# The opposite of mental math??? MM/CLE

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So, if MM and Singapore are mental math (Asian/Finland), then what are CLE and Saxon and others? How are these taught vs the mental way? Counting fingers? They still memorize facts, Of course, but just use a paper and write things out? I think I was taught that way. Can't add/subtract well in my head.

You would be even more helpful to me if you have used MM and CLE and could explain what you see as the differences in the way, say, addition and multiplication are taught. I'm amazed at how MM teaches stuff. I was not taught to do some of the math mind gymnastics like my daughter does.

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I just got CLE 302. We've used a bits of mammoth 2a and 3a. What I haven't seen yet in CLE is breaking down numbers to add. If I remember correctly, 36+18 in MM, they count to 20 to complete the 18 and subtract 2 from 36 to get 34, then 20 and 34 to get 54. Of course, I've only seen very little of both programs.

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One thing I have learned, is that different people are using the same vocabulary different ways. :glare:

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One thing I have learned, is that different people are using the same vocabulary different ways. :glare:

What do you mean?

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Regarding CLE, it really depends where you start. I have used CLE 1-5th. Facts are introduced as fact families and manipulives can be used if needed in first grade but sometimes they just offer pictures. They then move on to memorizing the facts. I think that this process is really dependent on the parent and how closely they are following the TM and can add in what their child needs. Then addition/subtraction is reviewed in 2nd grade with less of the conceptual side because they assume the students in their schools have already learned it. For multiplication it is taught with grouping using pictures and that builds from the end of 2nd grade all through 3rd until they move into memorization. They do something similar with division with remainders as well. It builds from the end of 3rd into 4th grade until they move into long division.

It's not as conceptual as Singapore but it isn't lacking either. It may seem that way at first but if you pay attention, you will see that they are really very incremental and break it down into small steps that gradually build on each other and those early steps explain the why. It isn't just about going through the motions.

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I just got CLE 302. We've used a bits of mammoth 2a and 3a. What I haven't seen yet in CLE is breaking down numbers to add. If I remember correctly, 36+18 in MM, they count to 20 to complete the 18 and subtract 2 from 36 to get 34, then 20 and 34 to get 54. Of course, I've only seen very little of both programs.

This is what I mean. Except my daughter would do (for 36+18) add 2 to 18 to make 20, so 36 plus 20 is 56 and now take away 2 getting an answer of 54.

Does CLE teach in that way?

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One thing I have learned, is that different people are using the same vocabulary different ways. :glare:

:iagree:and there are different degrees to all those definitions.

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This is what I mean. Except my daughter would do (for 36+18) add 2 to 18 to make 20, so 36 plus 20 is 56 and now take away 2 getting an answer of 54.

Does CLE teach in that way?

No, they do not (although, I think it was in a "just for fun lesson" and slightly covered in estimating and rounding). They teach the standard method of regrouping.

Edited by jannylynn
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I really think it is more of a continuum than one program does this and the other does not.

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LA Mom, I think what you are really asking about is Constructivism, but maybe you are unfamiliar with that term. (I have some info here on my blog.) Programs like Right Start are very Constructivist. Singapore is in the middle, but leaning towards the Constructivist side. Saxon and Horizon would be at the opposite end of the Constructivist spectrum.

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I agree that the terms are being used in different ways. What programs like Singapore and MM do is delve more into the conceptual side of math, the "why" approach. CLE and Saxon do less of this, but I wouldn't say they're not conceptual at all... I do find that some of their explanations are not as deep.

Mental math, at its base, is really just, can you do math in your head? Singapore and MM do tend to give more methods to do this, like the making 10's method. But Saxon and CLE do both have mental math components as well.

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So, if MM and Singapore are mental math (Asian/Finland)......

Singapore Math is not "mental math," rather "mental math" techniques are among the scope of topics and techniques that are covered (and get some emphasis) in Singapore Math.

There are a great number of mental math strategies that are employed. They are helpful in my POV not only because they make make doing math "in ones head" easier, but because the ability to manipulate numbers using mental math techniques helps children (and adults who encounter this style of math on their "second time around") really see how the number system works.

It would be wrong to call the using a "pencil and paper" based standard algorithm the "opposite" of mental math. It is not the opposite, just an alternative method of calculation. Each alternative has its place. Singapore math (and Math Mammoth) teach both mental math techniques and the standard algorithms—as they should.

