Before he passed away a few years ago, we would occasionally attend my father-in-law’s church. Attending someone else’s church is a strange thing. I almost always came away with a few things I agreed with, a few things I disagreed with, a few things I was curious about, and a few things that seriously offended me.

On May 1, I attended someone else’s church again. I joined (I think I was invited) a whole bunch of University Mathematics Professors from across Alberta at their 2014 Alberta Mathematics Dialogue, put on by PIMS at Augustana in Camrose, Alberta. Of particular interest to me was a series of talks on Number Theory, but I wasn’t there to engage my own mathematical interests. I was there to listen to these math professors analyze the Alberta K-12 math curriculum.

The sessions I attended were:

- Leif Stolee (Retired Edmonton Public Schools Principal) – NEW MATH: Building the Roof without laying the foundation.
- Vladimir Troitsky (University of Alberta) – Critique of Alberta K-4 math curriculum
- Cornelia Bica (Northern Alberta Institute of Technology) – Grade 5-6 Mathematics Curricula Around the World
- Christina Anton (Grant MacEwan University) – Mathematics Curriculum in Junior High School
- John Bowman (University of Alberta) – What is Missing in High School Mathematics Education?
- Mark Solomonovich (Grant MacEwan University) – Now and Then; Here and There: problems with what they call education.

For details beyond the titles: the full abstract of their talks.

Based on the four categories of reactions I had in my father-in-law’s church, I present, without commentary, my reactions to the sessions I attended.

**Things I heard that I agreed with…**

- All of the Math Professors emphasized that basic facts and problem solving are both important.
- Cornelia Bica wants students who persevere, solve problems, and do more than just follow recipes.
- Cornelia Bica talked about a blend of pedagogies. She said, “Singapore and China aren’t as hardcore as they used to be.”
- Cornelia Bica said that pedagogy should be left to the teachers.
- John Bowman stresses that he is not criticizing K-12 teachers. He is asking for leadership in making changes to curriculum.

**Things I heard that I disagreed with…**

- Troitsky insists that bar graphs, charts, patterns and shape and space are not mathematics.
- Cornelia Bica said that you need basics before you move on to problem solving.
- Christina Anton stated over and over again that concrete and pictorial representations need to be removed because they are a waste of time. Only the symbolic should be taught because it, alone, is real mathematics.
- John Bowman said, “Strategies and tricks are interesting for the brightest students. However, the average student (including future professionals) will be better served by learning the time-tested algorithms for arithmetic computation that we learned as children and continue to use in our daily lives.” (This one might be better placed in the category below. My gut says, “disagree”. My head says, “think about it.”)

**Things I heard that I want to think about some more…**

- Two of the speakers (Troitsky and Bowman) mentioned that they believe kids can’t develop personal strategies until they have first mastered a traditional strategy.
- Cornelia Bica carefully analyzed at what level different topics are introduced in different curricula around the world.
- Cornelia Bica’s analysis included comparing instructional time in mathematics as a percentage of total instructional time. Alberta was the lowest of the regions she showed.
- John Bowman spoke about Math 31 (HS Calculus) and whether we need it.

**Things I heard that really offended me…**

- Vladimir Troitsky said that kids who can’t multiply 7 x 8 or those who need 5 strategies to do it should go work at McDonalds.
- Christina Anton said that concrete material and pictorial representations should be reserved only for special needs students.
- Leif Stolee opened with Matthew 8:28-34. It’s a story about demons and swine and Jesus. As I understood him, Alberta Education is the demons, K-12 teachers are the swine, and he is Jesus. I not sure I followed him completely.
- Leif Stolee told a story about working in a school in which I would later work. Two of his English teachers applied for a job at Alberta Education. One of them was a superstar (Leif’s words). The other was mediocre. The mediocre teacher got the job. He extended this thought by pointing out that good teachers like himself can no longer advance in Edmonton.

Edit: After Dr. Troitsky’s comment below, I looked back at my notes from his talk and found that he had also included patterns and shape and space in the list of things he found in the curriculum that are not mathematics. I have added them in that bullet above.

