kitchen table math, the sequel: measurement in EM

## Wednesday, January 24, 2007

### measurement in EM

My first grader has been learning about measurement in 1st grade. In principle, this is a worthwhile topic. I understand that on most international comparisons, U.S. students tend to do quite badly on measurement. There is an interesting compilation of international studies by the National Center for Education Statistics. According to NCES, US 4th graders did their worst on measurement; it was the only content area where 4th graders couldn't even meet the international average.

So, what are we doing about it in the schools? My first grader spent a couple weeks now on measurement. First, their only homework for the week and the bulk of their classtime was spent on making up their own measurements. For example, she might measure a book to see how many forks long it was. She was to find common objects and use them to measure other objects. After a week or so of this, she segued into standard measures. It's hard for me to gauge the amount of class time devoted to this, but it does seem that she spent at least 1/2 her time on made up measurements. This might aid her conceptual understanding of the topic, but I'm not sure that it is the best use of her time.

Judging from my 5th grader, there is little effort spent between 1st and 5th grade on the tedious task of learning to carefully line up your ruler, or protractor, to get an accurate measurement.

In the end, conceptual understanding is topping careful accuracy. Based on what I am seeing of Everyday Math in the elementary schools, I doubt we'll see much improvement in this content area on the next international assessment.

Anonymous said...

I think first graders should be getting lots of practice with basic addition and subtraction facts.

It's fine to teach measurement also, but like you say, the way they teach it in EM is not the best use of the childrens' time.

I'm starting to see how important the issue of time is. They don't have time to drill the kids on math facts. That's outsourced to parents. They barely have time to teach the algorithms of arithmetic.

I'm amazed at how little time my fourth grader spent on division. But there's time to teach all this other nonsense.

I've looked at the EM 4th grade reference book. M.J. McDermott is right. Not much on long division, but pages and pages of maps and flags.

Where's the math, indeed.

Catherine Johnson said...

I've become simply infuriated on the subject of time.

My school treats Christopher's time with utter indifference.

He's a kid, he has time.

His scores will bounce back.

All "enrichment" activities are good.

No effort is made to measure where the kids are now, what they know and don't know, what needs to be learned today.

la-di-da

Catherine Johnson said...

"Everyone in Irvington is above average."

Catherine Johnson said...

The data on measurement are fascinating.

In 5th grade, when I was afterschooling math every single day in order to make sure Christopher got a '4' on the TONYSS, the one scale he flopped on was measurement.

I was appalled. I also learned from two teachers who posted on ktm-1 that measurement is a VERY difficult skill for kids to acquire.

There are lots of high school kids who can't use a ruler to measure.

Which makes sense because: MEASUREMENT INVOLVES FRACTIONS.

Of course measuring things with a ruler is so ingrained in my own knowledge base that I simply didn't see this.

LynnG said...

If there is one skill that my 5th grader has yet to master, it is measurement. She is pretty decent with a ruler, but the protractor is a mess. We have worked steadily on it through Singapore Math, and she is improving. Still, rather than finding the mode and median of yet another data set, a little more time at school on this would really pay off.

LynnG said...

As a reference, our 5th graders posted their worst scores in measurement on last year's CMT tests.

While 95% of them aced data handling, only 58% mastered approximating measurements and 61% mastered customary and metric measures.

I know the teachers see the same data I do. It is all posted online. Why is there so little effort made to use their classtime more wisely?

Accurate measurement is tedious. I'm voting for that.

Catherine Johnson said...

We've just narrowly averted a protractor debacle with Christopher.

I had taught him how to use a protractor back when we worked through Saxon 6/5. But that was just one lesson, I think, an Investigation (iirc).

We didn't do any distributed practice.

So the math class just "did" protractors.

Christopher mentioned they'd done protractors, and that using a protractor would probably be on the test.

He said he knew how to use a protractor.

He didn't.

Not even close.

So Ed re-taught protractors; then I had him practice several nights in a row.

Now he can use a protractor.

I'm going to WRITE IN ON MY CALENDAR NIGHTS TO PRACTICE PROTRACTOR USE.

Maybe ruler use, too.

