Following my recent posts on questions to ask your child’s primary school teacher (here and here), I had a request to expand on my comments about the teaching of mathematics. There are a few issues surrounding maths that I believe parents should know about but before I go into that, I wish to make two points. Firstly, a fundamentally misconceived maths program taught by dedicated, evaluative teachers will always be better than one that meets the highest standards of evidence but that is taught badly. In my school we have specialist mathematics teachers and I think this is far more important than the specific details of the program. Secondly, the intention of this post is to inform parents and not to have a go at primary school teachers. Some teachers were offended at my comment that time-tables songs were not the best way to memorise tables. They thought I was suggesting that this is what many primary school teachers do. No – I was just setting up two contrasting alternatives in order to explain my point.
1. Discovery learning
Discovery learning is ineffective and most people tend to recognise this. So you don’t see many schools advertising their programs as discovery learning (apart from in AITSL’s illustrations of their teaching standards, bizarrely). Yet if it looks like a duck and quacks like a duck then it probably is a duck. And there are two powerful fallacies that drive people towards discovery learning. The first is the idea that we understand something better if we discover it for ourselves. We don’t. Secondly, we tend to assume that by asking students to emulate the behaviour of experts then our students will themselves become experts. Experts in maths are research mathematicians who make new discoveries so we should get our students doing that. Yet this is also fallacious thinking.
In primary maths, discovery learning takes on the form of ‘multiple’, ‘alternative’ or ‘invented’ strategies. Students are intended to make-up their own ways of solving problems and to solve a single problem in several different ways. Explicitly teaching a standard approach, such as the standard algorithm for addition, is discouraged. Of course, many students don’t discover much and so they pick-up these strategies from others or are led toward them by the teacher.
Standard Algorithm for Addition
2. Big to little or little to big?
The kinds of alternative strategies that the students ‘discover’ are generally variations of strategies that we might use for mental arithmetic. Imagine I wanted to add 25 and 49. I would probably first add the 20 and the 40 to make 60. Then I can add the 5 and the 9 to get 14. Sometimes, little sub-moves will be encouraged as part of this e.g. take 1 from the 5 and add it to the 9 to get another 10 so that we have 70, then add the remaining 8.
Notice how this proceeds from big to little. We add the tens first and then the units. But when we add the units we find that we have yet another ten so we have to loop back and add this to the tens that we already had. This is inefficient when we get to larger numbers and is the reason why the standard algorithms generally start with the units first, then tens and so on. Indeed, students who use the standard approach seem to have more success, particular with larger and more complex calculations.
The objection to standard algorithms seems to be that kids can learn them as a process without understanding how they work. Presumably, they have to understand procedures they’ve invented themselves? This may be true if they really have invented them but I suspect such genuine invention is rare, with most children latching on to the ideas of others. In this case, these alternative procedures could also be replicated as a process without understanding.
In my view, students should be taught the standard approach and this approach should be explained to them. This requires the teachers to also understand how these processes work.
3. Words and pictures or actual maths?
Given that alternative strategies are meant to be ad hoc and contingent, there is no formal way for expressing them. You may see it done with pictures or even in words. Contrast this with the standard algorithms – their universality means that they follow a tightly defined set of notation. One way that alternative strategies may therefore be promoted is by insisting that students ‘explain their reasoning’ or draw diagrams when answering questions on homework or assessments. This is basically a way of marginalising the standard algorithms.
For instance, imagine the following question:
“A lottery syndicate of 13 people wins a total of $3 250 000. If the money is shared equally then how much would each member receive?”
A simple use of the long division algorithm is sufficient to explain what the student is doing, why they are doing it and to determine the right answer. If the student has gone wrong then the error will be easy to find. An insistence on words or pictures would be redundant unless you wish to privilege alternative strategies.
4. Maths anxiety and motivation
Maths anxiety is real. Some people struggle with maths – perhaps because they were not taught very well – and develop a fear of maths tests and maths more generally. It is complex and the chain of cause and effect is not entirely clear. Evidence does seem to point to timed tests as being associated with anxiety but perhaps better test preparation and framing would mitigate this. However, as well as advising us against timed tests, a whole raft of things that look and smell a lot like alternative strategies and related ideas such as the use of ‘authentic’ problems are proposed as possible solutions.
Authentic, real-world problem are considered good because the idea is that they will motivate children and so the children will learn more. In fact, a lot of discussion centres around motivating and engaging students. I am sceptical that many of the activities that are suggested as motivational are actually motivating for students and evidence suggests that motivation works the other way around. Maths achievement predicts motivation but motivation does not predict achievement. In other words, teach them maths, increase their competence and then they will start to feel more motivated about maths.