Edit: Based on Dr. Bowman’s comment below, I changed what I originally had above (strategies and tricks should only be for our brightest students, the other students are best served by learning algorithms) to the direct quote he provided.

on May 2, 2014 at 6:11 pm |xiousgeonzChristina Anton seems to think that in math, anybody not already speaking the language is *not* allowed into the club. I’m guessing she’s on the hyperlexic end of the ‘verbal before visual’ learning spectrum and is imposing her brain on others.

There are places where just learning an algorithm suffices — I agree, that goes in the “think about it” phase. HOWEVER, it’s a little bit like “separate but equal.” What actually *happens* is that expectations are kept low for everybody not branded young as “bright and capable,” and teachers justify/ rationalize trashing concept development because “at least I’m giving them the algorithm.”

on May 2, 2014 at 9:15 pm |Vladimir TroitskyDear Mr Scammell,

Let me comment on a few points in your intersting story. Sometimes a seemingly minor change of words may considerably change the meaning.

1. “(Troitsky and Bowman) mentioned that they strongly believe kids can’t develop personal strategies until they have first mastered a traditional strategy”. I do not think we said so. I said that a person cannot create his/her own strategies until he/she learns the basics of a subject. However, in order to MASTER a subject, one indeed has to (at least try to) create his/her own strategies.

2. “Vladimir Troitsky said that kids who can’t multiply 7 x 8 or those who need 5 strategies to do it should go work at McDonalds.” I hope I did not say that. I believe I said that I expect first year university students to know 7 x 8 instead of knowing 5 strategies to compute it.

3. “Christina Anton said that concrete material and pictorial representations should be reserved only for special needs students.” I believe she strongly critisized the use of sticks, pieces of fabrick, tiles and similar objects to study mathematics in higher grades. She gave an example where such objects were used to study addition of polynomials. I agree with her on that. On the other hand, I think she was not against using pictorial representations. Pictorial representations are used in mathematics at all levels.

Finally, even though it is a minor issue, I indeed think that bar graphs are not mathematics (but I said that children need to know them anyway). If you think that they are mathematics, could you please specify what to part of mathematics they belong (Algebra, Analysis, Geometry, etc)? Could you find them in any university math course?

I hope I answered some of your concerns.

Sincerely,

Vladimir Troitsky

on May 3, 2014 at 7:37 am |A. Stokkexlousgeonz:

No, I can assure you that Cristina does not think that “in math, anybody not already speaking the language is not allowed into the club”. The entire point of this discussion – the very reason mathematicians feel the need to speak out – is that we want to make sure that more students are able to speak the language of mathematics. If we wanted to have our own exclusive club, that few people could get into, we’d just stay quiet and let things go the way they’re going.

on May 4, 2014 at 5:02 pmxiousgeonzCan you explain how that is true when — from what I read here — she advocates against using tools & strategies (e.g., manipulatives) that help students understand the language of mathematics? (IDTS)

on May 4, 2014 at 5:14 pmxiousgeonzI appreciate your clarification… it seems (and I did mean seems strongly, since I wasn’t there) that she clearly did belittle using manipulatives. Perhaps she saw them being used ineffectively (I certainly have). However, even your description essentially said we need to leave all that bridge-from-concrete-to-abstract stuff behind whether the student has crossed the bridge or not.

Many, many, many students have not crossed that bridge by middle, secondary school and college, but I have seen expert teachers use manipulatives judiciously and students making progress they thought impossible as a result. I read http://commons.carnegiefoundation.org/wp-content/uploads/2013/05/stigler_dev-math.pdf and was really glad somebody had gone in and dug up just how far behind college students are in math concepts.

In my experience, simply speaking the language of math and stating that we want students to understand it do not, in fact, teach that understanding. Rather, it provides a “Mathew Effect” — those who speak it get further ahead of the ones trying to memorize the algorithms and survive.

Could you tell me what you do provide to students who don’t understand the concepts behind that math language, so that they can do more than try to make their equations look about the same as the sample problems and hope it’s enough to make a passing grade? I’m always interested in ideas.

on May 2, 2014 at 10:13 pm |John ScammellDr. Troitsky, Thank you for your reply. I hope math professors and math educators in Alberta can work together on this in a respectful manner. I looked back at my notes to check what I had written down.

1. When talking about the Grade 1 outcome about adding to 18 in their own ways, you said that they needed to know how to do it first, and then they could create. I took that to mean they needed a traditional algorithm before coming up with different personal strategies. If I misinterpreted you, I apologize.

