I have received more criticism of my curriculum design theories in the last four months than I have in the last four years combined. Much of it has been useful, especially insofar as I’m able to notice common questions and then formulate clear responses. If I’m then able to summarize those responses *visually*, all that’s left is for me to track down those critics one by one and shake their hands.

I imagine the questions that led to the panel above are self-evident. The answers are tricky, but I’m grateful, at least, that they now have a frame.

**BTW**: Add Michelle Bullard to the list of Commenters I Wish Would Just Get A Blog Already. Her response is pretty wonderful:

I think that students who can turn problems inside out like that, from concept to detailed numbers rather than from numbers to concept, can’t fail to pass any standardized test, but I really don’t care whether they do or not.

## 13 Comments

## David Cox

June 21, 2010 - 12:14 pm -Find that middle subset and you’re golden. The thing I’m finding is that many teachers don’t think there exists a subset of what’s valuable and the other two sets.

## Laura

June 21, 2010 - 1:43 pm -@ David:

I agree. The thing I love about this place is that it seems there are a lot of people who want to stay in the “valuable” section, but they do their damnedest to find/create ways for that to overlap with the other two.

A lot of teachers have trouble finding that marriage…I am so grateful for all of you for teasing this out and not giving up.

## Mr. K

June 21, 2010 - 3:23 pm -My summer project is to expand the bottom left leaf of the inner trefoil, and maybe try to move some of it into the center curvilinear triangle.

I might also try to figure out how to move more stuff from the top right circle to the left.

What was that you said about needing notation for this?

## Raymond Johnson

June 21, 2010 - 6:16 pm -Occasionally I’ll hear a teacher make a comment that makes me wonder if they feel that “what is valuable” and “what will improve our test scores” are disjoint sets. Generally I hear these things mid-rant during spring testing time, but I think those circles overlap more than some of us would like to admit. (In most states, hopefully.) The pressures teachers feel to deliver that content, however, is another matter.

Dan, I’ve read some of the criticism sent your way and part of it comes from a lack of distinction between “curriculum design” and comprehensive, point A-to-Z teaching and learning. When I see someone on the Math Forum ask “Where’s the math?” after seeing one of your WCYDWT posts (because you don’t show how a problem would be actually be solved), I can’t help but feel they’ve misunderstood what you’re trying to do (and not do). I wish more math teachers and mathematicians would as quickly criticize an equation and its solution by asking “Where’s the context?” Curriculum design is just part of the puzzle – an important piece for which you have considerable talent. I’m anxious to see how those talents develop in grad school. (Check out Freudenthal’s design work – there’s good stuff there!)

## vickie

June 21, 2010 - 6:42 pm -It looks good in theory, the problem is how individualized each venn diagram would be. Each teacher thinks their particular subject is what the students have time for and value, so, finding that middle space is certainly KEY for the best learning.

The whole test scores thing would just be a by-product of GREAT learning going on simultaneously with GREAT teaching.

vickie

## @mmeveilleux (twitter)

June 21, 2010 - 7:09 pm -This is one of the first times I have seen such an awesome Venn diagram with titles/labels only but no data set. I like the ‘insert your own data’ that allows one to project one’s own content into this.

I must admit that the bottom circle “what improves our test scores” does not strike me as terribly important. What do we do when that circle does not overlap with “what is valuable”?

A rich diagram.

## Christopher Roberts

June 21, 2010 - 10:28 pm -I disagree up to a point with saying that “what improves our test scores” is not too important. Given this question a couple of years ago, and I would have probably said the same thing. However, these days I would just say thats slightly naieve.

In am idealistic world, by all means, scrap testing, give everyone a job they love and a good future, and everyones happy. But this isn’t the reality.

Our students not only rely on us to educate them in way which makes them see world in a different light, but also to help them pass any tests with flying colours, so that doors to their future stay open. In theory though, if the teaching is good enough, and your formative-y type assessment is good enough, then the tests should be no problem. But, whichever way we look at it, the test scores should matter to some degree.

## Dan

June 21, 2010 - 10:54 pm -The “where’s the context” question seems like one of the main points this blog, implicitly and explicitly, raises, and it’s crucial. I would say that when you leave out the context you’ve left out the math.

Incidentally, with reference to Michell Bullard’s comment: she’s absolutely right, and if you check out Eric Mazur’s work (at the college level, in physics), he’s got the data to back it up. Specifically, he’s shown that students may understand the rote mechanics of very elaborate processes without understanding the concept, but those that understand the concepts fully can virtually always do the computations as well. He’s got a nice talk here: http://www.youtube.com/watch?v=WwslBPj8GgI.

## Reid Atcheson

June 22, 2010 - 7:42 am -Standardized tests are not an end in themselves. Before criticizing a method based on how it yields improved standardized test scores, ask yourself what the test measures and why that measurement is important.

The standardized tests I took as a kid in elementary/middle/high school measured my ability to perform a series of unrelated 30 second problems. The problems I solve now are not in any way related to this skill I learned precollege. I simply have something I need to create or improve, and I need to figure out how to do that. For anybody who will use mathematics in their career, they are not simply given a hundred word problems and paid money to solve them.

That is what school was meant to prepare a student for wasn’t it? So why do we put so much weight on the measurement of such a minor skill?

## Michelle

June 22, 2010 - 2:05 pm -Hi, thanks, I guess I’ll be ready one day. Right now I’m happy learning all of the ideas you have, and the other blogs also.

## Joe

June 24, 2010 - 6:21 pm -How about a fourth circle called “the math I will actually need in my life and in my career”?

## Joshua Fisher

June 25, 2010 - 7:58 pm -If you check out Eric Mazurâ€™s work (at the college level, in physics), heâ€™s got the data to back it up. Specifically, heâ€™s shown that students may understand the rote mechanics of very elaborate processes without understanding the concept, but those that understand the concepts fully can virtually always do the computations as well.Dr. Mazur’s “work” is on peer collaboration. And his data speak to that work. I’m not sure “those that understand the concepts fully can virtually always do the computations as well” is an appropriate take-away from that presentation. I could be wrong, of course. I’ll have to watch his present again.

## Dan

June 25, 2010 - 11:32 pm -He did work on peer collaboration, but he also compared student understanding of conceptual questions versus technical questions, and the takeaway was the correlation I described. It shows up as a blank spot in a corner of one of his charts: there were no students who mastered the conceptual while failing the technical (though some mastered the technical and failed the conceptual). That correlation was separate from any work on peer collaboration.