Marcy on math games and Asperger’s

Re Op-ed in yesterday's Philadelphia Inquirer, Marcy writes:

One of the things I have a real problem with is the math games. They are really pushed "all the kids have fun playing math games!"

But my AS son doesn't play turn-taking type games. They drive him crazy. So when all the other kids are working on their math facts by playing math games, my son is really left to fend for himself.

Since this is a boy who read Edwin Abbott's Flatland in 3rd grade, I know he has a good head for higher math. Why should he not be allowed to learn math in a more concrete way? Why should he sit around while everyone else plays games?

## Thursday, December 31, 2009

### Favorite comments of '09: Marcy on math games

## Wednesday, December 30, 2009

### Favorite comments of '09: Obi Wandreas the Funky Viking on “real world” problems

Re Why do our top math & science students defect?, Obi Wandreas writes:

The primary problem with "real world" problems, is that in order for them to require only the math you are trying to teach in the lesson, you have to set up a problem so contrived that it ceases to bear any actual resemblance to the real world.

The other issue, of course, is that it requires a great deal of time and effort to set up a problem and story only to find that your students will refuse to donate any airborne intercourse.

It's great to show students that math has real applicability. For the most part, however, what they are doing simply builds the foundation for the stuff that's really useful. A coach doesn't show athletes how they could use a sit-up to help them in life. When the athlete does something that requires the use of their abs, however, they will be glad to have done them.

## Tuesday, December 29, 2009

### Favorite comments of '09: Beth on the worst of both worlds

Re Myths about left-brain schooling, II: media complicity, Beth writes:

The tragic reality is that our schools don't do anything well. The fact that the humanities are suffering doesn't mean that mathandscience is taught well. The fact that the schools don't have the rigorous content and skills of the traditional approach doesn't mean that they encourage true creativity like the progressive approach.

They just don't do anything right. They've somehow managed to weld together the authoritarian, anti-creative, carrots-and-sticks approach of the worst kind of traditionalism, with the hollowed-out content and mushy thinking of the worst kind of progressivism.

## Monday, December 28, 2009

### Favorite comments of '09: Dawn on art activities

Re myths about left-brain schooling, Dawn writes:

I think even though art gets into math and English these days by way of projects it's degrading art as well as those other subjects. Art is as much based on skills and work and practice as math. Expressing yourself is something you can only do after you've acquired the skills to do this. And I say this as someone for whom drawing has always been tremendously important.

Perspective, light, structure, tools, exercises, etc. THAT's what makes art. This half-assed stuff they incorporate into math or english not only cheapens those subjects but cheapens art.

It's a strategy. Maybe if more kids WERE encouraged to bite their teachers when bored....

## Sunday, December 27, 2009

### Favorite comments of '09: vlorbik on two plus two

Re Postmodern math: does 2 + 2 always = 4, vlorbik writes:

the right answer to

"what is 2+2?"

is of course

"why do you ask?".

unless it's implicit.

a four-year old?

one thing maybe.

somebody testing a piece of computer code?

another maybe.

somebody trying to catch you out

in some pseudophilosophy?

maybe still another.

sure and it's empty to posit that

"properly" defined, correct symbol

manipulation yields correct results.

owen by the way

oh... and by the way...

i was created a doctor

of philosophy [for my

sins] in '92 and have

professed maths.

so here goes an experiment.

these SOB's downtown

have decreed that push

these damnable calculators.

let's see what happens when

i put "2+2=4" [the whole string]

into the command line.

wow. 1.

that means true.

okay. you win this one. sort of.

put it into google then.

let's see.

pretty interesting.

would't call it "true" though.

is google now "not mathematically

well-defined" or some such

weaselword dodge? of course not.

(everybody knows that there are

always *undefined* terms and

other such conventions; this

isn't at all when we're doing math

and seldom when we philosophize).

PS

2+2=5 is the symbol in 1984

of the kind of "truth" that can

only be beaten into you.

mathwarriors split on whether

2+2=4 is of the same type.

i myself do not claim

to know the answer to this.

## Saturday, December 26, 2009

### Favorite comments of '09: Niels Henrik Abel, Cheryl van Tilburg, and Beth on math projects

Re Summer math projects, grade 5 and Summer math projects, grade 4:

Niels Henrik Abel writes:

As you prepare your game "Package" be sure to include...A colorful and creative game board

So in other words, were a kid to design a game like chess that was challenging yet had a dull & boring game board, he would be marked down?

I guess that's an indication as to what's truly "important."

Cheryl van Tilburg writes:

Instead of ignoring the assignment (which just causes your child grief at the beginning of the school year), I alter the assignment to make it more appropriate for my child. For example, instead of the board game, I would have my student complete several multiplication worksheets and complete a weekly timed test of multiplication facts.

