Recently, Andrea Sands of the Edmonton Journal published an article outlining the current debate in Alberta around math curriculum. It was well-balanced and well researched. It presented both sides nicely. I sent her a short email pointing out my only concern with the article. She quotes University of Alberta Math Professor Gerda de Vries, who I have met and respect. de Vries says,

Alarm bells have been going off for a while. The students, in theory, are supposed to be better problem-solvers and that’s not what we’re seeing. And I’m getting that message from my colleagues in physics and chemistry is that the students just are not as well-prepared to solve problems.

Nowhere in Andrea’s article does she mention that de Vries and the rest of her colleagues at U of A have seen exactly one term’s worth of students who were taught using the new curriculum. Last September’s 12th grade students were the first batch of high school graduates to have any exposure at all to the revised curriculum, and they only had it for three years. If alarm bells have been going off for a while, then they were going off under our old curriculum.

Andrea replied quickly and assured me that de Vries was careful to say that she wasn’t sure whether the curriculum was the problem or whether there might be other factors at play. Unfortunately, the article fails to mention that. Andrea then asked about blending the approaches. Here’s what I sent her (slightly edited).

Andrea,

You’re bang on. This whole debate is a little silly. One camp thinks if we only teach algorithms, understanding will be lost. The other camp seems to think that if we only focus on understanding, proficiency will be lost. Why would we consider doing one without the other? Why is there the perception in the public that we are doing one without the other?

Bumpy is a great way to describe curriculum implementation, mostly from the teacher perspective. It takes a while to figure out what the new standards are. This curriculum was new, not so much in content, but in how we were being asked to teach. Older curriculums focused on algorithms (procedures) and hoped understanding would come along for the ride. This one flipped it so that we were to start with understanding, and have students build algorithms with that foundation of understanding. This is where things got bumpy for some of us (teachers).

The debate seems to center on two main things.

- At what point to we give students the algorithm if they are unable to develop it on their own?
- Do students have to do it one way that makes sense to them, show it to me in multiple ways regardless of their preferred method, or do it the one traditional way that many of us were taught?

The classes I taught (at the secondary level) were always based on the idea that I had to help my students get at the understanding in addition to the proficiency. I did this because when I went to University, I struggled with math at first despite doing quite well in high school. I was a great imitator. I didn’t have great understanding. My teacher would show a problem and then put the same one with different numbers on the test, which I would ace. When I hit university and had to think, I was in trouble. In addition, I had never struggled with math before, and didn’t know how to get out of trouble. This was in 1988, which is why I can say with some personal experience that this notion of students struggling in university math is not a new thing.

So when I became a teacher, I tried hard to make sure my students knew not only what to do to solve questions, but WHY that method worked. I wanted them to avoid a struggle at University like I had. Instead of starting with the algorithm, I tried hard to start with the understanding, and build the algorithm out of that understanding. In the end, whether the understanding was there or not, I had to give my kids an algorithm so they could do certain things. I just flipped it so that instead of starting with the algorithm, I started with understanding and built (or gave out) an algorithm from that understanding. To me, that’s the more effective way to teach math. I agree with your assertion that it’s not one or the other.

This may oversimply things but this debate is about proficiency (ability to DO math) and understanding (knowing WHY they are doing what they are doing and WHEN to do what they are doing). We can arrange those two things four ways.

**Proficiency with understanding** – I think we would all agree that this is what we are shooting for. If every kid got there, we’d be doing a great job.

**Lack of proficiency with lack of understanding** – Clearly, none of us want our students ending up here.

This whole debate seems to come down to how to rank the other two permutations of those two states, which is a little silly. Very few kids end up with one without some of the other.

**Lack of understanding with proficiency** – These kids would be able to multiply numbers quickly and efficiently. They’d struggle with why it worked, and would struggle with when to use multiplication. This was me. I could take complicated derivatives with the best of them. I had no idea at that time what a derivative represented. People who think this is the better of the other two states are happy kids can multiply without a calculator (and make change – It always seems to be about making change).

**Lack of proficiency with understanding** – These kids would struggle to multiply numbers quickly and efficiently on paper. They’d know what multiplication represents, and when to use it. People who think this is the better of the other two states think that kids can use a calculator for mundane calculations, and knowing why and when to multiply is the important skill.

That got long. That wasn’t my intention. Thanks for following up.

