Ed Begle’s First And Second Laws Of Mathematics Education

Ed Begle:

  1. The validity of an idea about mathematics education and the plausibility of that idea are uncorrelated.
  2. Mathematics education is much more complicated than you expected even though you expected it to be more complicated than you expected.

Begle coined those two laws in the latter half of the School Mathematics Study Group, a multi-decade project to figure this mathematics education thing out. I’ve heard those laws before but I hadn’t tracked down the original source until today. He seems weary in the speech. His list of tried-and-failed innovations is lengthy and disturbingly current.

Over forty years after Begle’s work with SMSG ended, those laws still offer us lots of comfort and at least a little humility. Math education is hard. My gut is probably wrong. Anybody who says differently is selling something.


Begle, E.G. Research and evaluation in mathematics education. In School Mathematics Study Group, Report on a conference on responsibilities for school mathematics in the 70’s. Stanford, CA: SMSG, 1971.

2016 Feb 26. Bowen Kerins’ links to a better copy of the entire proceedings. That site also contains links to some of the SMSG “New Math” curriculum, which I’m excited to investigate.

2016 Feb 28. Raymond Johnson cautions us not to read Begle too pessimistically:

I really do love the history of my subject and posts like Dan’s send me into hours of searching through old papers and citations. But, I must be mindful of our tendency to underestimate change when we read from our wisest predecessors. It’s too easy for us to throw our hands up and say things like, “Dewey knew it all along!” or “We’re stuck in the same damned place we were 25/50/100 years ago.” Is Begle’s 2nd law (“Mathematics education is much more complicated than you expected even though you expected it to be more complicated than you expected”) still true? I would agree it is. But, as a field, we’ve made enormous progress since Begle gave this talk in 1971. The danger, as individuals, is to not learn from this progress. To avoid reaching the same conclusions as Begle, we need to avoid starting in the same place as Begle. When I browse the pages of Begle’s final book, Critical Variables in Mathematics Education: Findings From a Survey of the Empirical Literature, I’m struck by the sheer number of things Begle and the field knew little or nothing about compared to what we know now. Don’t we owe it to ourselves, as individuals and as a field, to push past prior conclusions by starting farther ahead and taking more seriously work already done?

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Yes, sounds weary in his speech and I feel his pain. I love that you find comfort, and humility, because it made me re-frame my frustrations. Still…same song 6th verse…and sometimes I wonder why I think I can make a difference. WHICH is why the second law has become my favorite motto as of today! Please tweet this blog so I can re-tweet. We all need to acknowledge those two laws!

    Carry on.

  2. Thanks for tracking down this paper. I began teaching in 1962, was fortunate enough to have had Robert B. Davis as my undergraduate advisor (he was also part of the movement that Ed Begel describes), and used early SMSG materials with my eighth grade students in California for several years. I’m still working on sorting out how to best help students learn math.

    While Begel seemed weary, he was still looking forward. In his final paragraph, he pointed to a possible direction for progress. He wrote: “It seems likely that evaluation will play a more prominent role during the 70’s than it has in the past. We will need to not only continue the production of more refined measuring instruments, but also to educate school administrators, school boards, and concerned parents to the existence of more useful tests and more penetrating evaluation procedures.” I wonder what he would think now.

    In my more difficult and puzzling moments, I remind myself that futility is better than apathy.

  3. I’m not sure how to feel about not much having changed in the intervening 45 years.

    In part I’m heartened, because I have insecurities about both what and how I teach, and I muddle on through finding best practices where I can, and the vetting them through practice in my classroom.

    On the other hand, I have “experts” and administrators on all sides telling me that there is a proper, scientifically based method to teach all this stuff, if only I follow the book, and it totally dissonates with me.

  4. Well, one thing hasn’t changed at all – the quality of research into math education is still poor, and not helped by the axe grinding approach of “This method, which is my pet method, is better” as the starting point.
    Fascinating article, and I didn’t feel the weariness, only a very down-to-earth practical and honest assessment.
    Can you get a better scan?

  5. Begle’s text is extraordinarily sensible and honest, as his biography seems to say of the man. I guess his feeling of disappointment with math ed research is strengthened by his disappointment with the New Math movement he helped to create – his article is from 71, when most of the evaluation results of his movement where coming up and didn’t end up as expected.

    I’m just glad one of his last guesses on the article didn’t turned up real, at least not around here. For one side, of course teachers are (at least partially) responsible for the scores of the students, but accounting them that heavily for it isn’t reasonable! Around here (and I guess in any other places of the globe too) teachers just don’t have enough theoretic support, preparation, resources or even freedom to make the best teaching decisions, let alone knowing what the best teaching decision is!

    I think Begle’s own history is an example of his research. No doubt he, as all educators, gave his best on his work. But hitting the target is hard when you’re shooting in the dark.

  6. Thank you for sharing this. I love Begle’s wording of 2nd law. It is my opinion that this might be true for imparting knowledge period – not just math. Although math is highlighted because of the importance of the topic and the mindset that it is okay to not “be good” at math. Educating others is a very complicated process. It is so personal for both the teacher and the student. There is not one approach that works for every student or every teacher. When we find an approach that works for a topic and you think (and are very excited) that you have found THE Answer to teaching your students math, you wake up the next day to find that IT doesn’t work for next topic or situation. We just keep struggling to educate every child to the best of our ability. As Marilyn stated so wonderfully, “futility is better than apathy”. It is our job as educators to keep trying to find what works – our students deserve it.

