MATHY MOMS (and dads)- I need your HELP!

[Heather in Neverland]

First disclaimer: I am quite sure I will not use all the proper math terminology in this post. Please correct me when necessary as I need the lingo.

 

Second disclaimer: I had your typical public school math experience, got good grades because I am a "rule-follower". I did well in math because I learned and followed the patterns up through Algebra II. Calculus (in college) was beyond me. I did not know any "whys" behind anything and didn't realize it.

 

So I am using MUS Gamma right now with my very "visual" ds. We have been flying through it, no problems. Today we got to lesson 23- multiple digit multiplication. I was amazed with how Demme explained it with the blocks and place-value and what-not because it finally made sense to me

 

My ds got it too with the blocks. Then we did it on paper as 4 separate problems and it looked like this:

 

12

x13

------

6 (3x2)

30 (3x10)

20 (10x2)

+ 100 (10x10)

------

156 (my columns aren't lining up well but you get the idea...drats, I keep trying to edit this to make the columns straight but no luck. Pretend they are in their correct columns!)

 

My ds totally understood this concept using place value and the idea of it as 4 different problems. No problems so far.

 

Then in walks my dh. He thinks I am teaching him the hard way and I should just show him the pattern to follow that we learned as kids where you would say: 3x2 is 6, then 3x1 is 3 (put it to the left of the 6), then put a 0 under the 6 as a "place-holder" and so on...you know, the traditional way.

 

My son's eyes glazed over. He could not comprehend why we needed a place-holder among other things. My husband's response was "you don't need to know why, you just need to know how to do the pattern because it is faster than that other way."

 

OK, he has a point, it is faster. So I really have two questions:

 

1. How do I explain the traditional way of doing it while still enabling my son to know the why behind it and not have him just follow the pattern?

 

2. How do I explain to my dh who is very much a follow-the-rules kind of guy, WHY we should know the WHY behind the math?

 

I know there are a lot of people on this board who can explain this in a much better way than I can. I know in my head why it is important but I just can't put it into words.

 

p.s. ds did eventually comprehend the "pattern" way of doing it and is doing it that way with success but I can't help feeling like I lost out on some opportunity to take his math understanding to the next level and now he has ditched that in favor of "quick".

[chiguirre]

I don't use MUS, but the way they explained multiplication allows you to do it in your head. The "fast" way requires pencil and paper unless you've got a massive short term memory. I learned the pattern way and figured out the other way as an adult when I needed to do mental math in the supermarket.

[angela in ohio]

Well, I'm assuming MUS will have to explain the traditional way when they get to longer problems, right? Otherwise, it would take all day to write it out. I would look ahead and see when they teach it, and let dh know when that will be.

[Mama Lynx]

I am not familiar with MUS. However, many math programs that teach a concept in that way, will teach the standard algorithm later. Right Start does this; after RS presents the concepts, and the child has had enough practice with it to demonstrate understanding, it will teach the algorithm. At that point, the child is doing the math in the way that you and your dh are familiar with from school, but has a more thorough understanding of what's going on.

 

I don't know what you can say to your dh. It seems pretty self-evident to me that it's usually better to understand the concept, to know *why*, instead of just being able to crank out a formula. Ask him why he thinks it's not important to not know why? I'd ask him to defend that attitude. Does he feel that "you don't need to know" has helped him in math, in his life?

 

It's easy to teach an algorithm. It's especially easy to teach it after the concept is understood. It's not always easy to teach the other way around, because if we don't understand a concept in the first place, we tend to cling to the rote pattern, and be lost when we let go.

[WTMindy]

It is great to understand the why of math!! Now that your dd understands the why (and practices it a bit) you can show the old method and she will probably get it. Use the same problems that she has already done so she can see where the numbers are coming from. She will be happy to do the shortcut by then also. :001_smile:

 

Some kids need it and some don't, so I would go with your gut on this one. If it is making sense to her, then keep doing it.

[Sunshine State Sue]

12

x13

------

6 (3x2)

30 (3x10)

20 (10x2)

+ 100 (10x10)

------

156

 

I love MUS because it lays the foundation first, it doesn't skip it. What he is teaching is that you are not multiplying 3x1 and putting the answer in the 2nd column. You are multiplying 3x10 and writing the 3 in the 10's place. You are not multiplying 1x1 and putting the answer in the 3rd column. You are multiplying 10x10 and putting the answer in the 100's place. To me, it is brilliant. Your child will understand WHY you do what you are doing.