John

on December 29, 2013 at 6:54 pm |Robert CraigenHi John. You say you’re for teaching both understanding and proficiency. On that note, you sound like you’re ready to join us at WISE Math. Welcome aboard, grab a pitchfork — here’s the signup page: http://wisemath.org/join/

You write, “Why is there the perception in the public that we are doing one without the other?” Uh, let’s see … from the WNCP framework document itself? From the reports of innumerable parents and teachers all over western canada who complain that the standard algorithms are entirely missing from their children’s education, and paper-and-pencil skills “de-emphasized”? From the empirical evidence of sudden decline of said skills over the last decade — as this version of WNCP was implemented? From the textbooks that de-emphasize or lack altogether some of those fundamental skills or any evidence of practice materials necessary to reinforce them?

Let me tell you a story. It’s the story of how I ultimately decided to follow this path.

It was 2009. I was in one of the all-day meetings of the Manitoba Provincial Math Curriculum Steering Committee. Yes, John, I was one of those phantom mathematicians to whom you continually refer as having been “consulted” about the WNCP curriculum. I had been on that committee since 2005, as the sole university Math representative from our province. Though I have tried for two years now to coax the names of counterparts in other provinces from the provincial WNCP contacts and the Team Leader in Alberta (Christine Henzel) I have not, through such channels, yet been given the name of a single PhD in math who had comparable access during that period or earlier. This, by the way, is why I relentlessly mock your assertions about Mathematicians being part of the process. Not that I believe there were none, only that I’m agnostic. I actually believe that the WNCP folks are simply terrified that we might actually … you know .. get together and compare notes. Or something.

I had been put on that committee, BTW, largely by default. I was not from Manitoba and frankly had no idea what the curriculum was replacing. I was completely (at the time) conversant with much of the language, and had no frame of reference for evaluating stuff. Still much of what I saw in the framework and heard as they discussed it disturbed me, and we’d had numerous heated discussions about the importance of fundamental skills and in teaching basics to mastery before moving on (rather than “spiralling”, which was touted about like some miracle drug). Yet I still had only vague discomforts. Not knowing what to look for in the curriculum I could not see some of the things directly in front of my eyes. Or rather, I did not notice some of the things that were missing.

Further, the assumption seemed to be that I was there to provide input ONLY concerning the high school pathway courses. But all the stuff that disturbed me most was in the K-8 (mostly around grades 3-6) grade material. I would voice dissent, and they would roll their eyes. “Yes, yes, what else would you expect a university professor to say … obviously he has NO idea about teaching children”. And they’d pat me on the head and move on.

Anyway, that morning the committee was hearing a presentation by one of the provincial consultants who had been travelling around the province running training for teachers on how to teach the new curriculum. At the time teachers did not have textbooks in their hands — which was frustrating for them, but the consultants regarded it as a timely OPPORTUNITY. Word around the table at such meetings was that, with the earlier WNCP introduced in the 1990s teachers had, unfortunately, not gotten the message about HOW TO TEACH “the new way”, and they saw this as a grand chance to “bring them onside”, i.e., to explain the in-class dynamics of groupwork, discovery learning, and dealing with the rather attenuated kind of skills differentiation that put a strain on instruction when the curriculum itself assumed spiralling.

The consultants slogging about the province were regarded as the missionaries of this new doctrine (“fuzzy” or “reform”, depending on your perspective). And they addressed us as a breathless congregation eager for stories of the campaign amongst the natives: how many new converts won?

This consultant had one complaint — many of the rural middle-school teachers were having the same problem, which was voiced by one Grade 5 teacher, as she related: “But … I don’t get it — where do I introduce long division”. The consultant continued “I told her: You Don’t! She replied, Oh, has it been moved to Grade 6? No. It’s started in Grade 4 or Grade 7? Should I be working on some precursor … ” You see, she said to us — they just don’t seem to get the idea.

At that point I had enough, and my hand was in the air. “Wait a minute! Are you telling me that long division isn’t going to be taught in this curriculum??!” I began flipping frantically through the hundreds of stapled pages of the draft document in front of me. I could not believe this was the first time this had come to my attention.

The consultant could see I was upset and looked over to the ministry official who ran the committee, for some help. This official turned to me, and, in condescending tones, explained — Rob, were you unaware of this? You’ve been on the committee for over 4 years now. We don’t want them focussing on skills to the exclusion of understanding. We’re all about teaching understanding, you know.

Understanding? Understanding? How can you have understanding if you don’t have the framework of skills and contextual information? And besides, long division is an important precursor for so many things! How did long division get left out? Was that an accident? Don’t you know how important it is?

The ministry official: Rob, you should understand that this is an essential part of the underlying philosophy. This is fundamental in the new curriculum. NO STANDARD ALGORITHMS.

