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Gifted Math Education: Teaching to Learn

January 6, 2008
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You do not really understand something unless you can explain it to your grandmother.

Albert Einstein

Should Gifted Math Students Tutor Others?

So let’s start the new year with a little controversy. ;-)

Many schools ask gifted kids to “tutor” some of the other kids in class at some time or other in their school careers (particularly during the K-8 years). Most parents of gifted kids are wholeheartedly opposed to it, for a number of good reasons. However, I propose that, properly handled, this can be beneficial for kids. Let me say right up front that tutoring should never be the only thing an advanced child gets to do in class. But gifted students have time to go beyond the basic curriculum, and in my opinion, it is quite reasonable for them to spend some of that extra time doing some tutoring, in addition to other types of enrichment such as investigating topics outside the normal curriculum, doing non-routine problem solving, etc.

Typical objections to this practice include:

  • I think my child deserves a full-year’s growth in the math every year.
  • My child doesn’t like tutoring.
  • My child isn’t good at explaining how to do things that are so easy for him.
  • If learning to explain math to others is so important, why isn’t it on the curriculum for everyone?
  • It’s not fair to the child being tutored to be taught by a child instead of a qualified teacher.

I’ll address each of these in turn.

I think my child deserves a full-year’s growth in the math every year.

I agree that every child deserves to have the opportunity to obtain a full-year’s growth in every subject, and this is certainly an area where gifted kids are often let down. However, I think it’s important to examine what a math education is intended to provide, and what counts as “growth”. Is the point of a good math education only to learn a pre-defined bag of tricks listed in the math curriculum for a certain grade (and to cover as many grades worth of material as a student can handle as quickly as possible)? Or is it to become a better mathematician (which may include topics and skills beyond the basic curriculum)? In my opinion, it is very narrow-minded to view advancing by at least one grade level on the basic curriculum as the only worthwhile type of growth for a gifted math student.

If a child’s mathematical thinking and mathematical communication skills improve as a result of tutoring another child, that may well be worth a lot more, in the long run, than mastering another year’s worth of arithmetic or algebra topics. A mathematician who proves an important theorem is nowhere if she can’t communicate the essence of her proof to her community. (I realize, of course that the type of explanation that is required in the latter case is much different than that of a student helping classmates, but the skills are related and I firmly believe that the ability to do the mathematical communication required to tutor math concepts one already understands is truly a crucial skill.) A year’s worth of growth does not necessarily have to be linear along curricular lines, and there are many reasons why it is often better if it isn’t.

Even if the student doesn’t go on to become a mathematician, there are times in almost any career where a person is called upon to explain and/or defend his ideas or plans to others who are less familiar with the subject matter. This very skill of explaining something to others who don’t understand it is an important life skill in almost every discipline.

My child doesn’t like tutoring.

So, I’m not so sympathetic to this one. Some kids don’t like fractions either. They still have to learn them. School isn’t about only getting to do what you like.

There is a reasonable argument that a young gifted child who loves math and hates tutoring shouldn’t have to spend all his math time tutoring. But as I said above, tutoring should never be the only thing a gifted kid gets to do. And having to do something one doesn’t like so much part of the time is not, in my opinion, unreasonable.

My child isn’t good at explaining how to do things that are so easy for him.

Isn’t school about learning to do things you aren’t already good at? It is reasonable, in my opinion, for one of a child’s goals in math class to be learning to explain the things he knows clearly, in ways that others can understand. Referring back to the Einstein quote at the top of my post, clearly I’m not the only one who sees value in learning this skill.

But… here’s where one of my caveats comes in. Too many teachers assume that because a child is good at math, he is necessarily good at teaching math. If a child isn’t naturally good at explaining the things they know, someone has to teach him how to do this. It’s not appropriate to expect a child to learn how to explain what he knows without giving him any help or guidance in achieving this goal.

If learning to explain math to others is so important, why isn’t it on the curriculum for everyone?

