Charles Benoit's YA novel You is written in second person. According to the book jacket, Benoit is a former high school teacher and Edgar-award winning author of adult mysteries. I see the "former high school teacher" all over this book, which is unrelentingly grim, but worthwhile. The main character is a high school boy who has this to say about math:

Math.

It's your favorite subject. Which surprises you.

[snip]

You like it because either you're right or you're wrong. Not like social studies and definitely not like English, where you always have to explain your answers and support your opinions. With math it's right or it's wrong and you're done with it. But even that's changing, with Ms. Ortman up there at the whiteboard saying how this year you'll be writing something she calls Mental Notes, which explain how you solved the problem and support your answer, saying that having the right answer isn't as important as explaining how you got it and bam, just like that, you hate math.

If having to explain how you got your answer makes you hate math, then it wasn't math that you liked to begin with. Proofs that others (or at least others that are expert in the same field) can read, follow, and understand, are the core of mathematics.

However, I've often seen math ruined by having to write an explanation for something like why 5 + 2 = 7.

Back when my daughter was in 1st grade, I was ok with them having to draw a picture to show why 5+2+7. This year in 2nd, I was a little less understanding. If she is still doing this in 3rd grade, I will not be happy.

Or better yet, Envision math. Instead of asking the kids, 7*8 = ? (which my tutoring student knows), they make them do a problem like this:

Show how you can use 4 math facts to calculate 7*8 (which my little guy just doesn't get WHY he would have to do that, NOR does he understand, when he already knows the answer is 56).

What they are looking for is:

4*7 = 28 + 4*7 = 28 = 56

So what is ultimately more important (or satisfying as a student)?

If you can't explain how you got the solution, you probably don't understand it very well. My son, who is generally good at math, was having trouble with the explanations and with showing his work (which, btw has been around as long as I can remember). When we worked with him a bit, it became clear that he was shaky with the concepts. He could do the problem, plug and chug style, but I don't think that is good enough.

On the other side of this, he also had to do the "draw a picture for 2+5" in first grade. Every single day. They started with 1+1, went to 1+2, and on and on. Horrible, absolutely soul-deadening.

When my daughter was in elementary school, math homework (which consisted of simple addition or multiplication facts) always included a line for "How do you know?" My daughter's standard answer was "Because I am smart."

In multi-step problems, showing your work IS explaining why, to a great extent. In order to choose the next step, you have to know (at least on an intuitive level) why you are choosing it; otherwise, you'd be likely to choose something else.

Note that in the story excerpt, "having the right answer isn't as important as explaining how you got it...." so they aren't really talking about proofs in math. They are talking about bs.

The old "show your work battle" is one worth winning (on the teacher's side) exactly for the reason Bonnie has said. I try to get my home-schooled kids to write a short verbal tag or label for every expression they use, in story problems, as a way to keep them from just munging around with numbers. They don't have to write sentences explaining why simple facts are true, however.

Students' explanations are sometimes very enlightening. When my dad taught remedial math, he asked his students to explain their reasoning when doing some "story problems."

They explained: When a problem has three numbers, you add. When it has two numbers, you multiply.

Boys are typically 1-2 years behind girls in literacy skills up to at least middle school. Writing explanations can be an unwelcome obstacle to a boy's math studies, which has its own non-literary language. Writing may have the effect of turning boys off to math.

Explaining your work often means writing a paragraph (I think this is in the NY state tests, for example) or a journal entry. Often for the math-gifted boy, this can be a long and painful distraction from his enjoyment of math.

I generally support "showing your work" and explaining answers, but for the gifted math student I think it is often unnecessary for purposes other than testing. These kids "get it", and are sometimes too immature to appreciate the importance of explanations.

Katherine Beals has written about this topic, so maybe she'll chime in.

A math major I knew in college, a woman, said that her method for arriving at the right answer was "I just look at it, and I know it." There's no room for that anymore.

A lot of bright/gifted kids are like that. The gifted teacher at our grade school had to pull kids for two weeks to teach them how to draw charts for a simple multiplication word problem. The kids knew to just multiply, but they needed to draw the chart or they wouldn't get it right.

