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# How Much Mental Math Do You Need to Learn?

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### #1 EngOZ

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Posted 30 May 2017 - 07:29 PM

Last week we came across some mental math problems (CWP 3) that involved 4-5-6 digits problems involving multiplication, addition, subtraction and division.  Is it really necessary or even useful to spend time training you kids to solve mental math problems at this difficulty level?

Edited by EngOZ, 30 May 2017 - 07:30 PM.

### #2 elmerRex

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Posted 30 May 2017 - 09:21 PM

What was the problem?

Was it meant to have child use simplification/estimation or where they asked to be exact when they answered? ?

We do some years of training with the mentals maths with the goal of having children capable of doing flash anzan competition, but I don't feel it's necessary. But for our children its useful. For us, advanced mental maths comes to be useful in that it gives the child tremendous confidence and training them to juggle numbers in their head helps them. Also it teaches them good  habits to organize numbers mentally in their mind.

However I would not slow down other work to gain this skill. We are not going to compete internationally so being quick with 5 digits is enough. What does your currency go up to?

I want my children to be able to confidently use 5 digits because in this country you have 2-3 digits for dollars and 2 more digits for cents.

### #3 sunnyday

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Posted 31 May 2017 - 01:25 PM

I think working memory is a muscle that can be built up, one that weakens if left unused. I think that manipulating numbers abstractly requires a greater depth of understanding and number sense than just going through a rote algorithm to perform an operation. And I find that my kids get a great sense of accomplishment if they can come up with a strategy to tackle a large calculation and hold multiple steps in their head. So yes, I do think there's value in working on this skill.

"Training" the skill as a separate thing? Not really, not so much. I mean, the goal of getting educated in mathematics is to train the mind to work with patterns and abstractions. If all we needed from math was the ability to calculate, the marvels of this technological age mean that we are almost never more than an arms reach from some type of electronic calculator. Human calculations are largely redundant.

### #4 EngOZ

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Posted 31 May 2017 - 07:15 PM

What was the problem?

Was it meant to have child use simplification/estimation or where they asked to be exact when they answered? ?

We do some years of training with the mentals maths with the goal of having children capable of doing flash anzan competition, but I don't feel it's necessary. But for our children its useful. For us, advanced mental maths comes to be useful in that it gives the child tremendous confidence and training them to juggle numbers in their head helps them. Also it teaches them good  habits to organize numbers mentally in their mind.

However I would not slow down other work to gain this skill. We are not going to compete internationally so being quick with 5 digits is enough. What does your currency go up to?

I want my children to be able to confidently use 5 digits because in this country you have 2-3 digits for dollars and 2 more digits for cents.

Thanks for your reply elmerRex, great food for thought.

Here's a few of the problems from CWP - page 32-33:

158+93+42

997+605

126-75

163-92

85x5

462x5

620/5

905/5

There are a lot of suggested worked examples, but my initial thoughts were why not pull out the calculator and be done with it?

I think working memory is a muscle that can be built up, one that weakens if left unused. I think that manipulating numbers abstractly requires a greater depth of understanding and number sense than just going through a rote algorithm to perform an operation. And I find that my kids get a great sense of accomplishment if they can come up with a strategy to tackle a large calculation and hold multiple steps in their head. So yes, I do think there's value in working on this skill.

"Training" the skill as a separate thing? Not really, not so much. I mean, the goal of getting educated in mathematics is to train the mind to work with patterns and abstractions. If all we needed from math was the ability to calculate, the marvels of this technological age mean that we are almost never more than an arms reach from some type of electronic calculator. Human calculations are largely redundant.

Hi Sunnyday, thanks for your reply. Since that post, I decided, as far as mastery of mental maths, that I would only require mastery of multiplication (up to 1-12) and addition/subtraction (1-12).

We did finish multiplication, and we're now working through two books on addition and subtraction facts by Kate Snow. However, when I look at the worked solutions in CWP, it becomes a little overwhelming.  I just don't know if it's worth making the effort to go beyond these basics.

Edited by EngOZ, 31 May 2017 - 07:20 PM.

### #5 winterbaby

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Posted 31 May 2017 - 07:39 PM

Thanks for your reply elmerRex, great food for thought, particularly on holding numbers in your head  and in daily application with dollars and cents.

Here's a few of the problems from CWP - page 32-33:

158+93+42

997+605

126-75

163-92

85x5

462x5

620/5

905/5

There are a lot of suggested worked examples, but my initial thoughts were why not pull out the calculator and be done with it?

Well, I can do any of those much faster than I can reach into my pocket and bring up the calculator app. If someone never reaches that level of facility after trying, perhaps due to problems with working memory, that's one thing; but not to try to learn how to do these means missing out on important aspects of number sense as it relates to place value. The sort of breaking up and rearranging of elements you need to do these in your head - quite different from the vertical pencil and paper algorithms - is a powerful tool for relating math knowledge to the real world. It's rare that real life will require you to just recall exactly 8 times 7 or 11 times 11, so having facts to 12 memorized alone is of limited value. The facility with place value that makes problems like these easy (or at least doable, depending on the person) is the other half of what's needed to say one really understands arithmetic.

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### #6 sunnyday

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Posted 04 June 2017 - 10:46 AM

If your student doesn't have the stamina for these yet, then maybe you don't want to push their interest level by insisting on multiple problems like this every day. Pace yourselves. But these examples have all been chosen to demonstrate how, as humans, we can work SMARTER than a calculator and reduce the calculation required and the memory space needed. They aren't difficult at all to someone who's had any practice looking for the clever way.