2. Your exact quote was, and trust me, I wrote it down correctly and checked with the people around me to make sure I had it right was “If kids can’t multiply 7 times 8, or if they need 5 strategies to do it, they should go working at McDonald’s”

3. Again, I was careful to make sure I captured it correctly. She said, “Concretely, pictorially and symbolically is a terrible problem in this curriculum. This stuff is only appropriate for special needs students.”

I honestly don’t know how to address your last question.

on May 2, 2014 at 10:39 pm |Cornelia BicaHi John,

If we had the present Singapore math curriculum, this symposium at the AMD would not have taken place, and Wise Math would not have formed. Singapore modernized (for instance, they encourage various pedagogies and make careful use of technology) but they stayed strong in both basics and problem solving. Of course, they also allocate the resources necessary for success (more math hours than us, students receive timely help, teachers benefit of more prep time, …). The language they use in their math curriculum would also have pleased Vladimir

In regards to what Cristina said about concrete manipulatives, can you please explain why is it necessary for the students to get tested on the use of such methods in solving simple exercises such as multiplying 1.2×3.7 and (x+5)(2x+3)? Can’t you agree that by grade 7 many students get distributivity without concrete manipulatives? My son told me that most kids in his class completely understood what they were doing but they had to use these tools, which they perceived as a waste of time (Imagine 12 year old boys having to draw and shade in squares). The point is maybe more philosophical. In general mathematicians like to imagine abstract things concretely (such as matrices or functions as vectors let’s say), but we also want to be able to generalize and you can’t do that if you have always been taught in school that math is always concrete. Try to use a fabric or whatever for the Dirac distribution, or to express the idea that there are as many real numbers in the interval (0,1) as in the set of all real numbers. Maybe it’s even possible to do it; but like Cristina, and really like many children these days, I would have been bored to death if I had to listen to the teacher (or participate in a group discussion) about drawing and coloring squares on a test or homework. Is there a point in grade 7-12 math when we introduce a math concept abstractly, as an exercise of the mind, and then tie it to the concrete? Can we at least not push the manipulatives on everyone?

Sincerely,

Cornelia

on May 2, 2014 at 11:40 pm |John ScammellCornelia, I have a post on concrete and pictorial representations all ready to go. In my mind, though, it logically needs to follow one about the McDonald’s comment. I won’t post that one yet. It’s possible I misheard Dr. Troitsky. I think the session was recorded. I want to get it right.

When I publish that post, I think it will address what you are asking about above. Give me a day or two.

on May 5, 2014 at 3:19 pm |Yolanda ChangI just wanted to chime in, I am currently teaching a Grade 11 20-2 math class and my students often make errors when doing the distributive law correctly such as (x+5)(2x+3). They understood multiplying binomials much better when I showed them that it was simply a calculation of area by using a grid, which left us with 4 sections (4 terms) that could be simplified into a trinomial.

What may be simple for your son may not be simple for a vast number of our students. Yes, some students may be completely comfortable with using completely symbolic means to multiply binomials. But it may also be possible for them to be comfortable with the operation of multiplying without understanding the meaning behind multiplication.

For example, some students get lost when faced with (3x+2y-7)(3x+3) because the “FOIL” acronym that they learned falls apart.

Students are also more likely to grasp concepts if they tie to real life (concrete) situations. For example, I have students who have difficulty adding and subtracting integers (such as -2+3) because they forget the “rules” for these operations (or get them confused with multiplication). But they have a MUCH easier time of doing the calculation if I phrase the question as “If it’s -2 degrees outside and it gets 3 degrees warmer, how warm is it?”

on May 5, 2014 at 8:37 pmJohn ScammellI agree, Yolanda. The 10 frames they start with in Elementary school lead nicely to algebra tiles, which really do help students bridge to symbolic algebra.

on May 6, 2014 at 6:45 amR. CraigenHello Yolanda. One problem with physical models in algebra is that they are not explanatory — or rather that the explanation can become more complex than the thing they are explaining. Consider, for example, the use of algebra tiles to “explain” (x-2)(x-3). On the one hand using FOIL we have four terms, three of which involve negative signs (-2)x, x(-3) and (-2)(-3), each of which has a different “cognitive” explanation, but all of which should be established knowledge by entry into algebra, which the student should invoke without extra effort at this point and arrive at the answer.

