(From my book Math Class Redesigned, Chapter 29)
If someone were to give me the unlikely assignment of taking a class of thirty children and trying to frustrate and annoy them, ruin their experience of math, and slow down their learning tremendously, I would know exactly how to go about it.
I would assign a set of math problems (the same problems for all the children, regardless of their level of proficiency and speed) and tell the children they needed to explain on paper exactly how they arrived at their answers, using full sentences.
Most children hate doing this.
Why do children, including ones who love to write, recoil from explaining on paper where their math answers came from?
Their resistance to explaining their mathematical reasoning on paper is completely different from their resistance to a rather dry task like memorizing times tables. Most children will not take the initiative in memorizing math facts and would prefer to do something else, yet they can understand that it is useful to have their math facts memorized. If pressed to memorize them, children will, and after they have, they will be glad they did. I have never met anyone of any age who memorized the times tables and later wished they had not bothered.
Children generally loathe explaining mathematical processes on paper because it distracts them from learning and doing math. It’s not a simple matter of finding it tedious. Their resistance runs deeper than that, and forcing them to explain their reasoning on paper comes with a price.
Show Your Work Versus Explain How You Got Your Answer
A clear distinction needs to be made between asking children to show their work or show the steps of a formal procedure and asking them to explain where their answers came from. It is perfectly reasonable to ask a child to show her work, meaning that she must solve a math problem or set of problems on one page rather than using a separate piece of scratch paper to solve parts of the problems and then writing the final answers on the page she will hand in and discarding the scratch paper.
If a child does her computations or diagrams on a piece of paper that the teacher does not get to see, it is harder to pin down where the child went wrong if she arrives at a wrong answer. She might have understood exactly how to solve the problem but made a small arithmetic error — or she might have had no idea how to solve the problem and put down a guess.
Seeing the written steps a child took, all in one place, is useful for both the child and the teacher. It gives the teacher more insight into the child’s level of understanding and also allows the child to check her work more easily and find her errors and correct them.
It is also reasonable — at the right stage — to ask a child who is learning a new procedure to write down each step in a formal way. For example, if a child is working on 6x + 5 = 29, even though the child might be able to solve this problem in his head, it would be reasonable, at the right time, to ask him to show the steps of subtracting five from both sides and then dividing both sides by 6. This encourages the child to solve the problem in a systematic way which in turn makes it easier to solve similar problems involving bigger numbers, and to solve more complicated equations which have more steps. Following precise steps also tends to result in more consistently correct answers. Teaching children how to construct formal proofs (again, at the right stage) is also a very valuable skill.
On the other hand, insisting that a child explain on paper how he arrived at his answer after he has solved a problem, or making him explain his steps in sentences as he works on a problem, is far from helpful.
Requiring a child to explain his answers is rather like insisting that a child in an art class explain (in writing) his painting. If you give a class of children a stack of thick watercolor paper and some good watercolour paints and invite them to paint, they will be happy. You could show them techniques and set up a big table with art books they could look through for inspiration. You could invite them to set up arrangements of objects and paint a still life. They will develop their painting skills and all will be well. The more they paint and look at other people’s paintings and are shown specific techniques like how to draw a nose or how to create shadows and then practice these skills, the better they will get.
But suppose every time a child does a painting, you require him to write a paragraph explaining how he arrived at the painting. Very quickly, most children will lose momentum.
“But this is my painting. What do you mean explain it?”
“Well, explain why you added purple over in this corner. And why this dark patch here?”
“That’s just how it felt right. Can I just write that I painted what felt right?”
“No, no. I want you to write a full paragraph explaining your process.”
“Can I do another painting now and explain them both later?”
“No. You’re not allowed to take another piece of paper until you have explained your artistic process. How did you arrive at this painting?”
“I just painted it. And now I really want to try a still life using the apple and the olive oil bottle and the key. I want to try that shadowing you showed us …”
“All in good time. I’ll keep your next piece of paper here and you can have it when you have written your explanation of that first painting.”
