I’ve had the opportunity to engage in further conversation about math education with Dr. Robert Craigen, Assoc. Math Professor and co-founder of WISE Math. My response to Dr. Craigen’s most recent comment wouldn’t fit in comments so I’ve included it, along with the initial response, here.
Thanks for your reply, Deirdre; I look forward to your full response.
I address the one point you make here about memorization.
Memory is the seat of learning. No memory, no learning. Those opposing memory work would do well to — at minimum, acknowledge its importance (I presume you do not disagree, but I note that there is no hint of it in what you have written.) And, having acknowledged this, to explain why memorization of times tables is not a particularly good use of memory.
I can tell you why, to the contrary, it is a good use of memory.
Early-years learners are passing through a rapid acquisition phase during which important facts are committed to memory, establishing the foundation upon which later synthesis of cognitive understanding takes place.
The ease with which these particular basic facts are automatized at this age contrasts the difficulty faced by those who must do so later in life — students who fail to memorize these elementary facts at this stage may struggle mightily with them later, and many will simply fail in the effort and never master them.
The goal of education is UNDERSTANDING. And understanding cannot grow in a vacuum — it requires context. Context necessarily entails factual content about things, processes and relationships, committed to long-term memory.
Our long-term memories are very well suited to storage and retrieval of such trivia. (Long-term) memory is not the ANTITHESIS of understanding — it is an integral part of understanding. Understanding is functionally impossible without a well-stocked memory. I don’t think any of this is controversial in the least.
WNCP Math and the like forces children to adopt inefficient and ad-hoc mental approaches to even single-digit arithmetic, and in multiple ways.
How different is memorizing multiplication tables from WNCP’s rote work? “Rote” means “repetition”. The demand for children to revisit the same “strategies” by rote in WNCP is simply astonishing.
Grade 5 students are expected to “apply mental strategies” and use skip-counting, doubling, halving and looking for patterns, to “determine answers” (i.e. NOT facts they have yet mastered) “for basic multiplication facts to 81” (WNCP K-9 Framework, p104).
“to 81” = “the 9×9 times table”.
Five years is FAR too long to force children to obsess over the mechanics of single digit arithmetic without closure! The 2008 NMAP report warns against “any approach that continually revisits topics year after year without closure”, an indictment of WNCP Math, which specifies NO point at which these are to be mastered.
Now it doesn’t take a PhD in brain function to know that working memory — particularly for abstract facts — is extremely limited. Famously, humans can reliably hold about seven digits in their heads but few can do the same with fourteen digits. It is a very small working space.
A year ago a paper published in cognitive psychology quantified the rather obvious consequences. A team of researchers led by Dr. D. Ansari of UWO separated children according to whether they performed single-digit arithmetic with automatic recall or strategies. They collected longitudinal data on the same children years later from their SAT Math scores. Those who had, years earlier, committed the fundamentals to memory, significantly outperformed those who had been trained to use ad-hoc approaches.
Some contend that those ad-hoc approaches constitute “understanding”. In fact the WNCP Framework itself, in outcomes specifying that children “demonstrate understanding of” such-and-such a concept, the corresponding achivement indicators specify that the children are to show multiple strategies.
Yes, some call it “understanding”. But I call it “clutter”.
Thanks Dr. Craigen,
I wanted to begin by acknowledging that it seems we agree on several points. Learning is without a doubt the cultivation of memory. However I would argue that there is an important distinction between rote memorization and experiencing something memorable. Memory is formative and compositional. It requires judgement, connecting, shaping and reflecting. It is a creative, inventive process of making sense of experience.
You have written that memorizing times tables is a good use of memory. I would suggest that finding ways of making multiplicative concepts memorable is a much better use. My experience with “times table memorization” in schools is that it typically involves recitation with no conversation, no questioning, and no opportunity for students to recognize or identify patterns and connections between multiples. I have noticed that the WISE site lists John Mighton’s JUMP math program as a resource. I have used JUMP myself on occasion and recommended it to students, specifically because Mighton sets out to cultivate memory very differently to the way it is done in traditional math workbooks. As a basic example, he asks students to practice counting by 2s and 5s and guides parents to ask their children what they notice. He discusses multiples of 4 and multiples of 6 as connected to multiples of 2 and multiples of 2 and 3 respectively. The formulation of memory is connected to discovery and he creates opportunities for the students to notice patterns of fundamental importance and articulate them in their own words. The work the students are involved in goes beyond recitation in an effort to commit the basic facts to memory. This is worthwhile practice.
