Discovery learning in math: Exercises versus problems Part I

Barry Garelick, via email:

By way of introduction, I am neither mathematician nor mathematics teacher, but I majored in math and have used it throughout my career, especially in the last 17 years as an analyst for the U.S. Environmental Protection Agency. My love of and facility with math is due to good teaching and good textbooks. The teachers I had in primary and secondary school provided explicit instruction and answered students’ questions; they also posed challenging problems that required us to apply what we had learned. The textbooks I used also contained explanations of the material with examples that showed every step of the problem solving process.
I fully expected the same for my daughter, but after seeing what passed for mathematics in her elementary school, I became increasingly distressed over how math is currently taught in many schools.
Optimistically believing that I could make a difference in at least a few students’ lives, I decided to teach math when I retire. I enrolled in education school about two years ago, and have only a 15-week student teaching requirement to go. Although I had a fairly good idea of what I was in for with respect to educational theories, I was still dismayed at what I found in my mathematics education courses.
In class after class, I have heard that when students discover material for themselves, they supposedly learn it more deeply than when it is taught directly. Similarly, I have heard that although direct instruction is effective in helping students learn and use algorithms, it is allegedly ineffective in helping students develop mathematical thinking. Throughout these courses, a general belief has prevailed that answering students’ questions and providing explicit instruction are “handing it to the student” and preventing them from “constructing their own knowledge”–to use the appropriate terminology. Overall, however, I have found that there is general confusion about what “discovery learning” actually means. I hope to make clear in this article what it means, and to identify effective and ineffective methods to foster learning through discovery.

Garelick’s part ii on Discovery learning can be found here.
Related: The Madison School District purchases Singapore Math workbooks with no textbooks or teacher guides. Much more on math here.

3 thoughts on “Discovery learning in math: Exercises versus problems Part I”

  1. Spot on. We can talk about funding, pay, new buildings, bad parents, bad kids, video games, obese kids with restricted blood flow to the brain, influx of foreigners, Bush, Cheney, Rumsfeld, NCLB, etc…, but until this author is listened to and heeded, no change will occur.
    My son had “Discovering Geometry.” I have a few credits short of a math degree. I could not help him. The book was guess as you go with nothing concrete put into words.
    The author of this piece is much more thoughtful and kind than me. The schools that use discovery methods as described, with the brand names attached, are guilty of crime.

  2. Barry,
    Ditch the teaching degree. You’ll never find a school that will let you teach the right way. Even if you do, in a year or so some fuzzy headed curriculum director will convince the BOE to put in Everyday Math anyway. Instead, buy a few used computers on the cheap, line them up in your basement like a mini-classroom, fill the computers and shelves with Singapore math material, and advertise like heck. Get 30-40 desperate parents to pony up $100/mo to make their kid a math whiz or to remediate the damage done, and you’re at 48K per year no sweat. Minimal investment and you won’t have a curriculum director or superintendent calling you into their office to get your head right.

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