How to encourage young student to think in unusual ways?

Question

I tutor a young girl aged 11 (grade 4).

She is doing OK for her age, but I have observed that she has a tendency for rigid ways of thinking. She is usually more inclined to follow rules and stick to known ways of doing things which I think limits her learning.

Examples:

Calculating 2×7×3×5: she understands that she can multiply 2 and 5 first to get 10, and easily get the answer. However, when faced with similar tasks, she still prefers going from left to right and "does not like" to consider if there might be an easier way.

Or, consider the following exercise:

Let's add 3400 and 5700. Adam adds 34 hundred and 57 hundred and gets 9100. Ben adds 3400 and 5000 to get 8400, and adding the remaining 700 he gets 9100 as well. John adds 3400 and 700 to get 4100, then adds 5000 and also gets 9100. Explain how each of them did it. How would you do it?

When faced with such tasks, she always "likes" the way to split the numbers because she thinks working with hundreds is "strange".

I think it is normal to understand one way easier than others. But I observed that she has a tendency to put up resistance when I try to give her exercises that are more easily solved in the ways that challenges the "order" that she thinks things should be done. I think this limits her learning, because eg. to understand why certain procedures work, or to be able to check results etc. all depend on understanding things from different angles.

How can I encourage her to think in ways that she might at first find "strange" or seemingly defying "order"?

Answer 1

A lot of the answers here are suggesting giving examples where the alternatives you're interested in are necessary (or at least wildly easier than the "rigid" approach). That's a good option, but in my experience - depending on the student - it might have the opposite of the desired effect, because these examples will always feel "forced". She may feel that the examples you're giving are examples cooked up specifically to poke at her weaknesses, which can cause a student to retreat.

As an alternative suggestion: when I teach creative mathematical thinking, I present it as a freedom, not a task. The fact that we have many ways to solve the same problem means that we have the freedom to choose which approach to use; this is a good thing! Importantly, though, in order to get a student to a point where they actually do think of it as a freedom, I have to really give them the freedom to choose - and they may choose the "rigid" option. My priority, then, is to make sure that they're making an informed choice.

So, for example, looking at the problem $2 \times 7 \times 3 \times 5$, I might start by asking the student to write down an approach - not an answer, just an approach, like "multiply 2 by 7, then by 3, then by 5". Then ask her to write another approach, and then to decide which approach she wants to use and why. If the answer is "I want to use the 'rigid' approach because I like straightforward things better", then fine - the goal is to instill the habit of considering her options, not to get her to use the option I think is easiest. When I've used this strategy with students, they tend to gradually get better at spotting alternatives and more likely to spot those alternatives without prompting; and this then makes those alternatives feel "easier" to them, because they aren't so hard to think about.

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Relatedly, I've also had some success simply changing my language around students. It's natural to ask "how do we solve this problem" or "how do we start this problem"; but that language implicitly suggests that there's only one correct way. I tend to phrase it as "how can we solve this problem" or "how can we start this problem".

Answer 2

Caveat

Having spent a decade in education, and many of those years working with a body of students only 10% were at or above grade level (and I'm pretty sure that the metric didn't indicate any of the latter), I've seen the sort of rigidity you mention. Anyone who thinks education isn't primarily an exercise in psychology hasn't spent much time in the classroom. Given the poor way math is taught in the US, math is more psychological than most. The challenge to inculcate creative thinking in math is perhaps the challenge, so make sure you understand it yourself.

Why Math Students Are Rigid

In mathematics education, there has been a tension between fundamentalists (let's stick to the arithmetic/geometric fundamentals which generally means lots of rote memorization) and math education philosophies that look to enrich a child's conceptual understanding, like New Math and Common Core. I've found students often have psychological preferences much like the former, "fundamentalists" and "imaginative" thinkers. A "fundamentalist" tends to believe there's one way to do things, and that getting correct answers is important, whereas "imaginative" students are creative in their approaches (I had a secondary student essentially reinvent math pictographs) to do problems. The latter group of students likes to get right answers, but tends to prefer finding a way that makes sense to them.

The NCTM has criticized (rightfully IMNSHO) math education as a being a mile wide and an inch deep. Given the sequential nature of math, I think the rigid thinking of children (barring the occurrence in psychological disorders like OCPD) often hinges upon an emotional experience of math that might be described as a brutal series of rejections for students who show little talent for it or are deprived of a quality year of education, often at no fault of the student. A bad 3rd-grade math teacher may result in a struggle in math for the next 9 years, and for a child sensitive to failure, that's a lot of rejection. Standardized tests tend to exacerbate that rejection.

