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.