Catherine Johnson said...

Math-dad told me one thing you have to drill into kids' heads is: WHEN YOU ARE MEASURING, ALWAYS START AT 0.

Catherine Johnson said...

I didn't really follow what he said, but he told me that he doesn't even let his students touch a protractor until he's spent a lot of time waving his arms around forming different angles.

I think he may have them wave their arms around, too, forming angles.

(This is a 34-year teaching veteran, btw, who loathes TRAILBLAZERS.)

Barry Garelick said...

Yo, Catherine, how's that letter coming?

Catherine Johnson said...

I decided to throw the first draft out.

Catherine Johnson said...

2nd draft t/k

KathyIggy said...

Of course, my 1st grader and I did spend time tracing every family member's foot in a different color crayon and then "talking about" it, and we also measured how many "hands" long something was (and talked about that too). Pretty useless.
One thing I've found interesting are the differences when using EM with my autistic child when she was in 1st grade, and now with my "typical" 1st grader. (This could be the subject of a whole other post--the many problems with EM for different types of students.) With my older daughter, the "language-based" instruction in EM was the big problem, given her language delay. "Talking" about anything isn't her forte. But now with my 6 year-old, we don't have that difficulty. She has understood every concept so far pretty much the first time it's explained. She gets close to 100% on EM tests and finishes the "home links" in less than 5 mins. According to her report card, she "exceeds district expectations." But I don't see much challenge at all and am not sure what she is learning. I'm thinking of giving Singapore a try with her this summer, especially the "Challenging Word Problems."

Anonymous said...

"(This could be the subject of a whole other post--the many problems with EM for different types of students.)"

I agree. It's totally confusing for kids who are weak in math. But then it's too easy for the strong students. The average students muddle through, but have shaky foundations.

That's quite an accomplishment! How did the authors of EM manage to design a program that fails everyone, but in different ways?

LynnG said...

I am convinced kids in EM have no idea what they are doing with math tools.

Today my 5th grader brought home this multi-colored ruler type thing with various holes in it and a slidey thing in the middle. It looked totally cool and she had lots of fun with it. Her assignment was to make a circle with a diameter of 4 cm. She set the thing at 4 cm and made the worst looking circle I've ever seen with a diameter of 8 cm.

I pointed out the difference between radius and diameter, she erased, and tried again. Made the same mistake. More erasing, more holes in the paper.

The very cool slidey ruler thing giggles on the circle. She finally got the radius and diameter right, but it was the most wobbly looking circle ever created. As the ruler spins around, it kept hitting the spine of the math journal. This has got to be a design flaw.

A compass would have worked beautifully; the cool looking wobbly ruler thing was useless. But, if you are a school system, you can buy a compass anywhere. A slidey ruler with holes -- you have to buy that through EM.

Call me a cynic.

Anonymous said...

My daughter is in 4th grade this year. I taught her multidigit multiplication the traditional way. I told her under no circumstances was she to do the lattice method. And she told the teacher and no lattice method problems came home.

She actually can do it for fun. But she mastered the traditional algorithim before she saw lattice. So lattice is like a puzzle, but she checks it with the traditional algorithim.

Once students have learned an inefficient method, it's really hard to get them back.

Anne Dwyer

SteveH said...

"The very cool slidey ruler thing giggles on the circle."

Hey! My son is in fifth grade with EM. I haven't seen this slidey thing yet!

What I did see was a homework on fractions and decimals where they had to look at a picture of little boxes and figure out the fraction of the whole and its decimal equivalent. All nice even parts of one-hundred. Then there was the one where they used curved irregular areas. All you could do was estimate the fraction of the area. He really didn't like doing that. Once again, the teacher didn't explain how to do the homework before it went home.

They talked about something like a hundred grid(?), but he said that the teacher didn't hand them out. We get no manipulables! Is that a plus or a minus?

He was home sick today, so I spent some time comparing 5th grade EM with Singapore Math. Ugh! Forget multiplying and dividing. Look at fractions. Singapore is heavy into fractions in 5A, but EM has just spiraled (danced) past fractions a few times. Fraction sticks anyone?