In the algebra tiles model one obtains a “regular” large square tile representing x^2 and rectangle tiles in the alternative colour representing (-2)x and x(-3). In the visual/concrete metaphor these are “rectangles”, one of whose sides is a negative number … okay … we can think of these as “take-away” or “negative weight” tiles in our diagram. I get it. A little complex, but we’ll live with it.

Now we’re staring at the “regular” small rectangle tile representing (-2)(-3). The metaphor gives us a rectangle BOTH of whose sides are negative numbers. Which means … we … uh … take away twice … or maybe take away the taking-away, which is “giving back” … uh … I’m still trying to picture that rectangle …

This is not impossible. But it is a cognitive stretch for beginning learners. It also adds enormously to the complexity of what is being learned over the original objective. It requires far more mental gymnastics than the original task of multiplying monomials in which a symbol (x) represents an unknown number.

That’s a problem. But it is not THE problem. The key problem with over-reliance on visual/concrete models is that it violates the mission of school mathematics, which is to train minds into the processes and modes of reasoning appropriate for mathematics.

Mathematics is intrinsically abstract. That is the whole point of the subject. Abstraction is a uniquely human tool which enables us to transcend the limitations of intuition and visual and concrete models. The whole point is to assist the child to achieve “escape velocity” from the gravitational attraction of the visual/concrete and intuitive.

Some people call intuition “horse sense” for the simple reason that horses (like most other higher vertebrates) have it. This is a purely animal capacity that humans share. It is closely aligned with our purely physical senses and enable us to interact with concrete and visual models — whence our intuition arises.

It is on the level of abstraction that we are uniquely human. Mathematics is that discipline that empowers humans with our unique capacity for thought which transcends that of the beasts: the power of abstraction.

Early algebra is a key step in that process, the point at which students learn the critical skill of working with named numbers — in the ABSTRACT, in which the names stand as proxy for the numbers themselves. The same parts of the brain are involved in both the abstract manipulations of algebra and the manipulations of symbols in language for speech, cognition, reading and writing. The learning of algebra is the technical counterpart of learning to speak, understand, read and write. It is fundamental to what it means to be educated as an adult human being.

Now intuition and the base senses are important, and function as useful tools in leveraging abstract thought. These are the scaffolds, the launching pads, upon which we construct abstract understanding.

So it is helpful, when first learning concepts, to draw, to manipulate objects, and so on, and to do so in close tandem with the abstract models upon which the child will later depend in their mathematical journey. For at some point they must LEAVE the launching pad and take flight.

In grades 1 and 2 children leverage their understanding of number … symbolic expressions like 12, 33, and 6+7 … with blocks, bags of buttons and marks on a page. But if students are still doing arithmetic in this way in later grades, that is not a triumph of pedagogy — that is a failure to launch.

Where these models are helpful in leveraging students TOWARD the abstract processes and tools of mathematics, they are proper and useful elements of the math class. But we must NEVER regard them as the goal. And we must avoid, at all costs, training students to depend on them or teach them that these are equivalent substitutes for the abstract tools. They are not. They are toy versions of the real thing, training wheels whose purpose is purely to assist learning.

When I saw your example (3x+2y-7)(3x+3) it immediately went through my mind “Gosh, I hope she’s not providing students with multicoloured tiles for this!” If you find it absolutely necessary to use visual/concrete metaphors, then a 3×2 rectangle would be appropriate, but with labels 3x, 2y etc and products sitting inside the individual regions, (3x)(3x), (2y)(3x), (-7)(3x) and so on. Not as discrete blocks whose identities and physical properties represent complex ideas that distract from the simplicity of the algebra.

Of course the FOIL rule does not (directly) apply, and if students are trying to do so, this should be a flag for the teacher that something is missing in how they are being taught. FOIL is a rule for multiplying binomials. So the first thought a student should have when faced with this product should be “Hmmm, FOIL doesn’t apply because this involves a trinomial”. It is important, when teaching rules, to reinforce the domain of applicability. Failure of students to learn where a rule applies leads to common forehead-bangers like (a+b)^2=a^2+b^2 .

