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# Entries from FractionsDecimalsAndPercents

SwoopAndSwoop 07 Jul 2005 - 20:38 CarolynJohnston

This evening, we are working on long division with decimal divisors, and comparing the sizes of two fractions. We are working merely on getting these skills down: nothing too deep.

When I first showed Ben the cross-multiplication algorithm for comparing two fractions, I showed him why it works the way it works.

"It's easy to compare two fractions when they have the same denominator, right?" I said. "Well, it's easy to get two different fractions to be over the same denominator. Just multiply on each side by 1, written as the other fraction's denominator over itself. Then you notice what you get on the left side is the numerator times the right side's denominator, and vice versa on the other side. All you do is compare those numbers. That's called cross-multiplication because it makes a cross. Now you show me."

He tried to follow the steps in my first demonstration, and didn't get it right.

"It's like this. The numbers move in an x when you do cross-multiplication, like this. They just go swoop, and swoop, like this":

And that was it: he got it: those swooping moves with the pencil and the crossing numbers. That's what the standard algorithms are: they are moves that you learn how to make. Those moves get into your fingers, just like learning the piano or the violin or typing, and eventually you can do them completely mindlessly.

But that doesn't mean that nothing is going on in the kid's head. If a kid really has those moves down, it frees his mind to think about doing the next thing, and he becomes more receptive to learning why the moves need to be what they are, because the anxiety of not being able to handle the calculation is gone.

Learning the piano or the violin involves a lot of repetition, while your eyes and your mind and your fingers make the connections that allow you, eventually, to experience the music you're playing on a higher level, without calculating where your fingers need to go next. Math is just like that. Math is something you learn to do, like playing an instrument or riding a bike, not something you learn about remotely, like Magellan's circumnavigation. It has a huge kinesthetic component.

swoop and swoop
SlideRules
the craft of math
Wayne Wickelgren on why math is confusing, & Carolyn on procedural memory
KUMON & hands-on math

QuickThoughtAboutFractionManipulatives 27 Nov 2005 - 19:23 CarolynJohnston

Catherine mentioned that she is a fan of tile fraction manipulatives over the more usual 'pie' manipulatives:

She said that her daughter didn't get anywhere using the more-common circular, 'pie chart' fraction manipulatives; she needed to see rectangular fractions. I have no idea why this would be, but it 'felt' right to me, so I searched for rectangular manipulatives and found these.

I prefer tile manipulatives too, for what I think are solid pedagogical reasons, and here is why: if you want to talk about improper fractions -- fractions greater than one -- with your kid, then the pie-shaped manipulatives add potential for confusion because you can't make a single connected object that represents a quantity greater than one. If you want to represent, for example, 3/2 with pie manipulatives, then you'll have one whole circle and a half circle. You can tell a kid that that represents a single object, the quantity 3/2, all you like; but to him it will look like two objects. Fractions are confusing enough without that.

Conversely, you can make a single line of tiles that is as long as you like.

So unless your child is really off and running with the pie manipulatives, I'd recommend the tile manipulatives.

FriendlyFractions 01 Jun 2005 - 16:16 CatherineJohnson

Want to know what comes up when you Google friendly fractions?

``` Visit Fraction Town and meet Friendly Fractions and Fractions Not So Friendly, even see a Fraction Frenzy as students learn about fractional parts. Dividing and multiplying by one and two digits and determining the probability of events occurring finish up the school year. Have a great summer! ```

Day 151 sounds especially fun!

``` Fraction lesson created for day 152 of the 180 day sequence of lesson plans. Students will use their knowledge of fractions to create a map of Fraction Town and decorate their map using what they've learned about fractions. ```

DontRelyOnStateTests
PenfieldParents
NewYorkStateMathCurricula
CompareAndContrastPart3
PaperFractions
WorksheetsForSummer
AssessYourChildForFree
AssessYourChildForFreePart2
BonusOnlineAssessmentQuestions

TutoringAdvice 24 Jul 2005 - 01:55 CatherineJohnson

I'm probably going to spend some time working with a friend of Christopher's on his math.

They're the same age--both going into 6th grade--and my sense is that math is probably this boy's strong suit.

I just gave him the Saxon placement test, and he placed into Saxon 7/6, which is the 6th grade book.

That would be great, but here's the hitch: he has been taught almost nothing about fractions at all. (He had a good math teacher--he and Christopher were in the same Phase 3 class for the first half of this year--who left to have a baby. So it seems that the subject of fractions & decimals fell through the cracks.)

So.....if anyone has thoughts, I'd like to hear them. I'll probably go ahead with 7/6, but that means I'm starting a 6th grade book with a child who's been taught virtually nothing about fractions and decimals.

### update

Here's the fraction worksheets site Carolyn J found.

### whose job is it, anyway?

This is the kind of thing that I just don't get.

Why should I be the person figuring out that this boy hasn't been taught fractions & decimals?

Why shouldn't the school be figuring this out? (Yes, the school might say he was taught fractions and decimals, but didn't learn them. However, it's clear to me that there are certain topics he simply hasn't even heard of, because with some topics he'll say, 'I kind of remember that.' In other words, he can tell me which topics he failed to learn, or didn't learn well enough to retain, or whatever it is. With topics like adding fractions, he simply doesn't know anything about them, and has no memory of having been taught.)

So, yes, the school might say, 'He was taught, but he didn't learn.'

But so what?

If he was 'taught' and 'didn't learn,' then he wasn't taught as far as I'm concerned. It's the school's job to perform formative assessment to know what students have and have not learned.

Then it is the school's job to re-teach if a child has not learned.

Then, if the child still isn't learning, it's the school's job to figure out what else he needs.

### common sense from The Education Wonks

I don't want to take this too far, of course. Parents & students are responsible, too:

That's one of my major concerns with NCLB. When students don't do their homework or study for exams, or even attempt to do classwork, it's still considered to be the teacher's fault if the students don't achieve their federally-mandated level of proficiency in reading, math, and science.

And yet NCLB doesn't give me, as a teacher, the authority to require student's who aren't even attempting the work to stay after school and complete their assignments. Unless the kid has committed some breach of the school's disciplinary policy, I can't keep them any later than the school regular dismissal time.

The No Child Left Behind Act holds me solely accountable for my students' academic progress but doesn't give me the authority to help make that happen, especially for children that are considered to be "at risk" of failing to meet minimal standards of academic progress.

Sadly, under the law as it is now written, a large number of children are going to be left behind.

He's right. If a student doesn't do his work, and the parents don't require him to do his work, that isn't the teacher's responsiblity.

But that's not the case with Christopher's friend. This boy has done all of his work; he's a serious student; his parents are serious parents.

It's the school's responsibility to know whether this boy has or has not learned how to add, subtract, multiply, and divide fractions, and to teach or re-teach the subject if he hasn't.

Dan K and Carolyn Morgan have given me some incredibly helpful advice on teaching fractions 'in a hurry.' I'll get it pulled up front as soon as I can, but in the meantime take a look at the Comments thread.

MissingKnowledge 25 Jul 2005 - 22:46 CatherineJohnson

More good stuff from Education News:

Today's math lessons, Armbrecht said, focus much more on "inquiry-based learning" than the math of yore. Students are given a problem, then asked to use their understanding of number structure, logic and math concepts to solve it. In Armbrecht's generation, most students were told to memorize facts instead of being challenged to understand the underlying concepts, he said.

Furthermore, today's math students use calculators, computers and hands-on objects more often than their parents did. So, like Wilmington resident LaMere Henderson, even well-educated parents aren't equipped to help their children with math.

[snip]

But math teacher Dawn Olmstead, recently retired from Alexis I. du Pont High School, said so many reach high school unprepared that remediation can't be avoided.

"What we're seeing is the kids don't know how to add fractions," she said. "Some don't even know what fractions are.

"When they come into ninth grade, they're supposed to be prepared for algebra, and they're not."

There are so many topics to cover, she said, it's a burden to teach them all by the time of the test, which is given in March.

"How about probability?" she said. "Why would I teach that in an algebra class? Because it's on the test. I have to do both: algebra and what's on the test."

TwoMathEdBlogs 27 Jul 2005 - 19:26 CatherineJohnson

Stephanie just sent me a link to a fascinating list of prerequisites for college math, which includes a terrific Comments thread, at Tall Dark and Mysterious, a blog written by "Twentysomething curmudgeon seeking employment teaching college math in BC."

And btw, these are not prerequisites for a serious college math course:

A year ago, I would have posted that list under a heading more along the lines of “Things Students Should Know By Grade Nine”, but alas, experience as extinguished such optimism on my part.

This is long, but it's so valuable I'm quoting the entire list, which I'll probably 'archive' over on....the 'math lessons' page? Another Content Question for the folks at Information Architecture, Inc. (Definitely read the Comments section as well):

Based on my experiences, students graduating from high school should, in order to succeed in even the most basic college math classes:

1.Be able to add, subtract, multiply, and divide fractions. Moreover, they should understand that the horizontal bar in a fraction denotes division. (Seem obvious? I thought so, too, until I had a student tell me that she couldn’t give me a decimal approximation of (3/5)^8, because “my calculator doesn’t have a fraction button”.)

2.Have the times tables (single digit numbers) memorized. At minimum, they should understand what the basic operations mean. For instance, know that “times” means “groups of”, which will enable them to multiply, for instance, any number by 1 or 0 without a calculator, and without putting much thought into the matter. This would also enable those students who have not memorized their times tables to figure out what 3 times 8 was if they didn’t know it by heart.

3.Understand how to solve a linear (or reduces-to-linear) equation in a single variable. Recognize that the goal is to isolate the unknown quantity, and that doing so requires “undoing” the equation by reversing the order of operations. Know that that the equals sign means that both sides of the equation are the same, and that one can’t change the value of one side without changing the value of the other. (Aside: shortcuts such as “cross-multiplication” should be stricken from the high school algebra curriculum entirely - or at least until students understand where they come from. If I had a dollar for every student I ever tutored who was familiar with that phantom operation, and if I had to pay ten bucks for every student who actually got that cross-multiplication was just shorthand for multiplying both sides of an equation by the two denominators - I’d still be in the black.)

4.Be able to set up an equation, or set of equations, from a few sentences of text. (For instance, students should be able to translate simple geometric statements about perimeter and area into equations. ) Students should understand that (all together now!) an equation is a relationship among quantities, and that the goal in solving a word problem is to find the numerical value for one or more unknown quantities; and that the method for doing so involves analyzing how the given quantities are related. In order to measure whether students understand this, students must be presented, in a test setting, with word problems that differ more than superficially from the ones presented in class or in the textbook; requiring them only to parrot solutions to questions they have encountered exactly before, measures only their memorization skills.

5.Be able to interpret graphs, and to make transitions between algebraic and geometric presentations of data. For instance, students should know what an x- [y-]intercept means both geometrically (”the place where the graph crosses the x- [y-]axis”) and algebraically (”the value of x (y) when y [x] is set to zero in the function”).