Doug Sundseth said...

"I told her under no circumstances was she to do the lattice method."

Hear, hear!

Zero tolerance for scholastic enstupidation.

LynnG said...

It is my understanding that the middle school teachers met just before Christmas with the 5th grade teachers and told them, no more lattice, no more partial products, you have to teach the standard algorithms.

Yay! Now I hope someone tells the 3rd and 4th grade teachers.

The 5th grade EM math journal vol. 2 just came home last night. This is pretty depressing. Like Steve points out, pretty light on the fractions for the rest of the year. Most of the fraction work is finding equivalents, converting between mixed numbers and improper fractions, and a little addition and subtraction of fractions with different denominators.

A quick look through reveals some multiplication of mixed numbers. They will be covering this important focal point topic at the same time they are introduced to volume of prisms and cones. I'm not quite sure why these two topics should be paired. Some synergy I'm missing?

Thankfully, we've mastered mixed number and fraction mulitiplication thanks to SM.

LynnG said...

Steve,

Look ahead to Lesson 10 in the 2nd 5th grade math journal if you get a chance. I am seeing a whole bunch of "pan balance" problems. What is this? It looks like a pictorial representation of algebraic substitution with up to 3 variables.

Are they serious?

Barry Garelick said...

Pan balances are indeed pictorial reps of algebraic substitution.

Luckily in 5th grade, my daughter's teacher handed out problems from an old text book, and pretty much skipped EM, except for the "review boxes".

In 6th grade, I tutored my daughter and her friend with Singapore concentrating heavily on fractions because I could see problems ahead. It helped a great deal.

I noticed that in the 5th grade workbook, multiplication of fractions is explained through rectangles (i.e., the area of the rectangle taking fractions of a side; Singapore does this the same way, but they actually have some sequence and problems). In the sixth grade workbook, when they spiral back to multiplication of fractions, they use the ruler line method. So kids will be totally confused, and just plead in desparation "Just tell me how to do it!"

When it comes to division by fractions, for all the "understanding" EM says that students need to have, they really provide no explanation of invert and multiply. In the Singapore series, the pattern of dividing as multiplication by the reciprocal gets established in 5th grade (1/2 x 5 is the same as 5 divided by 2, 1/4 divided by 3 is the same as 1/4 x 1/3) so by the time they get to fraction divided by fraction in 6th grade, students make the cognitive leap. No such luck with EM. Ironically, they just teach the kids "invert and multiply" which reformers have held in disdain because it's considered rote.

Well, Andy Isaacs thinks it works well, and he also thinks EM has the research to support every stupid thing that EM does.

SteveH said...

"Look ahead to Lesson 10 in the 2nd 5th grade math journal if you get a chance."

He is still a ways from the end of book 1!!!!! Burn!

"When it comes to division by fractions, for all the "understanding" EM says that students need to have, they really provide no explanation of invert and multiply."

No mastery and no real understanding! What's left?

I mentioned before (when looking at the 5th grade EM reference book) that their idea of understanding is some sort of low-level or gut-level understanding of very simple (nice) problems. Then they dive right into teaching rote techniques for doing problems. They introduce different ways to look at the same things, don't require a lot of practice, and then quickly spiral on to something else.

The understanding they do teach is very low level (and not as good as in SM), and they don't take that understanding to the next level. Since they can't find some sort of low-level (non-math) way to have kids understand the invert and multiply fraction division method, they avoid it and stick with simple fractions. For those who have seen 6th grade EM, do they EVER talk about invert and multiply?

I know the H. Wu goes into detail about how this should be taught, but I like to use a simpler (less formal) approach. (it assumes that you know how to multiply fractions)

If you have something like:

3/4 divided by 5/7, then this is just one number (3/4) divided by another number (5/7). I write the division as a larger fraction:

(3/4)/(5/7)

I can multiply this without change by 1, and 1 can be replaced with any number divided by itself.