“I don’t really know what you mean but I’ll try …”
The child writes:
I decided to paint a picture because I wanted to. I picked colors I like. I put a purple blotch in the corner because it seemed like a good place for a purple blotch. I painted a chair in the middle because I thought of that. I made the chair green because green chairs are nice.
This explanation does not give the child any real insight into his art. It adds nothing. He created his painting by sinking into the activity of painting. Rather than helping him paint an even better picture next time, or understand his art or himself more deeply, the whole exercise of forcing a child to try to turn a nonverbal process into a verbal one feels awkward and contrived. If repeated over and over, it starts to form a barrier in the child’s mind between his urge to paint and his actually painting. The unpleasant thought of all the awkward explaining will loom up when he thinks of painting.
Furthermore, he can’t put his finger on why this thing the teacher keeps asking for feels all wrong. He knows writing is a legitimate thing, and he knows he is supposed to be able to reflect on things and learn to write better. After all, this is school and it’s not as if his teacher is asking him to turn his new painting into a doormat or tear it into strips to line the class hamster’s cage. On the surface, the teacher appears to be making a perfectly reasonable request — yet it feels completely wrong to the child in a way he cannot articulate to himself, let alone to the teacher.
The same analogy could be made about learning to play the piano. Suppose a child, each time he plays a piece — either at home or for the teacher — has to write a reflective paragraph on how he played it. How did he get his fingers to scale that octave? Through what process did his brain take in the notes and direct his fingers to play them? How and why did he remember to play the F sharp?
There would be no problem if the child were taught music theory in addition to learning to play pieces and was asked to answer questions on paper such as, “Draw a quarter note,” or, “Circle the arpeggios.”
The problem arises when the child is made to switch gears constantly from playing music to writing about the process of playing and explaining or defending how his brain is making sense of the sheets of music. The truth is, he probably doesn’t know. The fact that he cannot provide a clear explanation for how his brain makes sense of a sheet of music and directs his hands to play it does not mean he is failing to learn the piano or is learning in a superficial and sub-par manner.
A child who is very accomplished at painting or very talented at the piano or excelling in math is no more likely to want to explain the process on paper than a child who is struggling with these skills. The resistance in the child does not arise from ineptitude or confusion or laziness.
The resistance arises because it is distracting and jarring to jump back and forth between the task of solving a problem and the task of explaining how one’s brain is solving the problem. It interferes with the natural flow and momentum of the process.
Over the last forty years I have taught hundreds of children and closely observed what helps and what hinders their mathematical development. I have arrived at the conclusion that demanding that children explain their mathematical reasoning on paper undermines their mathematical development rather than supports it.
* * *
There is a state of ease a person can enter when completely absorbed in an activity like writing or doing math or playing an instrument or painting or drawing or teaching. In this state, which is typically highly pleasurable, we forget about time, worries, and almost anything but what we are working on.
This is the state we want children to enter when they are doing math: a state that is happy, peaceful, focused, deeply absorbed, and productive.
As teachers, we need to do everything we can to create the right environment and the right kind of assignments to increase the chances that children will fall into this state and stay in it for prolonged periods.
A key element of this state is focus. We need to set up the classroom in a way that is conducive to focus, and then allow children to sink into an activity.
Do not fragment a child’s focus
A math problem may be stated in words but when we ask children to solve it, we need to allow the children the freedom to enter into the realm of math and stay there while working instead of demanding that they keep coming back into the realm of English sentences (or whatever their language is).
The unravelling of a math problem, the sifting and the sorting, the recognition of similarities, the leaps, the falling back on familiar algorithms and memorized facts to deal with certain aspects of the problem so the brain can focus on the real core of the matter, the clarity that can come very suddenly … this is not primarily a verbal process. Words certainly come into it. A person solving a math problem may shift back and forth between thinking in clearly identifiable sentences or sentence fragments and thinking in a way that is precise and accurate but not verbal.
Math as its own language
In math, symbols and pictures hold information which the brain can make sense of without translating them into words. In this sense, math is like a language.
