inaccurate explanation in math book

[Grantmom]

I was looking at a book called "All the Math You'll Ever Need" the other day.  It was written by an economics professor because he found that students didn't have a solid math background and therefore struggled.  Right in the beginning, he is explaining multiplication, and he states, "Long multiplication is just simple multiplication combined with addition."  He goes on to give an example.  "When we ,multiply 89 x 57, we're multiplying 89 x 7, then 89 x 5, and then adding the two products."  

 

I thought to myself, that's not true, at all. You are multiplying 89 times FIFTY, not FIVE, and adding the two products.  He does go on to write out the steps of the problem, and when he gets to the second line, he "indents" and talks about indenting and why you indent on each line, which to me makes no sense at all.  I think this explanation is not just confusing but completely wrong.  You are not multiplying 89 x 7 and 89 x 5 and then adding those two answers.  That would be 89 x 12, not 89 x 57.

 

What do you think?

[Arcadia]

The indent was the lazy way of not writing a zero at that rightmost space. Okay for people who already understood place value/number sense well. I was taught to indent back in the late 70s/early 80s when writing out the working for long multiplication.

 

His explanation for the example is wrong

Edited December 29, 2015 by Arcadia

[fdrinca]

One can make a case for what he's saying, but it only applies to people who have a good understanding about place value (i.e. **not** those who are his target audience!) Multiplying 89 by 50 is the same as 89 by 5, adjusting for place value, which I'd assume he did using the indention method.

 

Still, a really tangled way to explain arithmetic to someone!

[Monica_in_Switzerland]

His algorithm works, but his explanation is ridiculous.  lol.  

 

When I was a kid, we were taught to place an X as opposed to an indent.  Equally ridiculous.  It's a zero people, because you're multiplying by TENS on the second line.  :-D  Burn the book before one of your children accidentally gets ahold of it!

 

If you ALREADY understand math, then you can choose to indent, place an x, or place a 0, because you already know what the heck you're doing.  

[daijobu]

 

 

What do you think?

 

I think I don't think highly of economics professors.

[Tap]

I grew up using the indent. If you are teaching this style of math, you do multiply by 5 (not 50) and let the indent take care of the zero.  You wouldn't multiply by 50, because that is completely wrong in this style.  The indents make it easier, not harder.  You can either understand place value and actually understand it, or you can just follow the indent rule and get the right answer.  My teacher taught us just that....'here is the why it works, and if that doesn't make sense, then just follow the steps and get the right answer'.

 

 

DD17 was really confused by the way her math book was teaching her to do long division and multiplication. Once I stopped using the book and taught her to indent, she was flawless at it. 

[Tap]

,

Edited January 1, 2016 by Tap

[regentrude]

I grew up using the indent. If you are teaching this style of math, you do multiply by 5 (not 50) and let the indent take care of the zero.  You wouldn't multiply by 50, because that is completely wrong in this style.  The indents make it easier, not harder.  You can either understand place value and actually understand it, or you can just follow the indent rule and get the right answer.  My teacher taught us just that....'here is the why it works, and if that doesn't make sense, then just follow the steps and get the right answer'.

 

Shudder on the bolded.

 

Plus, as soon as the quantities are symbolic and not numbers, those students will be completely lost.

[SparklyUnicorn]

That is how I was taught and hence why I was so lousy at math!

 

 

[Tap]

Shudder on the bolded.

 

Plus, as soon as the quantities are symbolic and not numbers, those students will be completely lost.

Not necessarily.  Not everyone knows why the quadratic formula works, but it does. You just plug your information into the formula and get an answer.  

 

I did great in math. I even tutored math.  Some people don't need to know the why behind things to excel in them.  They just need know how to do the steps.  