5. Cognitive load
Finally, it is worth mentioning that a lot of fashionable strategies are at odds with what we know about human cognition. Children should know their maths facts because that means that they don’t have to work out 5 x 8 whilst attending to other aspects of a complex problem. Those people who dismiss times-tables as ‘rote’ learning fail to take account of this. And so do those who propose big, messy, open-ended, real-world problems. Such problems have many facets and often contain information that is irrelevant to finding a solution. All of this needlessly increases cognitive load and makes learning less efficient.
Filling the pail 
I think you are missing the point of studying math. For the last 30 years or more people have been in possession of calculators, and have absolutely no need whatsoever for “the standard algorithms”. I wonder when you last used a standard algorithm. What is important, and what “playing with numbers” does for the kids is to help them get a clue as to what is going on, so that they can tell if the calculator answer is any good or not. And don’t get me going on “addition of fractions” !!!!!!!!!!!!!!!!!
Last Wednesday, I used the standard algorithm for multiplication. In the Year 12 VCE maths course there are two exam papers and one is calculator free. The questions invariably require a little number work including addition and subtraction of fractions and use of standard algorithms. Students also need to be able to do long division of polynomials. Which is hard if they don’t know long division.
But this is not the point.
If you want to understand how a calculator works then you are far better to understand the standard algorithms. Inefficient alternative strategies are not what calculators use. They crunch through the same procedure each time.
But this is not the point either.
Why do we define maths in terms of how ‘useful’ it is in some notional part of everyday life or for a future career? It’s as if we were to try to sell reading on the basis that if you learn to read then you’ll be able to understand letters from your electricity provider. It’s true, but it is far from being the only reason to learn to read. XKCD sums this up well:
https://xkcd.com/1050/
“Students also need to be able to do long division of polynomials. Which is hard if they don’t know long division.”
Do you have the slightest bit of research evidence to support that claim? Or is it, like much of what you advocate, simply your opinion?
I teach intermediate algebra to adults these days. I teach polynomial long division. On my view it is: 1) trivially easy compared with regular numerical long division, and 2) students can be weak in or ignorant of the latter and very successful with the former. And I bet if you stopped to think about it for 30 seconds, you could make the argument for the second claim. If you can’t figure it out, I’ll gladly return to elucidate.
My guess is that you’re so entrenched in your disdain for any approach to mathematics education (or education in any arena) that smacks of progressive thinking that you simply parrot a claim about why students “need” to learn regular long division (that so-frequently called-upon division of polynomials) that you deduce, absurdly, that the latter builds on the former.
It doesn’t. Only if you want to understand proofs of why algorithms are true or, perhaps more importantly, WHY the process you’re taught in the first place makes sense (harder but ultimately more satisfying than cranking through a hundred examples) is it important to master the first before studying the algebraic division.
Note that I make no assertion about the “usefulness” of learning either regular or polynomial long division. That’s the sort of thing that traditionalists enjoy. So don’t try to suggest I’ve commented on that and get away from my point so as to avoid dealing with it.
Yes. And when your nurse whips out a calculator when determining the equation for your medication dosage in the hospital, you better hope the batteries aren’t dead.
Opinions like these reflect why we now have up to 50% of elementary/primary aged school children attending tutoring centres to learn their math facts. And let’s see what methods these tutoring centres use…hmmm…lots of worksheets, the use of the 4 standard algorithms, and tons of daily practice. Why? Because they’re the most effective methods to ensure our kids have a firm grasp of their math facts!
Math hasn’t changed, and neither have kids. But the edubabble surrounding learning styles is straight out of cuckooland. If your child’s math homework looks goofy, and if the reasoning behind the convoluted work sounds like gibberish, it probably is. Steer clear of this madness, and enroll your kid in Kumon. Today. Before it’s too late.
Sure Tara Houle, that’s what happens at hospitals all the time. In your dreams.
It’s truly remarkable the horror scenarios educational conservatives can imagine and invent in order to scare children and gullible adults. Shall we call that “conservababble”? Note that it’s only “babble” when it’s progressive, child-friendly, and unlike the sort of things you underwent as a child. The untested, unresearched things done in your classrooms was, of course, mother’s milk and pure gold rolled into one.
Reblogged this on The Echo Chamber.
Thanks Greg, another great post. I’d like to see this one published in a primary teachers magazine…
Maybe students would be more moivted for maths if they could find a tutor like a phone sex line: http://boingboing.net/2015/11/10/math-tutoring-service-in-the-f.html
Yes, you hit five nails on the head. With Common Core going full tilt in the US, we are seeing that as throwing gasoline on the math reform fire that’s been raging for 20+ years.