Now, that was the first time I’d heard the phrase in that context, but I quickly grasped what it meant and so I outlined “the four”: adding down columns, with carrying (gone); subtracting similarly, with borrowing (gone), and vertically-arranged multiplication (gone). All replaced with what some call “horizontal algorithms”, diagrams and various special-purpose “strategies”.

Oh yes, and the capstone skill for the lot: long division.

She began to explain how teaching students skills short-circuited students’ opportunities to gain understanding…

The rest of the committee looked at me pityingly, as one might a primitive who had stumbled into a modern computer store, and had no comprehension of the technological marvels surrounding him.

The debate that ensued took up the rest of the morning, the only person jumping in on “my side” being (remarkably, actually) an Education Prof, who agreed that an unfortunate false dichotomy between skills and understanding had crept into the educational establishment, and — not to worry — he was on the case and working on it. Well, here he was amongst a bevy of potential converts, but he was making no more headway than me. In any case, aside from this philosophical point, he appeared to support wholeheartedly the WNCP elimination of standard algorithms — he did not speak against it, in any case.

That’s when I knew that I carried an important burden. Since that time, consultants and converts have, like you, been running interference on the skills versus understanding business, in most cases outright denying that the current WNCP has eliminated them and that early and middle-years teachers have been systematically indoctrinated into believing that teaching them (or teaching them “too early” — meaning in grades where they are conventionally taught) is educationally harmful. If you must maintain this, well … it’s a free world. But I’m too close to what’s going on to be fooled. Who’m I to believe — you, or my lying eyes and ears?

The WNCP document is an encoding of this doctrine that understanding is developed in a vacuum of skills. Indeed, my education prof friend was wrong. It is not merely that there is a false dichotomy between skills and understanding that has found its way into the educational establishment: the prevailing doctrine in some influential circles says that the two are antithetical. Incompatible. Emphasizing one will harm the other. And … hey … since “understanding” is the more important of the two …

In the face of outright denial (once the gurus realised that this doctrine cannot stand up in the light of day, there was a rapid effort to “rebrand” by simply changing the terms of discourse), I simply pulled out the Framework of Outcomes and compiled some statistics. Here’s a sample:

“Demonstrate an understand”: 108 occurences

“Strategy”: 106 occurences

versus

“Standard” (…method, approach, skill, etc — in any context) 0 occurences

“paper and pencil” 5 times, all diminishingly (as in “without the use of” or “placing less emphasis upon”)

“Skill” 8 occurrences (in the front matter, but none in the outcomes)

“Fluency” 4 occurrences (again, none in the outcomes)

“Algorithms” 3 occurrences (all in the front matter; one is not about the standard algorithms; the other two are diminishing statements about them)

“Drill” once, diminishingly and negatively

“Exercise” once, diminishingly

“Organize” or “organized” twice, neither referring to the nature of student work

“Structure” only once, in a title in the bibliography

Etc. There’s lots more. Almost any term I could think of to search revealed a lot about the WNCP philosophy.

Well, you got another of my massive comments. Sorry for the long read. But I’m all about full information. I’ll leave you with this telling exchange (November 29, 2005) between the WNCP framers and a publisher asking about the algorithms (Paraphrase: “we could take or leave yer stinkin’ standard algorithm, they’re of no consequence in WNCP. But support for non-convergent instruction is essential!”). There’s a lot more where this came from:

PUBLISHER QUESTION:

“At several grades, the Number outcomes for operations emphasize use of personal procedures. At the same time, the same outcomes also call for students to record their work symbolically. Does this mean that the introduction of a standard algorithm is:

-Prohibited in any form?

-Acceptable if preceded by appropriate conceptual development?

-Acceptable if presented in conjunction with multiple recording procedures?

I suppose there are other approaches, as well – publishers will need to know if any approach puts their submission at risk.”

WNCP RESPONSE:

“The introduction of a more standard algorithm is acceptable if presented in conjunction with multiple recording procedures. Using only the standard algorithm would put a publisher’s submission at risk.”

on December 29, 2013 at 7:12 pm |Robert CraigenSorry, just a question, about your category: “Lack of understanding with proficiency”.

Do you know of someone who promotes that approach? I haven’t met one yet, but I’m always open to new information.

on January 17, 2014 at 12:09 pm |Charles WillinghamI would have enjoyed hearing John’s rebuttal. It seems supporters of WCNP only want to hear the echos of their own voices. A lot of this nonsense has to do with big money for publishers, right? Who is working to make math that failed in other jurisdictions a staple in Canada? I give Robert credit for standing up to these people who do not want to here the suggestions of the real mathematicians.

on April 2, 2014 at 11:13 am |Joe Bower (@joe_bower)Great post, John. You make some stellar points that make this topic far more nuanced than some columnists would like.

Joe