[maize]

I grew up using the indent. If you are teaching this style of math, you do multiply by 5 (not 50) and let the indent take care of the zero. You wouldn't multiply by 50, because that is completely wrong in this style. The indents make it easier, not harder. You can either understand place value and actually understand it, or you can just follow the indent rule and get the right answer. My teacher taught us just that....'here is the why it works, and if that doesn't make sense, then just follow the steps and get the right answer'.

 

 

DD17 was really confused by the way her math book was teaching her to do long division and multiplication. Once I stopped using the book and taught her to indent, she was flawless at it.

But, Tap, when you multiply by a 5 that is in the tens place you are still multiplying by fifty--by five tens. I know you know that, but why wouldn't you want to make sure any student you are teaching knows that?

Sure it's fine to indent without the zero, but it's a disadvantage to kids if we don't help them understand that the 5 really is a fifty whether they write the zero in or not.

[regentrude]

Not necessarily.  Not everyone knows why the quadratic formula works, but it does. You just plug your information into the formula and get an answer. 

 

Every semester I have to deal with students who memorized the quadratic formula in high school algebra and cannot recall it four or more years later.

If they had understood where it came from, they could easily re-derive it from factoring within two minutes. Since they didn't understand it, they are unable of "just plugging information" into the formula - because they forgot the formula and have no way of reconstructing it.

Some try to look it up and can't remember what the symbols a,b, and c stand for and put their coefficients in backwards. Which is equally useless.

 

"plugging information into formulas" is not math.

 

 

 

Some people don't need to know the why behind things to excel in them.  They just need know how to do the steps.

 

We may have different ideas what it means to "excel".

Edited January 1, 2016 by regentrude

[Caroline]

Every semester I have to deal with students who memorized the quadratic formula in high school algebra and cannot recall it four or more years later.

If they had understood where it came from, they could easily re-derive it from factoring within two minutes. Since they didn't understand it, they are unable of "just plugging information" into the formula - because they forgot the formula and have no way of reconstructing it.

Some try to look it up and can't remember what the symbols a,b, and c stand for and put their coefficients in backwards. Which is equally useless.

 

"plugging information into formulas" is not math.

 

 

We may have different ideas what it means to "excel".

Even if they can remember and plug a, b, and c in correctly, they rarely remember what it means if they didn't get the math behind it. I had a student who very proudly sang the quadratic formula but when I drew a graph of a quadratic with two real roots, he couldn't tell me how the graph was related to the quadratic formula.

[Tanaqui]

I did great in math. I even tutored math.  Some people don't need to know the why behind things to excel in them.  They just need know how to do the steps.

 

But then they're not "excelling in math". They're performing a limited set of steps with no understanding.

 

[daijobu]

Even if they can remember and plug a, b, and c in correctly, they rarely remember what it means if they didn't get the math behind it. I had a student who very proudly sang the quadratic formula but when I drew a graph of a quadratic with two real roots, he couldn't tell me how the graph was related to the quadratic formula.

 

This reminds me of a conversation I had with a student we were carpooling.  It happened to be Pi Day and she was describing how the students in school competed to memorize the digits of pi.  I asked her to tell me what pi is (the ratio of circumference to diameter of any and every circle).  She couldn't tell me.  It was elementary school though.  Maybe she knows now.  

[Grantmom]

I disagree that with the indent you are multiplying by 5.  You are still multiplying by 50, because you are shifting the entire number over one place.  The author, word for word, said, that you are multiplying 89 x 7 and then 89 x 5 and then adding the two products.  But you are not doing that.  So, I just think it's more confusing to explain it that way.  

[SparklyUnicorn]

Not necessarily.  Not everyone knows why the quadratic formula works, but it does. You just plug your information into the formula and get an answer.  

 

I did great in math. I even tutored math.  Some people don't need to know the why behind things to excel in them.  They just need know how to do the steps.  

 

Yeah and you know what would happen to me when I got interrupted during the steps?  I had to start over.  I only knew the steps from start to finish, and I better not be interrupted.  If I got stuck midway though, I literally had no idea what I was doing and why so I just started over.  And I found that so fracking frustrating and thought something was wrong with me.

