In the July 2014 issue of Costco Connection I (R. Craigen) debate the merits of “discovery math” in Elementary school with Deirdre Bailey, an Alberta teacher who is an outspoken proponent this approach.

Her argument starts out by mischaracterizing the justification for foundation-first (or “back to basics”, or conventional) education. While we do argue that conventional approaches produce empirically better outcomes in computation and symbolic manipulation (and enquiry advocates admit as much) it is also true, as we repeatedly argue, that students well versed in the basics also do better on complex and information-rich tasks, and show more deeply connected knowledge — that is, “understanding” of the subject matter than those who have been taught to “think critically” in a vacuum of foundational knowledge. There is a great deal of empirical evidence to this effect, over several decades and in studies of many different types. However, her framing of that statement suggests that the merits of conventional instruction are limited to purely mechanical outcomes while (she implies) discovery stimulates deep learning and understanding. Actually, it is well established through numerous studies that conventional methods (direct instruction, foundation and basics before exploration and discovery, structured outcomes, hierarchical exposure to topics) is associated with better skills, better understanding and better emotional response to learning.

Finally, note the example of “inquiry learning” in her argument: finding the dimensions of a rectangle given information about its area and perimeter. Problems of the sort she describes are older than her grandparents. They are standard. Conventional. They are easily solved using straightforward procedures learned in early algebra (Junior High School)

Presumably this is “inquiry” for her students because they are years from starting algebra. Indeed, in that case a student’s toolbox for such problems is rather empty. It seems that “inquiry” is code for requiring students to solve problems for which they have not yet been exposed to appropriate methods.

The problem she states is easy enough to solve using guess-and-check. But even a simple variant would suddenly be unsolvable in that way (e.g., what are the dimensions of a rectangle whose area is 10 square cm and whose perimeter is 5 cm?). So not only is this “math-as-an-Easter-egg-hunt”, no general method of value is learned — that will still only come in high school. And the problem only “works” if it is contrived so that a solution is unnaturally easy to find.

This leaves untouched the cognitive complexity caused by the categorical problem of setting equal areas and lengths as if units of measurement didn’t matter. Classically this is resolved in a couple of different ways — by shifting to an abstract formulation in which the concrete meaning of those quantities is ignored by fiat, or by adjusting the meaning of the relation so that one ends up with matching units. Neither of which is appropriate for students to grapple with when still mastering the ideas of dimensions, areas and perimeters of rectangles!

You can see this teacher in action here, a promotional video for their school in which two teachers purport to guide students to “discover” a **definition**. You might ponder in what sense this is even possible, in principle. The interested viewer is directed to the discussions early in the video in which children are clearly echoing things they’ve been told by adults or older siblings in a sort of guessing-game in which students attempt to “get it right”, while the teachers dutifully record all contributions, helpful or not, on a board, avoiding accidentally actually … uh … teaching them anything. While the pretence is that the children jointly generate this knowledge out of the blue, in reality you see enough vicarious expertise in the room from children who’ve been versed in these ideas at home or elsewhere that eventually a serviceable definition emerges, probably the most gruelling possible way of defining a new term. On Deirdre’s blog she argues that this approach “makes learning memorable”, and given the amount of time and effort that went into this trivial outcome, it may be that she has something of a point here.

There certainly is a lot of positive energy in the room. It’s just too bad not to use all that energy for something more educationally productive. Surely dragging out the acquisition of a new definition into an hour-long socially complex exercise is not the only (or even the best) way of helping students enjoy math class.

You have to understand, watching this, that Connect! School is a Potemkin Village of sorts for discovery learning: a vaunted example or showcase to sell the idea. And the lesson shown is, for them, a sort of pinnacle of the art, or as their webpage puts it, an “exemplary example of teacher inquiry”. In other words … this is as good as it gets, folks.