This may also be worded as, “why is my child required to learn to do something she doesn’t enjoy, when no one else is?” My answer to this is that the standard curriculum is pared down to a bare minimum, since students of all levels of ability have to complete it. For a child who can complete the basic curriculum in considerably less time, there is time to expand and include other valuable skills and topics. And one of those skills is the ability to communicate what you know to someone with a lower level of understanding. The fact that not everyone gets the opportunity to learn and practice this skill is a necessary effect of the fact that most students take the full time allotted to learn the more basic items in the curriculum. For those who complete the basics quickly, learning new skills and exploring topics outside the basic curriculum is a wonderful opportunity that is wasted when people insist on just moving the student as quickly as possible through the basic curriculum.

It’s not fair to the child being tutored to be taught by a child instead of a qualified teacher.

Indeed, this is an important consideration. It is of course always the teacher’s responsibility to ensure that each child has the material presented to him in a way that is understandable. It is not acceptable to put this responsibility on another student. If a teacher asks another student to spend some time explaining the material, the teacher must always monitor to see whether the student being tutored is indeed getting what he needs. This is closely related to the point about students not necessarily knowing how to explain things — the teacher must monitor and see that the “tutor” is getting the support she needs in learning how to explain things, and the “tutee” is getting the support he needs in terms of improving his understanding of the subject matter.

Theory versus Reality

“Well, that all sounds very nice, but that’s not the way it is in my child’s school!”

Indeed, in too many cases what really happens is that the teacher assigns the top kids to tutor the bottom kids, so that the teacher can teach to the middle, and not have to deal with the kids at the extremes. The advanced kids aren’t getting any advice or instruction as to how to help the struggling kids, so unless the advanced kids happen to have a natural knack for it, everyone gets frustrated.

So, as much as this post supports the notion of having gifted kids doing some tutoring, it does not give the green light for teachers to do it lightly, or as a convenience for themselves. There are several important things that I feel must be addressed in order to do this “right”, which do make more work for the teacher who wishes to employ this strategy:

  1. The student being asked to tutor is taught how to do so;
  2. The teacher takes full responsibility for the learning of the student being tutored, and checks in regularly to ensure that the tutoring relationship is a fruitful one; and
  3. The gifted student is not expected to do this all the time — much of the time he must be given other appropriate enrichment activities.

Finally, I’ll add one additional caveat, which is social. There can be social consequences of placing a student in the role of “teacher”. It is important for the teacher to handle this well. In many cases this can work out fine with the right attitude and guidance from the teacher, but it is one more thing that a teacher needs to be aware of and keep an eye on.

Related post:
Gifted Math Education: acceleration, enrichment, and the Calculus Trap

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18 Comments leave one →
  1. January 10, 2008 9:04 pm

    Peer tutoring is sometimes the best way to go.

    I usually try to hook one of my AP calculus students up with needy Math A kids. The kids end up becoming friendsand e-mail and phonefor help. The AP kids often come to me to get additional assignments for the kid they are helping and stress when a test is given.

    I’ve seen students improve their English skills through tutoring. Of course this is always voluntarily done. If the kids don’t click, we don’t carry out the program.

    In return for tutoring, my AP kids get wonderful letters of recommendation and tutoring themselves when they need it. It is a no lose situation for everyone.

  2. January 10, 2008 9:15 pm

    That sounds like a great arrangement, PO’d teacher.

  3. January 18, 2008 7:09 am

    While reading the end of this post I found myself nodding vigorously in agreement. In my opinion the most important thing you can do for a potential tutor is to teach them how to teach!

    When I was in high school I was asked to be a tutor to a less able student who was a couple of years younger than I was. I was keen to help and flattered to be asked so I wanted to do my best but I had NO IDEA what I was letting myself in for. Without being given any support or guidance I was introduced to my new student and left to get on with it. I had to develop my own learning materials – everything.

    He hated maths and had learning difficulties whereas I loved the subject and found that it came naturally to me. So how on earth was I supposed to know how to teach him? In a nutshell I failed – in fact it was worse than failure. I have always felt that this kid had his nose rubbed in it – rather than being given someone who had the skills to help him he was introduced to someone who simply could not understand why anyone could have trouble with the basics. I really wanted to help this guy but did not know how and he had been shown exactly how bad he was compared to someone who was good.

