On multiplicative thinking and how *play to learn* in math class is often just pretense

(I’m starting a series with my reflections on numeracy teaching and learning, in particular, the cluster fraction-proportionality-multiplicative thinking, which is where everything breaks down in elementary school mathematics. It’s based off my grad school research, experience in the classroom, and observations of my own children’s learning)

In elementary school, children do not learn about the rational numbers of abstract algebra. If they did, then, e.g., to justify that 2/5 is less than 3/7 , it would be enough to say that 2×7 is less than 3×5.

That’s it. No need for drawing pizzas, playing with manipulatives, or tinkering on apps.

But this is not how children learn: they need to attach numbers to some concrete objects (or their representations) plugged into something that resembles a real life situation – abstract entities alone don’t cut it.

Playing with real objects is at the core of learning – say the progressive educators AND There’s so much beauty in abstraction – say the mathematicians. Yet some of the brightest in either category warn against the risk of indulging in either play or abstraction alone. The former remains just that, an indulged interest if it is not “realized through its direction”, says John Dewey, the philosopher of education whose view of learning was nothing if not of an experience-based, child-centered process. The latter runs the risk of “deforming the taste and creating snobs” if it is in and of itself the goal of teaching, says Henri Lebesgue, the mathematician who created a highly abstract theory of integration. The progressive philosopher of education is concerned with keeping the mathematics in the experience, the enlightened mathematician – with keeping the experience in mathematics.

But mathematics and experience are not properly connected in elementary math teaching, although there’s plenty of pretense to the contrary. And that’s why fractions teaching, and, as a result, multiplicative thinking in children is BAD.

Parents, teachers, policy makers, researchers would all agree that you can’t teach children about fractions as equivalent classes of pairs of integers, where (a,b)~(c,d) iff ad=bc. Yet, that’s exactly what they are being taught, only it’s only after they play a bit to make it “meaningful”.

*Play to learn* feigning goes largely unnoticed in classrooms: there’s objects and there’s math, but there’s no take-off to the abstract from the concrete. It stroke me when I first witnessed it in an undergraduate course at university offered as part of an elementary teacher-training program. I looked at a few such courses – they’re called math methods courses – to understand how future teachers do math in their teacher training: their curricula, their practices, their institutional features.

It turned out that students preparing to be teachers embrace progressive discourse about providing children with authentic experiences, where their learning is meaningfully rooted in “reality”, yet, they end up encouraging rote, algorithmic problem-solving where numbers are ultimately divorced from quantities.

Here’s an example from a unit on proportionality from a math methods course I observed.

The undergraduate students had to prepare a group activity for their peers, where they would teach a lesson on fractions inspired by a real-life experience. The goal was to get the students to provide meaningful, quantitative explanations – like the ones they would provide to children learning about proportionality.

One of the future teachers was passionate about baking, and proposed a problem-solving situation involving a recipe she had actually tried. She prepared a lesson on conversions between the metric and imperial systems of measurement. She tried this lesson with 8 classmates, acting as her students – the assumption being that this is how she would have run the lesson in an actual classroom, with children.

As a preamble, she outlined some features of the two systems for her classmates (playing the roles of pupils) and circulated various objects in the group: measuring cups, spoons, a scale, etc.

Then, handing out a worksheet, she asked them to “practice conversions” using the following conversion equations as givens (notice how the second and the third contain redundant information to the first): 

[1] 1 𝑐𝑢𝑝=240 𝑚𝑙=8 𝑓𝑙 𝑜𝑧=1/2 𝑝𝑖𝑛𝑡 (𝑙𝑖𝑞)=16 𝑡𝑏𝑠𝑝=48 𝑡𝑠𝑝 

[2] 15 𝑚𝑙=1 𝑡𝑏𝑠𝑝 

[3] 2 𝑡𝑏𝑠𝑝=1 𝑜𝑧 

The recipe called for quantities such as 480 𝑚𝑙, 360 𝑚𝑙, 240 𝑚𝑙, 180 𝑚𝑙, 60 𝑚𝑙, and 5 𝑚𝑙 to be measured in cups, tablespoons, teaspoons, and ounces. The members of the group were given some time to reflect and solve. After this, the teacher-student asked some of her pupil-classmates, to present their solutions; she praised them for their work, without commenting on the content of their proposals. She then introduced, as a recipe in its own right, what she called “the method of cross product”: setting up a proportion, with one unknown, 𝑥, and solving for it algebraically to find the desired imperial measures. She did this despite the fact that all of the quantities in the recipe were multiples of 60 𝑚𝑙 (480 𝑚𝑙, 360 𝑚𝑙, 240 𝑚𝑙, 180 𝑚𝑙, 60 𝑚𝑙) or a divisor of it (5 𝑚𝑙) thus suitable for more intuitive strategies for conversion, and without reflecting on the meaning of the proportion in light of the obvious multiplicative relations involved. Moreover, she ignored some participants’ ideas that alluded to such quantitative relations (e.g., 360 𝑚𝑙 is 1 1/2 𝑐𝑢𝑝𝑠 because it is made of of 240 𝑚𝑙 and 120 𝑚𝑙, which are 1 𝑐𝑢𝑝 and 1/2 𝑐𝑢𝑝, respectively). She also mentioned, in passing, that if decimals are obtained when solving the equation, they should be converted to fractions. More surprising was, however, the participants’ critique of her lesson: she was praised for the fun and “hands-on” activity (she had passed measuring utensils around), that would be, in their view, very appealing to children. One person from the group did notice that the method may be too “algebraic” for children, and proposed that the 𝑥 in the proportion be replaced by a question mark. They seemed oblivious to the missing link between the proposed activity and the ensuing calculation: a sensible explanation involving the quantities at hand. 

There was play and there was math, but they were not connected. As a trained mathematician and as mother of young children this approach felt alarmingly bad, and honestly, even depressing. I started getting why children enter school with hard wired quantitative intuitions but progressively learn to divorce these intuitions from their learning of math.

(to be continued)