Changing Math Thinking

When I earned my math degree, I saw that the attitudes about math didn't change much from K to graduate classes. There's an absolute lack of wonder about mathematics. It's tough to bring about wonder, and some students just aren't gifted in math and fewer want to learn. Gauss on math education:

I have a true aversion to teaching. The perennial business of a professor of mathematics is only to teach the ABC of his science; most of the few pupils who go a step further, and usually to keep the metaphor, remain in the process of gathering information, become only Halbwisser [one who has superficial knowledge of the subject], for the rarer talents do not want to have themselves educated by lecture courses, but train themselves. And with this thankless work the professor loses his precious time.

1. Depending on the age of the student, you need to get into the student's head. With a senior in high school, you can just have the conversation about how why they do what they do, how they feel when they get wrong answers, etc. On the other hand, with a younger student, you have to be a little more indirect and infer. For most students who are rigid in their approach, it's simply a combination of poor math skills and a dislike-to-fear of failure. A rigid student is just clinging to an algorithm to avoid appearing completely ignorant, more often than not, and sadly, they've often been rewarded for it in the past. That's why so many people hate the Common Core: it doesn't focus on answers so much as reasoning, and well, critical thinking is hard work.

2. Once you have a sense of why it's so important for a student to stick to an algorithm, then, if you have good rapport with a student, persuade them to broaden their thinking. Math understanding comes from having a dialog, even with yourself when working on a problem, and students are conditioned to execute an algorithm than applying a series of heuristics. But, one thing that has been made clear is that mathematical problem solving involves defeasible reasoning. Math solutions are very much based on intuitions.

3. To develop a student's inner dialog to engage in problem-solving, it often helps to teach problem-solving strategies. George Polya's work on the matter is a good start. Some students (those with ADHD or impoverished grammars and vocabularies) struggle to solve math because they have a hard time reasoning through it. But I suspect, since the matter is an empirical question, that most students simply haven't had practice at it. When I taught, in my math department, of the 25 teachers, only 2 of us actually had math degrees, and new teachers often had little insight of the material. I suspect that's the norm in the US. You have to make use of teachable moments and role model math problem-solving.

As you may have heard, there is no royal road to geometry, but there are things you can do:

I had three practices in the classroom to encourage a child to understand that making mistakes in math is the norm (not the exception), and my students would complain. First, I used to make my students use pens instead of pencils because and they would protest that I'd see all of their mistakes; second, I would give vocabulary and writing assignments and assessments devoid of calculation. Third, I would give problems AND the answer and ask for work and rationale. You would think I was pulling teeth without analgesics. The usual rejoinder was "this isn't how you do math!" I ultimately had to flip the classroom, and I encountered this strategy in the university as well.

More directly,

I'd say a good first start is to redesign your lesson and materials around the answer of getting from the givens to the conclusion. For instance, if your student is struggling with associativity in addition, give worksheets that ask for two ways to go from 3 + 6 + 9 to the answer 18. This way you're focusing on the path instead of the destination. Use graphic organizers. At the elementary level, if you have steps, you almost invariably have arrows and boxes meant to suggest what gets combined. Don't do a lot of problems, do a few problems repeatedly and well to encourage understanding. Teach non-standard algorithms. I can add three-digit numbers in my head, but that's because I've had a lot of practice with non-standard methods. Use lots of vocabulary including Q&A; when my child started addition, we talked about addends, adders, sums, and doubles as well as ten frames, groups, addition, combination, etc. Most of my at-risk high schoolers knew less math vocabulary than my 1st-grade daughter. Multiplication can be talked about as a product of factors, or more simply m groups of n. So, 3 x 2 = 6 is also three groups of two things is six things that are more concrete. Mathematics is terse, and it helps if you develop semantics to accompany that syntax.

I could go on, but there are lots of tricks to the trade, and most of them are specific to the type of math involved. Make problems concrete. For commutativity or transformations, I would talk about putting on socks and shoes (non-commutative). Methods like FOIL for products of binomials, mnemonic devices for memorization, games, etc. Teaching math is no different than repairing cars in that anyone can do it, but few can do it well. Ultimately, keep on developing your skills is the best long-term strategy. Being a math teacher is a lot like being a wizard, and you have to have a spell book of materials and techniques.

Answer 3

Let's add 3400 and 5700. Adam adds 34 hundred and 57 hundred and gets 9100. Ben adds 3400 and 5000 to get 8400, and adding the remaining 700 he gets 9100 as well. John adds 3400 and 700 to get 4100, then adds 5000 and also gets 9100. Explain how each of them did it. How would you do it?

When faced with such tasks at age 9, I would be bored out of my wits. As others have suggested, the problem may simply be that what you are giving her is just downright uninteresting. Why should I care what is the fastest way of adding ten numbers? If I can't be made to care, can a child?

How can I encourage her to think in ways that first seem "strange" or seemingly defying "order"?

