What does a teacher do for the smart and very smart kids in her class? It is a tricky question and, for a number of reasons, there is probably no single good answer. We must be able to do better, however, than the standard and astonishingly bad answer of simply pretending that these smart and very smart kids do not exist.
There is a lot to untangle here. To begin, when I write “the smart and very smart kids” I am also including, as appropriate, the industrious and very industrious kids; by whatever combination of nature and nurture, a classroom will typically contain a non-trivial number of kids who require less instruction and will be quickly ready for and will desire more meat, and it does not matter who is why. Secondly, I am distinguishing the smart and very smart kids from the advanced and very advanced kids; there will be plenty of overlap, of course, but, thanks to ignorant parents and idiotic tutors, the second group contains way too many kids who have hurtled along and for whom the best and good advice is to shut up and slow down. Thirdly, and as an important counter to the second point, any sensible discussion must deal with the reality of Australia’s puddle-deep and snail-slow curriculum; a kid, even a not-all-that-smart kid, can be “advanced” simply by having covered the standard mathematics to a decent depth.
It is not remotely a new problem for a teacher to have to deal with a broad range of abilities and backgrounds but it is now a mess as it has never been before. There was once a general and justified trust in the curriculum to offer a broadly satisfactory foundation for each next step and for this foundation to be delivered in decently functioning classrooms. This trust is gone, having been pissed away by decades of idiotic education academics and idiotic education bureaucrats. It is now every man for himself, leaving the poor beleaguered teacher to deal with her twenty-odd men running every which way.
A very important semi-solution to the problem is streaming, to have kids grouped in mathematics classes according to ability and/or prior success.1 We’ll leave for another time the genuine but weak objections to streaming.2 It suffices to note that streaming is not always possible or appropriate,3 and, even when implemented, streaming can at most mitigate the problem we’re considering here, of what to do with outliers in a given class.4 Similarly, the semi-demi-solution of “extension classes”, for which stronger students are withdrawn from their regular mathematics class for a period-ish per week-ish, is relevant and important but is also distracting from the main point here; this is particularly true since extension is typically voluntary, with extension students still having their regular classwork/homework to do and their being assessed in the exact same manner as non-extension students.
The problem we are concerned with here, and the problem that would remain even given streaming and extension and so forth, is how to cater to the different students within a given classroom. There is a standard answer to this, of course: differentiation. This standard answer is simply an appeal to the Gods and is as fruitful as any such appeal. The aitsl Standards for Teachers gives as one of its sub-standards,5
1.5 Differentiate teaching to meet the specific learning needs of students across the full range of abilities
Well, yeah. But the question is how to cater to the many different students in an achievable and sustainable manner, without the teacher going nuts after a week or two, and which still amounts to teaching. The answer, at least for the multi-differentiating seemingly envisaged, is that it cannot be done.6 As Tony Gardiner noted in an excellent comment,
Extreme differentiation (believing each student has to follow their own path) makes a kind of sense in the context of home-schooling; but not with institutions. The idea seems to me to be incompatible with the idea of “schools” and “teaching” – period. So once one accepts educational institutions, one has to find some way of keeping groups together – while modifying individual diets slightly. Hence we need a curriculum, and a mode of teaching that reflects this.
The class is the class, the teacher is the teacher, and if there is to be teaching then this one and only teacher must be spending a decent fraction of the class time dealing with the class as a whole, on the same page or very close. To the extent possible, one works to keep the group together and it is madness to ignore this reality.
Accepting this reality, however, does not negate the other realities, of the smart and very smart kids, and the problem of what to do for them. How does one keep the class together while also giving meaning and substance to “modifying individual diets slightly”? Gardiner had an answer for this as well, or at least the solid foundation for answers.
As regular readers may recall, Tony Gardiner wrote Mathsteasers, a series of extension books for primary and lower secondary mathematics. These excellent books, written as companions to the English Maths – No Problem! primary textbook series, comprise a range of beautiful and carefully chosen problems arranged by topic. The particular interest here is that Tony’s books also contain an excellent appendix essay, explaining the underlying philosophy of MNP and consequently of Mathsteasers.