6.Understand basic logic, such as the meaning of the “if…then” syllogism. They should know that if given a definition or rule of the form “if A, then B”, they need to check that the conditions of A are satisfied before they apply B. (Sound like a no-brainer? It should be. This is one of those things I completely took for granted when I started teaching at the college level. My illusions were shattered when I found that a simple statement such as “if A and B are disjoint sets, then the number of elements in (A union B) equals the number of elements in A plus the number of elements in B” caused confusion of epic proportions among a majority of my students. Many wouldn’t even check if A and B were disjoint before finding the cardinality of their union; others seemed to understand that they needed to see if A and B were disjoint, and they needed to find their cardinality - but they didn’t know how those things fit together. (They’d see that A and B were not disjoint, claim as much, and then apply the formula anyway.) It is a testament to the ridiculous extent to which mathematics is divorced from reality in students’ minds that three year olds can understand the implications of “If it’s raining, then you need an umbrella”, but that students graduating from high school are bewildered when the most elementary of mathematical concepts are juxtaposed in such a manner.)

7.More generally: students should know the basics of what it means to justify something mathematically. They should know that it is not enough to plug in a few values for x; you need to show that an identity, for instance, is true for all x. Conversely, they should understand that a single counterexample suffices to show that a claim is false. (Despite the affinity on the part of the high school text I am working for true/false questions, the students I am working with do not understand this.) Among the educational devices to be expunged from the classroom: textbooks that suggest that eyeballing the output of a graphing calculator is a legitimate method of showing, for instance, that a function has three zeroes or two asymptotes or what have you.

also added to the list by commenters:

I would add estimation and verification to that list. Students should know the difference between a sensible and nonsense answer.

Another blog by a college calculus professor: Learning Curves

ParkerAndBaldridgeOnFractions 13 Nov 2005 - 19:54 CarolynJohnston

I am especially fond of Parker and Baldridge's section on the basics of fractions. I've focused on this chapter, as it's appropriate for my son's age, and because fractions are a huge challenge to teach properly. Fractions are confusing for students from the day they're introduced, and for a lot of them (as we know) are brought down by it. As Catherine says, "lives are lost in the struggle to learn fractions".

Parker and Baldridge introduce the topic beautifully, emphasizing for their teacher-students the concepts that are most likely to trip up students. Kids tend to be stunned by the terms 'numerator' and 'denominator' at first, so P&B suggests that teachers eschew it at first, substituting 'top' and 'bottom'. Kids tend to think of a fraction as somehow representing two independent numbers, and so fail to see that a fraction is a single number. This results in the common misconception/error:

a/b + c/d = (a+b)/(c+d).

P&B encourage the teacher-student to think of a denominator, instead, as representing the fractional unit into which the whole has been divided. In fact, it can be useful to think of different denominators as being like different units entirely. It doesn't make sense to add quantities represented in feet and meters without doing a conversion to a common unit first; similarly, fractions must be converted to a common unit (i.e., denominator) before they can be added. The Singapore-style line drawings that illustrate this conversion in P&B are well-done and get the point across clearly.

One tidbit that I got out of P&B was an answer to Catherine's question: why is multiplying the numerator by the reciprocal when dividing fractions the right thing to do?

The easiest fraction-division case to understand, they point out, is the case where both the numerator and denominator have the denominator in common, as in the problem:

6/2 3/2.

In this case, one should envision a bar representing 1/2; if we have 6 of these bars, and divide them into groups of 3 such bars, clearly we will have two groups. But whether a bar represents 1/2 or a whole is irrelevant, as long as the bars represent the same quantity; and so this is really the same problem as

6 3.

Therefore, fraction division is pretty easy to understand if you have common denominators. But, of course, any two fractions can be converted to have common denominators. So now look at the general problem:

a/b c/d.

To figure out what we get, we convert both fractions to have a common denominator, b x d:

Now we have a common denominator bd, and so this problem is equivalent to the problem of dividing their numerators::

But note that this is exactly the usual multiplication-by-the-reciprocal formula:

a/b x d/c.

And that is why multiplying by the reciprocal is the right thing to do (incidentally, in a comment on this thread, J.D. Fisher mentioned that the key to understanding practically everything about fractions is to think about fractions that have a common denominator. Once again, he turns out to have been right).

FalsePositives 12 Sep 2005 - 03:15 CatherineJohnson

A couple of days ago, Carolyn explained the difference between frequentist statistics and Bayesian. She's a Bayesian, she said.

Well, that explained a lot, because it turns out I'm a Bayesian, too. I just didn't know it. Obviously, that's why Carolyn and I constantly find ourselves traveling the exact same thought path, even though we've never met, and didn't know each other until a year ago.

Of course, a real Bayesian (that would be a Bayesian who knew what she was doing, which would not be me) would probably not conclude that the reason she likes a person well enough to start a vast time-gobbling math-ed web site with her is that you both subscribe to the same school of statistical thought. I'll have to ask Carolyn.

I'm a Bayesian aspirant.

I'm having quite a little midlife run of Self-Discovery here, I must say. First I find out I'm Scots-Irish; next I'm hearing I'm a Bayesian.

I just hope no one's gonna tell me I'm adopted.

### I have a question

My question concerns a passage in a terrific book called What the Numbers Say: A Field Guide to Mastering Our Numerical World by Derrick Niederman & David Boyum. Boyum, it turns out, majored in applied mathematics at Harvard--I didn't know there was such a thing as a major in applied mathematics at Harvard!

Or anywhere else, for that matter.

I wish I'd know that when I was 17.

'Bayes Watch' is Niederman & Boyum's title for this passage:

Years ago a study asked the following question of students and doctors at Harvard Medical School:

If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5 percent, what is the chance that a person found to have a positive result actually has the disease, assuming that you know nothing about the person's symptoms or signs?

Ed and I both understand the answer now (neither of us got it right), but we still have a question about the precise calculations. (Don't hit this link unless you want to see the answer.)

### update

I've just checked Niederman & Boyum. They do not specify a zero rate for false negatives. They say nothing about false negatives one way or the other. (Neither does John Kay in false positives, part 2, assuming I'm understanding him correctly).

### Bayes & God

I actually bought this book a couple of years ago, though I haven't read it yet:

I believe it's intended to be a Bayesian proof of the existence of God, although I don't know how the word 'proof' is used either in the book or in the context of Bayesian statistics.

low birth weight paradox (& Monty Hall)
Monty Hall, part 2
Monty Hall, part 3
false positives
false positives, part 2
Doug Sundseth on Monty Hall
John Kay: We are likely to get probability wrong (subscription only)
Monty Hall diagram from Curious Incident
Bayes & the human mind
Bayesian reasoning, intuition, & the cognitive unconscious
most bell curves have thick tails
ECONOMIST explanation Bayesian statistics
Bayesian certainty scale
probability question from Saxon 8/7

Bayesianprobability

FalsePositivesPart2 21 Dec 2005 - 15:31 CatherineJohnson

Another version of the False Positives challenge. This one ran in John Kay's column in the Financial Times yesterday. (Probably only available to subscribers.)

...intuition does not correspond to the mathematics of probability. One person in a 1,000 suffers from a rare disease. A friend has just tested positive for this illness and the test gives a correct diagnosis in 99 per cent of cases. How likely is it that your friend has the disease? Not at all likely. In random groups of 1,000 people an average of 10 would display false positives and only one would be correctly diagnosed with the disease. But most people, including most doctors, think otherwise. “The human mind,” said science writer Stephen Jay Gould, “did not evolve to deal with probabilities.”

Hmmm. Let's see. This problem does give us false negatives, right???

OK, let me think.

[pause]

Good grief. Not only can the human mind not intuit Bayesian probability; apparently the human mind equally cannot produce consistently lucid prose. (Nothing wrong with Mr. Kay's lucidity on a normal day.)

Kay's example, too, appears to assume a false negative rate of 0.

As far as I can tell.

### update

This is funny. I was skimming Amazon reviews of Stephen Jay Gould's Mismeasure of Man, and I found this:

As Oxford academician Richard Dawkins says (see Bryson, "A Short History of Nearly Everything", pp. 330-332) "If only Stephen Gould could think as clearly as he writes!"

It's a Core Principle in the Writing Biz (& definitely in the Writing Instruction Biz) that you can't write clearly without thinking clearly. (True in my experience; that's for sure.)

low birth weight paradox (& Monty Hall)
Monty Hall, part 2
Monty Hall, part 3
false positives
false positives, part 2
Doug Sundseth on Monty Hall
John Kay: We are likely to get probability wrong (subscription only)
Monty Hall diagram from Curious Incident
probability question from Saxon 8/7

MontyHallPart2 17 Aug 2005 - 21:49 CatherineJohnson

Here is Kay on the Monty Hall problem:

The Monty Hall problem is named after the host of a 1970s quiz show, Let’s Make a Deal. The successful contestant chooses from three closed boxes. One contains the keys to a car and the other two a picture of a goat. The choice made, Monty opens one of the other doors to reveal – a goat. He taunts the guest to change the decision. Should the guest switch to the other closed box?

When the solution was published in an American magazine, thousands of readers – including professors of statistics – alleged an error. Paul Erdös, the great mathematician, reputedly died still musing on the Monty Hall problem. But the answer is, indeed, yes: you should change.

I'm happy to hear that Paul Erdos stumbled over Monty Hall, seeing as how I still don't understand it.

low birth weight paradox (& Monty Hall)
Monty Hall, part 2
false positives
false positives, part 2
Doug Sundseth on Monty Hall
John Kay: We are likely to get probability wrong (subscription only)
Monty Hall diagram from Curious Incident
probability question from Saxon 8/7

SaxonBarModel 25 Aug 2005 - 16:41 CatherineJohnson

We didn't get as much math done this summer as I would have liked. Yesterday we finished Lesson 22 in Saxon 8/7, and we've done 2 Investigations as well. So that's 24 out of 132.

We've also worked through the entire PRIMARY MATHEMATICS 3A & B lessons on fractions, and we've done all the workbook problems. Now we've started on the fraction lesson in PRIMARY MATHEMATICS 4A. I've been planning to post something about this, because we're doing the Singapore lessons with a friend of Christopher's.

We also did one bar model story problem from Primary Mathematics 3A almost every day, and we'll carry on doing one a day for good. As simple as the early 3rd grade problems are, there were plenty Christopher couldn't do.

When I say 'couldn't do' I mean that he specifically couldn't conceive of or draw the bar model.

He could do the problem alone, without the bar model. He could set it up and solve it. But he couldn't represent it visually or spatially at all.

Interestingly, he also can't even come close to doing perimeter problems of figures like this:

Asking him to figure an L-shaped perimeter is like asking him to do calculus. Completely beyond his capacities.

This is a kid whose one adult aspiration is to be an architect.

So we're gonna keep working on geometry. Nets are next!

### does geometry predict math ability?

Carolyn has said that mathematicians think a knack for geometry predicts math ability. Carolyn herself was good at geometry as a kid, while being not-very-tuned-in to algebra.

I think I'm noticing the same pattern in the 4 children whose math work I know best.

Two of these kids are Christopher & Lew. Lew lives next door and is one of Christopher's good friends. Christopher and Lew both seem to be strongest in verbal skills, and both are befuddled by perimeter problems once you get past finding the perimeter of a square or a rectangle.

Meanwhile, the two other boys I know & have worked with--I'll use their initials, 'G' and 'P'--are the exact opposite.

These two boys may have real math talent. I don't know how to spot Real Math Talent, but that's what I think I see in them.

