Education ∪ Math ∪ Technology

Tag: constructivism (page 1 of 1)

I did professional development all wrong

Last year, I presented a lot on the need to improve mathematics instruction. I had pictures, I had questions, I had effective arguments, and my audience was engaged. I could present like the best of them on some of the ways that we can improve mathematics instruction. What I did not have was effective teaching.

The role of someone involved in professional development for teachers is to help the audience, teachers, improve their practice. It may be that they take part of what you do and use it, and it may be that they attempt to copy your method exactly. The problem is that the typical presentation does little to improve someone’s practice. It may inspire them, it may anger them (I’ve done both), and it may provide some helpful tips, but effective change in practice does not come from someone presenting on their practice. The best you can hope for from a presentation is small, temporary, surface level changes.

Improving one’s practice requires thinking. It requires time spent looking at the context of one’s school, on the way that one approaches one’s own teaching, and on what other practices one can incorporate into one’s own pedagogy. It requires discussion so that the learner can take the ideas they are assimilating and seek clarification and direction.

So instead of spending the entire time I present talking, I give participants much more opportunity to talk. Instead of participants sitting around listening, I give them opportunities to do. The last few workshops I’ve done have been more about conversations. They’ve involved rich, mathematical problem solving activities. They’ve involved teachers having insights, and sharing those insights, often things that never would have occurred to me. I’ve learned much more from my workshop participants than when I was a presenter.

I spent an afternoon talking with my colleagues about computational thinking, how computational thinking really is mathematical thinking, and how if our students get opportunities to program, then they are doing mathematics. My colleagues were working on a particularly challenging problem, and one of them stopped and said, "Okay, I get it. Solving problems is hard. I can see why the kids struggle with this stuff." This kind of insight, not directly related to my objectives, was probably the most valuable insight to come out of that workshop. It never would have happened had I not given participants a chance to think and to do.

Converting degrees to radians

One of my students came up (with some help) this procedure for converting between degrees and radians.

  1. Memorize the fact that 60° is π/3 and that 30° is π/6.
  2. Note that 10° is therefore π/18 and that similarly 1° is π/180.
  3. You can then take any degree measure and convert it by converting the number of degrees into sums of degrees where you know the conversions. For example, 70° is equal to 60° + 10° = π/3 + π/18 = 6π/18 + π/18 = 7π/18.

Obviously this procedure is not by any means the most efficient way to convert between radians and degrees. Although I showed a much more efficient algorithm for converting between degrees and radians, it didn’t make sense for this student, and so he and I came up with this procedure (which I drew out of him by asking him questions about the angles), which he does understand.

In general, I’d prefer students use inefficient techniques that they understand completely than highly efficient techniques that they do not understand. Hopefully this student will continue to work on his procedure to make it more efficient as he has to use it over and over again, but if not, at least he will be thinking with something that makes more sense in his head.

Constructivist teaching is not “unassisted discovery”

I’ve been challenged recently to provide research which supports "unassisted discovery" over more traditional techniques for teaching math. This is not possible, as there are no teachers actually using "unassisted discovery" in their classrooms.

First, it is not possible to engage in the act of "unassisted discovery" as a student. Just knowing the language to describe what you are working on is a clear sign that at the very least you have the support of your language and culture in whatever you attempt.

Second, if a teacher has chosen the activity for you, or designed the learning objects you will be using, then they have given you an enormous amount of help by choosing the space in which you will be learning. Even Seymour Papert’s work with Logo was assisted discovery, after all, Logo is itself going to direct the inquiry toward what is possible to do with the language.

I can’t give examples of research which supports unassisted discovery, but I can give research which supports discovery learning in general. Without searching too hard, I found the following supportive research:

Bonawitza, Shaftob, Gweonc, Goodmand, Spelkee, Schulzc (2011) discovered that if you tell children how a toy works, they are less likely to discover additional capabilities of the toy than if you just give it to them, suggesting that direct instruction is efficient but comes at a cost: "children are less likely to perform potentially irrelevant actions but also less likely to discover novel information."

Chung (2004) discovered "no statistically signicant differences" between students who learned with a discovery based approach based on Constructivist learning principles as compared to a more traditionalist approach.

Cobb, Wood, Yackel, Nicholls, Wheatley, Trigatti, and Perlwitz (1991) discovered that students who learned mathematics through a project based approach for an entire year had similar computational fluency compared to a more traditional approach, but "students had higher levels of conceptual understanding in mathematics; held stronger beliefs about the importance of understanding and collaborating; and attributed less importance to conforming to the solution methods of others, competitiveness, and task-extrinsic reasons for success."

Downing, Ning, and Shin (2011) similarly found that a problem based learning approach to learning was more effective than traditional methods.

Wirkala and Kuhn (2011) very recently discovered that students who learned via problem based learning "showed superior mastery…relative to the lecture condition."

In a meta-study of nearly 200 other studies on student use of calculators in the classroom the NCTM concluded that "found that the body of research consistently shows that the use of calculators in the teaching and learning of mathematics does not contribute to any negative outcomes for skill development or procedural proficiency, but instead enhances the understanding of mathematics concepts and student orientation toward mathematics." (I’ve included this piece of research since many traditionalists oppose the use of calculators in mathematics education.)

Keith Devlin, in his book The Math Instinct, cited research by Jean Lave which found that people had highly accurate algorithms for doing supermarket math which were not at all related to the school math which they learned. In fact, people were able to solve supermarket math problems in the market itself with a 93% success rate, but when face with the exact same mathematics in a more traditional test format only answered 44% of the questions correctly. Later in the same chapter of his book, Devlin revealed more research suggesting that the longer people were out of school, the more successful they were at solving supermarket math questions.

It should also be noted that this discussion on what should be done to improve mathematics education shouldn’t be restricted to either traditional mathematics education, or discovery based methods, but that we should look at all of our possible options.