In April, I attended the NCTM conference in Boston. I was staying with Jake, a college friend, at his place in Cambridge a few blocks from Central Square. He was a terrific host; when I told him where I was headed, he explained that the public transit route was a bit tricky. To save me from getting lost, he began explaining my public transit route in exquisite detail: the number of blocks to walk, important landmarks, the side of the street to enter to make sure I was headed the right direction, and so forth. The direction was complicated, he explained: look to travel inbound, regardless of traveling to the conference or returning. This made no sense to me, but I wrote it down anyway and kept listening.
And then at one point – I forget at whose suggestion – we tried a much better approach. Jake simply told me, “On the Red Line, inbound and outbound are determined by whether you’re heading towards or away from Park Street.” This explained why I needed to head “inbound” in both directions, which had prompted the original need for the detailed directions. Jake realized that once I had that essential understanding, most of my steps would be more or less obvious. This seemed much more efficient.
And it was. He still gave me detailed directions too, and I mostly followed them, but the key understanding he shared with me became important once I deviated from his instructions. One day, I wanted to stop at a CVS on the way that led me to a different station; the following day, I wanted to venture off to Fenway Park – a move that involved the Red Line and a transfer to the Green Line. Thanks to my emerging understanding of the metro system, I required no new sets of detailed directions. Instead, I charted into new territory with confidence. Had Jake only equipped me with step-by-step directions for getting to and from the conference, I likely would have needed more of the same for my subsequent adventures.
Perhaps because I was attending a conference for math teachers, I saw an immediate parallel to the perils of teaching and memorizing of procedures in mathematics without understanding. Here’s why.
Jake and I both defaulted to a step-by-step explanation for my travel route, because I literally knew nothing about the Boston metro system. Many students I’ve worked with – especially ones who claim not to “get” math – are eager to watch, memorize and cling to a step-by-step procedure. It’s understandable to want to give in to this impulse – there’s (the perception of) a large gap that needs to be closed in a hurry, so giving a step-by-step process or a detailed set of processes to follow seems logical.
But here’s an example (from a session at that math conference called “Nix the Tricks”) of how this approach might manifest in an elementary math classroom. In an effort to get students to quickly memorize 3 x 4 = 12, a teacher might have students memorize this: TREE times DOOR equals ELF.
There’s plenty more where this came from. Every digit could have a similar sounding word to remember, and every product has an image like the one with the tree, door and elf:
Learning anything this way – transit directions or multiplication – must be exhausting. And the research shows that an approach like this is generally counterproductive. Let’s unpack why:
- Strikingly, the above approach to multiplication is completely removed from any understanding of the concept of multiplication. Understanding often gets left behind when there’s a rush to transfer a lot of knowledge quickly. Unfortunately, that necessitates the need for this extremely inefficient work-around involving memorization of dozens of bizarre images and words, each one in isolation, each one without rhyme or reason. An alternative could be to actually share the meaning of multiplication, perhaps using a visual like this:This is unquestionably more efficient, and also builds on an understanding of multiplication. That last piece is crucial, because memorized steps, procedures or mnemonics are often misremembered or forgotten – what happens when a student can’t remember whether tree times door is elf or chick? Or whether elf stands for eleven or twelve? With a method like the dots above, students have a lifeline based in reason and understanding.
- The alternate approach with the dots is preferable not only for retaining knowledge but also for building new knowledge. The concept inherent in the dots above is transferable. It can be applied to new, harder problems: like 4×6, 12×12, or even algebra if area models are used. Understanding is what allows transfer of knowledge; it’s how I was able to apply what I knew about the Red Line to the Green Line. Without understanding, knowledge only extends as far as the specific steps learned or memorized.
- Not only is memorizing like in the door-tree example an inefficient approach; it’s also ineffective. Jo Boaler explains why: “Data from the 13 million students who took PISA tests showed that the lowest achieving students worldwide were those who used a memorization strategy – those who thought of math as a set of methods to remember and who approached math by trying to memorize steps. The highest achieving students were those who thought of math as a set of connected, big ideas.”
- One final reason why memorization approaches are counterproductive: they distort the meaning of mathematics. In Boaler’s article, she argues that we have “produced a generation of students who are procedurally competent but cannot think their way out of a box.” Interestingly, this is a distinctly American phenomenon that has existed for some time. “The U.S. has more memorizers than most other countries in the world,” Boaler writes. James Hiebert and James Stigler found similar results a decade and a half ago. In their book The Teaching Gap, which serves as an analysis of math classes in multiple countries, they share one observer’s big takeaway from all the classrooms he visited (pg. 25): “In Japanese lessons… the students engage with the mathematics, and the teacher mediates the relationship between the two. In Germany… the teacher owns the mathematics and parcels it out to students as he sees fit, giving facts and explanations at just the right time… In U.S. lessons, there are the students and there is the teacher. I have trouble finding the mathematics.”
Well-intentioned teachers – myself included, I’m sure – are often guilty of overvaluing procedural knowledge while skimping on conceptual understanding. Procedures, shortcuts, tricks – they may help a student learn that day, but it’s easy to forget that without being pinned to some conceptual understanding, a memorized procedure doesn’t have much value. It won’t be long-lasting, and it isn’t a solid foundation on which to build and extend to new topics. Doing so is an exercise in cutting off your nose to spite your face.