I think perhaps the essence of our disagreement is not whether students should have basic facts committed to memory (it seems we agree, they should) but as to how this is best achieved. You wrote “understanding cannot grow in a vacuum — it requires context.” Absolutely, but I would add that committing factual content to long-term memory also requires context. This is what having students “skip-count, double, half and look for patterns” can provide. To be fair, I didn’t always think this way. Here’s some personal context because I think it’s an important part of the story.
I was successful in math class as a kid and through high school. My parents insisted I recite multiplication tables on long car rides and I had a gift for recall. Anyway, my high school IB scores exempted me from first year math in university. The four post-secondary math classes I did take boosted my overall average. I am not a mathematician, but I guess that makes me a relatively successful product of the system. Two years into my teaching career I found myself in a Grade 4 Math/Science classroom. Although my students had arrived from schools all over the city, my own success with classroom mathematics had made me pretty confident.
Half of my students came in overwhelmingly excited: “I already know my 9s tables,” while the other half were clearly dreading math. I had had them fill out an introductory survey and at least 50% had noted that they hated the subject. The first problem we introduced that year involved calculating costs of purchasing a variety of candies using either addition or multiplication. We just wanted to get an idea of where students were at with their understanding. It was a bit of a disaster.
Student: 3 x 32 cents = 15 cents
Me: The price of one candy is 32 cents. You decided to buy 3 candies and you have calculated that it will cost you 15 cents, does that make sense to you?
Student: Confused stare.
Me: How did you get 15 cents?
Student: Well 3 x 3 is 9 and 3 x 2 is 6 and 6 + 9 is 15.
Me: Why did you multiply 3 x 3?
Student: Because there’s a 3 on the bottom and a 3 on the top.
Me: But what does that 3 on the top represent?
Student: Blank stare.
Me: Does the “3 on the top” represent 3 cents or 30 cents?
Student: Blank stare.
“Alignment” mistakes from students using conventional multi-digit addition algorithms were also excessively common. It seemed they weren’t thinking, just executing.
Student: 24 cents + 13 cents = 154 cents
Me: How much is a quarter?
Student: 25 cents
Me: How many quarters in a dollar?
Me: You added less than a quarter to less than a quarter and got more than a dollar?
Student: Blank stare.
Some struggling students attempting to add double digit numbers were in tears because they just couldn’t remember what “carrying” meant and what column the number they were “carrying” was “meant to go in” (“my teacher just told me to put it there”). One student had successfully used multiplication to determine the cost of purchase of a whole variety of different candies but kept coming back to me to ask “is this right?” When I asked her if she could think of a way to verify whether she was correct using an alternative strategy, she couldn’t.
I’ll be honest, for the first few classes I did a lot of showing and telling. I figured, that’s how I had learned so it should probably work. I would be sitting next to students saying: “so these 3 candies are worth 10 cents each right? If you want all 3, you will have to add 10 cents 3 times to find out how much you owe. Since they all cost 10 cents though, you could also just multiply 10 by 3 right? Because that’s 3 tens. Three multiples of ten. Right?” The kids didn’t often have much to say though and I couldn’t shake the feeling that they still didn’t really “get it.” They were doing what I had asked alright, but relatively thoughtlessly. They also seemed determined that multiplication was a series of isolated concepts you memorized and then moved on from. “We’ve already done 5s and 6s so this year we do 7s.”
One weekend early on I was telling my husband about the “trick” I was going to use to teach my students how to remember multiples of 9. He responded with “why do they have to do it your way, that’s not how I do it.” Again, I am embarrassed to say that it had simply not occurred to me that there was a different way to remember multiples of 9. It did occur to me at that point however, that it might be worth reading a little bit more about how to teach math, so over the next few weeks I read as much as I could about how students construct mathematical understanding. I spent quite a few nights at the school until 8 or 9 with my teaching partner, reading and researching and thinking and we finally decided to have the students “review” multiplication with a hundreds chart. We had envisioned that the students would start by coloring multiples of one, two, three, four etc. on the hundreds chart, that it would help them recognize patterns and make connections between numbers and that it would help them remember.
We asked them to start by coloring in multiples of one and were shocked again when we were met with 50 blank stares. The students had no idea what the word multiple meant. So instead of coloring, we talked about what it means to multiply something and what constitutes a “multiple.” We used students as examples: “What would happen to Joe if I multiplied him by 2? If I had 3 Jessicas standing up here how many multiples of Jessica is that?” and then asked students to propose definitions for “a multiple of one” based on their understanding. As a group we reviewed their definitions and then added, verified or refuted their various conjectures through conversation. The result was a more engaged group of students than I’d seen all year. The conversations led to discussions and debates and lots of “Oooooh, now I get it!” I was hooked, the kids were hooked, it turns out math is really cool.