    Teaching Math is HARD work but so worth it when you see in your student’s eyes the light of understanding and/or the spark of love for Math.

  7. This is one of the giant black holes in the current wave of ed reform. They really acted like creating a great new generation of curriculum and instructional materials (and assessments) was not just an implementation detail, but a trivial one. So trivial that the same companies using the same processes could be relied upon to solve the problem, or alternately, organizations with extremely little experience could surpass the established players with a small team and extremely ambitious schedule.

    They literally just acted like they didn’t think any smart person had ever considered the problem of education before.

  8. The longer I am in education and the longer the challenges continue, I can’t help but think we need to make some systematic changes (specifically in mathematics education).

    First, we make a broad and, in my opinion, inaccurate assumption that all children are capable of learning a defined set of information solely because of their age. Many children never gain a mastery of skills before they are introduced to a new set of skills. In all likelihood, the new skills require a mastery of the previous set of skills.

    It is no wonder that the above described situation leads to two things; students who don’t feel confident in math and students who don’t like math. The first feeds into the second.

    We are stuck in a system that was developed at the dawn of the industrial revolution and we treat our students as though they are part of an assembly line.

    We need to think creatively about how we group children for success starting at an early age. This idea conjures up that I advocate “leveling”. This term developed its negative reputation as a result of how ineffectively it was used in classes prior to the heterogeneous classroom grouping. Children were leveled based on very little data. The delivery of math education that I envision is far more complex.

    What is vastly different now is that we have solid tools understand the needs of each learner. Let’s use progress monitoring to make sound decisions about what our students know and what they need more time to learn.

    My prediction is that if we created a system based on this approach, it would be extremely cost effective. A regular teacher could effectively meet the needs of his/her students. There would far less demand for special education thus saving districts money.

  9. My own experience in teaching mathematics, albeit only about 20 years worth, illustrates the constant Flux in ideas. Levelling-mainstreaming-discovery-mindset-adaptive-rinse and repeat shows that rarely are those people who really understand data given sufficient time to evaluate their own results. Neverending the truism that every group, nay every learner is unique in their own way of approaching the material as well as deficiencies. The idea, popular as it is with writers and administrators that all one needs to do is present a clear and good lesson and everything will fall right into place.

    Practice is good, exploration has its place and rarely (if ever) does any student truly benefit, at least in math from being socially promoted. Getting acceptance of these ideas is hard enough (on par with me getting 100% compliance from my students on their homework at least)

  10. @Tom, by “they” are you referring to SMSG or (more likely?) Achieve, Pearson, etc?

    Also: if established players can’t do it (old processes) and insurgents can’t do it (no political capital) — who can do it?

    Chris W:

    Let’s use progress monitoring to make sound decisions about what our students know and what they need more time to learn.

    It’s interesting to note that this isn’t a new idea, and it’s a tried idea, and it’s been found wanting in the era of the SMSG. (Read Begle on IPI in the linked speech.) Before I can get excited about another adaptive learning proposal, I’d want to understand how it differs from every other similar proposal that’s been tried.

    Scott Hill:

    Practice is good, exploration has its place and rarely (if ever) does any student truly benefit, at least in math from being socially promoted. Getting acceptance of these ideas is hard enough (on par with me getting 100% compliance from my students on their homework at least)

    Confusingly, neither do students benefit from grade retention. #begles2ndlaw

    Howard Phillips:

    Can you get a better scan?

    Short of driving to (I believe) UT Austin and tracking down the original microfiche, nope.

  11. Oh, by “they” I meant the corporate reform movement writ large — Common Core, RttT, etc. It seems clear to me that if you aren’t starting from a solid consensus — like, say, Japan’s curriculum — and steadily iterating on that… well, if you don’t have that, you have a very large problem. Throw in the partisan “math wars” part, and it is pretty much impossible.

  12. There seems to me to be somewhat of a pointlessness about the search for “The Best Method Of Teaching Math” when there is no agreement about the purpose of teaching math, and not a lot of agreement on the content. All I see at the present time, and earlier as well, is measurement of skill in carrying out procedures. Rather limited, isn’t it?

  13. My sense is that there are TONS of things that have been found to work, mostly in the category of teaching techniques, but that none of them have been found to be scalable. As a result, almost no strategies pan out in large-scale randomized trials.

    My personal example is a discussion technique called Accountable Talk, which I’ve been trying to learn to do for years. I’m not sure it’ll ever work for me. That said, each of these techniques can be mastered, and when they are, you usually get great results. (This is a falsifiable statement, in the sense that recognized experts in these techniques can reliably identify others teachers who are using them faithfully, and those other teachers’ students will generally be found to learn quite well).

  14. Mike Shaughnessy

    February 23, 2016 - 9:04 pm -


    Nice to see you tracing into your Stanford, and Math Ed, roots. As a high school student, my school was one of the ‘experimental’ trial schools for the SMSG curriculum. Oh, you should have seen (maybe you have) the mimeographed materials we worked from.