by Josh Rappaport
Everyone who teaches math knows it’s easy for students to make mistakes. After all, there’s often just one way to get a math problem right, but many ways to get it wrong.
And in addition, the more steps a problem has, the greater the chances for making a mistake. When you consider all of this, doesn’t it seem miraculous that anyone ever gets a math problem right?!
As easy as it is to make math mistakes —and despite the fact that children know that mistakes happen — kids often get quite upset when math mistakes “happen” to them. I’ve watched students break down in tears when they make a slew of mistakes. I’ve seen students get so frozen by the fear of mistakes that they refuse to do any but the simplest math problems. And so this fear of math mistakes, if we’re not careful, can stunt a child’s willingness to do math in general, as well as their openness to exploring mathematical ideas, which is crucial for enjoying math.
So what’s the solution? Through my work tutoring students, I’ve found that the way we talk to students about mistakes can have a big impact on their “math esteem,” the way they view themselves as young math students. To bolster students’ “math esteem,” I’ve developed a number of strategies that are really quite easy to use, and that can be employed by anyone in the teaching role.
This month I’d like to share a few of these approaches, in hopes that they’ll inspire and help homeschooling parents.
STRATEGY #1: NORMALIZING. One approach is to assure students that their mistakes are actually normal. “Oh, that’s a very common mistake,” I’ll say. Or even better: “That’s a really understandable mistake.” And then I’ll explain why it is that people might make this mistake.
EXAMPLE: Take the problem, 82 – 35. When students start work on this problem, they subtract in the 1s place. If students think about this correctly, they see 2 – 5 and realize they need to “borrow,” or “re-group.” But some students never get that far. Interestingly, their minds turn this part of the problem around, and they see it instead as 5 – 2. As a result, they get an answer of 3 at this stage. I tell students that it’s understandable to look at 2 – 5, re-cast it as 5 – 2, and mistakenly get 3 as the partial answer. And then I suggest a reason for this mental flip-flop. Students get flummoxed by the idea of subtracting a larger number from a smaller one (2 – 5), so their minds turn it around, changing it to the more familiar problem of subtracting a smaller number from a larger one (5 – 2). Once I tell my student this is a possible cause of his mistake, he usually relaxes and either agrees that this is what he did, or he offers a different explanation for how he made the mistake. The key is to make it clear that this kind of mistake is understandable and therefore nothing to be ashamed of.
Helping students relax around their math mistakes also gives parent and child the opportunity to brainstorm ways to avoid such mistakes. Brief example: I tutored a boy who would often say “Yikes!” whenever he noticed he was supposed to subtract a larger number from a smaller one. So together, we agreed he should continue saying “Yikes!” In fact, saying “Yikes!” served as a “mathematical alarm clock,” alerting the boy that he needs to “borrow” or (to use the more modern term) “re-group.” So together, we turned this boy’s natural reaction into a useful tool for avoiding mistakes.
To broaden the discussion a bit, a pre-algebraic mistake that falls into this “typical mistake” category involves simplifying terms like 2^3 (2 to the 3rd power). 2^3 means 2 x 2 x 2, so it equals 8. However, when students first learn this idea, they often fall back to thinking of multiplication, and as a result, they often view 2^3 as
2 x 3. As a result, they often get 6 when they simplify this term. A parent can reassure her child that this is a typical mistake, and point out that she made it because she is not yet used to working with exponents. To follow up, then, it would be helpful to give such a student a set of focused practice problems — perhaps 10 to 20 problems just like 2^3, so she can immediately grasp the concept of exponents and get practice simplifying these terms correctly.
STRATEGY #2: EXPRESSING INTEREST. Another response to mistakes that I’ve found useful involves telling students that that I find their mistakes interesting. I’ll say things like, “Hmmm, that’s a fascinating mistake!” Or, even better: “Wow, that’s an amazingly interesting mistake!” At first kids often look at me as if I ‘have a screw loose.’ That’s because they’ve developed an attitude that mistakes are simply “bad,” something to be thrown into the mental garbage can — not pondered with interest. Also there’s the prejudiced idea of: “Who but a nutcase could possibly find math fascinating?”
No matter, though. If you do take an interest in mistakes, your ability to reach your child and help her with math will improve in several ways.
