to get an account, which also has no cost. I can provide a copy of the book to you.

      For the Dedekind book, you can also get it free in English and also look at the David Joyce Notes.

      Link to Dedekind



      Around paragraph number 40, Dedekind becomes harder to read. If you think of the prime function as successor of natural number, and a closed set as a tail of natural numbers, i.e. all numbers from n onwards including n, then it is easier to follow at this point. These paragraphs are necessary for Dedekind to set up order of natural numbers, proof by induction, definition by induction in his recursion theorem. Dedekind understood this logic better than any following textbook. Order of natural numbers is done better by Dedekind than any other text.

  2. Michael Paul Goldenberg says:

    I don’t know if a definitive or comprehensive treatment exists in print. But you’ll do better with Zal Usiskin’s take than that of David Klein. Zal worked with Joe Payne at U of M and is older than I am, so I suspect he has more direct personal knowledge of what went on. I don’t think he has axes to grind about either of the modern reform periods, but I would bet that David Klein and his Mathematically Correct friends would strongly disagree with that claim (full disclosure: Zal is the head of the University of Chicago School Mathematics Project that produced some of the first materials to be considered “reform” (the UCSMP secondary books aren’t radical by current standards, and they predated all the stuff Mathematically Correct attacked in the mid-90s as “fuzzy.” But since UCSMP eventually produced EVERYDAY MATHEMATICS, which has been relentlessly attacked by the MC/HOLD people, I’m sure Zal is seen as at least as guilty as, say, Andy Isaacs (my contact with the EM project) and others. Nonetheless, I think Zal’s stuff on the New Math is good: http://lsc-net.terc.edu/do/conference_material/6857/show/use_set-oth_pres.html

    There is also Irving Adler’s book on the New Math, but it’s not a history, if memory serves, so much as a presentation of the modern ideas associated with it. Of course, Morris Kline’s attack on the New Math, WHY JOHNNY CAN’T ADD (playing off the popular and, to my mind, very wrong Rudoph Flesch book, WHY JOHNNY CAN’T READ) is a definitive contemporary (’73) attack.

    I still suggest you see if you can get your hands on some of the stuff Robert B. Davis was connected with, particularly in the ’60s, for kids and teachers. He was, if memory serves, the head of the Madison project branch of the New Math. I have access through Michigan’s interlibrary loan services to a lot of those materials. Not sure what your situation is in that regard.

    I will download the requisite software for reading your book if you send it in either form that is convenient. Much appreciated.


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