For the “fill-in rectangle” approach for multiplying this particular product I would recommend AVOIDING any emphasis on the area model from elementary school, because it is unhelpful here. The rectangle is useful primarily as an organizer for individual terms. It should be used until the general rule becomes obvious: Multiply all possible combinations of terms and then add. I would also use it to reinforce the two basic ways to systematically ensure that all terms appear: row-by-row (i.e., distribute the second factor over the first) and column-by-column (distribute the first factor over the second).

But in either case note that the rectangle is only a template for helping students establish a process for working purely symbolically. It is training wheels, and the objective should be to move away from it when its job is complete.

on May 2, 2014 at 11:17 pm |John BowmanTo set the record straight:

I was in the audience and what Professor Troitsky actually said is that if

we refuse to teach students what 7 x 8 is without the use of 5 personal

strategies, the only option that will be available to them will be to go

and work at McDonalds.

It’s surely a deliberate exaggeration, but hopefully now you understand his

point.

As you have probably gathered, this issue deeply concerns us as educators:

students should not be denied the opportunity to achieve their potential

just because of a desire to test some grand educational hypothesis…

What I actually said about strategies is:

“Strategies and tricks are interesting for the brightest students.

However, the average student (including future professionals) will be

better served by learning the time-tested algorithms for arithmetic

computation that we learned as children and continue to use in our daily lives.”

You can verify the above quote from my posted talk:

http://www.math.ualberta.ca/~bowman/talks/amd2014.pdf

As a parent, I know first-hand that the expectation that elementary-school

students develop their own methods, without first being taught at least one

reliable method to fall back upon, is leading to unnecessary frustration

and confusion. Just talk to the rising number of parents who have been

forced to seek after-school tutoring for their kids. Once students have

been taught arithmetic, it is fine to explore tricks that work in special

cases. However, please be careful to avoid confusing students in doing so.

— John Bowman

on May 2, 2014 at 11:57 pm |John ScammellDr. Bowman, Thanks for commenting here. Thank you, also, for your talk. I very much appreciated that you incorporated student voice. That’s who all this is about. I liked your talk a lot, but I’m probably biased here because my work is predominantly in high schools.

Thanks for providing me with the precise quote about strategies and tricks. I originally paraphrased and shortened from what I had in my notes. I have inserted your preferred quote in the post above. I was sincere in my post when I said I am considering this one and trying to reconcile it with what I know from my experience. I’m giving it a lot of thought.

I did not deliberately exaggerate what Dr. Troitsky said. I hope you are right and that I misheard. I am trying to find out.

I know that some frustration exists among elementary school parents. I presented on this topic at an elementary school parent night a couple weeks ago, and there were some frustrated parents in attendance. There were also a number of supportive parents in attendance. There seems to be some confusion about the intention of these strategies. I’m seeing different things happening in different schools. Should teachers show students a whole bunch of strategies and let the students choose which one works best for them individually? Should students develop their own strategies, and the teachers be prepared to accept that different students may do things differently? Should teachers insist that students all do things one way? Should students be asked to show multiple strategies on tests, or is one way that works for them sufficient? These are the things teachers are wrestling with right now, and as you said (I’m paraphrasing, here), we need strong leadership from ministry to clarify their intent.

I’m glad I saw your talk. We are in the same city. I work in high schools across the province. If you’d like to talk further about some of the things I’m seeing in my travels, I’d be happy to chat again.

on May 3, 2014 at 12:54 am |R. CraigenHi John, this is not my discussion, I’m an interested bystander, but I had to comment on your statement,

“There seems to be some confusion about the intention of these strategies.”

Perhaps there is some truth to this (I think I understand quite well what the intention is, and I think that they are misguided even on that basis alone). But, more to the point, regardless of the intention of these strategies … they inevitably lead to confusion.

on May 3, 2014 at 7:32 am |A. StokkeHi John,

I was also in the audience and would like to address the McDonald’s comment. I am not sure what wording Vladimir used exactly but it was clear to me that what he meant by that comment was that students who are unable to do basic math will have limited career options in the future.