When my son was in the third grade, I would have him do Singapore 5 or 6 word problems. He often would just put down the answer and it would be right. I asked him how he got to the answer and he would look at me like I was crazy. It was simple to him, in his head.

I certainly believe in showing one's work (in a mathematical way, not writing out paragraphs) to get credit, but our state tests gave more credit to kids who showed all the steps (in arithmetic) but got the answer wrong, than those who got the answer right, but didn't show the steps.

The gifted teacher basically had to teach her kids to go back two years, before they could think abstractly about solving a problem. They were beyond the picture drawing stage, but were penalized for it anyway.

Showing your work, when there is work to show, is an important part of math, but writing journal entries about math or explaining why 7*8=56 is just an attempt to turn mathophiles into mathophobes.

The Art of Problem Solving courses require explanations for the answers to their homework questions, but the questions are difficult multi-step questions for which the path from problem to solution is not obvious.

There has to be something to explain before explanation becomes a useful exercise.

Anonymous - I'd be interested in any other opinions on or experiences with Envision Math, since that's what our local elementary school uses.

I'd seen some red flags already, like pushing forward too fast at first then spiraling and the fifth grade teacher spending time on geometry and algebra topics but then reminding parents to work on their kids' multiplication facts, so your description doesn't add to my confidence.

...our state tests gave more credit to kids who showed all the steps (in arithmetic) but got the answer wrong, than those who got the answer right, but didn't show the steps.

That's always bothered me. And acceptable "steps" include guess & check. Thank you, constructivist math!

My boy was not behind in literacy. I really do not like the assumption that boys will always be behind. That assumption, on the part of several teachers, hurt my son's education. In any case, he couldn't explain his work because he didn't actually understand the algorithm he had memorized.

Showing your work and explaining it is very important in higher level math. If kids don't start practicing that skill early on, they won't learn how to do it. I wasn't taught, and was blown away in college when professors started asking that we show all our work. My chemistry professor, in a fit of exasperation, ended up spending a couple of classes tutoring us in the fine art of explaining ourselves.

I agree that journal entries are silly. That happens because teachers don't understand themselves how to explain steps in math.

Professors have given partial credit for wrong answers with the right steps since at least the days when my FATHER started teaching physics, which is why he always yelled at me to show my work. This is not some new constructivist thing.

One other advantage to carefully showing every step of your work (and labelling things ) - you are FAR less likely to make silly mistakes. That was another problem for my son.

Showing each line of your work goes back forever, but there's a difference between being careful and understanding. Go ahead and ask kids to explain 4 different ways to multiply 7*8, but at some point (soon!) you have to move on. For many kids, the torture lies in not moving on.

There is also the difference between a mathematical explanation and a verbal explanation. I remember showing each change in algebra and listing the identity used. That's a lot different than explaining using paragraphs. At some point, the math does the explaining all by itself. That's the whole point of math.

There are different levels of understanding and most K-6 math curricula get stuck at the basic level. You can see it fall apart in the later grades. The understandings of fractions are in terms of pictures that never translate into the mathematical understandings that allow students to deal with rational expressions. They get stuck at conceptual understanding and never get to mathematical understanding.

This would all be fine if kids ended up mastering the skills needed to get to algebra in 8th grade. It doesn't happen. K-6 schools go on and on and on about understanding, but they never get the job done, either via rote skills or via understanding. That's because they trivialize skills and their understandings are superficial. It strikes me that deep down, they don't like math. They want to change it into something else.

Haven't read all the comments, but wanted to point out that this excerpt is about a high school class in math.

To me, "mental notes" is a kind of mental debriefing akin to psychotherapy: why did I do what I just did?

People who are proficient in a subject - any subject at all - very often can't say why they did what they did. There's a long history of research into this phenomenon, which I believe was one of the reasons why the study of the 'cognitive unconscious' was founded.

It seems to me that a high school student who is good at math should be learning to do proofs, not writing "Mental Notes."