Both G & P, when I showed them a more-complicated perimeter problem, instantly got it. They SAW it; it 'popped' for them, the way a hidden figure in a hidden figures test 'pops' for Temple (Grandin).

I think that's pretty interesting.

Anyways...this is one of the reasons I love the bar models. I'm pretty sure bar models develop visual-spatial 'seeing' or 'understanding' in verbal kids (& in verbal adults like me).

I've mentioned before that the first girl ever to win an international Olympiad was a Singapore student brought up on Singapore bar models. Singapore, as far as I can tell, is the only country to use bar models systematically throughout the first 6 years of elementary school math.

### using bar models to teach fractions in Saxon 8/7

Lesson 22, Problems About a Fraction of a Group, uses bar models to teach fractions of a group.

Christopher spent the whole summer complaining about his daily bar model. Then, yesterday, when we used a bar model to find If 2/5 of the the 270 fans wore green to the game, how many fans wore green? he said, 'These bar models do help!'

They really do.

Here's an example of a Saxon bar model used to solve a diferent fraction-of-a-group problem:

From now on I'm going to have him draw the model and link it to the computation. I'm going to have him explain to me why & how 2/3 x 270 is the same thing as dividing 270 by 3 and then multiplying 90 by 2, as the bar model has you do.

Then we're going to keep doing that until it makes sense.

DougSundsethNumberLines 30 Sep 2005 - 21:37 CatherineJohnson

blank number lines (pdf file)

symmetric number lines (positive numbers, negatives numbers, 0 (pdf file)

number lines: all positive numbers (pdf file)

number lines: all negative numbers (pdf file)

### update

If anyone is interested in, or has time to, critique these study sheets, that would great. (There's no pressing need for this; I'm reasonably certain these are accurate, especially since the second document came straight from the pages of Mathematics 6.

addition & subtractions of integers review sheet

FractionPedagogyQuestion 13 Nov 2005 - 19:58 CarolynJohnston

Lone Ranger put this question on the Request Page:

Nuts and bolt help for my 8 year old daughter for all you math experts....So, we are working in book 4A in Singapore Math and are learning about fractions of a set. We move into problems such as 2/3 of 27. She can easily build this problem with beads or draw it so we move on to the the algorithm. My daughters asks me why 27 can be written as 27 over 1 and I am stumped. I cannot figure out how to "show " her like I can with improper fractions or mixed numbers. I told her it is a division problem but she wasn't ready to understand that. Any ideas?

This is one of those situations where I sputter a bit, trying to think up a good way to teach this to an 8 year-old. Here's my best shot at it (of course a good teacher has at least 3 ways to explain anything, so hopefully others out there will have other approaches):

You could say that the denominator represents the number of pieces that a unit is broken up into, and the numerator represents the number of such pieces that you have: she understands this intuitively when the denominator is a number greater than 1, I expect. Do comparative examples, perhaps, where you go from saying that 4/2 means you've split a unit into two pieces and you have four of them, to saying that 2/1 means you've split a unit into 1 piece and you have two of them.

So I guess that's one way to try to teach it; now we need two more.

RonAharoniOnTheFifthOperationOfArithmetic 14 Sep 2006 - 14:53 CatherineJohnson

Carolyn has kindly left my two favorite passages in Ron Aharoni's What I Learned in Elementary School for me to blooki.

Here's the first:

##### What Arithmetic Should Be Covered in Elementary School?

The embarrassingly simple answer is: the four basic operations—addition, subtraction, multiplication, and division.

Yet, this seemingly simple answer is deceptive in two ways. One is that there are actually five operations. In addition to the four classical operations, there is a fifth one that is even more fundamental and important. That is, forming a unit, taking a part of the world and declaring it to be the “whole.” This operation is at the base of much of the mathematics of elementary school. First of all, in counting, when you have another such unit you say you have “two,” and so on. The operation of multiplication is based on taking a set, declaring that this is the unit, and repeating it. The concept of a fraction starts from having a whole, from which parts are taken. The decimal system is based on gathering tens of objects into one unit called a “10,” then recursively repeating it.

The forming of a unit, and the assigning of a name to it, is something that has to be learned and stressed explicitly. I met children who, in fifth grade, knew how to find a quarter of a class of 20, but had difficulty understanding how to find “three-quarters” of the class, having missed the stage of the corresponding process of repeating a unit in multiplication.

I've thought about this observation every day since reading Aharoni's article. I probably can't explain why. At least, I can't at the moment. (Good thing I'm not taking the Regents, I guess.)

But it reminded me of a post Carolyn wrote early on:

Catherine mentioned that she is a fan of tile fraction manipulatives over the more usual 'pie' manipulatives:

She said that her daughter didn't get anywhere using the more-common circular, 'pie chart' fraction manipulatives; she needed to see rectangular fractions. I have no idea why this would be, but it 'felt' right to me, so I searched for rectangular manipulatives and found these.

I prefer tile manipulatives too, for what I think are solid pedagogical reasons, and here is why: if you want to talk about improper fractions -- fractions greater than one -- with your kid, then the pie-shaped manipulatives add potential for confusion because you can't make a single connected object that represents a quantity greater than one. If you want to represent, for example, 3/2 with pie manipulatives, then you'll have one whole circle and a half circle. You can tell a kid that that represents a single object, the quantity 3/2, all you like; but to him it will look like two objects. Fractions are confusing enough without that.

Conversely, you can make a single line of tiles that is as long as you like.

So unless your child is really off and running with the pie manipulatives, I'd recommend the tile manipulatives.

These are the fraction tiles I like:

You can order extra tiles, too, which I have done. I've used these over and over again, with Christopher, and with at least two of his friends.

Worth their weight in gold.

Aharoni article, part 1
Aharoni article, part 2: America's 'new math' goes to Israel
Aharoni on the fifth operation of arithmetic
Ron Aharoni on teaching fractions & forming units
What I Learned In Elementary School by Ron Aharoni (AMERICAN EDUCATOR)

RonAharoniOnTeachingFractions 27 Oct 2005 - 01:49 CatherineJohnson

Interesting observations via email from Ron Aharoni.

But first, you might want to re-read this post on the fifth operation of arithmetic:

In addition to the four classical operations, there is a fifth one that is even more fundamental and important. That is, forming a unit, taking a part of the world and declaring it to be the “whole.”

### against pizza

I'd sent him the link to that post, which also included an earlier post of Carolyn's about rectangular fraction tiles being superior to circular pizza-pies:

I agree that sticking to the pie representation of fractions is harmful. I also prefer parts of rectangles.

But: I believe that it is important to take, from the very beginning, fractions of sets. What is a half of 6 apples? A quarter of a set of 8 pencils? And then, immediately, WHAT IS TWO QUARTERS, and three quarters, of that set? This conveys the meaning of "three quarters" better than the manipulatives.

And, an elaboration, from a second email:

I try to start with fractions of all kinds of objects - shapes as well as groups. In first grade, I start fractions with division. I give groups of kids all kinds of objects: one group gets a rectangle, another a circle, another group two rectangles, another group an apple, and another A GROUP OF 4 APPLES, and ask them to divide the objects they got into two parts. Later, each group tells what they did. We then discuss the notion of "a half of". Then each kid is asked to do work on his own - take halves of shapes, and a half of say 4 objects drawn on paper.

Then we can divide into three parts, and discuss what is a "third of something".

Then "two thirds" (just repeat twice the one third), then a quarter - all this can be easily done with second graders, even first graders.

### Carolyn says: 3/4 is 3 '1/4's'

This tracks with a point Carolyn made in an email last night:

A unit is rather like the denominator part of a fraction. Many of the rules regarding their manipulation are the same. I intuitively understand why that is, and I am going to try to write it up, but right now words elude me.

Here's a quick try to convey the idea by analogy, though -- the correct way to think of fractions is as a unit -- of the form 1/3, 1/4, 1/5, 1/8, etc. -- occurring some number of times, where that number is given by the numerator.

So you should think of 3/4 as being "the unit 1/4, occurring 3 times".

### on not using a child's pre-existing knowledge

One of the common-sense themes of 'metacognitively-aware' teaching, with which I normally agree, is that one should use what's already there, inside a child's head.

When it comes to fractions, the 'friendly fraction' 1/2 is probably more or less innate; children figure it out without having to be taught. (quoting from memory; not fact-checked)

I'm thinking the 'naturalness' of friendly fractions like 1/2, 1/4, 1/3 and so on -- all representing, for children just starting out, one obvious, natural whole divided into parts -- may be a problem as much as an opportunity.

All textbooks begin teaching fractions with the fraction 1/2.

Always, this is illustrated as 1/2 of a pizza.

I think that's probably a mistake. I'm thinking the idea of 1/2-of-a-pizza may be so deeply ingrained in children's (and grown-ups') minds that the jump to 1/2 of a group is that much harder to make.

### don't laugh

OK, I finally looked up the page in Christopher's 5th grade textbook that utterly threw me last year.

It was 'Lesson 58 Fractions of a Whole.'

The lesson began:

We've looked at a fraction of a whole unit. Now let's review fractions of numbers greater than 1.

Take 1/4 of three identical sandwiches.

There followed a page of drawings showing that 1/4 of 3 sandwiches is the same thing as 3/4 of 1 sandwich.

I didn't get it.

I could see it was true.

I could see that the drawing was 'true,' and I knew, of course that 1/4 x 3 = 3/4 x 1.

That wasn't the problem.

The problem was, I didn't get it.

I was having an especially hard time with the pizza pie chart image that kept popping into my mind:

My problem with this mental image, which was very strong & vivid, was that I simply could not stop seeing THREEFOURTHS.

THREEFOURTHS, to me, is a highly overlearned mental THING; if you say 3/4 to me, I'm going to start seeing visions of circles divided into fourths with 3 of the fourths shaded in.

Period.

I have no choice. It's like a song that's stuck in your head. Only it's not a song. It's a textbook illustration.

So there I was, trying to think about ONEFOURTH of 3, and forget it. It wasn't happening; it wasn't going to happen.

I just could not make that bright, vivid, 3/4-of-1-whole-circle turn into 1/4 of 3 circles.

I could imagine 3 circles, side by side, each divided into 4.

But after that my brain instantly jumped to the THREEFOURTHS clumps. I kept imagining, in sequence:

• first, the shaded 3/4 of Circle Number One (on the left)
• second, the shaded 3/4 Circle Number Three (on the right)

then

• the left-over 1/4 from Circle Number One added to the top half of Circle Number 2 (in the middle) AND the left-over 1/4 from Circle Number Three added to the bottom half of Circle number 2.....

...which I bet at this point nobody can even follow.

I certainly couldn't follow it. Not because it's hard, but because working memory wasn't put together to perform a sequential circle-dividing task of that magnitude.

### the magical number 5

I was thinking.

3 circles, 2 THREEFOURTHS chunks, 2 ONE-FOURTH chunks, and 2 TWO-FOURTH CHUNKS ought to come out to the magical number 7, plus or minus 2.

Apparently I'm down to the magical number 5.

### rescue

Finally my friend Debbie came to the rescue. (I bet I can't find her email....nope, can't find it). Paraphrasing:

The way I always think of this is as three 'one-fourths.' There are 3 sandwiches, and you take 1/4 from each sandwich. That gives you 3/4, or 3 separate one-fourths.

