kitchen table math, the sequel: for Paula

Wednesday, September 12, 2007

for Paula

Trawling the web last night, I found a set of PowerPoint slides (pdf file) Thomas H. Parker used in two presentations to the Core Knowledge National Conference on March 5, 2004.

Here's what he had to say about place value, and in this color, too:

Place value causes MANY, MANY problems all through elementary school

Parker also has this to say:

This idea of using position to encode value is a secret code. Because this is second nature to adults, it is easy to overlook the fact that it is difficult to learn. It requires explicit teaching and must be repeatedly worked on. Place value ideas occur and present difficulties in topic after topic in elementary school.

I haven't read the slides yet, but they look helpful, and include place value exercises.

Sequencing in Elementary Mathematics
Thomas H. Paker


Me said...

According to "Dr. Math" place value was invented sometime between 2000 and 1000 BCE. In other words, it has been around at least 3000 years! Something so hard to understand would probably not have lasted this long.

I'm suspecting that a good part of the problem is that many teachers don't really understand place value themselves so they don't explain it well and don't answer questions accurately.

My sons were in a small single class school when the younger one was 5 and the older one 7. Their math teacher was a computer scientist who really, really understood math. Both of them "got" place value, including number bases other than 10 and fractional place values to the right of the "decimal" point, after ONE lesson. I admit they are both quite intelligent but, still, this suggests that it can be understood if presented correctly.

(I, on the other hand, had never heard of other number bases until IIRC I was a senior in college. And, even though I'm good at math, I was initially quite confused by the concept.)

Catherine Johnson said...

Something so hard to understand would probably not have lasted this long.

I'm trying to think whether this argument makes sense.....(that is to say, physics and calculus, which are hard to understand, will probably also last quite awhile...OR NOT, depending on whether our generation manages to pass the knowledge along...)

But I guess you're talking about general knowledge that everyone learns, right?

In that case, I have to agree...

I would also chime in with practice and procedural mastery question.

I don't think I understood place value well when I started re-teaching myself math; I may still not understand it all that well (though I'm probably not completely in the dark).

But I had zero problem using place value.

I had absolute proficiency in "procedural comprehension," to coin a term.

That is, I understood value in whatever fashion one needs to understand a tool.

Catherine Johnson said...

And when I say procedural understanding, I mean that I could also understand and use very large numbers and very small numbers.

Although I didn't quite grasp the fact, consciously grasp, that as numbers grow larger you are multiplying by ten with each digit to the....LEFT!....and dividing by 10 as you move one digit to the right.

However, when I needed to use a larger number, that is precisely what I did.

Catherine Johnson said...

Both of them "got" place value, including number bases other than 10 and fractional place values to the right of the "decimal" point, after ONE lesson.


that is VERY interesting


I actually did have a lesson on base 2, I think, in grade school, where I had some New Math.

I don't think I quite got it, but I remember liking it, thinking it was intriguing and neat.

Catherine Johnson said...

Parker and Baldridge have you do base 5 problems, I think. (And does Saxon have a base 2 problem at the end of 6/5??)

Both of those lessons were quite helpful.

Like you, I felt confused, but it seemed pretty obvious to me that I was confused because I'd spent so many years thinking in tens.

I didn't have the mental flexibility to shift back and forth amongst bases without some practice.

PaulaV said...

I reread the section in Liping Ma's book regarding place value. One particular passage jumped off the page for me: "Multiplying by a two digit number is the difficult point. Students need to learn a new mathematical concept as well as a computational skill. You have to make sure they get both."

Also, Catherine, I couldn't get the powerpoint slide presentation to open. Could you direct me to the website? Thanks!

SteveH said...

I get a 404 when I try to link to the PDF file. Did Parker give any examples of the MANY, MANY prolems? I'm a little skeptical because I've never seen this in my past teaching.

"But I had zero problem using place value."


Is there more? Perhaps there is when you start to convert to scientific notation, like

6.023 X 10^23

but I would like to see these MANY, MANY problems.

PaulaV said...

My son's homework from last night:

Write the ten, the hundred, or the thousand that is lower and that is higher than the number that is given to you.

Thousands example

1,400 1,417 1,500

We are talking about thousands here not hundreds. The example is in the middle...1,417. In my mind it should read 1,000 on the left and 2,000 on the right.

The hundreds example is correct:

200 259 300

There was no reference at the bottom to indicate where the sheet was copied. It looks as though someone made the sheet up. If so, at least the example could have been correct.

Me said...

Paula, I think you are right that the first answer is wrong.

I think the problem here is not so much place value but that these are very weird questions. What is being asked here is really "rounding" numbers. How do you round to the nearest 10, 100, 1000, etc. ?

Although rounding is based on place value, I have found that students need to be explicitly taught rounding as a procedure and that they need a lot of practice. It's a new concept that doesn't follow immediately from an understanding of place value.

For example, here's the procedure to round a number to the nearest 10. Change the last digit (the digit in the ones place) to zero. If the last digit is less than 5 you are done. Otherwise, increase the next digit (the digit in the tens place) by one. (There are several other options for rounding when you are dealing with 5 or 50, etc.)

Me said...

What I wrote above left out a point. Actually, there are three separate processes:

rounding up
rounding down
rounding to the nearest

And its really more than that because just because you can, say, round up to the nearest ten doesn't automatically mean you can round up to the nearest one hundred.

The sample problem required both rounding up and rounding down.

PaulaV said...


Yes, my husband and I figured out it was rounding numbers, but we thought the directions were strange. My son needs things spelled out (obviously so do my husband and I). My son became quite frustrated when my husband tried to explain how to round numbers.

I am afraid he is already becoming frustrated (again) with math. He just isn't getting the connection and he becomes frustrated when he can't. However, with directions like the ones on is homework, do you blame him?

Perhaps an email to the teacher is in order.

Me said...

Paula, I certainly didn't mean to imply that you didn't understand the problem :)

I was just trying to point out that if problems like the ones you showed are what people are using to decide whether kids understand place value, that is ridiculous. Just try writing down the six rules for rounding to the nearest, rounding up, and rounding down for 10 and 100. There's a lot to remember! In some sense this isn't math so much as understanding what "rounding" means.

I'm sorry your son has to deal with this totally illogical situation. I'm sure that if the teacher had spent some time on just rounding down and then rounding up to the nearest ten, he'd grasp it. (To round up to the nearest 10, change the last digit to zero and add 10 to the number.)

I can easily see spending a week on rounding when it is first encountered and then a lot of reinforcement.

It sounds as though the teacher didn't even use the "rounding" as terminology. No wonder your son didn't see the connection between what the teacher/instructions said and what his father was explaining to him.

PaulaV said...


No, no I didn't mean to imply anything by my comment. Trust me, no offense was taken. I was confused at first by the homework because just last week my son brought home a sheet on place value and suddenly he is rounding numbers.

Honestly, I need things spelled out! LOL! It is funny because I became confused immediately because there was no reference for rounding up and down, although that was exactly what it was and I knew it and my husband did to, however the wrong example threw us both off. At first, I thought is this some kind of new way of of explaining place value? Seriously. I need explicit instructions!

I am rusty on math terminology so I definitely need all the help I can get. Susan, your explainations will always be appreciated!