Purveyors of the inquiry-based and student-centered math agenda, which has persisted for more than 20 years, argue traditionally taught math has never worked for the vast majority of the nation’s student population, and they say mental math is the key to “understanding,” as opposed to what they consider to be rote procedures.
The mantras of “students shall understand” and “explain” appear in the Common Core standards and are, according to Tom Loveless of the Brookings Institution, “dog whistles” that alert proponents of the inquiry-based and student-centered math agenda to see Common Core’s math standards as aligning with the discovery-based math agenda.
It is ironic how inquiry-based math approaches seem to spend more time showing students strategies they might discover on their own than on teaching the standard algorithms they almost certainly won’t learn on their own.
It’s important to note Common Core does not prohibit the teaching of the standard algorithms prior to the grade level in which they appear in the standards. This has been confirmed by the lead writers of the Common Core math standards, Jason Zimba and Bill McCallum. Zimba recommends the standard algorithm for addition and subtraction be taught sooner. Zimba would introduce the standard algorithm for addition late in 1st grade, with two-digit addends. He would then increase the complexity of its use in subsequent grades, continually providing practice toward fluency. The goal would be complete fluency by 4th grade. Zimba also would introduce the subtraction algorithm in 2nd grade and increase its complexity until 4th grade.
Despite admonitions from those in the know, such as Zimba, the word does not seem to have reached school districts, administrators, policymakers, publishers, or test makers. Nor do such recommendations appear in the Common Core background discussion on its website.
Morgan Polikoff, assistant professor at the University of Southern California’s Rossier School of Education, was quoted in a recent article in The Florida Times-Union, and his comments show the widespread disdain for teaching the standard algorithms.
“Common Core … put[s] the algorithm last and the conceptual understanding first,” Polikoff said. “They want you to understand how the tool works before they give you the tool. If you understand the concepts, then when you get to higher levels of math problems, you’ll be thinking like a mathematician and you can do it.”
Such thinking manifests itself in many schools and school districts where teachers have gone so far as to issue warnings to parents not to teach their children the standard method at home because it would interfere with the student’s learning. Many parents are choosing to ignore such warnings. More power to them.
And don’t get me going on “addition of fractions”
I teach the addition of algebraic fractions every year and it is a nightmare. Since they don’t know how to add numeric fractions, which is so much easier, they have no base on which to build doing it algebraically.
Another skill that lack of basic skills burns is factorising. How do you expect a kid to factorise 42x + 35 if they don’t actually know that 6 x 7 = 42?
Then again, most of the 12 year olds that I get have to be (re)taught how to multiply by 10. They just don’t do it enough a primary levels for it to really burn in. Reliant on the calculator, they just lose any will to do even simple stuff in their heads.
Reblogged this on Century 21 Teaching and commented:
There is a lot to think about in here. Once I had a (very) basic grasp of the relationships between working memory and long-term memory, I realised how important the rote-learning of some basic number facts is. I also find it interesting that Kindergarten students are regularly drilled with basic number facts under names such as ‘friends of ten/twenty’ (this seems to be the most common label that I have heard), yet the same strategy, applied to higher level (yet still basic) number facts such as the times tables is seemingly actively discouraged. It fails to make sense to my mind.
Skip counting seems to be in favor nowadays. Want to know 5 x 7. Start with 5, then 10, 15, 20 25, 30, 35, 40, Oh dear I lost count !!!!!!!
But they have to learn it. So why not the “tables”. Crazy.
Where/when were you in school, Howard? I learned “skip counting” before I ever started kindergarten, and we definitely did it in the primary grades. I don’t recall anyone losing count, but then I don’t recall anyone saying “Oh, dear” either. Must have been a tougher neighborhood than yours.
Skip counting is fun, the first couple of times. But like any other rote practice, it can get old pretty quickly. Listen to kids reciting times tables as a group. If they’ve been doing it for a while, their tones will tell you how they feel about it. BORING!
I’m curious: if you put the words “popular” and “nowadays” into a sentence critical of some educational practice, is that tantamount to proving that it’s a bad idea? I’m not a huge fan of rote for the sake of rote, and generally there are lots of more effective ways, in my experience as a student, teacher, and parent. But I never concluded that it sufficed to dismiss something simply on the grounds that it was new and/or fashionable. I thought it had to be shown to actually be a bad idea or an inferior one. Not sure I’ve seen you do either.
My point was that those who decry the learning of tables do something very similar (skip counting), which is rote learned and not so good for getting familiar with 6 x 9. I like skip counting.