EXAMPLE: A child is just starting pre-algebra, so she is just beginning to solve simple equations. She needs to solve the mini-equation: 11c = 44. Her answer:
c = 33. Your response, “That is so interesting. I’m dying to know how you got that answer.” Your child tells you that she simply subtracted 11 from both sides. Aha! You have just learned something very important. Your daughter had not realized that 11c means 11 times c. She had been viewing 11c as 11 + c, and that explains why she subtracted 11 from both sides of the equation. She thought the 11 was connected by addition, and she was (correctly, in her mind) doing the opposite operation — subtraction — to eliminate this term. If you, the parent, had not taken an interest in her mistake, you possibly would have never realized that she held this misunderstanding. But now that you do know this, you can help her realize that in general, whenever a number and a variable stand side by side with nothing in between, those two terms are linked by multiplication, not by addition. Progress!
This leads to a point I like to make when I talk with students about mistakes and our approach toward them. Mistakes, I say, are like weeds. If you simply snip away at a weed, it will pop right up again in a few days. If you really want a weed to disappear, you need to rip it out by the roots. Likewise, if you tell a child that she simply made a math mistake, and that she simply needs to do things a different way, that very same mistake will reappear soon. Just as we need to get weeds out by the root, we need to uproot mistakes as well. And generally, the only way to do that is to trace the student’s incorrect thinking back till we find its conceptual “roots.” In the last example, the “root” of the girl’s problem was her misunderstanding about the meaning of a term like 11c. Without uprooting that misunderstanding and replacing it with the truth — that 11c means 11 times c — not 11 plus c — this mistake would have recurred.
STRATEGY #3: DOCUMENTING. It’s all well and good to use Strategies 1 and 2, but to give your work with mistakes that extra “oomph,” it’s critical that you also teach children how to document their mistakes. I do this by setting up and using what I call a “Mistake Journal.”
The “Mistake Journal” is quite simply, a bundle of identical sheets of paper on which students document their mistakes, over time. Each sheet has empty lines on which children respond to the following writing prompts:
1. What mistake did I make? [Here children write down the actual math mistake they made. They don’t need to copy down their entire answer if the mistake is just a small part of the answer, as it usually is. They need only copy down the line just before the mistake, and the line with the mistake.]
2. What’s the right way to work this problem? [Here children write down — always with your guidance — how they are supposed work the problem correctly, the part with the mistake.]
3. What was my misunderstanding? [In this section children track their thinking back and explain what lies at the root of their mistake. Students often need help wording their understanding of the root of the mistake, so stand by, ready to help, at this stage.]
4. How can I avoid this mistake? [Here children come up with creative ideas on how to avoid making this mistake again. At this stage, you help your child learn how to: a) spot the tricky situation when it comes up in a problem, and b) work the problem without making this same mistake.]
To make Mistake Journal templates, I simply write out these four question-prompts on a blank sheet of paper, and I leave lines for the students’ answers. Then I photocopy this sheet, and keep a sizeable (o.k., huge) stack of these empty Mistake Journal sheets close at hand. Then, whenever we come upon a mistake that warrants an entry, we grab one of the sheets, and the student answers the questions. I three-hole-punch the sheets and have students to keep their completed Mistake Journal entries in a three-ring binder, since that keeps the sheets in good shape.
But don’t stop there. Make sure that your students use their Mistake Journal sheets! Sometimes, while tutoring, I’ll just ask my young charge to read a recent Mistake Journal entry. To check whether or not my student has grasped this Mistake Journal lesson, I’ll give my student a few problems that require her to put her new understanding to use. For example, if she’s the student who was confused by the problem: 11c = 44, I’ll give her five problems just like that one, to see if she has really mastered the concept. I suggest that you do the same thing with your child.
Also, if you give your children tests, quizzes or math evaluations of any kind, encourage your kids to look over their Mistake Journal entries while studying for the evaluation. Reviewing those potential mistakes (and notes on how to avoid them) helps children feel more confident when they take the test.
So while children’s responses to math mistakes can be fraught with anxiety, the three techniques described in this article can help lower that level of anxiety. I encourage you to try out these approaches, to see if they make your math learning environment more relaxed, less frustrating, and more successful overall.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also the author of the Card Game Roundup books, and PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, www.mathchat.wordpress.com, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at homeschooling conferences., and tutors homeschoolers using SKYPE. You can reach Josh at: josh@SingingTurtle.com
Everyone who teaches math knows it’s easy for students to make mistakes. After all, there’s often just one way to get a math problem right, but many ways to get it wrong.