That one sentence clobbered my THREEFOURTHS image.

Suddenly I could 'see' separate little one-fourths pulled out of all 3 circles; I could see the individual one-fourths as.....units, I guess. Like Carolyn would say.

### in conclusion

This is what makes me wonder whether, in some cases, the 'natural math' a child (or adult) brings to class may not be the best hook.

In my case the problem wasn't just the probably-innate friendly fractions children & grownups understand without being taught. My problem was the image of the circle, which, as Carolyn points out, is not an easy thing to break into parts and then rearrange those parts in new configurations.

That's why circles represent things like 'eternal love' and the like, because we don't see circles as having beginnings, or ends—or pieces or parts.

Culturally speaking, at any rate, a circle is the Ultimate Whole.

All of this is a long way of saying that:

• I'm going to stick with rectangular fraction tiles
• Aharoni's idea of starting with 2/3 as well as 1/3 -- of teaching 2/3 in the same moment that you teach 1/3 -- is an excellent idea

### update

Here's a terrific example of why rectangular fraction tiles are superior to circles:

source:
Demonstrating division of fractions with pictures or manipulatives at Math Forum

Aharoni article, part 1
Aharoni article, part 2: America's 'new math' goes to Israel
Aharoni on the fifth operation of arithmetic
Ron Aharoni on teaching fractions & forming units
What I Learned In Elementary School by Ron Aharoni (AMERICAN EDUCATOR)

DimensionalAnalysisMathForum 27 Nov 2005 - 16:07 CatherineJohnson

### also at Math Forum

Dimensional Analysis and Unit Conversions
Dimensional Analysis and Temperature Conversion
Does My Fraction 1/1 Story Work?

DrMathOnFractionsAndUnits 13 Oct 2005 - 17:51 CarolynJohnston

In pursuit of more information about dimensional analysis, Catherine found a link to Dr. Ian on dimensional analysis at the Drexel Math Forum. He has some insights about fractions and units that are priceless. Here goes:

The ultimate way to think of a fraction

In the end, a fraction is just a division that you haven't bothered to do yet. -- Dr. Ian

It's true -- really, 5/7 just means '5 divided by 7' -- whatever number that is!

I remember reading that Wayne Wickelgren (I think it was, or maybe it was Hirsch -- my memory is shot) felt totally empowered and excited as a boy when he realized that a fraction was a division problem that he wouldn't have to actually do.

Dr. Ian also has a lot of cool insight into what makes a ratio a ratio (hint: it doesn't have any units), and how units can help you make sense of fractions:

```But let's say we want to divide 6 pies evenly among 3 people. Now we have

6 pies         6        pies
---------  =   ---  x   ------    =  2 (pies/person)
3 persons        3        persons

That is, the units stay around instead of cancelling. We end up, not
with '2', but with '2 pies per person', in much the same way that if
we travel 45 miles in 30 minutes, we end up with

45 miles        45         miles
--------   =    ---   x    -----   =   90 miles/hour
1/2 hour        1/2        hours

or 90 miles per hour, and not just '90'.

That is, when you introduce units, you end up with two results: a
number, and some combination of units that tells you how to interpret
the number.
```

It's worth repeating

When you introduce units, you end up with two results: a number, and a combination of units that tells you how to interpret the number. -- Dr. Ian

You need to know how to interpret the number, because quantities with units are "slippery". By that I mean that you can change the actual number by changing the units. For example, 8 feet/sec describes the same speed as 8.78 km/hr, even though 8 and 8.78 are obviously not the same number.

That's why it's so important for the units to stay with the quantity as part of the answer to the problem. Remember those good old hardnosed math and science teachers who used to mark us wrong if our answer was "8.78" and not "8.78 kilometers per hour"?

But ratios aren't slippery, because they don't have any units that you can change. If Catherine has 2 dollars and I have 1 dollar, then the ratio of her money to my money is 2, period -- no units are involved in the ratio, because the dollars cancelled. A number without a unit (such as the result of any ratio calculation) is called a 'dimensionless quantity", and those are the numbers you can really count on, because you can't change them with a unit conversion.

Finally, it's pretty easy to calculate using units -- the words just come along for the ride.

Terminator update

KDeRosa sent me an updated version of his essay on Fear And Loathing In Engineering School. It's got links in it, so that people can resolve those confusing references to the mysterious number 7. I've replaced the original esssay with it at its permalink location here. (For those who want to see the original essay, it's still here in the comments where he originally wrote it, about the fourth one down). Here's to its long life as an internet classic!

ProblemsRemedialCollegeFreshmenCantDo 27 Nov 2005 - 16:09 CatherineJohnson

Notice that all of these questions involve fractions. Fractions & percents.

I remember being shocked when Carolyn first told me that the barrier to college kids succeeding in algebra was fractions. I had no idea. Then Bernie told me the same thing, and since then every math professor I've ever talked to has said, 'Students can't do fractions.'

Of course, this made sense, since at the point where I met Carolyn I had recently discovered I couldn't do fractions, either.

That's not quite right.

I could do fractions.

I couldn't teach them.

Waukesha County Tech is short for Waukesha County Technical College. These are college freshmen we're talking about.

source:
Journal Sentinal 10-6-2003

FractionManipulativeLessonOnReciprocals 14 Sep 2006 - 14:26 CarolynJohnston

The other night, Ben was working on his math, and I was doing something else. He paused in his work and asked me: "Mom, what's a reciprocal?"

I guess it's no surprise that he doesn't know what a reciprocal is, since it wasn't taught in his Everyday Math classes in elementary school, and since (apparently) it's not introduced in the early part of Saxon 6/5, the curriculum I was supplementing from last year. But in Saxon 8/7, which he's using this year (he tested into it, I swear), knowledge about reciprocals, and the role they play in division of fractions, is assumed.

So I'm doing reactive teaching again, but at least this time I'm reacting to the curriculum of my own choosing.

Saxon has had problems in the mixed practice the last few nights that go straight to the heart of why the reciprocal gets involved in fraction division. The questions are like this: how many 3/8s are there in 1? How many 4/5ths are there in 1?

Here's a demonstration I devised for him on the 3/8ths problem, using the tile fraction manipulatives that Catherine and I have recommended here (warning: pies won't work for this very well).

This sort of question seems to throw him off, so I start by asking other questions that sound more familiar, like: How many 2s in 8? and How many 3s in 9? Then I point out that he is getting the answer by dividing, so by analogy, we'd want to divide 1 by 3/8.

Everyone knows the rule for fraction division: Ours is not to reason why, just invert and multiply. But of course, we are modern traditionalists here, and procedural knowledge is only the beginning of our demands. We want our kids to have an understanding, too, of why they are inverting and multiplying. I taught Ben the invert-and-multiply rule, but then I wanted to convince him that the answer that the invert-and-multiply rule gives you, 2 and 2/3rds, is the right answer.

We attacked the question of how many 3/8ths go into 1 directly, using the manipulatives. The manipulatives were all placed on a sheet of paper, so I could write curly braces and labels next to the tiles. I drew a diagram below of what we do with the tiles (note that the 3/8th tiles are not really single blocks, they are 3 1/8 blocks in a row; I have to tell him to think of them as a single unit.. The labels and curly braces help with this).

It's easy for him to see that two 3/8ths will fit into the 1; I stick them below the 1 tile, and label them as "2 3/8ths". A third 3/8th will overhang the end, though. So I take the extra 3/8ths unit and break it apart into thirds (pointing out that that's what I'm doing). Two of those thirds will fit into the rest of the space in the 1. So this gives us a total of 2 and 2/3rds 3/8-units that will fit into the 1.

I don't expect that this is the end of this; we'll do this a bunch more times and hopefully it will sink in. The trick is to get the kid thinking of the divisor (in this case 3/8ths), however weird a fraction it is, as being a unit. I hope Saxon keeps this sort of problem coming for a while.

FractionMultiplication 23 Oct 2005 - 01:52 CatherineJohnson

I think I have at least 4 different explanations of fraction multiplication from ktm Commenters — thank you!

I'll work through them all, and report back (and get them posted—I have some beautiful graphics from Dan and also from Rudbeckia Hirta).

InstructivistOnTeachingPercent 26 Oct 2005 - 14:46 CatherineJohnson

Math Disaster

Maybe the Instructivist will come to Irvington to teach Christopher & me:

I teach math to eighth graders and know that teaching math successfully need not be rocket science. Most of my students can now convert fractions to decimals and percents (and vice-versa) in their sleep. They can also solve the three different types of percent word problems (unkown rate, whole and part) in their sleep.

I'm actually not familiar with percent problems called 'rate' problems.

?  ?  ?

I think I'm reasonably adept at converting fractions to decimals to percents & back, and at solving percent problems....so I can't tell if this is a problem I haven't encountered, or if this is a problem I have encountered, but called something else.

### I like the chart

Instructivist's percent-fraction-decimal chart is a terrific idea, IMO.

It's the same principle as Saxon always printing the equivalent expressions on his fraction manipulatives, which Doug did, as well:

### metacognitive moment

Converting percents to fractions generally poses no problems when the percent is a whole number, e.g. 47% --> .47 --> 47/100. A special problem arises when the percent is a fraction like 5-1/4 %. This is where the students need to realize that an additional step is required. Many students want to enter 5.25 in the decimal column. The task of the teacher is to focus on this problem and to show that the students must first convert to 5.25%, then to the decimal .0525 and on to the fraction.

I love this!

Instructivist knows where his students are going to go off the rails.

This is ESSENTIAL.

I learned this lesson back when I was researching a book on happy marriages. (I may have 'blookied' this before...)

I was driving all over creation, interviewing folks in their homes.

Some people could give directions a person could actually follow; some people couldn't.

I call directions a person can actually follow 'good' directions.

I call directions that end up with a person driving 30 or 40 miles around in circles 'bad' directions.

The problem with the bad directions wasn't that they were wrong. They weren't wrong.

The problem with the bad directions was they made no allowances whatsoever for the fact that a human being was going to be attempting to get somewhere on the basis of those directions.

People who give good directions invariably say things like, "If you get to the Stop 'n Shop, you've gone too far."

Invariably.

People who give good directions know what mistakes a normal human being is guaranteed to make and they warn them off!

That's what Instructivist is doing here.

### J. D. Fisher on teaching fractions

It's necessary, I think, to make it instinctive in students to understand percent as a "number over 100."

From there, I've always liked the idea of moving the decimal point backwards.

5 1/4 = 5.25

5 1/4% = 5.25%

5.25% = 5.25/100

Move the decimal point to the left twice (the power of ten you are dividing by), write in the necessary zeros, and remove the percent symbol:

5.25% = 0.0525

### lots more math horror stories, too

One Commenter left this:

I am a high school chemistry teacher at Bowie High School in Bowie, MD. I have students in their junior and senior year who have received straight As in math for their entire middle and high school careers. They can't do simple arithmetic. They don't know the 'why' of a mathematical operation and so do not remember the 'how'. In fact, they have never been taught 'why' they had to learn anything in math other than counting for the use of money. I kid you not.

This part is incredible:

I have to teach math for 4 to 6 weeks every year. The students have been taught the wrong rounding rules by their math teachers.

The wrong rounding rules?????

The wrong rounding rules?