Gardiner emphasises the Concrete-Pictorial-Abstract progression underlying MNP‘s approach to school mathematics:
This transition (from the external “concrete” world, to an internal, abstract “mental universe”) is never done-and-dusted. Even when learners seem confident with abstract methods, we still need to check that the underlying mental abstractions are understood, and to make sure that pupils retain the crucial links between these abstractions and their underlying meaning.
The abstraction inherent in primary (and all) mathematics is hard, and students must establish the meaning, and the connections, underlying the concepts and techniques:
Basic techniques are the hardware — the visible building blocks — of elementary mathematics. They tend to be what gets listed in curricula, and what is tested in examinations. So it is important for pupils to learn to handle them with confidence. But there are two reasons why they need more than this.
• First, what gets tested … tend to be the easier, more predictable versions of these techniques; … what is generally needed for subsequent progression are certain harder, more elusive variations on these techniques.
• Second, if we restrict attention to mastering separate techniques … we risk neglecting the underlying logic that allows learners to organise these techniques into a coherent whole. … emphasising connections allows the subject to become more coherent as one progresses with the connections acting like “mortar”, that binds the separate building blocks together in the mind ….
Gardiner then explains the implications for “differentiation”:
Since the 1970s, there has been a tendency to think of elementary mathematics as a uniform “ladder”, up which all pupils have to climb, if at different speeds. ..
However, mathematics is more like a web than a ladder, with key ideas and methods linked together in complex ways. The power of this “web” depends both on mastery of each key idea, and on the number, and the robustness, of the connections between these key ideas and methods. …
Grasping such connections constitutes a significant challenge for all pupils! And those who rise to the challenge will generally be those who slowed down sufficiently to master the relevant variations in earlier material, rather than those who achieved superficial early success before rushing on ahead.
Slow down, slow down, slow down. Rushing the smart kids ahead is not simply a massive headache for the frazzled teacher, it is also a mess in the making for the kids. Their time is much better spent working on harder variations of the same material and on adding more strands to their web of understanding. This also maintains a common goal for the class:
ln short, effective education systems
• recognise that later progression depends on all pupils mastering current material in greater depth,
• and use whole-class teaching both to achieve this objective for all pupils, and to postpone for as long as possible the point at which classes begin to diverge.
That is what is to be done but how is it practically to be done? Obviously, it is reasonable and good to “differentiate” in the assigning of post-class exercises, and Mathsteasers was primarily written with this in mind.7 But that is “structured extension” for later, on top of what is to be offered in the classroom:
… standard classroom provision should
• develop ways of ensuring that pupils who grasp basic ideas quickly can routinely master the associated techniques in sufficient depth to support subsequent progression, and
• build up a repertoire of harder variations, which explore the multi-faceted aspects of each new idea or method.
The teacher must be offering more in the classroom than the basic mechanics for the smart and very smart kids, and for all kids. Can this be done? Yes, but it is not a science:
In England, after 30 years of rampant unthinking “differentiation” (with students being actively encouraged to gallop ahead: at one stage ~70% of students were taking the age 16 exam “early” – and either failing or achieving lousy grades, which they then tried to improve – with massively negative consequences) we had lost the art – and lost faith in the whole idea – of “whole-class teaching”.
Whole-class teaching is a difficult art. Yes, you keep the class together, but you do not keep the class together simply by drilling the lowest common denominator mechanics via pre-packaged lessons, by flipping fill-in-the-blanks PowerPoint slides. You will have to be cleverer, more nimble, less linear, much more weblike than that. Yes, the smart and very smart kids must learn to be patient, that at times they will be required to sit through five or ten minutes of (for them) very slow exposition. But it is has to be five or ten minutes, not forty-five minutes of sitting bored and silent, followed by some aimless “early finisher task” twaddle moments before the bell rings.
The Neo-Explicit-Teaching gang are, correctly and importantly, big on whole-class teaching. But if the N-E-T gang want to restore faith in the lost art of teaching they will have do to a hell of a lot more for the smart and very smart kids.8 As it stands, I cannot see that the N-E-T gang are offering the smart and very smart kids anything. I cannot see that they even care that are not offering anything.