And in addition, the more steps a problem has, the greater the chances for making a mistake. When you consider all of this, doesn’t it seem miraculous that anyone ever gets a math problem right?!
As easy as it is to make math mistakes —and despite the fact that children know that mistakes happen — kids often get quite upset when math mistakes “happen” to them. I’ve watched students break down in tears when they make a slew of mistakes. I’ve seen students get so frozen by the fear of mistakes that they refuse to do any but the simplest math problems. And so this fear of math mistakes, if we’re not careful, can stunt a child’s willingness to do math in general, as well as their openness to exploring mathematical ideas, which is crucial for enjoying math.
So what’s the solution? Through my work tutoring students, I’ve found that the way we talk to students about mistakes can have a big impact on their “math esteem,” the way they view themselves as young math students. To bolster students’ “math esteem,” I’ve developed a number of strategies that are really quite easy to use, and that can be employed by anyone in the teaching role.
This month I’d like to share a few of these approaches, in hopes that they’ll inspire and help homeschooling parents.
STRATEGY #1: NORMALIZING. One approach is to assure students that their mistakes are actually normal. “Oh, that’s a very common mistake,” I’ll say. Or even better: “That’s a really understandable mistake.” And then I’ll explain why it is that people might make this mistake.
EXAMPLE: Take the problem, 82 – 35. When students start work on this problem, they subtract in the 1s place. If students think about this correctly, they see 2 – 5 and realize they need to “borrow,” or “re-group.” But some students never get that far. Interestingly, their minds turn this part of the problem around, and they see it instead as 5 – 2. As a result, they get an answer of 3 at this stage. I tell students that it’s understandable to look at 2 – 5, re-cast it as 5 – 2, and mistakenly get 3 as the partial answer. And then I suggest a reason for this mental flip-flop. Students get flummoxed by the idea of subtracting a larger number from a smaller one (2 – 5), so their minds turn it around, changing it to the more familiar problem of subtracting a smaller number from a larger one (5 – 2). Once I tell my student this is a possible cause of his mistake, he usually relaxes and either agrees that this is what he did, or he offers a different explanation for how he made the mistake. The key is to make it clear that this kind of mistake is understandable and therefore nothing to be ashamed of.
Helping students relax around their math mistakes also gives parent and child the opportunity to brainstorm ways to avoid such mistakes. Brief example: I tutored a boy who would often say “Yikes!” whenever he noticed he was supposed to subtract a larger number from a smaller one. So together, we agreed he should continue saying “Yikes!” In fact, saying “Yikes!” served as a “mathematical alarm clock,” alerting the boy that he needs to “borrow” or (to use the more modern term) “re-group.” So together, we turned this boy’s natural reaction into a useful tool for avoiding mistakes.
To broaden the discussion a bit, a pre-algebraic mistake that falls into this “typical mistake” category involves simplifying terms like 2^3 (2 to the 3rd power). 2^3 means 2 x 2 x 2, so it equals 8. However, when students first learn this idea, they often fall back to thinking of multiplication, and as a result, they often view 2^3 as
2 x 3. As a result, they often get 6 when they simplify this term. A parent can reassure her child that this is a typical mistake, and point out that she made it because she is not yet used to working with exponents. To follow up, then, it would be helpful to give such a student a set of focused practice problems — perhaps 10 to 20 problems just like 2^3, so she can immediately grasp the concept of exponents and get practice simplifying these terms correctly.
STRATEGY #2: EXPRESSING INTEREST. Another response to mistakes that I’ve found useful involves telling students that that I find their mistakes interesting. I’ll say things like, “Hmmm, that’s a fascinating mistake!” Or, even better: “Wow, that’s an amazingly interesting mistake!” At first kids often look at me as if I ‘have a screw loose.’ That’s because they’ve developed an attitude that mistakes are simply “bad,” something to be thrown into the mental garbage can — not pondered with interest. Also there’s the prejudiced idea of: “Who but a nutcase could possibly find math fascinating?”
No matter, though. If you do take an interest in mistakes, your ability to reach your child and help her with math will improve in several ways.
