Last week, Dan Meyer wrote a brief reflection on Ed Beagle’s First and Second Laws of Mathematics Education:

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

The second law particularly resonated with me, a soon-to-be teacher. The more I learn about mathematics education, the more I realize that there is still so much to learn, and that anyone who says it’s simple is selling you something (Dan Meyer). My to-read list is growing longer and longer, even as I realize more and more fully that what matters most is not what I read, but what I *do *at the ground level with my students. (Side note: Last week I also began my foray into John Mason’s work – thanks, Danny Brown.)

At the same time, I worry that the talk that Dan linked to is too easily misinterpreted, especially if the date (1971) gets lost in translation. In the article, Begle provides a long list examples of beliefs about mathematics education that have proved to be false and innovations that have been found to be ineffective – almost enough to make a mathematics teacher, especially one on the cusp of beginning her career, throw her hands up in the air and give in to the kind of extreme relativism or skepticism that leads to complacency. In other words, as Raymond Johnson wrote, “the kind of attitude that verges on fatalism: ‘Nobody seems to know the ‘best’ way, so who cares if I just go with my way.'”

Now, I’m certainly not about to throw in the towel. I have certain beliefs about the teaching and learning of mathematics that aren’t shaken every time I see a meme about the supposed failure of Common Core, though I am open to reevaluating these beliefs should research and experience suggest it necessary. I know (or, at least, I feel) that Begle’s goal was to arouse some necessary humility in the field of mathematics education and to spur on further research. I agree with Dan Meyer that Begle’s words are still very relevant today, and that they may even offer a sort of comfort (though, again, not the sort of comfort that leads us to complacency).

What I am wondering, however, is this: What *have* we learned since Begle’s talk? Raymond Johnson suggests that we’ve made enormous progress, and I’m inclined to believe that this is true. Luckily, Danny Brown recently provided a list of key ideas he feels are important to teaching mathematics, and I’m very much looking forward to digging into it (my to-read list grew rather substantially after this post). I’m hoping other math educators could offer their own perspective:

**What do you feel are one or two key understandings about the teaching and learning of mathematics that we’ve gained since Begle’s talk in 1971? (Bonus: Where is our understanding still lacking?)**

I think our understanding of ways to teach mathematics has improved but like William Gibson says, “The future is already here — it’s just not very evenly distributed.”

Our issues today seem to be that there are two main very effective ways (based on limited empirical data) of teaching mathematics that have similarities between each other but that neither of these ways has any wide-spread following.

One way is the Jump Math strategy of carefully breaking down each mathematical idea into tiny pieces and having kids practice mathematics as a series of skills, and then discover the connections between different mathematical ideas mostly on their own. There is more to Jump Math than just the practice worksheets but that’s a better topic for a book John Mighton, the inventor of Jump Math, wrote. Jump Math has some examples of the limited empirical evidence they have that their program works on their website.

The other way is even less well known, and in fact most people who hear of it confuse it for ‘discovery learning’. It is what Magdalene Lampert (among others) calls ‘Ambitious Teaching’ wherein students do mathematics as mathematicians would within a mathematical community (again there are other important features of ambitious teaching outside the scope of a comment on a blog). Although there is a recent study out showing that elements of the inquiry-oriented ambitious teaching are correlated with student success on an empirical measure, there is not wide-spread enough use of ambitious teaching to be able to compare it to other forms of teaching.

Note that both of these ways of teaching take Cognitive Load theory (or some variant of it) very seriously but simultaneously are concerned about the internal mental models students develop for understanding mathematics. Jump Math aims to minimize cognitive load, especially extraneous cognitive load. Ambitious Teaching aims to minimize extraneous load but maximize the load (to whatever limit students can handle) for students in which they are thinking about mathematics.

Neither one of these ways is really well described enough in this post to be useful to you yet as a starting teacher, so unfortunately I’ve given you some more reading to do.

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