 

They do teach a more traditional approach in MUS, but the carrying is taught a bit differently. They are just drawing it out a bit so that they child understands the place value behind multiple digit multiplication. One of the strengths of MUS has to do with the student's understanding of place value. I love how they do it with the blocks and how easy it is to SEE the answer.

 

Multiple digit multiplication and long division are the 2 most difficult concepts before Algebra. Kudos to your son for picking it up so quickly.

 

And about your husband. You paid good money for Mr. Demme's expertise in teaching math. Why throw it away by not taking his counsel? I see again and again the wisdom to his method. If it makes a difference to your dh, I have a degree in math. I think Steve Demme is a genius math teacher.

[Heather in Neverland]

I guess I should clarify that my dh is not being a jerk about the whole thing...more like he is just used to the way we were taught, can't see a reason for doing it any differently, and my inability to express in words why it is important isn't helping things.

 

For instance, dh asked me for an example of WHEN it would be important for our son to know the "why" behind multiple digit multiplication, under what circumstances will he need to know the why behind the math. Dh is very good at math in the traditional sense and doesn't get my need to do it differently for our son.

 

I know it has to do with abstract vs. concrete, and problem-solving skills, etc., but I can't really explain it and his premise is that as long as you get the right answer, who cares?

 

Am I making any sense?

[Carol in Cal.]

I would say that if your son gets the wrong answer, he will have no way of knowing this because without quickly mentally 'estimating' the answer, he can make mistakes and is unlikely to catch them. People who are good at math do this mental stuff all the time without necessarily even being all that aware of it. They will say things like, "Wow, that just doesn't look right," and this will trigger them to check their work.

 

Also, frankly, (though I wouldn't tell him this), we are trying for a better education than we had; and if getting the answer was the only criteria, why not just use a calculator and be done with it? That is taking the question to its logical extreme. And the answer is, because people are better off all through math when they are fast, proficient, and also understand it well.

 

The other thing I would say to him, in private, is that you would really, really appreciate it if he doesn't question your teaching in front of your son. That makes your life a lot harder, and it undermines you as a teacher. Your family will be way better off when your son reaches his teens or puberty if your DH has established his respect for you AND his expectation that his son respect you in advance.

[Sunshine State Sue]

Tell your dh you will get to the "short-cut" after ds is fully understanding the "long way". Then don't do math when dad is around until ds has mastered it and moved on to using the "short cut" with full understanding and fluency.

 

:iagree::iagree::iagree:

 

Your ds won't be doing it the long way very long anyway.

[Heather in Neverland]

Tell your dh you will get to the "short-cut" after ds is fully understanding the "long way". Then don't do math when dad is around until ds has mastered it and moved on to using the "short cut" with full understanding and fluency.

 

:lol:

 

How funny! What's even funnier is that he is NEVER around when we do math lessons but today's was a doozy.

 

The other thing is that after showing ds the MUS way of doing it, I did show him the traditional way but the difference was I couldn't explain as well WHY it worked...it just does. With the MUS way of doing it I could easily explain why. So obviously, my own math learning has some gaps to fill! :tongue_smilie:

[Charon]

I guess I should clarify that my dh is not being a jerk about the whole thing...more like he is just used to the way we were taught, can't see a reason for doing it any differently, and my inability to express in words why it is important isn't helping things.

 

For instance, dh asked me for an example of WHEN it would be important for our son to know the "why" behind multiple digit multiplication, under what circumstances will he need to know the why behind the math. Dh is very good at math in the traditional sense and doesn't get my need to do it differently for our son.

 

I know it has to do with abstract vs. concrete, and problem-solving skills, etc., but I can't really explain it and his premise is that as long as you get the right answer, who cares?

 

Am I making any sense?

 

 

When would you need to know the why? How about multiply (x+3)(x+2)? Multiplying it out you get: 6+3x+2x+(x^2). Replacing x with 10, what did you have?

 

6 (3x2)

30 (3x10)

20 (10x2)

+ 100 (10x10)

------

156

 

"The long way" is getting you ready for polynomials. The short way is just some algorithm.

[Heather in Neverland]

When would you need to know the why? How about multiply (x+3)(x+2)? Multiplying it out you get: 6+3x+2x+(x^2). Replacing x with 10, what did you have?

 

6 (3x2)

30 (3x10)

20 (10x2)

+ 100 (10x10)

------

156

 

"The long way" is getting you ready for polynomials. The short way is just some algorithm.

 

Now we're getting somewhere..."just some algorithm" is kind of the direction I am coming from when I say I don't want him to just learn to follow the pattern.