EXAMPLE: A child is just starting pre-algebra, so she is just beginning to solve simple equations. She needs to solve the mini-equation: 11c = 44. Her answer:
c = 33. Your response, “That is so interesting. I’m dying to know how you got that answer.” Your child tells you that she simply subtracted 11 from both sides. Aha! You have just learned something very important. Your daughter had not realized that 11c means 11 times c. She had been viewing 11c as 11 + c, and that explains why she subtracted 11 from both sides of the equation. She thought the 11 was connected by addition, and she was (correctly, in her mind) doing the opposite operation — subtraction — to eliminate this term. If you, the parent, had not taken an interest in her mistake, you possibly would have never realized that she held this misunderstanding. But now that you do know this, you can help her realize that in general, whenever a number and a variable stand side by side with nothing in between, those two terms are linked by multiplication, not by addition. Progress!
This leads to a point I like to make when I talk with students about mistakes and our approach toward them. Mistakes, I say, are like weeds. If you simply snip away at a weed, it will pop right up again in a few days. If you really want a weed to disappear, you need to rip it out by the roots. Likewise, if you tell a child that she simply made a math mistake, and that she simply needs to do things a different way, that very same mistake will reappear soon. Just as we need to get weeds out by the root, we need to uproot mistakes as well. And generally, the only way to do that is to trace the student’s incorrect thinking back till we find its conceptual “roots.” In the last example, the “root” of the girl’s problem was her misunderstanding about the meaning of a term like 11c. Without uprooting that misunderstanding and replacing it with the truth — that 11c means 11 times c — not 11 plus c — this mistake would have recurred.
STRATEGY #3: DOCUMENTING. It’s all well and good to use Strategies 1 and 2, but to give your work with mistakes that extra “oomph,” it’s critical that you also teach children how to document their mistakes. I do this by setting up and using what I call a “Mistake Journal.”
The “Mistake Journal” is quite simply, a bundle of identical sheets of paper on which students document their mistakes, over time. Each sheet has empty lines on which children respond to the following writing prompts:
1. What mistake did I make? [Here children write down the actual math mistake they made. They don’t need to copy down their entire answer if the mistake is just a small part of the answer, as it usually is. They need only copy down the line just before the mistake, and the line with the mistake.]
2. What’s the right way to work this problem? [Here children write down — always with your guidance — how they are supposed work the problem correctly, the part with the mistake.]
3. What was my misunderstanding? [In this section children track their thinking back and explain what lies at the root of their mistake. Students often need help wording their understanding of the root of the mistake, so stand by, ready to help, at this stage.]
4. How can I avoid this mistake? [Here children come up with creative ideas on how to avoid making this mistake again. At this stage, you help your child learn how to: a) spot the tricky situation when it comes up in a problem, and b) work the problem without making this same mistake.]
To make Mistake Journal templates, I simply write out these four question-prompts on a blank sheet of paper, and I leave lines for the students’ answers. Then I photocopy this sheet, and keep a sizeable (o.k., huge) stack of these empty Mistake Journal sheets close at hand. Then, whenever we come upon a mistake that warrants an entry, we grab one of the sheets, and the student answers the questions. I three-hole-punch the sheets and have students to keep their completed Mistake Journal entries in a three-ring binder, since that keeps the sheets in good shape.
But don’t stop there. Make sure that your students use their Mistake Journal sheets! Sometimes, while tutoring, I’ll just ask my young charge to read a recent Mistake Journal entry. To check whether or not my student has grasped this Mistake Journal lesson, I’ll give my student a few problems that require her to put her new understanding to use. For example, if she’s the student who was confused by the problem: 11c = 44, I’ll give her five problems just like that one, to see if she has really mastered the concept. I suggest that you do the same thing with your child.
Also, if you give your children tests, quizzes or math evaluations of any kind, encourage your kids to look over their Mistake Journal entries while studying for the evaluation. Reviewing those potential mistakes (and notes on how to avoid them) helps children feel more confident when they take the test.
So while children’s responses to math mistakes can be fraught with anxiety, the three techniques described in this article can help lower that level of anxiety. I encourage you to try out these approaches, to see if they make your math learning environment more relaxed, less frustrating, and more successful overall.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also the author of the Card Game Roundup books, and PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, www.mathchat.wordpress.com, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at homeschooling conferences., and tutors homeschoolers using SKYPE. You can reach Josh at: josh@SingingTurtle.com