Keeping a class together, teaching a class together, offering and expecting of them rich and coherent mathematics is a difficult but achievable art. We cannot be snowed by the blinkered ideologues, of all stripes, who act as if it were impossible.
1. Simon the Likeable insists that the proper term is “setting”, and that streaming is the separation of students entirely, in all subjects, for a given year. Simon the Likeable is almost always correct, which is why I have learned to ignore him, as I am doing here.
2. See this post for a response to some typical anti-streaming nonsense. For less typical anti-streaming nonsense, see many posts by Ashman, who loves to harp on the problem of “making decisions about what group a student is in and what they need”. The solution, apparently, is to not even bother trying and to just have all students receive “the same curriculum”.
3. Streaming in primary school, for example, seems a bad idea. On the other hand, it is foolish to pretend that some younger students are going to make it to harder mathematics when it is already clear that they have Buckley’s hope.
4. There is the polar companion problem, of course, of what to do with the weak and very weak students. This companion problem is semi-addressed and only semi-addressed in a similar manner. I am concerned in this post with the catering to strong students, but that is not to imply the companion problem is of less importance.
5. Pun intended.
6. At least in mathematics. One of the problems with the differentiation debate, if one is foolish enough to participate, is that the term “differentiation” covers such a variety of practices, with such widely varying degrees of differentiating, there is seldom one clear thing that people are debating. Ashman plays games with this, citing a study to make a rather slippery claim that “the evidence for differentiation is weak”. As usual, the “evidence” rhetoric seems pretty silly, even if I am agreeing with Ashman on the substantive point (which is better and more honestly arrived at by plain common sense).
7. The comparable, and much more comprehensive, resource for secondary mathematics is David and Gareth’s excellent Enrichment Mathematics. Plenty of exercises from both texts would be entirely suitable for a regular classroom setting.
8. My focus here is on the smart and very smart kids but, for the reasons Gardiner indicates, I think the N-E-T gang are shortchanging all students.
This is a vital question, Marty. Also the polar question. You are right, there are no easy answers.
I have grappled with this since I started Masters (M.Sc. — a specialised course-work plus minor-thesis for school teachers with education components — at Monash — 1975) and Ken Clements presented a unit on Giftedness. At the time this was an emerging topic for education research, although it is an issue that has been with us forever — the very able and the highly able. Then in 1976 I started working at Burwood State College training teachers, and I was soon also looking at the polar question, as well as the middle-of-the-road (or stream) question — how to teach most of the class most of the time all together.
No easy answers there, either.
One of the big questions is how can teachers know which students are really struggling and who are the advanced or potentially advanced students who already know, or learn fast and easily, whatever the mainstream are plodding through as fast as they can, but far more slowly.
The answer for that is “assessment”. And that does not just mean testing (or other observations) what students have learned from the last month or six months or year of mainstream classes. It needs “tests” that have questions (tasks, challenges) that start simple and easy and get progressively harder and more complicated. You see how far up through the “graded” questions any of the students can go, and then wherever that is, that is where the next lessons resume.
I call that kind of “test” a profile. And that kind of how-far-can-they-go “test” is diagnostic.
It is also pre-testing, before the next stage of teaching begins. And unfortunately most assessment in schools is post-testing, and you never know if the students who got high scores (“mastery”) on a post-test needed all the mainstream lessons, or even if they needed any. (Could they already handle this level of mathematical challenge? Could they spontaneously figure out how to handle the challenge without needing to be taught?)
One of the things that Ken Clements reported were cases and institutions (Johns Hopkins University in the USA, as I recall) where young teenagers studied undergraduate mathematics alongside older teenagers. (Jody Foster was in a film about that kind of accelerated mathematics program: “Little Man Tate”)
Ken also worked as a personal tutor for Terence Tao when he was discovered to be extremely able as a child.
Far more importantly, the history of mathematics has many examples of extremely able students — prodigies — who began university studies when they were young teens. The current age-barriers to starting university in Australia are fundamentally wrong.