 

So at his age (he'll be 10 next month), is it OK to go with "the long way" and teach him the algorithm later? Teach both at the same time? Algorithm first and concept behind it later?

 

For instance, I learned in school that in order to divide fractions you multiply the first fraction by the inverse of the second fraction (or something like that). So 1/2 divided by 1/3 becomes 1/2 times 3/1 which equals 3/2 or 1 1/2.

 

But I have no idea WHY that is right (other than I followed the algorithm). And when we get to fractions and my son asks my why do you have to multiply the inverse in order to divide (and he WILL ask, unlike me when I was his age)? My answer will be...duhh... actually, I am hoping Steve Demme has an answer when we get to that lesson!

 

This is why I am in such a mood about this today. I think I am mad at my own math education and that fact that I never knew what I was missing until now!!

[Storm Bay]

Now we're getting somewhere..."just some algorithm" is kind of the direction I am coming from when I say I don't want him to just learn to follow the pattern.

 

So at his age (he'll be 10 next month), is it OK to go with "the long way" and teach him the algorithm later? Teach both at the same time? Algorithm first and concept behind it later?

 

For instance, I learned in school that in order to divide fractions you multiply the first fraction by the inverse of the second fraction (or something like that). So 1/2 divided by 1/3 becomes 1/2 times 3/1 which equals 3/2 or 1 1/2.

 

But I have no idea WHY that is right (other than I followed the algorithm). And when we get to fractions and my son asks my why do you have to multiply the inverse in order to divide (and he WILL ask, unlike me when I was his age)? My answer will be...duhh... actually, I am hoping Steve Demme has an answer when we get to that lesson!

 

This is why I am in such a mood about this today. I think I am mad at my own math education and that fact that I never knew what I was missing until now!!

 

Charon said it. I want to add that even I, who was good at math naturally and learned and "got" Algebra from my textbooks, never truly understood what I was doing when dividing fractions until I saw Mr. Demme do it. Not that that's the only good teaching of it, of course, but no one explained that when I was a kid and it was the first good one I saw.

 

fwiw, Singapore Math also teaches by showing the long way, then the algorithm, so it's not limited to MUS.

[Sunshine State Sue]

But I have no idea WHY that is right (other than I followed the algorithm). And when we get to fractions and my son asks my why do you have to multiply the inverse in order to divide (and he WILL ask, unlike me when I was his age)? My answer will be...duhh... actually, I am hoping Steve Demme has an answer when we get to that lesson!

 

Yes, Steve Demme does explain that. Not that I remember it...:D But, you won't start dividing fractions by multiplying by the inverse. No, you'll do it the long way first. You'll have to hide from dh for a few weeks ;)

[Capt_Uhura]

You can always forget the standard, quick algorithm after years of using a calculator. HOWEVER, if you understand the WHY of it, you'll never forget that and will be able to re-derive the standard, quick algorithm. Ask me how I know? :001_smile: When I was in my postdoc, I gave up a calculator for a whole week. I had to do long division, work w/ huge exponents and on paper/pencil or my head b/c I was so shocked at how rustry my math skills were on pencil/paper. Once I convinced myself I was proficient at math, I took out the calculator again.

[Dani n Monies Mom]

A side note:

 

DD12 just did that lesson two weeks ago and she had the same question, why would I flip the fraction? Steve does teach the long way to divide fractions first, and you do it that way for several weeks, however, when it got to the why of flipping the fractions and then multiplying, he didn't actually tell you why. He just introduced the concept as 'multiplicative inverse', did a few examples and never explained why it worked.

 

So dd and I took a problem and wrote it out 'the long way' and took the same problem and did it the flipped way and discovered that flipping the fraction to divide is just a shortcut for 'the rule of four', it's easier because you don't have to find the GCF or LCD for the denominator first. He does explain, very early in the program, how to use 'the rule of four' and why it works. So that explanation cleared it up for dd. (and me for the first time ever)

 

Anyway, I was surprised that he didn't explain it. From Gamma to now Epsilon, he's always had great explanations. That was the first time we experienced that. (I will add that this happened in lesson 23/24, which is the point in every MUS we've done that things get challenging.)

 

Ava

 

 

 

 

For instance, I learned in school that in order to divide fractions you multiply the first fraction by the inverse of the second fraction (or something like that). So 1/2 divided by 1/3 becomes 1/2 times 3/1 which equals 3/2 or 1 1/2.

 

But I have no idea WHY that is right (other than I followed the algorithm). And when we get to fractions and my son asks my why do you have to multiply the inverse in order to divide (and he WILL ask, unlike me when I was his age)? My answer will be...duhh... actually, I am hoping Steve Demme has an answer when we get to that lesson!