We didn’t do skip counting in 1947-53.
Congratulations on having been educated in the Age of Enlightenment. Less than a decade later, the darkness fell on American math education once and for all.
Sorry, I missed the “where”. Drayton, Berks, England.
I really don’t understand people’s beef with algorithms (point 2). Algorithms work because of the beauty of the ‘mathematical structure’. So learning/knowing them is an excellent way to understand maths as well. I don’t really see how algorithms and understanding exclude each other. Actually, they reinforce each other.
On the post: one thing that needs unpicking I think is what actually an ‘expert’ is (point 1). If we (and others) compare experts with professional mathematicians then of course noone in school is an expert. I’m not sure if that is the case. One can easily be an expert in something within school and without being a maths professional. So although I agree that novices can benefit from. This also touches on (point 5): I’ve always found that an absolute understanding of ‘Cognitive Load’ is not very useful. Even CLT literature does not say that ‘less load is better’ (I know I know, you are not saying that but humour me). Dual coding says that images and text smartly put together might be more than just one of the media (so more is better), and also the role of schema forming must not be underestimated. I mention this in relation to ‘experts’ because I think that one characteristic of being expert is an optimal prior knowledge and schemas having been formed for the topic at hand.
In general I agree with your (point 3). Certainly when it comes to marking down explanations, and also the beauty of (correct) calculations. I wouldn’t downplay all images (but I think you don’t say that any way). Take a formula and its graph representation. I can see a clear role for multiple representations.
Well, then (point 4). I see you have completely embraced that *one*, limitations and all, article from your previous post. I think you are overly emphasising this one article, but you know that. From a ‘debate’ point of view in a world that emphasises motivation over achievement I can understand that, but from a ‘knowledge base’ perspective I think this is a bit too simplistic. But it’s your party.
Ask anyone you consider to be “properly educated” (surely free from edubabble and all the things Barry Garelick would warn us about) to explain the long division process: that is, explain why we do each step and why it makes sense logically/mathematically. Don’t be too shocked if you mostly get attempts to simply DO the steps (that is, describe them like a recipe) or blank stares, or even a few epiphanies when folks realize they have never been taught that or contemplated it once they mastered the algorithm (if they ever thought about it at all). That’s the marvel of traditional schooling and other failures of real education: you get “good” at something without the first clue as to what you’re doing or why. Or you don’t get good at it, and then it’s your fault for being dim.
I was once a reader for a boy during KS2 maths. He had had traditional column methods drummed into him for more than a year. When he had to subtract 7 from 12, as part of a problem solve, he wrote it as a column calculation, realised he couldn’t take 7 from 2 so borrowed a 10 and then had to take 7 from 12. He stared at it for ages, tried a bit of jiggery pokery with his fingers, stared a bit more, rubbed it all out, repeated all of the previous, and then moved on to the next question.
I’m American, Pat: I don’t know what KS2 is. How old was this boy?
That said, what, if anything, are you inferring from this experience?
Michael Paul Goldenberg, he was 10 / 11. KS2 is key stage 2 in UK – for UK grades three to six, I think US 2 – 5?
I was telling you an anecdote that I thought supported your point that it is not always helpful learn methods without understanding.
@Pat, thanks for the quick explanation. Very helpful indeed.
I’ve often wondered why teachers in the earliest grades tell their students things like “you can’t take a bigger number from a smaller number” without any attempt to limit that assertion. Depending on the specific claim (we soon hear them stating things, intended to be helpful, no doubt, like “addition/multiplication always makes things bigger,’ “subtraction always makes things smaller,” which aren’t even true within the whole numbers!), students will very soon encounter counterexamples to these claims. Is it any wonder that some become very confused about just what is absolutely true about arithmetic in particular and mathematics in general?
And therein lies some of what’s wrong with teaching without regard to some degree of conceptual understanding. Of course, I think many lower-grade teachers lack that understanding themselves. And the things I mention above are hardly a function of progressive education, “New Math,” “New-New Math,” the NCTM Standards,” the Common Core Standards,” or any other development in the last 60 years or so. I know that for a fact because I started school in 1955, and teachers were telling us those sorts of things unhesitatingly. I very much doubt that it occurred to them how quickly their pronouncements would be given the lie or that some of them were already factually incorrect within just the set of whole numbers.
We’d all be much better off if our self-appointed math warriors would stop trying to defend practices that were indefensible in 1955 and remain so now. And if they’d lay off trying to twist reality so as to try to fend off any changes to whichever fantasy of the Golden Age of math teaching and learning they hold dear.
Who is advocating ‘teaching without regard to some degree of conceptual understanding’?