1 = a/a

So I'm going to multiply my big fraction by

(7/5)/(7/5) = 1

Then I get:

((7/5)*(3/4)) / ((7/5)*(5/7))

The denominator cancels (oops, I used that word) out to 1 and I'm left with

(7/5)*(3/4)

Invert and multiply. This works for any complicated rational expression you could find in algebra. If one of the numbers is not a fraction, then make it into a fraction by dividing by 1.

a = a/1

SteveH said...

"Well, Andy Isaacs thinks it works well, and he also thinks EM has the research to support every stupid thing that EM does."

I have had web-based discussions with Andy Isaacs and have commented on his algorithmic justification paper for EM. It's a simple after-the-fact rationalization for their pre-conceived ideas of teaching math. I think it was on Math-Learn where he once said that EM was not for the "elite". I don't know exactly what that means, but all of the "understanding" I have seen in EM is low level and non-mathematical. It's the sort of understanding that will fail you when you get to algebra.

I also notice that after EM makes their low-level attempts at understanding of simple examples, they plow right ahead with their own collection of rote algorithms. There is often big gaps between their feeble attempts at understanding and the problems they expect students to do.

If you look at the EM books, their low-level idea of understanding never gets translated into any sort of formal mathematical understanding. I expect EM has to teach invert and multiply as some point, but do they ever teach any sort of mathematical basis for doing so? On top of it all, they are quickly falling behind SM in fifth grade.

Superficial understanding, no mastery, and slower coverage of the material. Well, not everyone is going to grow up to major in math in college. EM guarantees that.

Barry Garelick said...

"For those who have seen 6th grade EM, do they EVER talk about invert and multiply?"

They mention it by making some semblance of showing a pattern and then stating the rule. So for all their bravado about how students need to "understand" math rather than memorize it, they state a rule for memorization with no explanation.

Speaking of Andy, I had an exchange with him a few years ago which I'm thinking of reproducing here. For someone who officially represents EM, he certainly doesn't control his temper in public forums!

LynnG said...

Steve, it is always worse than you think.

Here's what EM's Teacher's Reference Manual (Grades 4 - 6) has to say about fraction division.

"Indeed, few adults ever need to divide fractions once they leave school. Therefore, the main goal of division of fractions in Everyday Math is not to give students practical skills . . ."

and then,

"The common denominator method for dividing fractions is less mysterious than the traditional "invert-and-multiply" rule."

EM teaches a "quick denominator" approach that encourages kids to find common denominators and then divide the numerators.

So in your example, Steve, 3/4 / 5/7 converts to

(21/20) / (28/28)

(21/20) / 1 = 21/20

or 1 1/20

EM admits this is less efficient and not as easy as invert and multiply. But they prefer it because "few people understand why" the invert-and-multiply rule works.

And they are more likely to remember how the common-denominator-divide-the-numerator rule works?

LynnG said...

Oh Wait--

I messed up the EM common denominator rule.

Your problem -- 3/4 divided by 5/7 should convert to

21/28 divided by 20/28

then you get to

21/20
------
28/28

My conceptual understanding is slipping.

Catherine Johnson said...

It is my understanding that the middle school teachers met just before Christmas with the 5th grade teachers and told them, no more lattice, no more partial products, you have to teach the standard algorithms.

I love it!

Catherine Johnson said...

How did the authors of EM manage to design a program that fails everyone, but in different ways?

They seem almost PLANNED.

Catherine Johnson said...

ktm 1 post on Christopher not being able to measure in 5th grade

This post has advice from a high school teacher, Carl L, and from InterestedTeacher who was teaching measurement to 5th graders.

Invaluable.

Here's InterestedTeacher:

Learning to read/measure from an 'inch' ruler has to be incremental. Younger students can't look at a ruler and automatically discern what all of those marks mean. They have to be taught to find the 'half' mark and measure using the 'half' marks. Then add the 'fourth' marks, (Don't be surprised that students don't automatically know that the 'half' mark also becomes a 'fourth' mark.) Then have students measure using the 'fouth' and half' marks. And so on, going into 'eighth' marks, etc. Practice between each incremental step.

Practice is necessary so students develop the skill of disregarding the smaller (16ths and 32nds) marks. For some students, with visual discrimination problems, this is horribly difficult.

Saxon 6/5 covers through 'fourths' and I add a little 'eighths' for more advanced students.