Very bad that your son's teachers assumed he was behind in lit. skills. It's hard to imagine how that could happen, with testing and reading assignments. I can only guess these teachers were not paying attention.

I was probably not too clear, but I was referring to the "guess & check" part in thanking constructivism. There's a recent emphasis on that as educators have downplayed the value of learning "standard" algorithms.

When my kids were in elementary school, a big poster in the classroom that listed "problem solving strategies" has guess & check up at the top.

There is also the difference between a mathematical explanation and a verbal explanation. I remember showing each change in algebra and listing the identity used.

Right - exactly!

"Russian Math" has 6th graders do that. ("Mathematics 6")

My son languished in kindergarten for months because they didn't do the reading tests until late November. My son was reading simple chapter books at that point. When I had my PT conference with the teacher, after the test had been given, the teacher told me how well he had done, and then said she was completely surprised because "boys don't read".

Now, in that particular class, it turned out that the three top readers were all boys, all of whom were already reading when they entered kindergarten. I knew those boys parents pretty well. We all had this same experience - the teacher just ignored these boys because she couldn't believe that boys would be readers. At the spring conference, the teacher commented again that she was so surprised to have these boys who were readers because "boys don't read".

It is true that *on average* boys lag behind girls in verbal skills. However, those are averages. Variation within each gender is greater than between the genders. Most kids (and adults) are really mixed of both male and female stereotypical traits. Treating boys as automatically behind is as damaging as treating girls as automatically bad at math (and I was one of those girls who was good at math, back in the pre-feminist 60's and 70's, so I do remember how awful it was). I often wonder if it becomes a self-fulfilling prophecy - we expect boys to be behind so they are.

"Showing your work and explaining it is very important in higher level math. If kids don't start practicing that skill early on, they won't learn how to do it. I wasn't taught, and was blown away in college when professors started asking that we show all our work."

And it is important to "show your work" in even simple multi-step math when you are outside of school.

At work, we may do some sort of computation to figure out how much RAM we need to allocate for something. We can't just present a number to the guys in the review ("hey, we need 36GB"). They want to see the calculation to verify that (a) the model is correct, and (b) that we didn't screw up the calculations using that model.

I find that my kid is fairly okay with the idea that he has to show his steps because this is something that Dad does at work. I think that this makes it much less artificial.

I'd like to see schools emphasize that often in the real world, people will want to see *HOW* you got your answer, too. It isn't just the teachers adding arbitrary busywork.

Thank you Mark Roulo, I completely agree. The biggest complaint we get from employers who want to hire computer science graduates is that they can't communicate. And often, communicating means convincing a less-technical higher up (often someone who is technical but just not as immersed in the details) that your solution is going to work. I make my computer science students explain their programs, and no, I do not consider it busywork.

How are they explaining their programs, with words or using something like UML? Is this for documentation purposes or analysis purposes? Who is the audience, another technical person or a lay person?

K-6 math curriculum techniques do not necessarily translate into any sort of understanding or communication fix when students become adults. The other choice is not some sort of rote, no understanding approach to math.

K-6 math curricula talk on and on about understanding in vague terms as if whatever they do is good. Compared to what, exactly, bad teaching? My traditional math classes drilled line-by-line algebraic rigor into us. The communication was in the math, not paragraphs. If adults have difficulty communicating, exactly what does that mean? How is that directly related to what they do in K-6 math? Fuzzy math has been around for decades, so one should see an improvement by now in CS majors.

The issue is not about whether understanding or explaining is good or bad. The question is exactly what that means. K-6 math curricula hide behind the talk, but fail to show any results. The point was not that kids like math only if the don't have to explain things. It's about how many K-6 math curricula do that in terms of words and not math.

I find that UML works best as a visual aid while verbally explaining the system. If you have to do the explanation via report, UML may be one component, but there is usually a lot of text around it.

I worked in the software industry for years. I think that I often spent more time writing and presenting than coding. We never liked to hire people fresh out of college because they were so poor at communication.

## 31 comments:

If having to explain how you got your answer makes you hate math, then it wasn't math that you liked to begin with. Proofs that others (or at least others that are expert in the same field) can read, follow, and understand, are the core of mathematics.