 

Now that I know, it is never a problem.  And it was such a simple thing to learn!  So I do not understand why I wasn't taught something that is so ridiculously simple.

[Farrar]

I grew up using the indent. If you are teaching this style of math, you do multiply by 5 (not 50) and let the indent take care of the zero.  You wouldn't multiply by 50, because that is completely wrong in this style.  

 

What you're saying here is just incorrect. The indent *is* the zero. If I understand this "other method" (which is just the same method but without using zeros!) then you're just putting the numbers into columns and those represent the place values. You are still multiplying by 50 and not by 5, you're just leaving out the zero because the numbers are in columns.

[justasque]

As an aside, I've come across kids who were taught to use an "X" to hold the place, instead of the zero/indent.  I have mixed feelings about it.  On the one hand, it helps kids to keep straight which is the "placeholder" zero (not a zero in this case, but the X), especially when they also get a zero for the next digit.  On the other hand, using the x can obscure the underlying concept; a student writing 20x might not realize it is actually 200.  I usually dialog with the student to be sure they know the math behind it, and then encourage them to use the approach that works best for them.  Dyslexic students in particular find the "X" approach useful.

 

As another aside, I've always thought of it not as multiplying 89x5 (which it isn't), or multiplying 89x50 (which it ultimately is), but as multiplying 89x5x10 - you write down the 0 from the x10 part, then you do 89x5 to figure out the leading digits. 

 

[Grantmom]

But I mean, the placeholder zero really isn't even a placeholder, right?  I mean, you actually are multiplying by a factor of 10, so a zero is the digit that belongs in the ones place.  Is that right?

[Cake and Pi]

 

 

As another aside, I've always thought of it not as multiplying 89x5 (which it isn't), or multiplying 89x50 (which it ultimately is), but as multiplying 89x5x10 - you write down the 0 from the x10 part, then you do 89x5 to figure out the leading digits. 

 

^^ This is exactly what goes on in my head.

 

 

I vote the book is wrong.  It's teaching an effective algorithm, but it's explaining the algorithm like it's math... which it's just not, it's only some steps to follow to get the right answer, which, IMO, is not math.

[Garga]

Does he ever go on in the book to explain that it was really 50? Was this first lesson some sort of "here's a little formula, but later we'll explain what that worked?"

 

I only ask because sometimes in my son's CLE book we learn some things that I think are just busy work, and then 5 lessons later that "busy work" was actually laying the groundwork for something. And usually that busy work makes whatever new concept easier to understand. But it takes a few lessons to built up to the payoff and if I didn't look ahead I'd wonder why they were having the kids do some things.

 

If there's no further explanation in the book that you're using, then I'd be leery of using it.

Edited January 2, 2016 by Garga

[Grantmom]

To be honest, I was just looking through it.  It's not something I am using, and after seeing that, I flipped a little more through the book but didn't look much more.  I had already drawn my conclusions about the book.  I don't know if the topic was revisited again.  The book title explains the content, and it really was just a basic overview of math skills.  In this chapter, he was explaining multiplication.  It just seemed so way off-base.

[Monica_in_Switzerland]

I think we are touching on a flaw of mass education that may be inevitable.  It goes like this:

 

Teach the concept --> Show how the concept relates to a simplified algorithm --> Memorize the algorithm.  

 

Because ultimately, only part 3 can be assessed/graded en masse, the weight goes to part 3.  So often, children can "excel" in math and still have absolutely no idea what they are doing.  When your face time with students is limited and you (the teacher) will also be graded on their results, the "sensible" thing to do is memorize the quadratic formula, not to spend hours deriving it and re-deriving it until everyone can do it.  When I learned it in high school, I remember my teacher saying "Memorize this.  If you want to know where it comes from, it's derived on page xyz."  My geometry teacher, on the other hand, led us all through a cut&paste art activity to prove a^2*b^2=c^2.  She was cool.  :-)  

 

 

 

[regentrude]

I think we are touching on a flaw of mass education that may be inevitable.  It goes like this:

 

Teach the concept --> Show how the concept relates to a simplified algorithm --> Memorize the algorithm.  

