A Follow-up to Math Education: An Inconvenient Truth
Commenter Hoosier alerted me to another video from the Where’s the Math people. This one is from a professor of atmospheric sciences, possibly the professor mentioned in the previous film, adding to the pile-on.
But first, I think some clarification is in order. This video and most people use the terms “standards-based,” “reform,” and “constructivist” interchangeably. However, as I mentioned in the previous post, these terms aren’t quite interchangeable.
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Standards-based describes a curriculum that is designed to teach a specific set of math standards. These standards could be the ones specified by the NCTM, or they could be state- or district-specific standards designed by whoever got saddled with the job (legislators, parents, math teachers, math professors, school board members, some combination of the above). These standards are also the ones that are tested by the No Child Left Behind tests.
The math standards are often correctly criticized for being too vague, sprawling, and incoherent. On top of that, standards-based curriculums are often implemented with extreme mathematical idiocy. For example, some districts see 54 math standards on their list, 180 days in the school year, and decide that each standard will get about 3 days of class time, completely disregarding the relative difficulty of mastering some of the standards. Having standards is not inherently bad, but the current system leaves much to be desired.
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Reform is largely a marketing term. Currently, it’s being used to refer to standards-based curricula, but almost anything can be labeled as a reform curriculum if they need to sell textbooks.
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Constructivist describes a teaching technique in which the instructor guides the students through discovering principles for themselves, rather than just lecturing it at students.* These principles can come from a published set of standards, but they don’t have to be.
Many people, including teachers and curriculum developers, mistake constructivist to mean that they shouldn’t ever actually teach anything. That’s wrong. A main difference between good constructivist teaching and traditional teaching is the placement of the description of definitions and principles. In traditional teaching, teachers often lead with the principle of the day, and then start the students on an activity to practice application of the principle. For example, a teacher might demonstrate the algorithm for long multiplication on the board, and then give students a worksheet of multiplication problems to practice it.
In constructivist teaching, teachers often lead with the activity, and then gather the class for a discussion of what they discovered. Then they close the lesson by translating the class’s discoveries into the accepted terminology.
For example, in a lesson about commutativity a class may discover that “turn-arounds” always work. At the end of the class, the teacher will teach that mathematicians talk about “turn-arounds” using the term “commutativity” and provide the mathematical formalism for it.
There is a lot of research that supports constructivism as a superior teaching technique. The problem with constructivism is that you cannot teach a constructivist lesson on the fly; if you do that your students are likely to learn nothing. In contrast, anybody who can read can give a traditional lesson straight out of a book. Constructivist lessons need to be planned carefully, right down to the specific examples and terminology used, and the teacher needs to know the material deeply enough and have enough pedagogical skill to be able to anticipate students’ often mistaken and bizarre responses and weave them into a coherent lesson so that students arrive at the principle they need to learn. The set of lessons I describe at the end of this post is a good example of constructivist teaching.
When you conflate all these terms and then dismiss the entire bit, you wind up rightly dismissing something that sucks—the current math standards—and disregarding something that can be more effective than traditional teaching techniques right along with it.
This sort of sloppy rhetoric and sloppy thinking is extremely apparent in the video linked above. The professor trashes standorefoructivist thinking, but then highlights the improvements gained by adopting curricula based on those from leading math nations.
Well, guess what! Japan, the leading math nation across many, many studies uses a) standards, albeit significantly better ones, and b) constructivist teaching, including heavy use of those manipulatives he mocked.
A big part of the difference (other than the much-improved standards) is that the Japanese give teachers hours and hours of prep time and the ability to network with other teachers to plan, critique, and refine lessons. This is what is necessary to make the constructivist approach work. In contrast, American teachers are relatively isolated in their own classrooms and get maybe an hour of prep time a day, which isn’t even enough time to grade papers**. If the constructivist approach is failing in the schools, it should be attributed to this.
But it’s much easier to incoherently rail at a few textbooks than to build a support structure for teachers so that they can teach more effectively.
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*There is this common claim that constructivist approaches rely on calculator use. I strongly suspect this claim comes from the conflation of the terms—some sets of standards do advocate for calculator use—because calculator use is anti-constructivist. You cannot guide students through deriving the distributive property by using calculators!
**For reference, it takes me about 12 hours to grade and give feedback on 60 1-hour short-answer/essay exams (the worse they are, the longer it takes), about 8 hours to do 25 multiple-choice + short-answer/essay exams. Granted, college students are amazingly prolific under a time limit, but the point is, grading (and giving feedback) takes forever.
Chipanglish
CogKnition
Gustatory Circuit
In my math methods course we actually did a Math Lesson study, using the Japanese model. A team of 6 teachers taught a traditional straight out the book (2 taught, 4 observed pedagogy, lesson organization, and student response). Then we met and picked apart the lesson. Revised it, and taught the revised lesson (ideally, it’d be with the same group of kids). Again, observations were made.
While we only had to do 1 cycle of teach, revise, reteach, and analyze, the Japanese will often do two or three cycles of one lesson.
That said, this type of lesson planning and writing takes lots of time. We worked about 12 hours on it, outside of instruction time, revising and analyzing the lesson.