We might look at how MUSIC (playing an instrument) is taught, in Australia. There are no age-limits. Individual students learn as fast as they and their teachers can go, and everybody thinks this is great, and out of that we get the stars, as well as a lot of very good but not-quite-stellar musicians, and others who move across to other studies (Medicine, Law, Mathematics, Physics — Einstein was a good violinist, as an example) where they excel, while retaining very high music skills just for their own enjoyment.
Thanks very much, John.
I’ll read it all carefully but I am much more sceptical about the acceleration business. I have seen a number of mid-teens and low-teens kids at university, and it seldom worked out well. Yes, there are Taos who must be catered for, but the overwhelming majority of very bright kids do not remotely need nor will benefit from such acceleration, either mathematically or personally.
You compare to the age-free approach to music but the reality is that music exams are a much more rigourous test of musical skills and musicianship than school tests are of anything. There are *tons* of kids who could perform “well” on school maths three years above their age level, but there is no way that such acceleration is preferable to doing much earlier work much more solidly.
Marty. You have written a very important essay which needs a wider audience than this blog. However, as in the case of Tony Gardiner, whom I greatly admire, it is likely to get lost in the thicket of “educationists” who have their own schtick and are incapable or unwilling to embrace the complexities of educating the young. As I university professor, I have only dealt with school students on class visits, math camps and contest preparation, and teachers through talks at mathematical meetings and the occasional course at university. My approach is generally to start with something accessible, but which has strands that involve more thinking. To give an example, in one grade 3/4 class I had students take the digits from 1 to 8, make up two numbers and substract the smaller from the larger. Various kids came to the blackboard and wrote out examples (kids like to make stuff up and/or show off). Then I asked if they could make the difference as small as possible. So now there is a target. What ensued was actually quite interesting. One kid came to the board frequently, but did not make any progress. Another kid put up something like 8642-7531, eventually someone arrived at the answer. I asked the kid who got it how he got there and he could not say. However, the kid beside him (who had not participated to this point) gave an explanation. When you think about it, there is quite a bit involved. For example, the kids seemed to take it for granted that each number should have four digits. Then how do you approach it — from the left digit and work in?
Kids on the whole are brighter than one might give them credit for. What you need to do is to find a way to make them pay attention and notice what is going on. Often they will rise to the occasion, and I have had teachers tell me afterwards that an insightful comment or answer came from a kid that they would not have expected it. Unfortunately, when you do this sort of stuff, something different may happen in each class and I think a lot of teachers are nervous about this aspect. As one said to me, “I don’t know where to take it.” Part of the answer has to come in the formation of teachers and giving them similar exercises and then talking about it from a pedagogical view afterwards.
My role as a university professor is to provide the raw materials. But it is the very devil to try to get teachers to try stuff out. One difficulty is the fear that they will get distracted from “delivering” the regular syllabus and cannot afford the time. I think that this acts against many students as well. I had a friend who was the math curator at the Ontario Science Centre, who left to teach as a high school when the inevitable 1970s budget cuts came in. I used to visit his school occasionally and the students were quite keen. Somewhat later, I found that he had moved to a different school. When I asked why, he said that the demographics of the neighbourhood had changed and that the school was now full of kids who parents obsessed about marks and “it just wasn’t fun anymore”.
If anyone is interested in some raw material, please visit my website https://www.math.utoronto.ca/barbeau . I give no guarantees and it is up to you what you take from it. But I think you can find something that can keep some kids in the class profitably engaged without compromising the fortunes of the less able or interested.
Thank you, Ed. In case it matters, I *hate* writing these posts. They are so much work and I know they won’t do a damn bit of difference. I do it only because I feel I have no choice. (God knows why I feel that.)
No one is listening, and people simply lurch from Cult A to Cult B. I still probably prefer Cult B but their obtuseness and their arrogance is making it a damn close call.