I'm going to have to pause.

[pause]

There.

Breathing normally again.

Where were we?

Oh, yes.

The students have been taught the wrong rounding rules by their math teachers.

And that's not all—

They also can't find a percentage or know what it means, understand significant figures (significant figures!), do scientific notation, know anything about the laws of exponents, find the equation of a line or graph an equation without their \$90 graphic calculators (which they don't understand how to use), solve for a variable in an equation (density = mass / volume is an example), convert inside the metric system, covert in the English system, understand the concept of conversion factors, or do anything without extensive direct instruction with days of practice and activities in the mathematical concept.

[snip]

What in the ever loving heck is going on in the math classes in my country? This is insane. You ought to see when I have a parent conference of an honor student and point out that the kid they say is not good at math (a backdoor plea to lower the standards) had received As and Bs in all their math classes and I pretend to act incredulous that they would offer such an excuse for their obviously highly mathematically competent child. The parent and admin damn near chew a hole in their cheek. I just sit there and look all innocent. Admin is way on to me...but they never say anything. They do glare and act cold.

### what do you make of this?

another Comment:
It goes all the way up the line. When I was a college professor (at an expensive, selective school) teaching an upper-level molecular genetics course, I observed juniors and seniors, who had somehow gotten As and Bs in freshman calculus, who were mystified by the simple Algebra I equation-solving involved in teaching DNA renaturation kinetics. It's a wonder we manage to produce any scientists and engineers at all in this country.

Do these students really not know algebra 1 equation-solving, or have they......forgotten(?)

Last, but not least, we hear from teacher Bill D.:

I use my blog to combat ignoramuses like "instructivist" who are destroying education. Dumbing down education...please. Its the test nazis and the traditionalis who refuse to allow education achieve higher potential "Destructivist' is more like it. They destroy student's souls. As yo can see., I often use the blog to argue I also see great potetial for student/teacher interaction through blogging.

What is it with these people and their Nazi talk?

test nazis?

Where does this stuff come from?

And why is this person (presumably) teaching in our public schools?

TeachingFractionsWithMoney 31 Oct 2005 - 17:31 CatherineJohnson

I just noticed this Comment from a ktm guest:

I'm trying out something new that has been amazingly successful, alhtough it is an extension of an ubiquitous practice, which is using money to teach fractions to decimal conversions.

I've extended that use to, when we practive converting fractions to decimals, I say, "Put it in money," not just when it is abstract, i.e. pure math, but when we are doing story problems.

For example, if the question involved, say, dividing up three whole pies into five equal amounts, I would say, "Put it in money." Immediately, she puts down "\$3.00" and splits up five ways into 60 cents. This has enabled her to grasp with amazing speed and comprehension what was previously a concept she struggled with (Yes, we are Everyday Math victims).

I think that's pretty interesting; I especially like the fact that the child gets practice converting one entity (pies) into another (money). I found a fascinating study of children's cognition & perception this weekend that I interpret to mean this is a good idea. (Will get around to 'blooki-ing' it one of these days.)

I'm logging this into the book-style index.

DanOnFractionPreTest 28 Nov 2005 - 17:27 CatherineJohnson

Last week sometime I was asking people whether this online fraction pre-test was OK.

The website has some glitches, so the question became: would this test be OK if the website worked?

Here's Dan's response, which I'm filing in the Book-style index:

I think the test is too brief. If they ask two fraction addition questions and you get one wrong, was it a careless error or do you fail to grasp the concept? Four questions of each type would be a little more telling.

It should clearly state whether it wants improper fractions or mixed numbers as answers, or if it doesn't matter.

I also think the fraction addition problems are too easy. The denominators are too similar; there's no difficulty in finding a common denominator. Also, you could solve these by simply drawing a square cut into eighths. I want a problem that requires you to convert the fractions to a meaningfully different common denominator.

A problem with a negative answer might also be nice.

You could argue that it belongs in a decimals or percents section instead, but it might be good to ask for 25% to be written as a fraction, or 0.4 written as a fraction in lowest terms.

And I can't resist singing my favorite note: they should include problems that explicitly ask for cancelation of common factors in the numerator and denominator to prove that the student gets it. Then, this should be extended to units, i.e. dimensional analysis! (I couldn't resist)

trust but verify redux

I bring this up because Christopher missed the fraction pre-test his teacher gave last week, and I had a gnawing suspicion he's not remotely where he should be on the subject.

But more than that, especially in the wake of reading Engelmann, I think we parents need our own set of assessment tools. (The link above includes Lone Ranger's advice on using the Iowa Test of Basic Skills at home.)

Ideally, I would like to see every curricula used in the schools publish pre- and post- unit tests parents can administer at home if they choose; I would also like to see the results of any and all such tests the school administers. (The middle school—and, I assume, the other schools as well—administered pre-tests in every subject this fall. I think that's excellent, but I'd like to know how the kids fared. Another question for the TEAM MEETING.)

Given the fact that in my experience schools aren't especially forthcoming on these questions, I want access to such tests myself.

Beyond that, I would like to be able to administer the TIMSS test, or a valid TIMSS equivalent, to my own children. (You can administer a small portion of it.)

And because TIMSS is given only to 4th, 8th, and 12th graders, I need a valid, norm-referenced standardized test to use each year if I so choose.

I don't know what an 'A' or a 'B' means in the larger world, and I certainly don't know what a canned comment like 'Making satisfactory progress' indicates.

We need to introduce some checks and balances into the system—or more than we have now, at any rate.

pattern training redux

I gave the online test to Christopher. It was a disaster.

He answered only 6 out of 10 questions, and 1 of his answers was wrong.

I went on a frenzy of workbook/worksheet acquisition before I realized that I might be looking at pattern training. The online fractions aren't written with fraction bars, and the run-on word 'dividedby' was used instead of a division sign.

So I wrote out the problems in standard form. Christopher did every problem correctly.

Pattern training lives.

I think I have a fairly good sense of where he is with fractions at this point.

His knowledge is higly inflexible, and shaky to boot. He's nowhere near procedural fluency, although he does have the basic procedures down.

He does not know how to add and subtract with borrowing. That information has gone missing.

Ed said, 'Can you talk Mr. Liu into giving you the fraction worksheets out of sequence?'

I'm thinking that one over.

In the meantime, I have 3 very good workbooks, all purchased at Lakeshore Learning:

• if Pre-Algebra Math Grades 5-8

• Spectrum Math Grades 6, 7, & 8 (separate books)

The Spectrum series is organized by chapters, and includes a pre-test and a chapter test for each one.

another fraction pre-test

Cure Your Math Anxiety: Basic Math Skills-fractions

This site includes 8 lessons plus a fraction pre-test:

Fractions Pretest and Terminology

answers to fractions Pretest and Terminology

AnneDwyerOnTutoring 16 Dec 2005 - 21:44 CatherineJohnson

What I've noticed with my tutoring students is this: if they don't understand something in math class, they try to find a procedure or "trick" that works everytime.

Since they don't really understand it, when they have to go back and do it on a test or later, they don't remember the "trick" exactly and their answers are consistent, but wrong.

For example, I was tutoring a student in basic math. He didn't really understand that a whole number has an implied decimal after the number (e.g. 3 is really 3. for a decimal problem)

When he first learned to divide decimals and he was following the teacher's examples, he was doing the problems right: So if he was dividing .045 into 15, he moved the decimal over three places for the .045 and three places for the 15. He even managed to get it right on the first test.

But he did them wrong on every test after that. When we were studying for the final, I was able to watch him do the problems.

Since he really didn't understand, he made up his own "trick". In the problem above, he would move the decimal over for the .045 correctly, but he put the decimal point in front of any number inside the divisor sign. So .045 into 15 became 45 into 150 instead of 15,000. And, because he had taught himself this trick, he ignored all decimal points inside the divisor sign. So even .045 into 1.5 became 45 into 150.

Needless to say, it took a while to find the problem and then to correct it.

IMO, with Christopher, because the class is going so fast and he doesn't always understand what he is doing, he will figure out his own rule and then apply it. You have to go back and see exactly what he is doing when he does the problems so you can identify the error he is making.

We are in fraction & decimal he** around here, which is annoying because I don't think we would be with Saxon or Primary Mathematics—and we weren't going into this course.

This is part of what I mean when I say Christopher is 'losing knowledge' he already had, or experiencing 'math regression,' or just......getting all jumbled up. I think he is becoming uncertain of procedures and knowledge he used to have fairly well nailed-down. (Though I don't know.)

Anyway, both of the ideas here strike me as excellent ideas.

First of all, I'm going to start writing whole numbers with a decimal point and some zeros to the right. I know that will help.

And second, I'm going to keep my eye open for 'invented shortcuts.'

One strategy I've begun, which I think is going to improve matters, is that I'm continually telling him that 'math shortcuts' come from the longer equations he's learned in the past. His teacher seems to be teaching only the shortcuts—either that, or he's only picking up the shortcuts, not the explanation for why they work. Either way, the result is the same: he's learning math tricks.

Last night, when I insisted on showing him why you could invert and multiply, he got his 'eureka' smile.

I'm sure he will have forgotten what I told him by today, but I'm going to keep hammering away at this.

I do think that the basic principle—that math shortcuts come from general principles he already knows—will stay with him, and will help.

source:
reciprocals

FractionQuagmire 01 Feb 2006 - 03:34 CatherineJohnson

I am now officially sick of doing fraction worksheets.

I did 200 fraction worksheets in KUMON Math level E.

I have done 109 fraction worksheets in KUMON Math level F.

I just leafed through Russian Math, trying to see how many fraction calculations I did in that book. There were so many I gave up.

Let me put it this way.

In Chapter 3, Fractions, Decimals, and Percents, I did 303 fraction calculations, give or take a few, in Lessons 3.1 - 3.3.

The title of Lesson 3.4 was: 'More Multiplying Fractions.'

Meanwhile, I'm not getting better.

In fact, I'm pretty sure I'm getting worse.

Used to be, when I missed an answer, there would be some Minor Detail I'd messed up on.

A careless error, as they say in the math-ed business.

Those days are gone.

These days I'll do a fraction calculation, come up with an answer of 80, check the Answer Key, & find that the correct answer is 26 2/3. *

This happens on every worksheet, without fail.

Fortunately, Level F is it for fractions - assuming I manage to pass the Level F test & move on to Level G, that is.

I may not.

At this point all I have to do is LOOK at a fraction calculation to start thinking things like 7 x 6 = 43 or worse.

* true story

-- CatherineJohnson - 30 Jan 2006

NewYorkStateMathTestGrade6Part2 30 Jun 2006 - 11:07 CatherineJohnson

update: oops

Ms. Kahl did send home state test prep material (see below). Apparently, Christopher has a PACKET.

Good!

He and his dad are working on the scale drawing right now. (see below) They're having a blast.

fyi, I think scale drawing is a fabulous assignment. Christopher is finally getting some extended practice using a ruler, and of course a scale drawing means fractions, ratios, & proportions.

It's true Christopher couldn't do this assignment on his own. (I'm feeling smug today because my fiercest competitor-mom, aka the 'Homework Nazi,' could not do this assignment. She told Ed, 'I didn't even know where to start.' Hah! I say Hah! because this woman is good. She's blowing me out of the water.)