 

This is why I am in such a mood about this today. I think I am mad at my own math education and that fact that I never knew what I was missing until now!!

[Heather in Neverland]

When I was in my postdoc, I gave up a calculator for a whole week.

 

Now THAT'S scary!!!! :D

[Heather in Neverland]

A side note:

 

He does explain, very early in the program, how to use 'the rule of four' and why it works.

 

I have never heard of the "rule of four". ???

[Myrtle]

A side note:

 

DD12 just did that lesson two weeks ago and she had the same question, why would I flip the fraction? Steve does teach the long way to divide fractions first, and you do it that way for several weeks, however, when it got to the why of flipping the fractions and then multiplying, he didn't actually tell you why. He just introduced the concept as 'multiplicative inverse', did a few examples and never explained why it worked.

 

So dd and I took a problem and wrote it out 'the long way' and took the same problem and did it the flipped way and discovered that flipping the fraction to divide is just a shortcut for 'the rule of four', it's easier because you don't have to find the GCF or LCD for the denominator first. He does explain, very early in the program, how to use 'the rule of four' and why it works. So that explanation cleared it up for dd. (and me for the first time ever)

 

Anyway, I was surprised that he didn't explain it. From Gamma to now Epsilon, he's always had great explanations. That was the first time we experienced that. (I will add that this happened in lesson 23/24, which is the point in every MUS we've done that things get challenging.)

 

Ava

 

 

I did a blog entry on why invert and multiply works some time ago.

 

 

It's sort of an interesting philosophical question. So we have these three ideas. We have the idea of division as in 1 ÷ 3, we have the idea of multiplicative inverse, "There is some number such that if you multiply 3 by this number you will get one" and then you have the idea of a fraction as in 1/3. And why exactly 1/3 is the multiplicative inverse is a whole nother ball of wax so we'll just assume it for now and worry about the relationship betewen the ÷ and the fraction 1/3.

 

So how exactly do we know with metaphysical certainty ;-) that each of these three all lead to the exact same point on the number line? That is the underlying issue of why invert and multiply works.

 

Here is an example using numbers instead of letters:

 

I want to figure out why it is that 2 ÷ 3 = 2 x 1/3.

 

So here is our fact family:

 

a. 2 ÷ 3 = 2/3

b. 2 ÷ 2/3 = 3

c. 3 x 2/3 = 2

d. 2/3 x 3 = 2

 

 

So let's start with,

 

3 x 2/3 = 2. But if you look at the equation in "a" you can see that 2 ÷ 3 = 2/3 so I am going to replace the 2/3 in expression c with 2 ÷ 3 using substitution so I end up with this.

 

3 x (2 ÷ 3) = 2

 

And now I'm going to multiply both sides by 1/3:

 

1/3 x 3 x (2 ÷ 3)= 2 x 1/3

 

and I can use the associative law to move the parentheses

 

(1/3 x 3) x 2 ÷ 3 = 2 x 1/3

 

But we already know that 1/3 is the multiplicative inverse of three since

1/3 x 3 = 1

 

So, I can then substitute the expression 1/3 x 3 with 1 like this,

 

1 x 2 ÷ 3 = 2 x 1/3

 

and since anything multiplied by 1 is itself (another one of those field axioms)

 

I'm left with tah dum 2 ÷ 3 = 2 x 1/3

 

I don't know that a child in arithemetic would be able to follow such an argument. Because algebraic principles are needed to explain why, it's probably better left for algebra.

[Sunshine State Sue]

I have never heard of the "rule of four". ???

Don't worry. You'll get to it early in Epsilon. It has to do with fractions. You will remember it when you come to it. You probably just never learned a name for it.

[Storm Bay]

A side note:

 

DD12 just did that lesson two weeks ago and she had the same question, why would I flip the fraction? Steve does teach the long way to divide fractions first, and you do it that way for several weeks, however, when it got to the why of flipping the fractions and then multiplying, he didn't actually tell you why. He just introduced the concept as 'multiplicative inverse', did a few examples and never explained why it worked.

Ava

 

Hmm, I think I understood it when I watched him use the manipulatives, so I remembered him as teaching it. It could be that I also saw it explained somewhere else (does Gelfand's discuss this?) We were checking it out for my eldest who hated the manipulatives and stuck with the math she was already doing. At any rate, I agree with Myrtle that sometimes the theoretical understanding has to come later. I enjoy the way Gelfand discusses theory, but that's definitely a book for later on. Some of the problems are too hard, so we're doing it with our other Algebra.