However, I've often seen math ruined by having to write an explanation for something like why 5 + 2 = 7.

Back when my daughter was in 1st grade, I was ok with them having to draw a picture to show why 5+2+7. This year in 2nd, I was a little less understanding. If she is still doing this in 3rd grade, I will not be happy.

Or better yet, Envision math. Instead of asking the kids, 7*8 = ? (which my tutoring student knows), they make them do a problem like this:

Show how you can use 4 math facts to calculate 7*8 (which my little guy just doesn't get WHY he would have to do that, NOR does he understand, when he already knows the answer is 56).

What they are looking for is:

4*7 = 28 +

4*7 = 28

= 56

So what is ultimately more important (or satisfying as a student)?

Sigh.

Envision Math - a mile wide and one mm deep.

Its a nice post.Well i like maths a lot.Its very interesting subject.

If you can't explain how you got the solution, you probably don't understand it very well. My son, who is generally good at math, was having trouble with the explanations and with showing his work (which, btw has been around as long as I can remember). When we worked with him a bit, it became clear that he was shaky with the concepts. He could do the problem, plug and chug style, but I don't think that is good enough.

On the other side of this, he also had to do the "draw a picture for 2+5" in first grade. Every single day. They started with 1+1, went to 1+2, and on and on. Horrible, absolutely soul-deadening.

When my daughter was in elementary school, math homework (which consisted of simple addition or multiplication facts) always included a line for "How do you know?" My daughter's standard answer was "Because I am smart."

In multi-step problems, showing your work IS explaining why, to a great extent. In order to choose the next step, you have to know (at least on an intuitive level) why you are choosing it; otherwise, you'd be likely to choose something else.

Note that in the story excerpt, "having the right answer isn't as important as explaining how you got it...." so they aren't really talking about proofs in math. They are talking about bs.

The old "show your work battle" is one worth winning (on the teacher's side) exactly for the reason Bonnie has said. I try to get my home-schooled kids to write a short verbal tag or label for every expression they use, in story problems, as a way to keep them from just munging around with numbers. They don't have to write sentences explaining why simple facts are true, however.

Students' explanations are sometimes very enlightening. When my dad taught remedial math, he asked his students to explain their reasoning when doing some "story problems."

They explained: When a problem has three numbers, you add. When it has two numbers, you multiply.

Some observations:

Boys are typically 1-2 years behind girls in literacy skills up to at least middle school. Writing explanations can be an unwelcome obstacle to a boy's math studies, which has its own non-literary language. Writing may have the effect of turning boys off to math.

Explaining your work often means writing a paragraph (I think this is in the NY state tests, for example) or a journal entry. Often for the math-gifted boy, this can be a long and painful distraction from his enjoyment of math.

I generally support "showing your work" and explaining answers, but for the gifted math student I think it is often unnecessary for purposes other than testing. These kids "get it", and are sometimes too immature to appreciate the importance of explanations.

Katherine Beals has written about this topic, so maybe she'll chime in.

A math major I knew in college, a woman, said that her method for arriving at the right answer was "I just look at it, and I know it." There's no room for that anymore.

Hainish,

A lot of bright/gifted kids are like that. The gifted teacher at our grade school had to pull kids for two weeks to teach them how to draw charts for a simple multiplication word problem. The kids knew to just multiply, but they needed to draw the chart or they wouldn't get it right.

When my son was in the third grade, I would have him do Singapore 5 or 6 word problems. He often would just put down the answer and it would be right. I asked him how he got to the answer and he would look at me like I was crazy. It was simple to him, in his head.

I certainly believe in showing one's work (in a mathematical way, not writing out paragraphs) to get credit, but our state tests gave more credit to kids who showed all the steps (in arithmetic) but got the answer wrong, than those who got the answer right, but didn't show the steps.

The gifted teacher basically had to teach her kids to go back two years, before they could think abstractly about solving a problem. They were beyond the picture drawing stage, but were penalized for it anyway.