The last thing we want to do when a child is creating a neural pathway for, say, solving a set of two equations in two unknowns, is to fragment the path and add in little detours. We need to allow the child’s brain to work smoothly and efficiently. If I see a child struggling, then of course I will offer to help him and will need to use words — but if I see a child happily and productively working on a problem, the last thing I would want to do would be to interfere and break his focus.
Consider, for example, a simple equation like 2b + 5 = 11. If a person is just learning to make sense of these symbols, they may translate the problem into a sentence in their mind. But when one is proficient in this type of problem, no sentence forms in the mind that says, “Hmmm, two b’s plus five is eleven … so the two b’s must equal six … and one b must be three.” If I were trying to teach the problem to someone else, I would put my finger over the 2b, obscuring the part of the problem that is likely the stumbling part for the child, and ask, “What plus five is eleven?” Naturally, one reaches for language when trying to explain something, but if I were solving the problem by myself I would simply look at it and understand the symbols without translating them into English sentences and then write down the answer.
When a child is becoming fluent in a second language, we want them to connect the new word directly with what it represents, not circle it back to their original language first. If we were trying to get a young English-speaking child to learn the French word “chien”, ideally we would connect the new word chien to a picture of a dog or to a real dog if one were handy, not to the English word dog. Yes, we want the child to know that chien means dog — but when the child is conversing in French we do not want him to have to turn every French word he hears into an English word, think of an answer in English, and then translate his response word by word into French, finally answering the other person. We want the child to have a direct experience in French.
In the same way, we need to allow math to be its own language and not force a constant translation into English (or Spanish, etc) on the part of the child.
Encouraging children to experience the beauty of math
If we were to say to a child, “Paint a portrait of someone you love,” we would not demand accountability, on the page, for every nuance of colour or texture, or the decision to paint a deep blue background. Once the child enters “painting mode”, we let her remain there rather than dragging her back, over and over, across the threshold of language. And we certainly would not deem the painting incomplete until we had forced the child to scrawl words all over her painting explaining how and why she had painted it. We recognize the painting as an object of beauty and meaning in its own right.
That’s exactly what we should be doing when children are solving math problems. We need to let children go into the realm of math and peacefully work there, coming back on their own time, not artificially hauling them in based on an unproven educational theory that altering their thought process in an unnatural way will help them learn.
The child’s mathematical workings, if the child has successfully solved the problem, should be allowed to rest on the page as something aesthetically pleasing in its own right without our insisting the child lard the work with unnecessary sentences of “process” explanation.
If the child has solved the problem correctly and any steps that have been shown are mathematically sound and true, we need to let the math remain just as it is and let the child savour the victory or pleasure of having solved the problem.
If the child has solved the problem incorrectly, we need to isolate the error, acknowledge what the child has done right, and draw the child’s attention to the mistake— but in a respectful and helpful manner, not slashing away at their work with broad strokes of a red pen and certainly not demanding they write down how they arrived at the incorrect answer.
Children should be encouraged to see the beauty in math and to create beauty. It is highly unlikely that a child will develop a brand-new proof and this should not be the standard for beauty. A six-year-old, solving a problem in a way that is aesthetically pleasing to her on the page, is creating beauty and this beauty, no matter how simple it might seem to an adult who may have lost their sense of wonder about numbers and patterns and shapes, should be appreciated and respected.
Trying to force a nonverbal process into a verbal one is not helpful. It is distracting, frustrating, and impeding. It wastes valuable time and robs children of experiencing the joy and elegance and flow of math. We want children to know this joy, partly for its own sake but also because joy is the ultimate motivator in math. Grades and rewards and parental expectations and threats of not getting into a good college can never spur on a child as much as a deep love of learning and an intimate familiarity with the beauty and precision and power of a subject.
* * *
The false dichotomy of rote learning versus deep conceptual understanding
One excuse that is commonly given for demanding that children explain how they got an answer is to reveal whether the child actually understands what he is doing or is just doing it “by rote”. Doing something “by rote”, when it is mentioned in this context, is considered to be a bad thing.