However, Ed isn't doing this assignment for Christopher. He's helping.

update update

Ed is grumpy.

The scale drawing was fun for the first two hours.

The last two hours weren't fun at all.

"This is vacation."

"I don't see why they're giving this much homework on vacation."

"I have a huge amount of work to do myself; this took 4 hours."

"Christopher doesn't know anything about ratio."

"He doesn't have any conceptual understanding at all."

"He kept looking for formulas to do things."

"He didn't even know where to begin."

"He doesn't have a lot of natural ability in math." [ed.: Any assignment that ends with the parent deciding his child doesn't have any natural ability in math is the wrong assignment a far as I'm concerned]

"She has no idea how to structure an assignment." [ed.: ditto]

Over dinner Ed was pondering the 'packet,' which turns out to be a special Glencoe-produced 58-page booklet called "Mastering the Intermediate Level Mathematics Test: Diagnose — Prescribe — Practice Workbook."

Fifty-eight pages of items aligned to the New York state test, with no answers or solutions.

Apparently our job over 'break' is to Diagnose — Prescribe — Practice and also create our own answer key.

Well, thank God I've got Smartest Tractor lighting the way (pdf file).

back again

I've been off doing Career Stuff that's actually been quasi-fun.

I say quasi because my particular career seems to involve heaping loads of crapola* (not a nice word on Sunday!), not to mention the occasional bolt from the blue.

The other day I called my agent and, when her assistant answered the telephone, said, 'Hi, this is Catherine.'

The assistant said, 'Who?'

That's the crapola aspect; I'll spare you & me both an extended account of the bolt from the blue part (though poor Caroline is slated to get an earful today....)

Anyway, I've been off because I was doing Career Stuff that was actually a blast.

This involved going into the city to meet with our kids' psychiatrist, Eric Hollander (that was the fun part), after which I decided to surprise Ed in his lair. (What is that woman in white doing in the picture?)

It had to be a surprise, because I didn't have my cel phone with me. I didn't have my cel phone with me, because I forgot my cel phone.

I need WAY more exercise.

So I decided to drop in unannounced.

Naturally that didn't work out; Ed wasn't there, and when he finally did get there he had five minutes to get to a faculty meeting.

So there was nothing left to do but visit the NYU bookstore and look at every single education title on both floors.

Nothing by Diane Ravitch, that's for sure.

It was all constructivism all the time. Every last textbook.

That and feel-good books about heroic white teachers teaching poor black children — books like Small Victories: The Real World of a Teacher, Her Students, and Their High School. (I'm thinking a 'small' victory probably doesn't include teaching kids enough algebra to graduate from high school, but I don't know.) There were many of these books.

Until that visit, I hadn't realized that heroic white teacher saving poor black children must be an important fantasy element in ed schools today. I say fantasy element, because I'm pretty sure all of the teachers in all of the books were white, while all of the kids were black. Certainly Jaime Escalante was nowhere to be seen. (Of course, neither was Rafe Esquith, and I don't expect to see Our School turn up on the assigned reading lists any time soon, either.)

Perusing the offerings, you wouldn't know teachers teach math. Everything was about 'literacy' and 'authentic assessments' of literacy and the like. Which is probably just as well, considering.

There was one book that stuck out like a sore thumb: Techniques for Managing Verbally and Physically Aggressive Students. I think that was the title. This book was so unadorned by photos of Beaming White Teachers surrounded by Adoring Black Children that it was refreshing.

Leafing through the pages I found instructions on what a teacher should do when he is being strangled by a student.

The 2 or 3 books that did address math were constructivist all the way. Liping Ma was absent; John Van de Walle's now-classic hundred-dollar tome Elementary and Middle School Mathematics: Teaching Developmentally, Fifth Edition was present in abundance.

The funny thing was, the store management had stocked a bunch of food business textbooks just across the aisle from the ed books. There was a book on restaurant math — I think it was Math Principles for Food Service — that was pure direct instruction. No photos of smiling white teachers surrounded by black students yearning to succeed in food services, just stuff you need to know. Chapters on 'weights and measures,' 'portion control,' 'converting and yielding recipes,' 'production and baking formulas,' and 'using the metric system of measure.' If you're going to make it in food service, you're going to need some math. Seeing as how the first chapters cover addition, subtraction, multiplication, and division, apparently you're not going to learn any of this math in grade school.

Part 1 is titled: Using the Calculator.

upstairs, downstairs

So those were the texbooks, which are housed downstairs in the basement.

Upstairs, in the 'commercial' section, I found:

AND

A clean sweep.

New York state tests coming right up

In March.

Christopher's class took a sample test (pdf file) this week; only 2 kids scored a 4. Christopher thinks he got a 3. Apparently the teacher told them that any kids scoring a 1 or 2 would be moved down to Phase 2-3.

This is the Highly Accelerated, Algebra-in-the-6th-grade, Death March to Algebra-in-the-Eighth Grade Phase 4 extravaganza I've been banging on about. Only two of 19 children can score a 4 on the sample test and apparently there are enough kids in danger of scoring 1s and 2s that the teacher is talking about it in class.

So here's the scoop.

Christopher is studying algebra in the 6th grade, but he can't do percent. I pulled the Sample Test, which turned out to be the test Christopher's class took this week, and asked him about problem number 26:

On Friday and Saturday, there were a total of 200 cars in the parking lot of a movie theater. On Friday, 120 cars were in the parking lot.

Part A

What percent of the total number of cars were in the parking lot on Friday?

Part B

What percent of the total number of cars were in the parking lot on Saturday?

Christopher has no idea how to do this problem, in spite of the fact that he's just 'finished' the chapter on ratio, proportion, and percent in Prentice-Hall. (Says he 'froze up' on the test; expecting another D; etc.)

my vacation and welcome to it

We are on mid-winter vacation this week.

For my vacation, I will be teaching Christopher how to do percent.

I know how I'm going to do it. I'm going to use the Singapore-Saxon bar models and the Saxon-Dolciani percent charts.

I think I'm starting to get a feel for teaching-to-crammery, which is the skill middle school parents need most. If I've got 5 days to teach percent word problems to proto-mastery, I'm going to need bar models & charts (& possibly Saxon's brilliant starter WP variables to boot).

If that were all I had to do this week, I'd be cool.

It's not.

I'm also going to have to figure out what's on the freaking test.

I read some guy last week complaining that Most Parents don't have the Sense of Responsibility it takes to find out what the state standards are.

Sure, sure; we all know about those Parents who don't have a Sense of Responsibility as defined by the people who write state standards.

How many parents fall in this category?

I'd estimate, conservatively, that perhaps 99% of all parents have zero interest in what the State Standards are.

The reason 99% of all parents have zero interest in what the State Standards are is that their Bayesian priors are telling them the State Standards are likely to be:

a) impossible to find

b) bunk

Given my household's limited common sense-y, my own attitude can be characterized as: 'Damn the Bayesian priors, I want those standards!'

Thus, I have now attempted to a) locate and b) comprehend my state standards.

Which means I am now qualified to tell you that all those irresponsible parents are correct. Spending your Sunday morning tracking down New York state standards (pdf file) is what Carolyn calls a FWOT.

See?

a visit to the mathematical reasoning strand!

1. Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.

Students:

• apply a variety of reasoning strategies.
• make and evaluate conjectures and arguments using appropriate language.
• make conclusions based on inductive reasoning.
• justify conclusions involving simple and compound (i.e., and/or) statements.

This is evident, for example, when students:

• use trial and error and work backwards to solve a problem.
• identify patterns in a number sequence.
• are asked to find numbers that satisfy two conditions, such as
n > -4 and
n < 6.

That certainly clarifies things.

source:
New York state standards

the return of common sense-y

So forget about the New York state standards. If I need standards — and I do — I'll use California's.

My job now is to go through every page of the Sample New York state test, pull out the problem genres, and teach them to crammery.

I have one week to do this.

We're going to have to pedal, because we also have to help Christopher with the massive scale drawing exercise Ms. Kahl has sent home for the kids to do over vacation:

This project requires you to be creative and draw up the floor plans of your ideal bedroom. Will you have a big screen television, a walk in closet, or even a king sized bed? You will map out the blueprint for your room and show the furniture and items contained in our room from an aerial view in the form of a scaled blueprint.

The blueprints must contain at least two of each of the following geometrical figures:

• square

• rectangle

• triangle

• trapezoid

• paralleleogram

• circle

Oookaaaayyy!

Two trapezoids coming right up!

And two parallelograms!

In a 6th grader's dream bedroom!

Making those real world connections!

my vacation and welcome to it, part 2

Getting Things Done:

• analyze sample state test

• find out if Christopher can do any of the problems on it

• teach to crammery

• re-vamp book proposal, deal with inevitable assorted mishegoss sp?

• help Christopher construct highly complex scale drawing he can't possibly do on his own

question

You are teaching accelerated 6th grade math.

You give your class of 19 students a sample New York state standards test.

Only two children score a 4, 'exceeds state standards.'

Many of the children, who have just taken a test on ratio, proportions and percent, miss the percent question.

For mid-winter vacation you assign:

• daily 20-item problem sets of percent, ratio, and proportion problems ranging from simple calculations to word problem applications

or

• a complicated scale drawing requiring two trapezoids and two parallelograms

Alright. It's 2:28, and I must go for my 45-minute aerobic walk-run. If I do this 6 days a week until I'm dead I'll be younger next year and, even better, I'll stop screaming at my kids.

So I'm going to do it.

Because I am a responsible parent!

When I get back I'll analyze the test. Then I'll break the news to Christopher that we're going to spend mid-winter break cramming math.

Farewell, Ms. Kahl! We who are about to die salute you!

update, update, update: Verghis speaks!

For the blueprint, why not have a square study desk whose top is decorated with (a) 2 trapezoids, (b) 2 parallellograms, (c)....

This should (1) satisfy Ms. Kahl's requirements and (b) blend nicely with the surrealism that pervades the math curriculum.

Don't you think?

Yes! I do think!

* heaping loads of cr***** probably doesn't distinguish my job from most other people's jobs, I realize

pre-algebra is bunk
death march to algebra
NYU ed textbooks; NY math test
state test impending doom

teachtocrammery

-- CatherineJohnson - 19 Feb 2006

MsKTeachesDimensionalAnalysis 19 Mar 2006 - 06:46 CatherineJohnson

This is hilarious.

Christopher came home today and said that in math they 'did measurement.'

I asked him if they did any measurements not starting from the 0 end of the ruler, and he said scornfully, 'We didn't do rulers. We did cups.'

Then he said, as an afterthought, 'Oh, Mommy! She taught us those unit multipliers!'

I love it!

I guess after nobody could convert 48 ounces to quarts she had the same thought I did:

UNIT MULTIPLIERS!

NOW!

BEFORE IT'S TOO LATE!