I agree with what Cornelia said above about manipulatives and Cristina is correct that there is an overreliance on concrete materials (even in Grades 7-9, which were the grade levels that Cristina was discussing in her talk). For K-6, I think it’s a good idea to use some manipulatives to explain how certain operations work (for instance, you might explain regrouping using base 10 blocks). I use base 10 blocks myself when explaining some new concepts to young kids but I make sure they move past this and onto the symbolic. To be clear, I think that manipulatives are useful for giving explanations in K-6 but using concrete materials for doing mathematics should certainly not be the goal. The goal should be to move to symbolic mathematics. I have been surprised to find that using concrete materials for doing simple arithmetic is actually treated as a reliable METHOD to do arithmetic in some schools and in some teaching resources I’ve looked at. This is regressive! Think: Roman numerals. The entire point of mathematics is to move past pictorial and concrete materials and onto symbolic mathematics, which makes mathematical operations easier to deal with. Students need to move beyond these manipulatives but I think many are getting stuck at this manipulative stage that’s overemphasized in many schools.

Anna Stokke

on May 4, 2014 at 3:32 pm |Cornelia BicaHi John,

There are problems with the math curriculum and the sooner the minister, Alberta Ed, and you acknowledge this fact the better.

1. First, it’s divisive, and this is ethically wrong. It’s creating friction (to say the least) between schools and parents and distrust between parents and their children. Times have changed and while parents have always complained to a degree, this new curriculum managed to give this a boost in the wrong direction. When 14000 parents sign a petition and they can’t help their children with their math (even grade 3 math), then something is wrong with the curriculum, not with the parents. You can’t claim that the math it’s so much better when you need information sessions for parents to be able to help their children with math. The parents signed the petition because of their children; if the children were getting the math and were happy and achieving in math, then the petition would not have started. (This petition didn’t start because of me complaining of lagging standards). I could go on and on but I’m sure you get my point. It’s time to acknowledge that maybe some things are wrong, or wrongly implemented, and start a dialogue. Alberta Ed is still denying that there is a problem.

2. The new math curriculum is lagging too much behind other countries; meanwhile, the reduction in content hasn’t been replaced by awesome problem solving skills.

3. If there was a problem with the old curriculum, as far as I could tell from my position at NAIT, it wasn’t with basic arithmetic. It was with fractions, order of operations, equations with fractions, radicals, exponents, understanding of logs, and in general seeing connections between different math ideas and reluctance to persevere in the face of a new problem. Most students were ok though, and the better they had been in mastering procedures in hs, the better they were at tackling new ideas and problems in college. I fail to see how introducing fractions arithemtic so much later and doing less in polynomials and logs will change things for the better. Considering 10 different cases for the linear equation doesn’t mean more depth, it means lack of zooming out. I’ll wait and see how this new curriculum will affect what we do at Nait, but meanwhile I’m speaking out because of my children. I have first hand experience with these strategies and I have issues with the way this has been handled. There is lots to talk about.

The prejudice against mathematicians has to stop. We’re not condescending, and the reason most mathematicians are not speaking their mind is because of the backlash they receive if they do. Also, stop focusing on the occasional mistakes that we do, and start focusing on the points we make.

Thank you,

Cornelia

on May 4, 2014 at 4:18 pm |John ScammellCornelia, I have no prejudice against mathematicians (see my newest post). I am doing my best on this series of blog posts to use respectful tones and avoid logical fallacies (Thanks, Indy). I have enjoyed my interactions with you, and I do not find you in the least bit condescending. I can’t say the same of all of your colleagues, though. I am only addressing the “occasional mistakes” first, because I think they are indicative of attitudes (on both sides) that are hindering this discussion. Shortly, I’ll address (with appropriate research) the points in this post that I indicate that I disagree with.

It’s interesting to me that you feel that mathematicians are afraid to speak out for fear of backlash. I have yet to see a mathematics professor blasted in the media for speaking out against the curriculum. The only backlash I see is against math educators who speak out in support of it. I’m terrified to hit “post” when I write blogs on this topic. I hold my breath every time I go to the comments section to see if I’ve been attacked. Some people thrive on controversy and heated exchange. I’m not one of them. I’m just a guy who likes math and likes kids and wants to go quietly back to my job. The reason I didn’t go for coffee with you a few weeks ago when you suggested it was that I was trying to do just that. I wanted to let other people sort this out while I quietly did my job. I intended to stay completely out of it. Going to AMD has drawn me back in. It’s probably not good for my health.

on May 4, 2014 at 7:08 pm |suevanhattumI do not have a phd in math, just a masters, so you may not consider me a mathematician, but I do. And I didn’t *really* know 7×8 until I started teaching. I had to figure it out (using a very idiosyncratic strategy) each time, which students never realized, because it took less than ten seconds. That was the only “multiplication fact” that I didn’t have in my memory. I tell students this, because I think it’s important for them to understand that math is not about memorization.