Along with mental math techniques the better math programs teach for a depth of mathematical understanding. Mental math is part of that because to do mental math one has to have a good basic understanding of what one is trying to accomplish.

Unfortunately some of the math programs that only rely on teaching the standard algorithms (with a plug and chug approach) and emphasize the memorization of "math facts" can create an impression of false competence, because students can work the algorithms without necessarily understanding what they are doing with any sort of depth.

Bill

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Singapore Math is not "mental math," rather "mental math" techniques are among the scope of topics and techniques that are covered (and get some emphasis) in Singapore Math.

There are a great number of mental math strategies that are employed. They are helpful in my POV not only because they make make doing math "in ones head" easier, but because the ability to manipulate numbers using mental math techniques helps children (and adults who encounter this style of math on their "second time around") really see how the number system works.

It would be wrong to call the using a "pencil and paper" based standard algorithm the "opposite" of mental math. It is not the opposite, just an alternative method of calculation. Each alternative has its place. Singapore math (and Math Mammoth) teach both mental math techniques and the standard algorithmsâ€”as they should.

Along with mental math techniques the better math programs teach for a depth of mathematical understanding. Mental math is part of that because to do mental math one has to have a good basic understanding of what one is trying to accomplish.

Unfortunately some of the math programs that only rely on teaching the standard algorithms (with a plug and chug approach) and emphasize the memorization of "math facts" can create an impression of false competence, because students can work the algorithms without necessarily understanding what they are doing with any sort of depth.

Bill

Yeah that. Not feeling particularly eloquent today, Bill - thanks for speaking my mind ;)

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It turns out they do teach that method of mental addition but they teach the regrouping algorithm too. It just came up with my DS today. He said he adds that way all the time irl.

Pictured is the review in 4th grade from 3rd.

Edited by jannylynn
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When I think of the mental math in MM that I see, the difference from when I was learning as a kid and from the math my kids were doing with Harcourt, is that the mental math is explicitly taught. When I was a kid, nobody taught us to add 5+8 by thinking of it as 10+3. We just added 5+8. I'm sure some kids figured out they could think of it as 10+3, but I never thought to do it. I was excellent at math and understood it very well. I knew why the standard algorithms worked and I never thought to go outside of them. Maybe I would have come up with mental math tricks if the standard algorithm had been harder for me or I had been slower that way. As an adult who has taught MM to my son, I can see that I was disadvantaged. He's faster than me because he doesn't need a pencil. I didn't score as well as I could have on things like the SAT and GRE because even though I was accurate, I was slower.

I think the explicit instruction in mental math is what makes people call MM and Singapore and other Asian programs different. They do stress standard algorithms, but they are taught together with about an equal focus.

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This is what I mean. Except my daughter would do (for 36+18) add 2 to 18 to make 20, so 36 plus 20 is 56 and now take away 2 getting an answer of 54.

Does CLE teach in that way?

I, and also my son have found this method to be very confusing, for 36+18, he would add (30+10) + (6+8) Can anyone comment on the reason behind the two methods?

We are currently using CLE as well as Miquon, and I plan to add in Beast Academy a couple times a week to work on problem solving and thinking skills.

CLE is fitting my need for spiral, incremental and VERY simple instruction, and Miquon really helps him understand the conceptual side, and how things work.

He came out of PS, totally upside down about math, and Miquon has made the biggest difference in re-learning how to think about numbers and math in general.

We tried MM here and it was an utter failure, he did not at all understand what he was doing, or why he was doing it.

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Singapore Math is not "mental math," rather "mental math" techniques are among the scope of topics and techniques that are covered (and get some emphasis) in Singapore Math.

There are a great number of mental math strategies that are employed. They are helpful in my POV not only because they make make doing math "in ones head" easier, but because the ability to manipulate numbers using mental math techniques helps children (and adults who encounter this style of math on their "second time around") really see how the number system works.

It would be wrong to call the using a "pencil and paper" based standard algorithm the "opposite" of mental math. It is not the opposite, just an alternative method of calculation. Each alternative has its place. Singapore math (and Math Mammoth) teach both mental math techniques and the standard algorithmsâ€”as they should.

Along with mental math techniques the better math programs teach for a depth of mathematical understanding. Mental math is part of that because to do mental math one has to have a good basic understanding of what one is trying to accomplish.

