My post yesterday has generated much discussion. I’m trying hard to understand where WISE is coming from, so I sought out more information. One of the people behind WISE is Dr. Robert Craigen.

Dr. Craigen appeared on QR77 with Rob Breakenridge this week to talk about their movement. The full conversation between Breakenridge and Craigen can be found here. It’s about 16 minutes long, so let me summarize some of what I heard.

First of all, I should say that Rob Breakenridge impressed me. He seemed aware of what the revised program of studies is about, and he asked good questions to facilitate the conversation without making apparent any of his own biases. It was a good interview.

Secondly, Dr. Craigen did a good job explaining the WISE position. In fact, he presented his position in a much better way than it is presented on their website. I disagree with much of what is written on their site, but if Dr. Craigen truly believes what he said on the radio, then we have some common ground.

Dr. Craigen states that it is important for young kids to have algorithms to perform calculation because those algorithms will help them in later courses. I agree. I suspect we disagree on how these algorithms should be developed. I suspect Dr. Craigen wants teachers to show the students the algorithm that he thinks is best. I want teachers to help students develop their own (correct) algorithms so that the students understand the algorithms better. Students who can’t develop their own (correct) algorithms need to be steered to one. We are not leaving kids completely on their own. I have never suggested to teachers that we send kids home with no algorithms at all.

Dr. Craigen states that there is a false dichotomy between skills and understanding. He says that both skills and understanding are important. I agree with this statement. The struggle we face, though, is that when we teach in a more traditional way, for most of our students, all we get at are the skills. Dr. Craigen states that some students get the understanding in spite of this method. For years, I taught kids to factor polynomials. They could do it, and I could prove it because they could do it on tests. I’m not as confident they understood factoring. The revised curriculum and its more constructivist approach helps get at this understanding. I believe that kids can demonstrate math skills without understanding. I don’t think the converse is true. I don’t think they can demonstrate understanding without the skills. I agree that both are important, and we need to teach so that both happen.

I have less common ground with Dr. Craigen’s beliefs about automatization. He states that repetition and exercises are important so that students no longer need to think about the math. He wants students to be able to do the basic calculations and practice them so they become automatic. I would argue that too much automatization leads to imitation without understanding, particularly in the higher grades. If students can be made to truly understand their division algorithm, they don’t need to practice it 100 times. Some practice is important. Excessive practice is not worth it. The problem we face with asking kids to practice in K-12 education, that Dr. Craigen may not experience in his setting, is that students who would really benefit from practice don’t do it. Students who don’t need the practice faithfully do every single exercise we assign.

I love Dr. Craigen’s observation that in the old curriculum, there was a great deal of teaching without understanding going on. This comment is certainly true. Even with this new curriculum, it is still true in a lot of places. It’s true in nearly every University mathematics course I took. It isn’t a new problem. I fail to see how going back to more of that kind of teaching, as Dr. Craigen advocates, will alleviate this problem.

Dr. Craigen and I both want our students to come out with all the skills, understanding, and problem solving abilities that will benefit them in future studies and in life. He admits that a discovery and exploratory approach is good, and should be maintained, but this approach should be balanced with some direct instruction. I agree with this statement entirely. I see a typical math lesson following a patter that goes like: Exploratory activity, students share what they learned, teacher fills in gaps, students practice what they learned. I’m starting to wonder if teachers are leaving out the important third step. Are they sending students home without filling in the gaps? Are they sending students home with no understanding or skills? If so, then they have misinterpreted the intent and philosophy of the revised curriculum.

What Dr. Craigen doesn’t address is a passion and love for mathematics. I have it. He has it. The students he teaches mostly have it. The students I teach mostly don’t have it. A discovery and problem solving approach can show these kids that mathematics can be interesting, relevant, and fun.

A stand and deliver approach whereby proofs and algorithms are explicitly presented works well for him, me, and about half of the students he teaches. It works for far fewer of the students I teach. I’m not convinced that from his world of academia, Dr. Craigen understands my world. The bottom line is that in my world, I need to be a better teacher than the one he suggests I should be.

on December 23, 2011 at 1:11 pm |WISE MathYou’re quite right, John, that we appear to have considerable common ground. We’re getting into some nitty gritty now in the differences, and I won’t have time to pursue them at length in a setting like this for quite a while. I want to add that I don’t reject exploration, or the infusion of constructivist methodology, or the socratic approach, all related ideas. I use the same in my own teaching. But I am certain that these cannot be the meat and potatoes of everyday instruction; they are supplementary to the main delivery of content, as guides to understanding (and it’s not just because I’m a professor who lectures all the time and haven’t a clue what activities grade 3 students need; lets not oversimplify — we’ll go into this in more detail some other time). The term I use with my students is “owning the material” — they must take the knowledge and make it their own instead of passively taking notes and “receiving” it, and activities are designed to accomplish this. As you are perceiving we do have a “big tent” approach to methodology; we’re against narrow approaches or what I believe I called in the interview “educational Atkins diets”.

Something else missing, or sorely de-emphasized, in WNCP Is the deductive paradigm in key transitional ideas, often replaced by inductive explorations. This is a sad commentary on the developers; I suspect that, on the whole, they aren’t even aware of the mathematical issues involved.

Some things the authors of WNCP wrote to textbook publishers would be funny if they weren’t so sad, like their response to a question about how students should be explaining why division by zero isn’t permitted. WNCP writers explain with an example of a child dividing cookies among their friends. It is asserted that when there are no cookies, you can’t divide them among friends and when there are no friends, you can’t divide cookies among them, QED. The stupidity of the serious category error contained in this example astonishes me on several levels, and I certainly hope the textbook writers knew better, or we’ll have children believing that 0/4 is not defined and just as problematic (or of the same category) as 4/0. If the curriculum developer can’t develop this simple idea clearly and correctly, how is a rank-and-file teacher supposed to do so, or even worse, to evaluate and guide a student’s own explanation?

It would be nice if the above were simply an isolated example, but it’s not, and there are similar problems in the CCF itself. This sort of thing in the corresponding American curricula was what convinced many mathematicians there to get involved in the design and vetting of curricula. In our case we have been centrally concerned with the systematic, radical changes that were being sneaked in under the radar without any discussion in public or with expert and client disciplines, and these fine-grained errors are side issues though some, like this one, indicate deeper problems.

Anyway, I do look forward to taking these discussions up with you sometime in 2012, John. In the meantime have a happy and safe Christmas and New Year!

on December 23, 2011 at 1:12 pm |WISE MathOh, sorry, I forgot to sign that comment:

— WISE Math (R. Craigen)

(Since the other founders could also post under “WISE Math”)