    It effect it was like an educational version of the Dilbert principal – because I was good at something I was ‘promoted’ to a level where I was completely incompetent.

    It took me years to gain enough confidence to try teaching again and it turns out that, in certain circumstances at least, I am not bad at it but that is only thanks to the input of many people along the way who taught me how to teach.

  4. January 18, 2008 9:27 am

    Hi Mike,

    I think that one of the problems is that for many teachers, teaching does come naturally. So it may not occur to them that it doesn’t for other very bright kids.

    It’s ironic, because they will be the first to argue that people who haven’t had the same kind of official teacher training as they have shouldn’t be allowed to compete against them for teaching jobs, but will put kids in a position to do it with no training and no support.

    I’m sorry you had such a bad experience! I was lucky in that my mother was a teacher, and was able to support me when I was doing tutoring, so that I actually did have a clue as to what to do! But that’s when I was in high school and officially tutoring. All those times in elementary when I was given another kid to help with something I had already mastered, I’m sure I flubbed it more times than most. Some things were so obvious to me that it was hard to see what it was about them that other people would find confusing or non-obvious. And I suspect that that is more the rule than the exception with very gifted math students tutoring with no support.

  5. January 18, 2008 11:41 am

    The best two a-level teachers I had (subjects were maths and physics) were (by their own admission) not naturals in their subjects. They had to struggle every step of the way to get their own A-Levels and degrees (but did get very good ones in the end) and so they understood exactly how difficult these subjects could be. They had all kinds of tricks and tactics that they passed onto us – tricks that I still use to this day.

  6. January 18, 2008 11:46 am

    That’s very interesting, Mike. I wonder why they each chose to become teachers in subjects that did not come naturally to them. Clearly it worked out very well, but I suspect that it wouldn’t occur to most people to do that.

  7. Jan permalink
    January 26, 2008 8:24 am

    I have been “peer”-tutoring since I was 11, either on request by those peers or on request by the teacher.

    I found the experience of peer-tutoring to be quite varying. On one hand, there is pride you are considered being gifted, there is anxiety over your teaching skills and the joy of transferring skills to other persons.

    However, when I was tutoring on request of the teacher, especially when I was in primary school (over here this runs to about 12 years old) there was the problem that I was already a bit on the “outside” of my class, and this peer tutoring totally made my name as “the nerd”. This was a problem associated to my primary school (many of my teachers though not actively promoted bullying, but where visibly condoning it), but it can become a problem even in the ideal case. You are placing some kid, very likely already someone who has problems fitting in, above and outside the group.

    I don’t think peer-tutoring is bad, but you should really watch the group dynamics around this. The level of difference of competence should not be a problem, I have tutored students of my age who were still struggling through high school, didn’t know how to compute percentages or multiply by 10, when I was doing differential geometry, and though you have to work hard, it is really enlightening to both student and tutor.

  8. March 12, 2008 8:05 am

    “School isn’t about only getting to do what you like.”

    Indeed! In fact, it’s mostly about having to do what you don’t like (often when you don’t want to), isn’t it?

    It would really help if schools became places that truly respected kids. If you want kids to learn, you’d do well to not deliberately make school someplace they don’t want to be. You’d do well to not make it like finishing their spinach. You’d do well to not make kids who already might like math (or whatever subject) begin to dislike it because you’re turning it into something else (compulsory tutoring).

    Students aren’t school employees. They don’t have an obligation to teach their peers. Why is “because I don’t want to” not good enough in this case? If kids want to tutor, fine. But to force them to? That’s just completely disrespectful and yet another good way to make kids dislike school and possibly the subject itself.

  9. March 12, 2008 9:22 am

    Even a school that respects students (my kids are fortunate to go to one that honestly does) makes them do things they don’t like some of the time. My kids don’t like to write. They still have to learn to do so, and practice doing it in various contexts. (One uses an AlphaSmart to make the physical task of writing easier, but he still doesn’t enjoy it.) There are things that are worth learning in life, and kids may not realize at the time that something is one of them. They key is to achieve a balance between the student’s preferences and what the school feels they should learn.