Unfortunately some of the math programs that only rely on teaching the standard algorithms (with a plug and chug approach) and emphasize the memorization of "math facts" can create an impression of false competence, because students can work the algorithms without necessarily understanding what they are doing with any sort of depth.

Bill

:iagree:

I have used CLE and MM from grades 1-4. I have CLE 5 here, ready to start in a few weeks, and we have started MM 5.

For the most part, CLE teaches the same mental math strategies as MM but on a different timetable. I would say roughly 1-2 year later than MM does. I could see an argument for earlier being better, but they are there.

The conceptual teaching in CLE is a similar story. It is usually there, but it is harder to see. For one, it is sometimes in the TM. Since you don't really *need* the TM after Grade 2, it would be easy to miss. The second reason is because it is spiral, it can be hard to put the pieces together. They usually start with the formula, then add deeper and deeper understanding as they go. Still, they don't spend near the effort that a program like MM does on really understanding.

I :001_wub: CLE. My dd did a complete 180 after starting CLE. Math time went from tears to complete confidence. Because I like the explanations in MM much better and CLE is closer to "plug and chug" on a spectrum, I use a combo of CLE and MM.

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Wow! Thanks for all the great input. Very helpful. I never really knew the phrase mental math meant other things.

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:iagree:

I have used CLE and MM from grades 1-4. I have CLE 5 here, ready to start in a few weeks, and we have started MM 5.

For the most part, CLE teaches the same mental math strategies as MM but on a different timetable. I would say roughly 1-2 year later than MM does. I could see an argument for earlier being better, but they are there.

The conceptual teaching in CLE is a similar story. It is usually there, but it is harder to see. For one, it is sometimes in the TM. Since you don't really *need* the TM after Grade 2, it would be easy to miss. The second reason is because it is spiral, it can be hard to put the pieces together. They usually start with the formula, then add deeper and deeper understanding as they go. Still, they don't spend near the effort that a program like MM does on really understanding.

I :001_wub: CLE. My dd did a complete 180 after starting CLE. Math time went from tears to complete confidence. Because I like the explanations in MM much better and CLE is closer to "plug and chug" on a spectrum, I use a combo of CLE and MM.

How do you figure out what CLE to use with MM? We currently are in MM4a for my oldest. Do I just start CLE 4? Do you use both as complete programs? Seems like for us that would be too much. Ideally, I'd like to go with CLE and somehow supplement with MM.

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... Unfortunately some of the math programs that only rely on teaching the standard algorithms (with a plug and chug approach) and emphasize the memorization of "math facts" can create an impression of false competence, because students can work the algorithms without necessarily understanding what they are doing with any sort of depth.

The whole point of the standard algorithm is that one can do it without conceptual understanding. That is why it was created, and that is why it is valuable, so that accountants (in the days before calculators) could crank through lots of calculations with minimal mental effort and so that computers (in our day) can be programmed to do calculations.

Unfortunately, many people (adults and students both) think that being able to crank through calculations in this way is "understanding math", and a math curriculum that focuses on standard calculations may never challenge that assumption. Math programs that also teach mental math techniques tend to build deeper conceptual understanding, because those techniques require continual and flexible use of the fundamental properties of arithmetic: the distributive property, the commutative property, etc.

I think that all math programs have the goal that students will understand math. But programs that focus on memorization and standard algorithms (pencil-and-paper methods) let students get by with just learning to follow steps. Programs that teach mental math force students (and teachers) to grapple with deeper ideas.

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The whole point of the standard algorithm is that one can do it without conceptual understanding. That is why it was created, and that is why it is valuable, so that accountants (in the days before calculators) could crank through lots of calculations with minimal mental effort and so that computers (in our day) can be programmed to do calculations.

Unfortunately, many people (adults and students both) think that being able to crank through calculations in this way is "understanding math", and a math curriculum that focuses on standard calculations may never challenge that assumption. Math programs that also teach mental math techniques tend to build deeper conceptual understanding, because those techniques require continual and flexible use of the fundamental properties of arithmetic: the distributive property, the commutative property, etc.

I think that all math programs have the goal that students will understand math. But programs that focus on memorization and standard algorithms (pencil-and-paper methods) let students get by with just learning to follow steps. Programs that teach mental math force students (and teachers) to grapple with deeper ideas.