Then, my son brings the completed worksheets and time tests to school, along with a note from me explaining the "differentiated assignment." I include my phone number so that the teacher can call if he/she has any questions. To date, I haven't received any calls.... (I use this same approach when it comes to poster projects in my kids' English class.)

Is this a pain in the neck? Yes. But the way I figure it, things won't magically change without input from parents and other educators who understand that projects based on creativity aren't appropriate for all children.

It's also important to talk about this option with other parents in your child's class/grade. Other families also struggle with these creative summer projects and would welcome some advice how to handle them. (And there's something comforting about knowing that other parents are in the same boat!)

Things won't change until parents band together and demonstrate that they mean business. It's hard to be the only one to speak up (you and Catherine over at KTM are great role models in this regard -- and many others!) If five children come to class with a "differentiated" summer project, that sends a powerful message to the teacher (that hopefully will be passed on to administrators).

And Beth writes:

How do I hate this project? Let me count the ways:

1.) Like a lot of elementary school homework, it's really Mom-work. There's no way your average 9-year-old could complete this without massive assistance from Mom. From locating the grocery store, to buying the poster board, to getting a flyer, to nagging the kid into doing the work, this is one more headache for Mom.

2.) Most of the effort involved is pointless. For a bright child who balks at pointless work, "imagine you're planning a picnic ... now make a poster ..." is the beginning of existential despair. I wish I was kidding, but I'm not. I've seen this happen.

3.) This is a huge time waster. An extremely well-organized Mom might be able to get her child through this in about 3 hours, but that's a bare minimum. Remember there's a trip to the grocery store, and a trip to buy poster board.

4.) Public school is supposed to provide a free education. As soon as you require poster board that you don't provide, you've violated that. In this recession, people are really pinching pennies, and for some families, this is asking too much.

5.) It's called summer vacation because it's supposed to be a vacation! Hello!

I don't see this as a left- or right-brain problem. This is about the schools thinking they have a right to tell me what to do with my kids in our own time. I don't agree.

## Friday, December 25, 2009

### Favorite comments of '09: Beth on “be creative”

Re Summer Reading Project, 3rd grade, Beth writes:

"Be creative!" No. A child can either be creative or follow the teacher's directions, but he can't possibly do both at the same time.

It's like another old favorite, "Have fun!" usually encountered at the end of a long list of instructions.

So much of the stuff that goes on in school just seems half-baked. The teachers got the message that learning should be creative and fun, which I agree with, but nobody was willing to do the deep thinking and reform that would actually make that possible. So we wind up with stale porridge which is not creative, or fun, or learning.

## Thursday, December 24, 2009

### Favorite comments of '09: Joanne Jacobs on art projects

Re: Portfolios are coming home: insights into grade rationing, Joanne Jacobs writes:

A friend of mine hired a high school student to help her fourth-grade son produce dioramas. She reasoned that no educational purpose had been advanced for the art projects so it didn't matter who did them. She has a lot of artistic skills herself, unlike her son, but has a full-time job.

## Wednesday, December 23, 2009

### Favorite comments of '09: ChemProf on pressure for “innovative teaching” in college

Re: Pressure for "innovative" teaching in colleges, ChemProf writes:

It's definitely ed school related. My college is on an assessment kick, driven in part by the accreditation agency. The problem is that the assessment system is being driven by the ed schools and social sciences, so we get lots of blather about "rubrics", but my numerical assessment system (based on percentages) doesn't count. They've yet to try to dictate how I teach my class, but I wouldn't be surprised if it comes to that eventually.

## Tuesday, December 22, 2009

### Favorite comments of '09: VickyS and Obi-Wandreas the Funky Viking on cooperative learning

Re Cooperative learning?, Vicky S writes:

My son had a similar experience. In 5th grade math they did a lot of cooperative work, and this particular day he was appointed to give the group's answer. They haggled over the answer, and the kids insisted a wrong answer was right. My son knew the right answer so he stood up and reported the right answer. He was given a poor grade because he did not present the group answer. When challenged, the teacher said my son failed to convince the group of the rightness of his answer, so failed in that way as well.

And Obi-Wandreas the Funky Viking writes:

This has indeed been a very valuable teaching experience for your daughter. She has learned several crucial life lessons, including:

1) Who you can, and can't trust

2) What happens when responsibility for your success is placed in hands other than your own, and you sacrifice your individuality for a group

3) The difference between theory and practice

4) How not to run a lesson.

## Monday, December 21, 2009

### Favorite comments of '09: bky on invented algorithms

Re: Math problems of the week: 6th grade Connected Math vs. Singapore Math, bky writes:

When the curriculum has students invent algorithms for basic mathematical operations, to me the message is this: none of this really matters -- that's why we're letting 10-year olds make the decisions.