SusanS

Showing your work, when there is work to show, is an important part of math, but writing journal entries about math or explaining why 7*8=56 is just an attempt to turn mathophiles into mathophobes.

The Art of Problem Solving courses require explanations for the answers to their homework questions, but the questions are difficult multi-step questions for which the path from problem to solution is not obvious.

There has to be something to explain before explanation becomes a useful exercise.

Anonymous - I'd be interested in any other opinions on or experiences with Envision Math, since that's what our local elementary school uses.

I'd seen some red flags already, like pushing forward too fast at first then spiraling and the fifth grade teacher spending time on geometry and algebra topics but then reminding parents to work on their kids' multiplication facts, so your description doesn't add to my confidence.

...our state tests gave more credit to kids who showed all the steps (in arithmetic) but got the answer wrong, than those who got the answer right, but didn't show the steps.That's always bothered me. And acceptable "steps" include guess & check. Thank you, constructivist math!

My boy was not behind in literacy. I really do not like the assumption that boys will always be behind. That assumption, on the part of several teachers, hurt my son's education. In any case, he couldn't explain his work because he didn't actually understand the algorithm he had memorized.

Showing your work and explaining it is very important in higher level math. If kids don't start practicing that skill early on, they won't learn how to do it. I wasn't taught, and was blown away in college when professors started asking that we show all our work. My chemistry professor, in a fit of exasperation, ended up spending a couple of classes tutoring us in the fine art of explaining ourselves.

I agree that journal entries are silly. That happens because teachers don't understand themselves how to explain steps in math.

Professors have given partial credit for wrong answers with the right steps since at least the days when my FATHER started teaching physics, which is why he always yelled at me to show my work. This is not some new constructivist thing.

One other advantage to carefully showing every step of your work (and labelling things ) - you are FAR less likely to make silly mistakes. That was another problem for my son.

Showing each line of your work goes back forever, but there's a difference between being careful and understanding. Go ahead and ask kids to explain 4 different ways to multiply 7*8, but at some point (soon!) you have to move on. For many kids, the torture lies in not moving on.

There is also the difference between a mathematical explanation and a verbal explanation. I remember showing each change in algebra and listing the identity used. That's a lot different than explaining using paragraphs. At some point, the math does the explaining all by itself. That's the whole point of math.

There are different levels of understanding and most K-6 math curricula get stuck at the basic level. You can see it fall apart in the later grades. The understandings of fractions are in terms of pictures that never translate into the mathematical understandings that allow students to deal with rational expressions. They get stuck at conceptual understanding and never get to mathematical understanding.

This would all be fine if kids ended up mastering the skills needed to get to algebra in 8th grade. It doesn't happen. K-6 schools go on and on and on about understanding, but they never get the job done, either via rote skills or via understanding. That's because they trivialize skills and their understandings are superficial. It strikes me that deep down, they don't like math. They want to change it into something else.

Envision Math - a mile wide and one mm deep.Love it!

Haven't read all the comments, but wanted to point out that this excerpt is about a high school class in math.

To me, "mental notes" is a kind of mental debriefing akin to psychotherapy: why did I do what I just did?

People who are proficient in a subject - any subject at all - very often can't say why they did what they did. There's a long history of research into this phenomenon, which I believe was one of the reasons why the study of the 'cognitive unconscious' was founded.

It seems to me that a high school student who is good at math should be learning to do proofs, not writing "Mental Notes."

My daughter's standard answer was "Because I am smart."Oh my gosh!

Didn't she get in trouble for that??

This reminds me of the coaching we did with C. back when he had to explain his math work.

We made him memorize a stock answer: "I looked for a pattern, and then I used guess and check."

They explained: When a problem has three numbers, you add. When it has two numbers, you multiply.I've heard that, too!

In multi-step problems, showing your work IS explaining why, to a great extent.YES!

Bonnie --

Very bad that your son's teachers assumed he was behind in lit. skills. It's hard to imagine how that could happen, with testing and reading assignments. I can only guess these teachers were not paying attention.