I was looking through Passport to Mathematics,Book 1, a text that I am previewing for personal reasons, and I see lots and lots of metric work, but little with feet and inches. On pg. 32, students measure to the nearest inch, and nothing else that I can see until pg. 318. With no review of 'half' and 'fourth' inches, it jumps to 'eights' -- there is one problem.

I'm glad to hear that Saxon 7/6 has some in it. This makes me sad that our school has dropped Saxon 7/6.

Here's Carl L:

Re: Measurement

My first year teaching high school freshman (I just finished my 3rd year at a urban neighborhood school) I was completely shocked that none, and I mean none, of the kids could measure using an inches ruler.

How can they get out of middle school, or even grade school, not knowing how to measure? I still have no clue. I doubt its the constructivists fault due to their fondess for hands-on, manipulatives, and project, which all lend themselves to measurement.

What I have observed:

* Metric OK, Inches Not -- While the kids can't (or won't) measure in inches, many (but not all) can measure using a centimeter ruler. Fractions rear their ugly head again.

* Estimation, Schmestimation -- The kids do not know when it is, or is not, appropriate to estimate. The kids have trouble estimating measurements between the lines of the ruler. But the kids are very willing to make bad estimates to avoid having to figure out what the little lines mean. 2 5/16 inevitably becomes 2 1/2.

* What is a protractor? -- The kids REALLY don't know how to use a protractor (except as a frisbee). Most don't even know that its purpose is to measure angles.

A side note related, I believe, to measurement. Each year I do a lesson where we compare the kids height in inches to their shoe size. The majority of the kids do not know how tall they are, let alone how to convert the height in inches.

So by all means get a ruler, protractor, some measuring cups and spoons, and a kitchen scale (or even better a pan balance) and start measuring everything around the house!

Earthbox versus Earth: a measuring investigation!

Catherine Johnson said...

I'm thinking of giving Singapore a try with her this summer, especially the "Challenging Word Problems."

DO IT!

Catherine Johnson said...

I told her under no circumstances was she to do the lattice method. And she told the teacher and no lattice method problems came home.

yeah, that can work

I'm think we're not going to see any more poster assignments

at least so far so good

Catherine Johnson said...

A compass would have worked beautifully; the cool looking wobbly ruler thing was useless. But, if you are a school system, you can buy a compass anywhere. A slidey ruler with holes -- you have to buy that through EM.

Call me a cynic.

You're a cynic.

Catherine Johnson said...

Once students have learned an inefficient method, it's really hard to get them back.

Anne - are you definitely seeing this?

Can you tell us more?

I find this interesting, because it (probably) speaks to a difference between procedural and declarative knowledge.

Procedural knowledge is "how-to," like riding a bike.

Most sports skills are procedural.

Declarative knowledge is "academic" as in "I declare that George Washington was the father of our country."

(These are my own off-the-cuff definitions; not sure whether a cognitive scientist would sign off on them.)

What I understand about procedural knowledge - and this I know I've got right - is that it's VERY bad to learn the procedure the wrong way.

Once you've developed bad habits in athletics that's it. You can't "erase" them. (Maybe that's too strong a statement, but the jist is correct.)

Declarative knowledge doesn't seem to work that way I don't think. When you acquire wrong declarative knowledge, which we do all the time (and which you guys may be doing right now, reading this comment!) we can correct it.

It's possible that with declarative knowledge learning-from-mistakes is actually a useful approach. (I don't know.)

Your observation implies that these procedures REALLY ARE PROCEDURES.

When you learn them the wrong way it's difficult to relearn them the right way.

Is that what you're seeing?

Catherine Johnson said...

They will be covering this important focal point topic at the same time they are introduced to volume of prisms and cones.

Yup.

That's a Ms. K specialty, and she isn't even using a constructivist textbook.

Teach everything at once!

(Engelmann says one of the first lessons he learned was "Teach one thing at a time.")

Anonymous said...

Teach everything at once!

That is very funny. And so true.

Whose keeping track of your gems, Catherine?

I also love the changing titles. It's always good to start this subject with a little guffaw beforehand.