Frankly, this is likely to be an unproductive way of teaching. Mathematics is not about trying to find strange or disordered ways of solving problems! Rather, it is about appreciating and understanding the underlying beauty of mathematics. Have you shown her anything in mathematics that she finds beautiful? If not, then maybe you should reconsider what your intended goal is.

Note that even if your goal is for students to solve arithmetic problems fast, you should not forbid your students from using any methods that they like, and instead reward the speed of solving. More generally, mathematics olympiad problems at the appropriate level (e.g. AMC8) are a good source of questions that encourage thinking without forcing the student to use any particular method. All that matters is how many questions they can solve within the time limit!

But if your goal is to convey the beauty of mathematics, then you need to actually demonstrate that. Ad-hoc tricks to speed up arithmetic calculations are only very slightly interesting, unless you can show significant benefit or beauty in the 'non-standard' methods. For example:

• $(1+2+3+\cdots+99) + (99+98+97+\cdots+1) = 100×99$.

• $\frac1{1×2}+\frac1{2×3}+\frac1{3×4}+\cdots\frac1{9×10} = (\frac11-\frac12)+(\frac12-\frac13)+(\frac13-\frac14)+\cdots+(\frac19-\frac1{10})$.

• Let $S = \frac13+\frac1{3^2}+\frac1{3^3}+\cdots+\frac1{3^9}$. Then $S×3 = 1+\frac13+\frac1{3^2}+\cdots+\frac1{3^8} = 1+S-\frac1{3^9}$. Thus ...

These are the real tricks that students should be shown. Paltry tricks like $2×7×3×5 = 21×10$ are not convincingly a big deal, unless your goal is merely to be able to finish a list of arithmetic exercises faster than her.

Answer 4

My feeling here is that she might not be much interested in math, so simply doesn't enjoy looking for alternate ways of solving problems.

On the other hand - at least according to your examples - there is not much benefit of finding alternate ways.

I'd look for problems being hard solved the traditional way and very easy after some reorganisation.

E.g.
5 / 9 * 4 * 9
37 * 67 + 63 * 67 {= 100 * 67}

Maybe also give limited time to even more highlight difference.

Also, what helped me to look for alternative ways is trying calculate in my head as there I really felt forced to look for the easiest way.

Once she experience benefit of finding alternative ways, she probably will try to apply it also on her own.

Answer 5

In a comment I wrote "Perhaps her need for autonomy is part of this." I was asked to elaborate in an answer. I'll try.

People learn what they want to learn. Real motivation cannot come from the outside. (It has been shown that extrinsic motivation doesn't last and boomerangs.) If a student has little interest in math, then a tutor can make their time together more enjoyable (or at least less painful), but cannot make the student care about mathematics.

If we look at how young children learn before they encounter schooling, they pick what they learn, and throw themselves into it with gusto. Learning to walk and talk takes amazing dedication, and they all do it. There are not kids who just didn't learn that.

Schooling teaches kids to work by the clock and to obey authority far more effectively than it teaches content matter.

So a student comes to a tutor, most likely because their parents required it. The student is not going to be happy about losing some of the little bit of free time they have. They are not going to come into it feeling competent. And they are not going to have burning questions about the mathematics.

I have tutored students like this. I would say I had almost no success in 2 of the 3 most recent cases. In the 3rd, I asked the student if there was anything he liked that was at all related to math. (Not sure now how I framed that question.) He liked logic puzzles. So that was a big part of how we spent our time. And I "taught" very little there. If I had "taught" more, I would have stolen his sense of competence, and possibly his love of logic puzzles. I did tell him that logic was my first love in math, and that I had studied logic in grad school.

Humans have deep needs. After food, water, and shelter, one of our greatest needs is autonomy. Another is learning/discovery/making sense. I believe that autonomy is required for deep learning to take place.

Answer 6

(Disclaimer: I'm not an educator, though I did study maths to university level. But this draws on a memory from the pupil's point of view, so I hope it may still be helpful.)

When I was in junior school, aged around 9 or 10, my teacher gave me an open-ended project to look at over the holiday.

She drew a very simple grid of numbers:

1  2  3  4  5  6  …
2  3  4  5  6  7  …
3  4  5  6  7  8  …
4  5  6  7  8  9  …
5  6  7  8  9 10  …
…  …  …  …  …  …

And she indicated a square of numbers within the grid. I can't remember which one; say for example it was:

    3  4
    4  5

She pointed out that if you add up the top-left and bottom-right numbers (3 + 5) you get the same result as if you add up the top-right and bottom-left numbers (4 + 4).

And I think she told me to investigate that — no further direction or clarification; she just left me to look at that over the holiday and see what I could come up with.

So I went on to prove that the same is true for every possible square of 4 adjacent numbers, anywhere on the grid. And that it's true even if the numbers aren't adjacent, as long as they form the corners of any square, or any rectangle. I also found and proved a result or two from multiplying the numbers instead of adding. (I've forgotten how much more I did; this was a long time ago!)