A few weeks into our conversation about multiples I had a group of students trying to figure out how many different ways they could make a box filled with 3, 5, 7 or 9 truffles and one little boy came rushing over brimming with excitement..
J: “Mrs. Bailey I discovered something!!! You can’t fill a box with two rows if you have an odd number of truffles!!”
Me: “YEAH J! Why do you think that is? Do you think it’s connected to something we noticed about multiples of 2?”
He paused, then his eyes lit up: “YA! Because multiples of 2 are always even, not odd.”
Me: “Awesome why don’t you go and add that conjecture to the board.”
J rushed off delighted with himself to add his conjecture to the ever increasing list of mathematical “conjectures” that we would review and debate at the end of class.
I’ve told this story about J a few times in the last several years for a couple of reasons. J came into grade 4 terrified of math. Every time we talked about numbers he’d nod frantically while blinking back tears. Most of J’s classmates had recognized ages ago (Grade 2 maybe) that odd numbers couldn’t fill a box with two rows. Much earlier that year we had, as a class, approved the conjecture that “all multiples of 2 are even” and J had nodded, writing it down in his book. J had even practiced his “two times tables,” at home. He was proud that he could remember “all twelve”. But he was remembering twelve different multiplication equations as isolated facts and they meant very little to him in a mathematical context. In a traditional math classroom this would have been overlooked. In the inquiry-based classroom however, we designed mathematical investigations such that it came up again and again until J finally had an opportunity to understand. While some of our students were writing conjectures about the frequency of prime numbers and how to identify least common multiples, J was discovering, really discovering, that odd numbers can’t be split in half evenly.
Someone once told me that every time a child has an “A-ha!” moment, a fundamental brain shift has occurred. I’m sure there’s no empirical evidence to support this statement. I have only taught math to a couple hundred students thus far in my career so my sample size is low. But everything I’ve seen in the classroom and everything I’ve experienced in my own learning supports that statement. “A-ha!” moments are fundamental to the cultivation of memory.
School is a lot more interesting and seems an awful lot more real and personal when kids are rushing around bursting with discoveries than when they are sitting at desks rushing through math facts. I guess maybe what I’m saying isn’t so much “don’t teach memorization” as “make math memorable.” Situating the development of basic skills more intentionally within mathematical conversation, encouraging student formulation and articulation of ideas and guiding students toward discovery has been, in my experience, way more powerful, more exciting, and more memorable, than having them practice reciting and writing their multiplication tables.
I agree that students should be provided with the opportunity to study multiplication early on. I also agree that by Grade 5, if these opportunities have been effectively designed, students should have already committed most multiplication facts to memory. My grade 4 students were able to point out that if you “knew” multiples of 2 you knew multiples of 4 and 6 and 8, and that if you knew multiples of 3 you knew multiples of 6 and 9, and 5s were fairly straightforward so that really only left 7×7 (“and by the way Mrs. Bailey, 49 is one of those numbers and that’s why you have to work harder to remember it..” “One of what numbers?” “You know, it’s not really connected to anything. Kind of like a prime number, but not, because it’s a multiple of 7”).
The word “rote” is associated with “mechanical or habitual repetition” of something to be learned. Synonyms include: mechanically, unthinkingly and mindlessly. I cannot subscribe to the idea that students should be taught to remember things “unthinkingly”. Yes to memory. No to “mindless”. I haven’t read the word “rote” in the Alberta Education curriculum. If I had, I would have thoughtfully ignored it. You mention in your critique the curriculum’s acknowledgement of multiple strategies but Alberta Ed uses the words: “such as” to suggest the different strategies students might describe or demonstrate. It is not prescriptive or prohibitive. It doesn’t mean that children need to memorize multiple different strategies for multiplying 6 x 7. It means that they need to be able to recognize that (5 x 7) + 7 and (3 x 7) x 2 are equal to 6 x 7 and that both of those are more efficient ways to calculate 6 x 7 than counting up by 7. If they work with these concepts regularly enough in meaningful ways, are engaged in conversations about what they mean, and are held accountable for what they know, where they struggle and what they should practice, 6 x 7 is easily committed to memory and the rest just follows because it makes sense. It’s not clutter.