:iagree:

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The whole point of the standard algorithm is that one can do it without conceptual understanding. That is why it was created, and that is why it is valuable, so that accountants (in the days before calculators) could crank through lots of calculations with minimal mental effort and so that computers (in our day) can be programmed to do calculations.

Unfortunately, many people (adults and students both) think that being able to crank through calculations in this way is "understanding math", and a math curriculum that focuses on standard calculations may never challenge that assumption. Math programs that also teach mental math techniques tend to build deeper conceptual understanding, because those techniques require continual and flexible use of the fundamental properties of arithmetic: the distributive property, the commutative property, etc.

I think that all math programs have the goal that students will understand math. But programs that focus on memorization and standard algorithms (pencil-and-paper methods) let students get by with just learning to follow steps. Programs that teach mental math force students (and teachers) to grapple with deeper ideas.

Right. As you say learning the standard algorithms is valuable because they are often efficient ways to calculate (especially as numbers get larger), so programs that ignore them and/or don't develop "procedural competence" fail to teach a valuable skill. But alone, learning the standard algorithms without understanding mathematical reasoning is not a sufficient math education.

The other element that is usually given short-shrift in "algorithm-only" type math education is cultivating critical thinking, reasoning, and logic. Memorizing involves low level cognitive skills—not that these should be neglected or one might have weak low level cognitive abilities—but the brain also needs work that stretches the mind if one hope to build a strong and thriving neural network. The mind needs exercise if it is going to be strong.

If a child never deals with a math approach that goes beyond plugging provided numbers into provided formulas, or memorizing sums and products, they are not getting much of a mental workout. And it will show.

Bill

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Singapore Math is not "mental math," rather "mental math" techniques are among the scope of topics and techniques that are covered (and get some emphasis) in Singapore Math.

There are a great number of mental math strategies that are employed. They are helpful in my POV not only because they make make doing math "in ones head" easier, but because the ability to manipulate numbers using mental math techniques helps children (and adults who encounter this style of math on their "second time around") really see how the number system works.

It would be wrong to call the using a "pencil and paper" based standard algorithm the "opposite" of mental math. It is not the opposite, just an alternative method of calculation. Each alternative has its place. Singapore math (and Math Mammoth) teach both mental math techniques and the standard algorithmsâ€”as they should.

Along with mental math techniques the better math programs teach for a depth of mathematical understanding. Mental math is part of that because to do mental math one has to have a good basic understanding of what one is trying to accomplish.

Unfortunately some of the math programs that only rely on teaching the standard algorithms (with a plug and chug approach) and emphasize the memorization of "math facts" can create an impression of false competence, because students can work the algorithms without necessarily understanding what they are doing with any sort of depth.

Bill

This.

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Unfortunately some of the math programs that only rely on teaching the standard algorithms (with a plug and chug approach) and emphasize the memorization of "math facts" can create an impression of false competence, because students can work the algorithms without necessarily understanding what they are doing with any sort of depth.

Bill

Word. Lol.

We used CLE for quite a while and it seemed to work beautifully - until I realized that while DD11 could correctly and quickly plug in with formulas, she absolutely could NOT apply that math to real life (or any scenario outside of a given equation/problem). I could already see this as a problem coming up on algebra. Singapore was a bust for us, but we have happily settled into Math Mammoth 6. It is certainly a challenge for her, but I can already see the reward. Some of it is "review" from CLE, but the difference in wording and the way it makes her "think" and not simply "plug" makes it seem entirely NEW. Lol.

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How do you figure out what CLE to use with MM? We currently are in MM4a for my oldest. Do I just start CLE 4? Do you use both as complete programs? Seems like for us that would be too much. Ideally, I'd like to go with CLE and somehow supplement with MM.

I think you could start with the 400's in CLE. The first LU is a review unit, so that will help you get up to speed if there are a few things that need review. I found CLE and MM to run pretty close to each other at that level.

I typically move through both programs at the same time without trying to match anything up. I skip the MM chapters on time, measurement, and geometry. I skip the quizzes and tests in CLE. This helps us move through both programs at a reasonable pace. On a daily basis we work together for about 30 min on MM. We usually do between 1-3 pages. I save the word problem pages and review pages and throw those in her independent work(in place of a CLE lesson). I then assign a CLE lesson to be completed independently. Each lesson takes her about 20-30 min. We then meet for corrections so she spends about an hour on math each day.

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