    If math class has been “turned into” tutoring, then the school is overusing it in a way that’s not respectful of the gifted student’s needs. But if the student is expected to tutor occasionally as a way to learn to explain things clearly, that’s a reasonable use of the gifted student’s time, IMO, even if they’d rather not “have to” learn such a thing.

    If a student loved arithmetic, but hated geometry, should they just get to keep doing arithmetic when the class gets to geometry?

  10. March 12, 2008 9:37 am

    “If a student loved arithmetic, but hated geometry, should they just get to keep doing arithmetic when the class gets to geometry?”

    Tutoring is not comparable to studying geometry. In traditional schools, kids can’t move at their own pace or choose what to learn when. In some schools, kids do have these options. Montessori comes to mind.

    My point is that in public schools, kids should not be compelled to do extracurricular work. Should gifted kids also be forced to join the math club because an adult thinks it would “be good for them”? Where do you draw the line at making decisions for kids? Why don’t kids get to decide if tutoring is good for them or not? That’s the respectful way to handle it in an environment where they have so few individual freedoms and choices in the first place.

    If they want to tutor, that’s different. But why should they be forced to along with all the other things they’re forced to do each day? If kids know they can become a tutor if they want to, they’ll gain much more from the experience when it is freely chosen. And they won’t resent it.

  11. March 12, 2008 12:00 pm

    Tutoring is not comparable to studying geometry.

    Why not? Why do we require kids to study geometry if they hate it? Why do we require kids to learn to write if they hate it? Why do we require kids to go to PE if they hate it? And why is that different from expecting kids to do other things they may not prefer, such as learning to clearly explain the things they know.

    Should gifted kids also be forced to join the math club because an adult thinks it would “be good for them”?

    No, they shouldn’t be forced to join the math club outside of school time, nor should they be required to tutor students outside of school time. But, I believe strongly that gifted kids should be given math club type problems to work on in class once they’ve mastered the basic curriculum. And, yes, I think they should be expected to work on such problems, at least some of the time, whether they prefer them or not. Because learning to think in these ways is important to becoming a well-educated math student. (Yes, I think non-gifted kids should also be expected to do problems like this, and I think schools should make time for them to do it, but realistically, gifted kids have more opportunity to do this.)

    Yes, I think respect for the students preferences should be included. A student should have plenty of opportunity to work on the things they enjoy and prefer to work on. But I support schools expecting kids to learn things they don’t enjoy as well.

    It seems to me that you support the idea of “imposing” a very basic core curriculum, but not of requiring a more rigorous curriculum of those who are capable. For math, parts of that more rigorous curriculum include additional topics, additional depth, solving non-routine “math club style” problems, and, yes, learning to explain what they’ve learned. And I think that all of these things should be expected of gifted students who are capable of completing a much more rigorous curriculum than some of their peers.

  12. March 12, 2008 1:58 pm

    Why not? Why do we require kids to study geometry if they hate it? Why do we require kids to learn to write if they hate it? Why do we require kids to go to PE if they hate it? And why is that different from expecting kids to do other things they may not prefer, such as learning to clearly explain the things they know.

    Those are good questions. Some people say we shouldn’t require those things, but we all know that public schools, for the most part, operate that way and so that’s the reality we deal with. Content standards, curriculum frameworks, etc. Everybody has to cover x, y, z.

    But tutoring is different because if the goal is to have the student explain what they know, they can explain what they know to the teacher, whose job it is to determine how well they’re learning. So there is another way to explain what you know without being forced to teach another kid.

    It seems to me that you support the idea of “imposing” a very basic core curriculum, but not of requiring a more rigorous curriculum of those who are capable.

    Not exactly, so I’ll try to explain. I actually support opportunities for all children to learn at their own pace, although I really don’t like the label “gifted.” Nor do I like the term “rigorous.” Both imply that the kids who don’t move the quickest through the material are somehow lacking. They set up an arbitrary hierarchy.

    Anyway, I support activities that give kids opportunities to continue moving forward even if the majority of the kids in the class aren’t ready. I’ve heard too many parents say that their kids are bored and the teachers aren’t permitted to allow them to do anything the entire class isn’t doing. That’s just dumb.