I found this exercise really interesting at that age because it was open-ended. The algebra is of course trivial, but I was probably fairly new to the idea of proving general results, and of investigating and discovering for myself — so this was far more about imagination and curiosity than anything else I'd done before, and far less about trying to identify the solutions that the teacher expected.

Maybe a similarly open-ended task, pregnant with interesting results to be discovered, might help students like that in the question look at something mathematical without any rules to follow or previous examples to imitate — and without any pressure to find predetermined ‘right answers’?

Answer 7

Geometry

The more different ways one has of looking at a problem, the better the data for training the internal model. Instead of giving a symbolic multiplication problem, try a geometric one. Get some small blocks, form a rectangular solid with it, and ask the student how many blocks are in the solid. Rearrange the solid in different ways so that the "easy" multiplications occur on different axes, and ask the student each time why they chose the strategy they did.

After she has solved the blocks problem, write down the exact same problem on paper. Show multiple permutations of each problem, so she can see how they are equivalent, but that the "best" strategy that she chose depended not on following a prescribed rigid rule, but rather by making a value judgment about what might be the path of least resistance (the "easiest" route). Hopefully, she will recognize that the "rule" she ought to follow is one that she discovers herself, and that the rules she is given can be improved upon by looking at the problems differently.

Fermi Estimation

This might seem a premature for an 11 year old, but I think not. Learning that math can be done approximately and still be useful should be a universal lesson that all humans learn explicitly (especially since most of us use this fact implicitly anyway). Get several jars and marbles (or beads, or whatever has a consistent size). Put different numbers in them, and ask her how many marbles are in each one. Then, collect a few jars and ask her how many are in both together.

She will realize that she can guess there are 50 marbles in one jar, but she won't really know if there are 55 or 60 or 77 marbles in it. And when she goes to add the jars together, she will be forced to give up on precision in the lower digits and just do the math on the higher ones. She won't spell this out explicitly (unless she displays a sudden jump in precocity), but you can explain it to her.

If she insists on trying to make precise estimates, put time bounds on the problem. Tell her you are going to show her jars, and she has to add them in 5 seconds or whatever puts her on the edge of her calculating/estimating abilities. This will force her to estimate to only the highest digit of precision. After the 5 seconds are up, cover the jars with a towel so she can't see them any more.

Then write down the same marble problems on paper, with the real numbers, and show her how close she got by using different strategies.

Toys

If you know something about the toys the student plays with, you can connect your math lessons to her world even more directly. For instance, LEGO bricks are perfect for forming geometric solids. Most likely, if she plays with them at all, she will already have some intuition for how to do math with them. If she does any kind of crafts involving beads, buttons, sticker sheets, etc., you can use these objects as the things-to-be-counted. She probably already does some basic arithmetic on them, and if she is faster at solving problems using these objects, then you can ask what rules/strategies she is using to show that she has already invented her own rules which might be different from the textbook strategies.

If she doesn't play with LEGOs, you can introduce them as a good exercise for learning math while also being a fun way to relax or exercise creativity.

Answer 8

games and puzzles? logic games? math based games?

could you model using anchor numbers (sums of 5 and 10s?) de/recomposing numbers into place value?

multiply in this manner? add in this manner?

when doing multiplication reason (double or triple or half)

i wonder how much she thinks of math as rigid rules versus conceptual / relational (how math / numbers are related to each other?)

even empty number lines and showing jumps (again, decomposing numbers by groups of 10)

MODEL this way of thinking?

Answer 9

I have often observed my daughter doing the same thing.

I sometimes play a game with her, where I will give her a problem like yours 2×7×3×5 (albeit with addition because she is young). Then I ask her how fast we can solve a bunch of these problems. Gameifying it works for her.

A bunch or all of the problems are just permutations on the original. This reinforces the idea with her that she has already solved the subproblem. Maybe such a strategy will help you.

Answer 10

I'll comment on a point of view, perhaps it's useful.

You want to induce and encourage flexible ways of thinking.

The value of that, IMHO, lies in that flexible ways of thinking increase the chances of understanding (real) problems more deeply and finding solutions to (real) problems, and/or finding new solutions, and/or better in some metric.

I don't think your end goal is for her to always use the easiest way to do multiplication. Or addition.

Because, doing arithmetic by a fixed method is in itself easy: you don't have to think (which is "expensive"), just do. A fixed method gives you a tool to use easily to solve larger problems, so that you don't have to think about every little detail.

So, perhaps, it might help you if you tell her something like this:

"Look, I know you like/prefer doing it that way. I'm not trying to make you change your way. I'm just asking you to find alternative ways to help you learn that: the ability to find alternative ways, thinking of options, of possibilities. This is just an example.