@gregashman: No one is ADVOCATING that. But ask the real question: how many teachers are in fact teaching as if that’s their position? I’ve spent most of my professional time in real schools with real math teachers. Lots of teaching of procedures going on. Not a lot of teaching for understanding, teaching with understanding, teaching why along with what & how.
Your mileage may vary, Greg, but I’m not sure what it is you do or where or with whom. You’re adept at dismissing in what strikes me as a shallow and cavalier (and sickeningly familiar) manner anything outside a narrow band of ideas about and methods of math pedagogy. You’re good at defining things that you dislike in ways that make them seem facile, making it easy thereafter to dismiss those who advocate such things as fools or faddists. Reality isn’t quite as simple as your rhetoric would have it, and I’m really wondering what experience in your own classroom or those of others you bring to the table.
I suppose a short answer to your query would be: must be the mirror image of people like Straw Jo Boaler who advocate not teaching arithmetic facts.
http://www.telegraph.co.uk/education/educationopinion/12045884/Times-tables-are-not-how-you-teach-maths.html
Wow, you know how to cut and paste article links. And so? Please point to specific statements by Boaler that support your arguments. I’ve already gone to some length to refute your contentions about what she really said about memorizing. That has gone unchallenged here so far, though I assume at some point you or others will try to rescue your dead-in-the-water thesis. Is that link supposed be doing the task? Bad idea: you need to be able to point to actual statements and then show that they reasonably can be construed to mean what you think they mean. Thus far, I have yet to see you do that.
I await information about your experience in front of K-12 students and/or researching in the classrooms of other teachers.
I think the title is a quote: ‘Times tables are not how you teach maths.’
Say it is, for arguments sake. Obviously, you disagree.
But like your blog title, there’s an enormous amount of room for interpretation. You appear to be someone who thinks that it suffices to point to something for the implications to be obvious and clear to every reasonable person. If that’s your belief, I suggest you rethink that viewpoint.
I’ve been engaged in the Math Wars for a long time. I find that many players use very similar rhetorical tactics to try to achieve the same ends. After a quarter century or so, it gets a little dull trying to point out the holes, but I am usually game with someone new on the off-chance that s/he isn’t hopelessly entrenched in utterly rigid notions.
That said, it’s good to have SOME idea of who’s playing. I’m a veteran educator, having taught and worked in education professionally in various areas and subjects on and off since 1973. I’ve got graduate degrees in English and mathematics education and have taught or worked with students informally, as well as with teachers and teacher education students at nearly every level from 3rd grade through graduate school. I’ve done field supervision of elementary and secondary student teachers in math for the University of Michigan. I’ve been doing professional development work with mathematics teachers for over twenty years and have coached teachers in primary and secondary mathematics since 2004. And most of that work has been in high-needs, impoverished districts with mostly minority students and teachers.
You?
On Sun, Apr 10, 2016 at 10:49 PM, Filling the pail wrote:
> gregashman commented: “I think the title is a quote: ‘Times tables are not > how you teach maths.'” >
I am not interested in ad hominem or argument from authority.
No, at least not with people who answer back.
And I’m not interested in letting people who pretend they have relevant experience to start trashing the thinking and work of their betters with nothing more to bring to the game than their egos and emotional attachment to the what may have worked for them but doesn’t necessarily work well for others. One of the keys to being an effective classroom teacher in real schools is the ability to understand the thinking of kids who aren’t JUST like the teacher was (or believe s/he was) as a kid.
I don’t argue from authority, Greg. I argue from experience. Mine, that of many teachers with whom I’ve worked, and from kids I’ve taught or seen taught over a period of time. That’s not “authority,” but it’s not shooting from the hip, either. What’s your excuse?
Oh, and Greg, you’re making the same error in quoting that headline as if it’s a direct quotation from Jo Boaler, as you made in your attempted takedown of her ideas about memorization.
Here’s what the article quotes her as saying: “Research has pinpointed the onset of ‘maths anxiety’ around the age of eight,” she explains, “when they start doing times tables tests. They are all about speed and memory. If someone isn’t fast at doing them, they get the idea they aren’t good at maths and they lose confidence.”
That’s a very clear contention, but it isn’t opposing the learning of math facts. If you think otherwise, I’d like you to try to convince a neutral reader. Right now, I think you’re choosing to distort her words to beg the question towards a false conclusion.
Is that your usual approach? Take it from a veteran: it works with those already in your camp and those who aren’t interested in thinking an angstrom past where you’re leading them by the nose. Anyone who actually is interested in the truth or brings any skepticism whatsoever to the table might just not be terribly impressed. And so go the Math Wars.