    I also support activities that enable the kids who aren’t moving quickly to achieve mastery instead of having to move on to the next concept whether they’re ready for it or not.

    And I also really love the idea of collaborative learning and mentoring in the classroom. Again, Montessori classrooms do this very well, giving kids the option of working alone or in pairs/groups and giving older kids the chance to help younger ones. The difference is that in Montessori, kids have the choice most of the time. The motto is “follow the child.” They have a lot of freedom in the classroom.

    A student should have plenty of opportunity to work on the things they enjoy and prefer to work on.

    Honestly, though, when does this happen in a typical public school? How much of the day is spent in child-chosen activities? How many opportunities in a given day do kids have to say “No thanks, I’d rather read my book than do my social studies report right now”? Or, “I’m really not ready to stop working on math, so I’d rather stay here and finish before I go to science class”?

    So when it comes to mandatory tutoring, I think kids should be able to opt out. I just honestly feel that kids have so little say in what they learn in school or how/when they learn that giving them choice in the matter is the best thing the schools can do for them. So what if nobody takes the school up on the offer? Maybe that just means it’s not such a great opportunity after all.

  13. March 12, 2008 2:32 pm

    Lori,

    I agree with you that there are lots of issues with the way a typical public school day is structured. I’m just not sure why you are choosing tutoring as the straw that broke the camel’s back, the one thing kids should be allowed to opt out of, while more or less accepting “everything else” as “just the way public school is”.

    You say that students can explain what they know to the teacher, and that is certainly better than nothing, but frankly not nearly as useful as a genuine experience with trying to explain something to someone who doesn’t already know it. And I’m not clear why you think it would be ok to require a student to explain what he or she knows to a teacher, but not to another child?

    I think we’re at the point of having to agree to disagree on this. And I think that we really don’t disagree all that much anyhow. I suspect that our ideal schools/classrooms would look quite similar to one another’s.

  14. March 12, 2008 3:29 pm

    Mathmom, I hear you. Just a couple of final comments to clarify, if you don’t mind. (Is the horse dead yet?) 8-)

    I’m just not sure why you are choosing tutoring as the straw that broke the camel’s back, the one thing kids should be allowed to opt out of, while more or less accepting “everything else” as “just the way public school is”.

    Oh, that’s not the only thing I think kids should be able to opt out of. It just caught my eye in your post. As far as the structure of public school, the standardized curricula, the standardized tests, the movement of same-aged kids on the same path no matter their interests/abilities, etc., I wish there were any hope of changing those (or at least allowing change for those who wanted it) in the near term. But they’re here to stay for a while, I think, so the best places to make an impact on respect for kids and enabling choice is in the “little” things, like peer tutoring, that are at a teacher’s discretion. If it doesn’t have to be mandatory, then let make it a choice. Empower the kids. Anywhere else it can choice can be implemented, do it there, too. That’s where I’m coming from on that. Tutoring is just an example.

    And I’m not clear why you think it would be ok to require a student to explain what he or she knows to a teacher, but not to another child?

    Because the teacher is the one who is supposed to evaluate whether or not the kids have learned, not their peers. Give a test or ask them to explain it to the teacher — it’s still an evaluation. I also don’t think kids should have to get up in front of the rest of the class and do problems on the board if they don’t want to. (I’m sure you’re shocked.)

    Thanks for listening! Didn’t mean to hog your comment thread….

  15. March 12, 2008 3:31 pm

    If it doesn’t have to be mandatory, then let make it a choice. Empower the kids. Anywhere else it can choice can be implemented, do it there, too.

    Ugh. Please fill in the missing words and remove the extra words at your reading leisure.

    And no, this is not a test.
    8-)

  16. March 12, 2008 3:32 pm

    No problem on “hogging” the comment thread. I’ve enjoyed the discussion.

Trackbacks

  1. Walking Randomly » Carnival of Mathematics - Silver Jubilee Edition
  2. Gifted Education — Riding on their Coattails « Ramblings of a Math Mom

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