Naturally I'm feeling smug that my child was the only student in the class who'd heard of the things & knew how to use them.

update 7-16-2006: Smug is one thing; mastery is another. Christopher knew what unit multipliers were & how to use them, but he wasn't anywhere near mastery. We're starting unit multipliers again today. I'd like him to reach mastery/automaticity by fall, which I think should be possible. If not, this summer's unit multipliers will put him in good shape for the unit multiplier lessons he'll have in Saxon Algebra 1/2, which he'll be working his way through this summer and next school year.

update 9-14-2006: Christopher worked his way through 12 lessons of Algebra 1/2 and numerous pages of Vocabulary Workshop this summer. And that was about it. New target date: mastery of unit multipliers before college.

dimensional analysis word problems and answers
another cool dimensional analysis problem Saxon 8/7
Dr. Ian at Math Forum on dimensional analysis

Dan K's dimensional dominoes
Dan's dimensional analysis page

a nice dimensional analysis problem for printing

Carolyn on dimensional analysis & when to teach it
Carolyn teaches dimensional analysis
rough script for dimensional analysis lesson at home (needs revising)

another triumph for dimensional analysis

the sad life of a teen who failed to learn dimensional analysis

-- CatherineJohnson - 10 Mar 2006

KumonFractionLine 17 Mar 2006 - 00:45 CatherineJohnson

from H. Wu's Some remarks on the teaching of fractions in elementary school (pdf file):

It is widely recognized that there are at least two major bottlenecks in the mathematics education of grades K–8: the teaching of fractions and the introduction of algebra. Both are in need of an overhaul. I hope to make a contribution to the former problem by devising a new approach to elevate teachers’ understanding of fractions. The need for a better knowledge of fractions among teachers has no better illustration than the the following story related by Herbert Clemens (1995):

Last August, I began a week of fractions classes at a workshop for elementary teachers with a graph paper explanation of why 2/7 ÷ 1/9 = 18/7. The reaction of my audience astounded me. Several of the teachers present were simply terrified. None of my protestations about this being a preview, none of my “Don’t worry” statements had any effect.

This situation cries out for improvement.

I'll say.

Here's one of my favorite passages from the 'Math Warrior' literature I've consumed thus far:

Let us look more closely at the way fractions are introduced in the classroom. Children are told that a fraction c/d, with positive integers c and d, is simultaneously at least five different objects (cf. Lamon 1999 and Reys et al. 1998):

(a) parts of a whole: when an object is equally divided into d parts,
then c/d denotes c of those d parts.
(b) the size of a portion when an object of size c
is divided into d equal portions.
(c) the quotient of the integer c divided by d.
(d) the ratio of c to d.
(e) an operator: an instruction that carries out a process,
such as “2/3 of."

It is quite mystifying to me how this glaring “crisis of confidence” in fractions among children could have been been consistently overlooked. Clearly, even those children endowed with an overabundance of faith would find it hard to believe that a concept could be so versatile as to fit all these descriptions. More importantly, such an introduction to a new topic in mathematics is contrary to every mode of mathematical exposition that is deemed acceptable by modern standards. Yet, even Hans Freudenthal, a good mathematician before he switched over to mathematics education, made no mention of this central credibility problem in his Olympian ruminations on fractions (Freudenthal 1983).
I love his writing — very droll.

He goes on to say that:

...there are recurrent reports of students at the University of California at Berkeley and at Stanford University claiming in their homework and exam papers that

I have yet to make my way through Hu's 183-page opus (pdf file) on fractions & how they should be taught to elementary schoolchildren and elementary school teachers, but this point has stayed with me:

[Hu's approach] starts with the definition of a fraction as a number (a point on the number line, to be exact), and deduces all other common properies ascribed to fractions...from this definition alone.

I've been using this idea ever since I came across it.

I point out to Christopher that fractions are numbers that can be located on a number line, just like whole numbers.

I also remind him that whole numbers can be expressed as fractions. At some point in the near future I'll start bugging him about the meaning of the word 'rational' in rational numbers: a rational number is a number that can be expressed as a ratio.

Hence any whole number, fraction, or integer can be expressed as a ratio.

I've noticed that Saxon also approaches fractions as numbers-on-a-number line, although Saxon does this a bit differently: Saxon has kids use fractions for mental-math skip-counting, like so:

Count up from 0 to 5 and back down again by thirds.

I don't remember whether that particular exercise is in Saxon 6/5 or 7/6, but it could have been.

I like it.

It turns out that KUMON Level G has fraction & decimal number lines:

Here's the whole sheet:

number line worksheets

2 KUMON fraction & decimal lines

Chapter 2: Fractions by H. Wu (pdf file, 183 pages)

-- CatherineJohnson - 14 Mar 2006

VintageBooks 24 Mar 2006 - 00:50 CatherineJohnson

for the ktm brain trust:

Does anyone know whether and by how much the Richard Brown updates of Mary Dolciani's high school books differ from the originals?

Here is what I assume is Brown's update:

Algebra Structure and Method Book 1
ISBN: 0395977223

I have this book, and, if I'm lucky, I've just ordered the Teacher's Edition, too.

Unfortunately, Kathy Iggy has got me started down the path of BUYING VINTAGE ARITHMETIC BOOKS (it's all Kathy's fault), beginning with my purchase of Holt Arithmetic 1 last week.

Here's a picture of Book 2, which I've also ordered:

So now of course I'm looking to get into lots more trouble.

Algebra 1
ISBN: 0395535891

Algebra 2 and Trigonometry
ISBN: 0395535921

The one hope I have of not purchasing these books rests with you, dear ktm readers.

If someone in the know were to tell me these books are virtually identical to the later versions, I would have pause.

how to make money without really trying

This is something many of you may not know.

The way to make zillions of dollars writing a textbook is to rewrite a bestselling textbook after its original author has died.

I may have to have this

Algebra: Structure and Method Solution Key Mary P. Dolciani, Brown, Robert H. Sorgenfrey, William L. Cole
Paperback / Houghton Mifflin College Div / May 1993 / 0395677645

vintage arithmetic books at ebay

you are among friends

Aaaaaagghhh! Please. Stop. Or else I will be FORCED, I say, FORCED! to go to the used book store and buy more vintage math books.

(me): "Hi, my name is Google Master, and I'm addicted to buying old math and engineering books."

-- CatherineJohnson - 22 Mar 2006

UnderstandingReciprocals 27 Mar 2006 - 20:23 CatherineJohnson

[I've finally read the thread on teaching reciprocals - great stuff. I'll get it all pulled into this post so it shows in the archives tomorrow...]

I was bewildered by reciprocals when I started re-teaching myself arithmetic. I could divide fractions easily; I also knew when to divide fractions; I could even make up a proper word problem to illustrate the division problem 3 1/4 ÷ 3/4, something the American teachers in Liping Ma's study couldn't do.

But I had no idea why reciprocals worked.

Reciprocals seemed like magic.

Part of the problem, I now realize, is that I saw so many visual illustrations of fractions that I thought I was supposed to be able to visualize reciprocals, too. And I couldn't. Try as I might, I couldn't conjure up a Reciprocal Picture in my mind's eye.

So I was stumped.

Then one day, cruising math sites on the web, I saw this explanation of reciprocals and the whole thing became blindingly clear and obvious at once:

The minute I saw a fraction division problem rewritten as a "complex fraction" (is that the term?) I saw why you start with a division problem and you end up with a multiplication-by-the-reciprocal problem:

Et voila!

Division of a fraction turns into invert-and-multiply.

This is why I keep saying that 'Math Brains,' math teachers, and math textbooks alike all need to tell us rookies when a procedure or a formula is a 'shortcut' for a longer procedure.

That observation never seems to make much sense to people who are savvy at math, but I think it's important for novices. I know it's important for me, or was. I was 'stuck' in the idea that reciprocals ought to be 'obvious' the same way, say, adding 2 plus 2 and getting 4 is obvious. A reciprocal ought to be a 'thing' for me, in the way a 2 or a 3 is a 'thing.'

I think I'll stop now. I'm probably scandalizing all of you folks who Actually Know Math.

However, this is a good thing to know about students.

When you're learning, or relearning, arithmetic, you're constantly shown procedures and concepts that 'make sense' in a concrete, you-can-pick-it-up-and-hold-it-in-your-hand kind of way. Even place value makes sense that way, although I've come to feel that it probably shouldn't.

The addition and subtraction of fractions continues to make sense in this way — i.e. you can draw fraction addition & subtraction easily. You can 'see' it.

Fraction multiplication was, for me, the first operation & concept that didn't readily lend itself to visualization, in spite of the fact that everyone does produce drawings of fraction multiplication all the time in every textbook.

But fraction division was impossible. With the division of fractions, you hit the visual wall, or maybe the visual cliff. I suspect lots of arithmetic students have the same experience.

My feeling about teaching fraction division is this:

• For most students, you should delay teaching the 'why and how' of invert-and-multiply until students have mastered equivalent fractions and the multiplicative identity property of 1

• You should also tell students directly that invert-and-multiply is a shortcut, the same way cross-multiplying is a shortcut. You end up with invert-and-multiply after a couple of intermediate steps.

-- CatherineJohnson - 23 Mar 2006

DougFractionTilesRevisited 05 Jul 2006 - 23:07 CatherineJohnson

Here's a cool lesson to do with Doug's fraction manipulatives (pdf file):

source:
Primary Mathematics 2B US Edition

Doug's fraction tiles & number lines (pdf files)

RealWorldMathProblem 09 Oct 2006 - 17:59 CatherineJohnson

After a 2-week plateau, Jimmy is again losing weight. 204.5 this morning.

Christopher is losing, too, and is, according to the CDC, officially "not overweight." (Which, translated, means he's at the 94th percentile for his age. The CDC defines anything from the 85th to the 95th percentiles as "at risk for becoming overweight." This is all pretty silly, seeing as how Christopher's height is at the 90th percentile for his age, but still. Having the CDC officially define him as "not overweight" is good for his motivation. I went through all the various websites and figures the other day, and decided that my goal weight for Christopher is 120. He's got 10 lbs to go.)

UPDATE 10-6-2006 10 pounds may sound like a lot for a child, and of course it is, but visually he's good now. He's "tipped" back across the line that separates "fat" or "chubby" to "normal." So that's good. If he didn't lose any more weight at all, but simply maintained this weight and continued to grow, that would be fine. Also, be sure to read the Comments!

Ed told Christopher his job this weekend is to figure out which one of them has lost more as a percentage of his total body weight to start.

UPDATE 10-8-2006 Jimmy is down to 204 this morning. (5'10" or 5'11")

UPDATE 10-16-2006: Jimmy down to 201; Christopher down to 129.5. I haven't been telling Christopher's weight - just the weight loss - because I didn't want him to be embarrassed in case any of his friends happened to come across the blooki. But now that he's into the 120s, I think it's OK. He is at 90th percentile height and roughly 92-93th percentile weight. None of the kids call him "fat" anymore, which in terms of the social issue of being overweight in middle school tells you all you need to know.

I've lost 5 lbs, but there's no way to "disaggregate" the causes, what with our mad-bad doctor-originated cancer scare over the past few weeks and all. (Haven't posted about it; it's happily resolved; mention it only to say that I can't reach conclusions about Shangri-La based on my own quite dramatic - for me - weight loss.)

I found Jimmy's weight loss chart from the last time I managed to put him on & keep him on a diet back in 2003.

July 25, 2003: 205 lbs
November 15, 2003: 198.5 lbs.
6.5 lbs in 16 weeks. Compared to 15 lbs in 11 weeks this go-round.

By March of '04 he was up to 208.5.

