Hans Freudenthal (Wikimedia Commons, CC-BY-SA) |

*Major Problems of Mathematics Education*. I've briefly summarized the article on the MathEd Wiki and here I'll note the progress I think we've made on Freudenthal's 11 problems.

- Freudenthal believed we "need[ed] more pardigmatic cases, paradigms of diagnosis and prescription, for the benefit of practitioners and as bricks for theory builders" (p. 135). In the case of arithmetic, which was Freudenthal's example, I think Cognitively Guided Instruction (CGI) is very much the kind of thing Hans was looking for.
- Freudenthal wanted us to more carefully consider how people learn and observe their learning processes. I think several decades of teachers' awareness of constructivist theories of learning has changed how most people think of learning, and newer work in the area of teacher noticing puts fine points on what teachers notice and why.
- How do we design curriculum and instruction around progressive formalization? There is always more to learn, but the Freudenthal Institute in the Netherlands has now worked on this for decades and the frameworks for curriculum design are well-established.
- How do we retain and leverage mathematical insight? Freudenthal wrapped this into the conceptual vs. procedural debate, one that's still very much alive. However, I think we have better examples of productive approaches to this problem, and some research results (the BEAR project work at Berkeley comes to mind) showed that more focus on the conceptual didn't come at the expense of procedural facility. Still, this problem gets wrapped up in people's beliefs about mathematics and the teaching and learning of mathematics, and those beliefs sometimes aren't swayed by current evidence.
- How do we reflect on our learning? This is another problem we now know much more about, particularly due to Schoenfeld and his work on metacognition.
- How do we develop a mathematical attitude? This is still a challenge, and not just because some students say they don't like math. I think this problem might be closest to what Jo Boaler is currently trying to change with her focus on mindsets in learning mathematics.
- How do we coordinate students working together when the are at different levels of learning? Many teachers and scholars have worked quite hard on this problem and I feel like most teachers now see the benefit of heterogeneous ability groups. For more, I'd suggest Ilana Horn's book,
*Strength in Numbers*. - How do we create contexts for mathematizing? I think there's been a wealth of work in this area, from work based in Realistic Mathematics Education, work on word problems like that from Verschaffel, Greer, and de Corte, and, most recently, Dan Meyer's work. I could go on, as there are many more examples, and perhaps future work will give us a clearer picture about which contexts work best and why.
- Can we teach geometry by having the learner reflect on spatial intuitions? Maybe it's my lack of expertise in geometry education research, but I really don't know where we stand on this problem. Freudenthal seemed to be reaching in his article on this problem, and maybe a more tangible articulation of the problem would have helped me better judge any solutions we might have.
- How can technology increase mathematical understanding? Freudenthal admitted not being tech-savvy even in 1981 (he used "the ballpoint" as an example of technology that changed instruction, and not in an obviously historical way), but I think we now have numerous examples of tech that helps increase understanding. We also have a lot of examples of tech that doesn't, and I'm sure Freudenthal would have seen problems in our ability to judge the good from bad.
- How do we use a holistic approach to educational development for change? In his native Netherlands, Freudenthal would likely be pleased today to see his colleagues' commitment to design-based, participatory approaches to research. We have some of that here in the U.S., too, but we also struggle for a "scientific" approach to finding "what works" based on experimental studies. We also have too much faith in how standards affect change; if Freudenthal thought curriculum development for change was a wrong perspective, surely he'd think the same about standards. Those things are just part of a much bigger picture.