Personally I prefer Cult M 😉
I just wanted to echo Ed, I appreciated this post very much. Good read and on the money (as always)
This year, with my methods class, I’ve been running into an issue most likely created by this problem. I have kids who have had barely any challenges throughout school in maths, really struggling with methods content, a lot of the time because they lack fluency with algebra and other aspects that they should be stronger in. I wonder if because the testing is often done on the more basic, easy techniques, the students have appeared as highly capable compared to their peers, but were never properly challenged. Which leads me to finding them out of their depth in methods (at least, that’s what I have observed).
Well, at least Cult M has (intentional) jokes.
Let me offer a few suggestions.
* Convince the school that these students have “special needs” and deserve appropriate teaching. My first job at my last school was based on this idea. This was a government school and not wealthy in the least. I had small classes of above average students and pretty free reign on what to do in class, although I roughly followed the curriculum. Sadly, such a dream job could not last forever. Still, the students and I had a couple of good years.
* Get students involved in Australian Mathematics Competition and organise training for them based on past papers. There are papers from similar competitions held in many countries available on-line.
* Encourage such students to focus on other subjects where they might not have yet shown their talents.
Thank you, Terry. In response:
a) Some schools need little convincing, but the majority of schools will be listening to Sullivan and/or Ashman much, much more than Terry and Marty. They’re the experts, after all.
b) The AMC and other AMT offerings are great, but it is really off the point. Good mathematics should not be this added extra done at lunchtimes or whatever.
c) Huh?
Let me elaborate on my last suggestion to encourage such students to focus on other subjects where they might not have yet shown their talents. I have often met students who are very able in mathematics, but not so able in other subjects. So instead of fuelling their interest in mathematics, they could consider devoting more time to the other subjects.
I recall meeting a graduate who studied IT at university. He was very good at mathematics and computing. After graduating, he got an excellent job in IT with a major firm. I met him a few years later and asked him how his university studies had helped him. He said that he wished that he had chosen some elective subjects in humanities. This would have improved his writing skills. Every six months he has to meet with a manager to discuss his progress, and invariably he gets some negative feedback about his written expression.
I had a student in Year 9 who was very good at mathematics; best in the class. We had a chat one day and I told him that clearly he was the best in the class – but his handwriting was appalling. He needed to do something about it. He said “I know. My parents have bought me three books on handwriting.” Enough said!
My suggestion was that one might concentrate on those areas where we need improvement. If you are getting 95% in one subject and 45% in another, it might be easier to improve the latter by 5% than to improve the former by 5%.
BTW, I do dislike the comment that I have often heard. “I’m no good at mathematics: I am more of a creative person.”
OK, I see. Thanks, Terry. As usual, a very good suggestion.
Especially that 95% increased by 5% is? Whereas 45% increased by 5% is?
That’s as I became aware of % and p. p. as different concepts, about a decade ago , and I tend to include them in smarter classes since.
What is p. p. ?
Or may be pp. Percentage point. So if inflation increases from 2% to 3 % it is 1pp increase or 50% increase.
Ah. So a question of “percentage of *what*”.
Differentiating: how many levels of ability can you run in a class? The more time you spend on one, the less on other(s). I can not imagine leaving students to do their own work, without checking, monitoring.
Actually, in my school years, lessons were for theory, proof (I think no thing went without a proof – yes, hips of epsilons, deltas, thetas). Then presentation of an application. Exercises were the thing of homework. Hips of problems to solve/practice. Never was there a lesson of “quiet work”, ever. May be once – our teacher (our mathematical god) lost his voice. Mathematics extended to 1st year uni: months on limits, Cauchy’s definition, Weierstrass, Taylor’s series, de l’Hopital rule (each with a formal proof), just to mention a few. Calculus (serious) started towards the end of Y10. I consider myself blessed. Only felt lost in probability central limit theorem.
Plus extension, after hours, weekends sometimes. Very serious business.
But one day I could not remember it, classmates reminded me, our teacher’s private comment, reflection: “who is it for, 5% of students?”
Today’s practice, as a teacher: problemo and similart stuff , more than enough.
You can’t do ten “levels”, but you can easily do minor variations with not so minor implications. “What if that 4 were a 5?”