No, I do not believe the bolded to be true. Other countries manage to have a vastly better mass education in mathematics - because they use qualified teachers.

I am not sure that all US math teachers can  derive the quadratic formula.

 

Because ultimately, only part 3 can be assessed/graded en masse, the weight goes to part 3.  So often, children can "excel" in math and still have absolutely no idea what they are doing.

 

No to the bolded. It would be entirely possible to construct a test question - even presented in a multiple choice format - that assesses the student's conceptual understanding.

 

the "sensible" thing to do is memorize the quadratic formula, not to spend hours deriving it and re-deriving it until everyone can do it.  When I learned it in high school, I remember my teacher saying "Memorize this.  If you want to know where it comes from, it's derived on page xyz."

 

Of course there will always be some students who don't "get" it. And the teacher will of course not drill the derivation until that is memorized. But math can be taught with the underlying philosophy that everything is proved or derived, and a math teacher would  derive the formula in class, because everything would be derived. It is a difference in approach.

Edited January 2, 2016 by regentrude

[SparklyUnicorn]

I think we are touching on a flaw of mass education that may be inevitable.  It goes like this:

 

Teach the concept --> Show how the concept relates to a simplified algorithm --> Memorize the algorithm.  

 

Because ultimately, only part 3 can be assessed/graded en masse, the weight goes to part 3.  So often, children can "excel" in math and still have absolutely no idea what they are doing.  When your face time with students is limited and you (the teacher) will also be graded on their results, the "sensible" thing to do is memorize the quadratic formula, not to spend hours deriving it and re-deriving it until everyone can do it.  When I learned it in high school, I remember my teacher saying "Memorize this.  If you want to know where it comes from, it's derived on page xyz."  My geometry teacher, on the other hand, led us all through a cut&paste art activity to prove a^2*b^2=c^2.  She was cool.  :-)  

 

I don't know.  I do think there is some attempt to assess understanding of the concepts.  For example, tests now often require a student to explain the concepts and steps.

 

When I was in school the state was developing new standardized tests that were very different than anything I had taken in the past.  I was part of a group that took the exam as test subjects.  Instead of just straight up multiple choice questions they had parts that would create a scenario where one would have to explain the various steps to solve the problem.  For example, talk about how one would go about building a specific book shelf with x amount of money.  So you had to work through the various parts from cost of materials, to dimensions of the shelf, etc. 

[4KookieKids]

My feeling on this in math is similar to my feelings on history, politics, and religion: it's ok to teach an incomplete concept, but it's not ok to teach an incorrect concept. It's up to the person teaching to simplify a complicated situation in such a way so that their "simplifications" are both understandable for the target audience AND still true. Otherwise, it's an automatic fail in my book.

[TracyP]

But I mean, the placeholder zero really isn't even a placeholder, right?  I mean, you actually are multiplying by a factor of 10, so a zero is the digit that belongs in the ones place.  Is that right?

 

That is how a placeholder is defined. The number 50 becomes a 5 without the placeholder 0. The number 204 becomes 24 without the placeholder 0. The point is that a zero belongs in that place (in our place value system) to build the correct number.

 

I hope that explanation makes sense. Personally, I'm not fond of the term placeholder and don't use it with my kids precisely because of what you said. It is not hard to understand that a 0 is the digit that belongs in the ones place when there is 0 ones. Using 'placeholder' just muddies the waters.

[daijobu]

This has gotten me so curious that I put the book on hold from the library.  Can't wait to hear his explanation of long division, lol!  

[Grantmom]

I didn't get that far so you'll have to report back!