I was probably not too clear, but I was referring to the "guess & check" part in thanking constructivism. There's a recent emphasis on that as educators have downplayed the value of learning "standard" algorithms.

When my kids were in elementary school, a big poster in the classroom that listed "problem solving strategies" has guess & check up at the top.

There is also the difference between a mathematical explanation and a verbal explanation. I remember showing each change in algebra and listing the identity used.Right - exactly!

"Russian Math" has 6th graders do that. ("Mathematics 6")

My son languished in kindergarten for months because they didn't do the reading tests until late November. My son was reading simple chapter books at that point. When I had my PT conference with the teacher, after the test had been given, the teacher told me how well he had done, and then said she was completely surprised because "boys don't read".

Now, in that particular class, it turned out that the three top readers were all boys, all of whom were already reading when they entered kindergarten. I knew those boys parents pretty well. We all had this same experience - the teacher just ignored these boys because she couldn't believe that boys would be readers. At the spring conference, the teacher commented again that she was so surprised to have these boys who were readers because "boys don't read".

It is true that *on average* boys lag behind girls in verbal skills. However, those are averages. Variation within each gender is greater than between the genders. Most kids (and adults) are really mixed of both male and female stereotypical traits. Treating boys as automatically behind is as damaging as treating girls as automatically bad at math (and I was one of those girls who was good at math, back in the pre-feminist 60's and 70's, so I do remember how awful it was). I often wonder if it becomes a self-fulfilling prophecy - we expect boys to be behind so they are.

When my kids were in elementary school, a big poster in the classroom that listed "problem solving strategies" has guess & check up at the top.

When I first started writing kitchen table math, "Guess and Check" logos were all over the web.

I wonder if they still are?

I think whole language/balanced literacy is in some ways a form of guess and check reading.

"Showing your work and explaining it is very important in higher level math. If kids don't start practicing that skill early on, they won't learn how to do it. I wasn't taught, and was blown away in college when professors started asking that we show all our work."And it is important to "show your work" in even simple multi-step math when you are outside of school.

At work, we may do some sort of computation to figure out how much RAM we need to allocate for something. We can't just present a number to the guys in the review ("hey, we need 36GB"). They want to see the calculation to verify that (a) the model is correct, and (b) that we didn't screw up the calculations using that model.

I find that my kid is fairly okay with the idea that he has to show his steps because this is something that Dad does at work. I think that this makes it much less artificial.

I'd like to see schools emphasize that often in the real world, people will want to see *HOW* you got your answer, too. It isn't just the teachers adding arbitrary busywork.

-Mark Roulo

Thank you Mark Roulo, I completely agree. The biggest complaint we get from employers who want to hire computer science graduates is that they can't communicate. And often, communicating means convincing a less-technical higher up (often someone who is technical but just not as immersed in the details) that your solution is going to work. I make my computer science students explain their programs, and no, I do not consider it busywork.

How are they explaining their programs, with words or using something like UML? Is this for documentation purposes or analysis purposes? Who is the audience, another technical person or a lay person?

K-6 math curriculum techniques do not necessarily translate into any sort of understanding or communication fix when students become adults. The other choice is not some sort of rote, no understanding approach to math.

K-6 math curricula talk on and on about understanding in vague terms as if whatever they do is good. Compared to what, exactly, bad teaching? My traditional math classes drilled line-by-line algebraic rigor into us. The communication was in the math, not paragraphs. If adults have difficulty communicating, exactly what does that mean? How is that directly related to what they do in K-6 math? Fuzzy math has been around for decades, so one should see an improvement by now in CS majors.

The issue is not about whether understanding or explaining is good or bad. The question is exactly what that means. K-6 math curricula hide behind the talk, but fail to show any results. The point was not that kids like math only if the don't have to explain things. It's about how many K-6 math curricula do that in terms of words and not math.

I find that UML works best as a visual aid while verbally explaining the system. If you have to do the explanation via report, UML may be one component, but there is usually a lot of text around it.

I worked in the software industry for years. I think that I often spent more time writing and presenting than coding. We never liked to hire people fresh out of college because they were so poor at communication.

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