A student who's come all the way up through Singapore, for example, shouldn't have any problem at all seeing a thousand in 997 + 605 and near-instantly seeing the outcome of the addition problem. If they haven't had a lot of practice composing and decomposing units of higher value (eg. making tens) then that's the skill level you need to be working more on, not just brute-force chugging through these problems without the benefit of paper.

My DD is working on third grade math but we haven't built those skills and she would be very intimidated by these problems. Our summer plans will include games to build her subitizing skills.

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### #7 elmerRex

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Posted 04 June 2017 - 11:37 AM

Thanks for your reply elmerRex, great food for thought.

Here's a few of the problems from CWP - page 32-33:

158+93+42

997+605

126-75

163-92

85x5

462x5

620/5

905/5

There are a lot of suggested worked examples, but my initial thoughts were why not pull out the calculator and be done with it?

Is the child allowed to see the problems while they work them? Or must they go by hearing only? When they are not mastered with the technique, my children find it easier to look at the problem while they figure it in their head until they gain more fluency with the technique. Only my oldest can do these problems by ear only and that is after a couple of years of regular learning and practice.

Here is how I would be coaching a child to do each of those problems mentally.

Always find tidy numbers where you can.

158+93+42 becomes 158 + 93 + 40 + 2 becomes 160 + 40 + 93 becomes 200 + 93 = 293

997+605 becomes 997 + 602 + 3 becomes 1000 + 602 = 1602

126-75 becomes (100 + 26) - 75 becomes 100-75 + 26 becomes 25 + 26 = (25 + 25) + 1 = 51

163-92 becomes  100 + 63 - 92 becomes 100-92 + 63 becomes 8 + 62 + 1 becomes 70 + 1 = 71

85x5 becomes (80+5) x 5 becomes 400 + 25 = 425

462x5 becomes (400 + 60 + 2) x 5 becomes 2000 + 300 + 10 = 2310

620/5 becomes (600 + 20) / 5 becomes 120 + 4 = 124

905/5 becomes (900 + 5) / 5 becomes 180 + 1 = 181 and here, since my child wouldn't know 5*18 rote, we'd work back from 5*20 (Which they know by rote) 5*20=100, so that's too big. 5*19=100-5 or 95, so obviously 5*18=19, or 5*180=190.

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### #8 MistyMountain

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Posted 04 June 2017 - 02:01 PM

I think the examples you gave can be handled mentally if they have built up their mental math skills and have a good sense of place value and decomposition and I do think it comes in handy to have those skills. I have one kid who can handle them and one I need to really work on that skill with and it will be harder for.

Edited by MistyMountain, 04 June 2017 - 02:07 PM.

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### #9 calbear

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Posted 05 June 2017 - 09:39 PM

If you have not worked mental math strategies then those problems are going to look overwhelming. The Fan Math Express Math Strategies workbooks are pretty good for teaching these skills incrementally with practice. The problem pages in CWP assume that you have been learning the strategies that lead up to those problems. If you haven't been doing it, then it will seem too much.

If you have not done any mental math strategies, you may want to start at level one. IMO, addition/subtraction/multiplication/division facts aren't really mental math strategies, but part of your tool box you use to do mental math if that makes sense. I found my son gained a much better sense of mathematical relationships and a much better working understanding of the commutative properties of addition and multiplication working on this skill set of these mental math strategies (if they are not just memorizing the strategy but understand why it works that way).

Edited by calbear, 06 June 2017 - 09:43 PM.

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### #10 EngOZ

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Posted 06 June 2017 - 09:43 PM

Thanks that makes complete sense. I'll have a look into Fan Math

If you have not worked mental math strategies then those problems are going to look overwhelming. The Fan Math Express Math Strategies workbooks are pretty good for teaching these skills incrementally with practice. The problem pages in CWP assume that you have been learning the strategies that lead up to those problems. If you haven't been doing it, then it will seem too much.

If you have not done any mental math strategies, you may want to start at level one. IMO, addition/subtraction/multiplication/division facts aren't really mental math strategies, but part of your tool box you use to do mental math if that makes sense. I found my son gained a much better sense of mathematical relationships and a much better working understanding of the commutative properties of addition and multiplication working on this skill set as these mental math strategies if they are not just memorizing the strategy but understand why it works that way.

### #11 underthebridge

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Posted 07 June 2017 - 08:25 AM

Mental math is important because it encourages the use of number sense and conceptual thinking over procedural thinking. There was an interesting study by Eddie Gray and David Tall that examined the correlation between students' math achievement and their use of memorized math facts, derived math facts, and counting. In a nutshell, higher achieving students mostly used a combination of memorized facts and derived facts while lower achieving students mostly used memorized facts and counting.

Higher achieving students used number sense to derive facts - for example,

15 + 8

would be solved by thinking of it as

10 + 5 + 8 = 10 + 13

or

13 + 2 + 8 = 13 + 10

or

15 + 5 + 3 = 20 + 3

The authors argued that using number sense to compose and decompose in arithmetic also carried over to understanding how to compose and decompose in algebraic functions and more complex math.

I think of mental math not as a computation tool, but as a way to develop flexible number sense. If remembering numbers and digits are difficult, I would write the problems down horizontally (not stacked) and take turns finding different ways to solve the problem.

For example

339 + 494

Method 1: Swap the ones digits
334 + 499 = 333 + 500

Method 2 : Make a 10
333 + 500

Method 3 : Add left to right
700 + 120 + 13

Good luck OP!

Edited by underthebridge, 07 June 2017 - 08:26 AM.

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