At some point in there (I think it was in '04) we were able to persuade our pediatrician to put him on glucophage, which stabilized the weight gain. He didn't lose, but he stopped gaining.

Then his liver enzymes (iirc) got screwy and that was the end of the glucophage. By this summer he was up to 222.

Taking Risperdal, he's been gaining weight every year, year in, year out, without growing taller. (Eric Hollander managed to get him off the Risperdal, but his new med, Depakote, which he must take for seizures, also produces weight gain.)

When I say he's been gaining every year, I'm talking about a young, strong male, much bigger than I am, who is ravenously hungry, screaming for food, and biting himself until he draws blood.

The diet we managed to do 3 years ago had to be constantly monitored and thought-through; we had to do everything in our power to feed him foods that would take the edge off his unceasing hunger so we could withhold food when his nighttime binge hour arrived and he started screaming and biting himself.

A nightmare.

The nightmare wasn't just that he was screaming, bingeing, and biting himself.

The nightmare was that Ed and I were chained to the refrigerator. To this day we have locks on every cabinet and a bike lock on the fridge. We can only buy side-by-side refrigerators, because those are the only ones you can use a bike cable on.

It would be 10 at night, or 11; we'd be dog-tired; and Jimmy would need to consume a thousand calories of parent-prepared food before the screaming stopped. This after an entire day of manning the kitchen because he was hungry all day long, too.

Dog-tired is wrong.

Bone-tired.

The Shangri-La diet has made all of that go away. All of it. There is no nighttime binge. There's not much daytime noshing, either. We have meals.

We got a little sloppy the last two weeks, and weren't always giving Jimmy his oil. Because you have to give the oil one hour after eating and one hour before eating, you can easily end up without that two-hour window of time, especially when a child has school or weekend programs during the day.

The nighttime binges came back.

I'm a believer.

Meanwhile Ed, who spends a lot of time pooh-poohing whatever my latest idea for self-and-other improvement happens to be, is a huge believer.

He's practically a Shangri-La acolyte.

I weighed myself this morning.

I gained a pound.

I don't think it's a real pound.

Since July I seem to have lost 2 stable, definitely-gone pounds on Shangri-La.

BMI 21.5.

I want a BMI of 20.5.

At most.

another olive oil diet

I went to see Erika yesterday.

We met Erika here in Irvington a couple of weeks after we moved in. She had been the on-call physician for the schools, and had a practice in a little house on Main Street in downtown Irvington.

She was wild. She was the original n-of-1 self-experimenter, and was then writing her book on energy and the mitochondria. She took one look at Ed & packed him off to an acupuncturist to treat his back.

The acupuncture didn't work, but everything else she said made sense and I've been seeing her ever since.

Now I have to trek into Manhattan to get to her office, but it's worth it.

Erika is a female Arnold Schwarzenegger.

I'm not sure how she'd feel about that, but if I had to bet I'd say she'd like it.

She's Austrian, and has the "Arnold" accent, but what's Schwarzeneggerian about her is the inexhaustible energy, the optimism, and the utter indifference to the received views of her betters. A massively publicized NIH study proving once and for all the mortal danger of hormone replacement therapy has no more dampening effect on Erika than a spring drizzle has on a Labrador retriever. She simply goes bounding along evangelizing her precious "bioidenticals."

When I say "bounding along," what I mean is that at the precise moment the NIH death-by-estrogen study hit the papers, Erika decided to stop being an internist and start being a full-time hormone replacement specialist. The mere fact that the entire US media universe is seething with stories of middle-aged women dying of breast cancer because they took estrogen is simply not a factor in Erika's decision-making calculus because, obviously, the study is bunk.

That's Erika.

The other Schwarzeneggerian thing about Erika is that she makes this kind of thing work.

Erika can decide that the NIH and the New York Times are wrong, and she can base a major business decision on the fact that the NIH and the New York Times are wrong, and, for Erika, this turns out to be a good call.

I lost track of Erika for a couple of years when her practice in Hawthorne disappeared. She had by then moved from Irvington to a suite of offices in the Dobbs Ferry Hospital, and then from Dobbs to another suite of offices over in Tarrytown, and after that she set up shop in a huge suite of rooms up in Hawthorne.

But suddenly I couldn't get them on the telephone. The number had been disconnected and nobody had notifed me about a new number some place else.

I felt bad about that. The practice must have failed I figured.

It didn't fail. She'd had some kind of falling out with the folks she was partnered with so she stopped practicing in Hawthorne and started practicing full-time out of the Manhattan office she'd had all along, unbeknownst to me. She is mobbed by patients.

Yesterday I told her the Wall Street Journal had just run an article (\$) on a new study showing that hormone replacement therapy is good for the brain (big surprise there) and Erika said, "In 5 years no one will talk about that NIH study any more. Everything will be bioidenticals."

That's Erika.

In 5 years all the bad, wrong ideology will be gone!

Because that's the natural fate of bad, wrong ideology!

I just wish I could train Erika's attention on Lucy Calkins.

Then again, Lucy Calkins is tougher than the NIH.

Or the New York Times.

wasn't I talking about olive oil?

Yes.

I was.

So I sat down on Erika's sofa and said, "I've found a diet that works."

"Tell me."

"It's called the Shangri-La diet."

She started laughing. "You're my funniest patient," she said.

I'm sure that's true.

Then again, Erika is my funniest doctor.

I started filling her in on Shangri-La, but she interrupted me.

She'd never heard of Seth Roberts, but for awhile now she's had 10 patients on an olive oil diet of her own devising and they've all lost lots of weight.

That's 10 out of 10.

Ten out of 10 patients using Erika's made-up olive oil diet have lost weight. She reports this like it's nothing.

Her regimen is to have them take 1 Tbsp of olive oil 3x a day, midway between meals.

She takes at least a Tbsp of olive oil a day herself, I believe, and has been doing so for years.

Erika doesn't have a Unifying Theory of hunger and anti-hunger on the Serengetti plains. She just figures a Tbsp of olive oil 3x a day keeps people full.

I said, "How'd you come up with that?"

Erika shrugged and said, "It's obvious."

more on mitochondria:
Bruce Ames in Discovery
Bruce Ames in REASON
Bruce Ames at fumento.com

The Shangri-La Diet at Amazon
Seth Roberts website

Shangri La diet in freakonomics
Shangri La diet part 2
diet, evolution of the brain, & McDonalds
Marginal Revolution on Shangri La
progress report 7-23-06
Jimmy 7-24-06
mind hacks & Shangri-La 7-26-06
7-29-06 update
my life and welcome to it - 8-6-06 - success
compare and contrast photo op 8-12-06
9-12-06 update
9-17-06 Jimmy is melting
10-4-2006 Dr. Erika's olive oil diet works, too

shangrila

-- CatherineJohnson - 04 Oct 2006

MoreFunWithFractions 12 Oct 2006 - 01:10 CatherineJohnson

available to order from:
Prank Place

-- CatherineJohnson - 08 Oct 2006

FractionDecimalPercentAssessmentTests 25 Oct 2006 - 13:51 CatherineJohnson

Let's review.

Fractions, decimals, and percent are the bottleneck in math K-12.

Before I met Carolyn, I didn't know this.

Since meeting Carolyn, I have been told, by every high school and college math teacher/professor with whom I've spoken, that the big hold-up is fractions, decimals and percent.

Lately, it's occurred to some of us Phase 4 parents to ask whether our own kids, all of whom will be expected to master algebra in the 8th grade, can do fractions, decimals and percent.

Guess what?

One mom, whose child is now in 8th grade studying algebra, said child could not figure a 20% tip on a restaurant meal. I don't remember all the details, but I think she had to actually show child how such a problem could be set up and solved as a ratio.

20/100 = X/\$18.50

Another mom gave child a series of fraction/decimal/percent word problems in the car. Child couldn't begin to solve them.

"I went to the store and spent some money. Now I have \$50 left, which is 50% of what I started out with. How much money did I have before I went shopping?"

Child had no idea what to do with this. Child's mom is now ordering Saxon Algebra 1/2, the "fractional part of a number" book in the Saxon series that Christopher is working his way through.

When I asked Christopher this problem, he said, "Can I use the calculator" — not because he couldn't do the computation, but because the problem sounded much harder to him than it was. When I said "No," he pretty quickly realized that 50% was 1/2, so if the lady had \$50 dollars left and that was 1/2 of her original amount she started out with \$100.

Still, he didn't see instantly that this was a super-simple question, and I'm not confident he could have done the same problem with a less "friendly" number like, say, 25%.

He definitely can't do "work backwards" fraction/decimal/percent problems such as:

The final price of a pair of jeans with 5% tax is \$42.00. What is the price without tax?"

At least, he can't begin to do such problems mentally; I'm pretty sure he can't do this using pencil and paper, either. I'll check again.

online fraction/decimal/percent assessments

Rather than write a bunch of problems (which maybe I ought to do, to make sure I cover the bases) I went Googling for fraction/decimal/percent placement tests.

I found a couple that look pretty good:

These placement test study guides are probably all over the web, and are terrifically helpful, I think.

does our school know whether accelerated math students have mastered fractions, decimal, and percent word problems?

This may be the single most important truth for parents to grasp about United States public schools: decades of case law and custom agree that the school is under no obligation to find out what students know and don't know and then make sure they learn the content they missed.

None.

Parents have been taking school districts that failed to teach their children to court for many, many years; all courts have ruled against them.

All courts have held that if the child didn't learn, "that could be something about the child."

If your teacher or school is checking to see whether your child actually knows the content that has been covered in class, that teacher or school is going out of his/her/its way to perform a function many, many other teachers and schools do not perform.*

You can't count on the next teacher or school doing the same.

Our schools cover material, give tests, and move on: assessment of learning, not assessment for learning.

Inputs, not outputs.

It's up to us to find out what our kids do and don't know and reteach what they missed.

Of course, when your school does not provide parents with course syllabi, scope and sequence, or topic matrix it's a little hard to assess whether your child has learned the material covered in school.

I remember last year one of my Phase 4 chums saying, "I'd just like a list of what they were supposed to learn this year. Then I could work on it with him."

We'd all like that.

Two years ago Lisa Urban, legendary middle school math teacher now retired, told me the math department was "working on" such a document.

I don't see parents getting their hands on it any time soon.

While they're working on the Irvington Math Scope and Sequence, it might also be nice to give the K-5 parents a sequence showing them where their TRAILBLAZERS kids will be at the end of Grade 5.

Are they on track to take and master algebra in 8th grade?

Are they on track to take calculus in high school?

Do we know?

Inside the Black Box Black & Wiliam
Assessment Crisis: The Absence Of Assessment FOR Learning
Rainbow Resource least expensive source I've found for Saxon Math books

* If your child attends a school that is failing to make "satisfactory yearly progress" under NCLB, it's a different story.

-- CatherineJohnson - 24 Oct 2006

LindaMoranListserv 11 Dec 2006 - 19:25 CatherineJohnson

I think everyone here knows about Linda Moran's Teens and Tweens blog.

I've recently (re)discovered that she has a listserv attached to the blog.

I joined last week, and I think some of you might like to join as well. There have been some very interesting posts to the listserv that I don't believe have been posted to the blog itself — and that I don't expect to see posted to the blog itself.

-- CatherineJohnson - 09 Dec 2006