You may also know the story of Ernst Kummer (http://mathforum.org/kb/message.jspa?messageID=764296, if you don’t). I always saw it as illustrating that real math is not about whether you know details like this.

on May 5, 2014 at 8:39 am |John ScammellThanks for your comment, Sue. I appreciate you sharing your story. I will read about Ernst.

I have been trying in these posts to refer to math professors, rather than mathematicians. There are days I’m a mathematician (at the amateur, non-paid, non-professional level). Is a guy in a beer league a hockey player, or is that term reserved for the NHL players? Sorry for the Canadian analogy.

on May 5, 2014 at 9:13 amR. CraigenHi John. No problem with such informal use of “mathematician” if the meaning is made clear. Where we have problems is people who claim that mantle in order to make authoritative proclamations about the subject as if they have expertise. We see plenty of this in the world of math consultants.

Sue has a BA and Master of Arts in Mathematics from U East Michigan. It is an education-oriented program and so has a different basic philosophy than we find in many of the Science-faculty Mathematics programs. Nevertheless the program requires study of appropriate material at the appropriate level and I have no problem with her saying she is a mathematician in the full sense on that basis.

A Master’s degree in math is a very respectable qualification in the subject, even though it does not rise to the same level as PhD. It is certainly enough of a credential for the purpose of a discussion like this. I certainly agree that “math is not about memorization”. And I empathize with her story about not memorizing 7×8; as one who never memorized a few of the intermediate trig identities I can personally attest to the presence of what I call “mental clutter” every time the need for these arises and I have to perform mental gymnastics to pull them out in the middle of a larger task, which obviously happens more often in my career than for others.

But memorization is easy and beneficial for later work because of the way it de-clutters working memory, freeing it up for more sophisticated and analytical work. Children’s brains are optimized for memory work. A good teacher understands this and takes advantage of the rapid acquisition phase while it lasts — because what is put there at an early age becomes a resource for life. It is harder to do so later on.

We want memory work done as early as possible precisely BECAUSE math is not about memory. It is about understanding. And understanding is impossible without a well-stocked context of skill and factual knowledge. Understanding does not exist in a vacuum. It arises in a substrate of raw material stored in long-term memory.

If you are really committed to seeing children understand math … as all mathematicians whom I know, including Sue, seem to be … then you ought to understand the importance of using long-term memory, which for all intents and purposes, to boost the capacity of short-term memory, which is famously capable of juggling approximately 7 abstract symbols at a time. Since it is short-term memory where “cognitive understanding” and analytic work must be carried out it is absolutely essential that we provide students early in their educations with ways to ensure that such trivial clutter stays out of the way when they “dive deep” into more advanced material later on.

on May 5, 2014 at 3:43 pm |xiousgeonzRE: your comments about thinking about traditional algorithms being necessary before student-invented ones. In my experience, *most* students are better served with the traditional algorithm (not necessarily because it’s best, but if it’s the ‘traditional’ one they’ll see it more often), but occasionally one will come along who is really designing their own algorithm from their own sense of the world. When (again, from the students I see) they try to make up their own, MISCONCEPTIONS RULE.

Most of the people who tell me (working with struggling students) they have “their own way” of doing things are terrified that they’ll be discovered for Not Knowing Anything. Of course, the longer they get away with that, the bigger the gap gets… “their own way” is often fascinating, based on making problems look the same as the sample and all sorts of misconceptions…

on May 5, 2014 at 9:55 pm |Vladimir TroitskyDear Mr Scammel,

You write “After Dr. Troitsky’s comment below, I looked back at my notes from his talk and found that he had also included patterns and shape and space in the list of things he found in the curriculum that are not mathematics.” I am sorry but I did not say so. Your interpretation of my words is not completely accurate. To put the record straigh, I have the slides of my talk. It is phrased very differently there, and the meaning is quite different.

Best regards,

Vladimir