Let’s examine this. First of all, being able to consistently come up with the right answer to a class of problems, even if one could not give a lecture on exactly why this works, is actually an accomplishment, not a failure. If I can safely drive my car to all the places I need to go every day and understand and respect the rules of the road, the fact that I cannot explain how a carburetor works is not a great failing. It’s a separate matter. If a person can look at her bill in a restaurant and calculate a 15% tip within a few seconds, this is a valuable skill even if the person could not provide a deep explanation for why her method worked.
A child who can add, subtract, multiply and divide fractions but cannot explain why you invert and multiply the second fraction when dividing has still mastered something very useful and this is not something for us to dismiss as, “Oh sure, he got 100%, but does he know what it all means?”
We need to make a distinction between the valuable skill of being able to execute a procedure with speed and accuracy and the higher-level skill of understanding exactly why this procedure works and being able to articulate this.
When a child is learning a new skill, we want her attention fully on mastering it. If we were teaching a child to ride a bike, we would not stop her every few seconds to get her to explain why she didn’t just fall over. The actual experience of learning to ride a bike is a rich and valuable thing in and of itself. The child will soon become fully aware that she needs to go a bit faster in order to keep the bike balanced without being able to explain the physics of this.
Arithmetic is rather like this. The more a child does it, the better she gets at it and the more intuitive sense she develops about it, even if she cannot put this into words yet. Deep conceptual understanding is something that grows as one does increasingly difficult problems and matures.
One way to gauge if the time is right for it is to see if the child spontaneously starts to generalize principles and is eager to engage in a discussion of how and why a concept works. If the child is obviously still wanting to do the problems without explaining them, then the time is not ripe. If a second grader is perfectly willing to do subtraction problems, either on paper or using Cuisenaire rods, but balks at the request to explain what he is doing, he should be allowed to get on with his work in peace.
If I wanted to measure the depth of a child’s understanding, there are far better ways to measure this than asking a student to write full sentences explaining how she arrived at an answer.
Suppose, for example, I wanted to know if a group of eight-year-olds understood the concept of place value. I would offer a series of questions (on paper) like this:
Circle the tens digit in 457.
Put a square around the ones digit in 6738.
Underline the hundreds digit in 7469.
How many digits does 326 have?
How many digits does 36,271 have?
Arrange these numbers from smallest to largest: 724 274 472 742 247 427
What is the largest number you can make using the digits 1, 5, and 2?
What is the smallest number you can make using the same digits?
How many tens are needed to make 100?
Write three different numbers which have the same thousands digit.
Give an example of a pair of 2-digit numbers which add up to a 3 digit number:
__ __ + __ __ = __ __ __
Give an example of a pair of 2-digit numbers which add up to a 2 digit number:
__ __ + __ __ = __ __
Is it possible to find a pair of 2-digit numbers which add up to a 4 digit number?
__ __ + __ __ = __ __ __ __
These questions actively engage the child in doing math and thinking about place value. Asking a child to answer a variety of kinds of questions and getting the child to make use of the concept to figure something out is a much better way to gauge his understanding than making him explain it in words on paper. Also, children like answering these kinds of questions and so the measuring of their understanding does not come with the price of creating resistance and boredom in the child.
A child’s answers to the questions listed above would give me a far more accurate sense of a child’s understanding of place value than a question like:
Simon has three hundred and fifty-two marbles. Mary has four hundred and six marbles. Describe the math situation and show how to arrive at the total number of marbles using a place value math picture. How do you know your answer is correct? Explain.
A child’s response to this overly-wordy question which demands still more words is going to give far less precise information about her level of understanding. The question will also likely annoy the child because there is something contrived and artificial about it which the child will quickly sense. It’s a question with an agenda, rather than a question designed to let a child get on with enjoying and learning math. Children heartily dislike this sort of math problem.
Frustration, boredom, and resistance are all deterrents to learning. We want children to spend as much time as possible cheerfully doing math and as little time as possible thinking, “Oh not this again!”
I can’t recall ever encountering a child who wanted to write sentences explaining her math process but if I were to meet such a child, I wouldn’t dream of discouraging her. If this method was helping the child, I would welcome it (for that child). I would not, however, impose it on hundreds of other children for whom it was not helpful.
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