What if you teach them how to add fractions and build it up by: (1) giving a good foundation of what a/b means (b partitions of [0,1], count up a of them), (2) using that foundation to show, very common sensibly at this point, how to add fractions with like denominators, then (3) give a good foundation for understanding equivalent representation of fractions (e.g. why 2/3 = 4/6), and then (4) very naturally lead to the general algorithm for adding fractions with different denominators ...? (that sentence started out as a question) Then the message is that adding fractions is so important (and the students are so important) that we want the kids to really be able to do it and understand how it works.

## Sunday, December 20, 2009

### Favorite comments of '09: vlorbik on computer programming

re Interactive computer programming environments: essential yet elusive, vlorbik writes:

everything got hugely harder with "windows".

in the DOS environment, one had BASIC

right there, ready to go,

on every box on every desk.

you'd open it up, find a program that runs,

and start banging on to see what happens.

also "BAT" files. anything you kept doing

over and over the same way? just write it out

once and run it whenever you need.

right there at the top level directory

where you can't miss it. and self-explanatory.

i can't do the exercises you set a few posts

from now at all now; noplace to start.

..."but you *just* have to..."

download this, install that,

buy the other.

yes. if you happen to *own*

the computer you're working on

and have a fast connection

(and probably some free access

to competent tech support

and god knows what else).

so it looks easy to whoever it is

telling me it's easy.

meanwhile, if i want a list

of pythagorean triples

(beyond the famous 3,4,5

and 5, 12, 13 that everybody knows)

i'll have to (re)-write the code

on my f--ing graphing calculator to do it.

thanks a lot, GUIs.

## Saturday, December 19, 2009

### Favorite comments of '09: Anonymous and bky on key words in word problems

Re Ideas on Helping Children with Hard Word Problems, Anonymous writes:

Looking for key words is not a good idea. Things are not necessarily going to be written with key words meaning the same thing.

There are 15 candies altogether. Two are outside the jar. How many are in the jar?

Students taught to look for the "key" word altogether might add. What makes them key words anyway? the textbook writer?

...

If a student understands the problem and what it is asking, "key" words are irrelevant. If a student is dependent on key words, and cannot solve the problem correctly without them, then any time in life when a problem is worded without those so-called key words, or the words are used differently, as in the examples above, that student will not be able to solve the problem. It is an ineffective "tool". Better to give students tools they can use in all circumstances.

and bky writes:

I think that Anonymous' last remark about keywords is on the money: if kids know to read a problem and get the mathematical content, keywords are irrelevant. Therefore the goal should be to get kids to read for understanding -- it is as much about literacy as about arithmetic. It is also not something that can be done in one lesson, it needs to be the ongoing framework in which word problems are addressed.

## Friday, December 18, 2009

### Favorite comments of '09: TerriW on open-ended assignments

Re: Against open ended assignments: evidence from psychology, TerriW writes:

Well, as a parent of little ones still, this is a no-brainer.

"Go get ready for bed."

vs.

"Go pee on the potty"

"Okay, now wash your hands."

"Okay, now brush your teeth."

"Okay, now put on your jammies."

"Okay, now pick out your nighttime book."

Which version gets the job done quickly and which one causes the parent to get angry and drives the child to tears?

### Parent-teacher conference!

I just had my first ever parent-teacher conference from which I did not come away with the sense that there was something wrong with my daughter. No mention of IEPs; of uncooperative, fidgety behavior; of deficiencies in expressive writing. All good. All wonderful, in fact.

Dd has made tremendous developmental strides over the years; I wonder how many others like her are simply following their own idiosyncratic time tables.

But there's also this year's wonderful teacher. All of her teachers, in fact, have been wonderful-- but in different ways. And what for some people is pathology for others is quirky creativity. Perhaps it depends on who you are, what you grew up with, or what you hold nearest and dearest. Whatever it is, would that other left-brain children were as lucky as dd is this year.

## Thursday, December 17, 2009

### Favorite comments of '09: Obi-Wandreas the Funky Viking on science and science fairs

Re School Science Fairs: Right-brained obstacles to left-brainer recognition, Obi-Wandreas writes:

Given the fact that sensationalism has trumped fact and inquiry in some of the most famous fields today, this would seem to be an accurate representation of today's scientific climate.

Although left-brain work in science tends to be that which endures, that's a small consolation to those having to deal with science as a popularity contest today.

## Wednesday, December 16, 2009

### Favorite comments of '09: bky on why teach long division

Re: why teach long division?, bky writes:

There are two good reasons for teaching the standard algorithm for dividing integers, if that's what you mean by long division:

(1) so kids can calculate the quotient of two numbers(and have an exact form for the remainder if that is wanted, rather than a decimal expansion), and

(2) it is an introduction to algorithms. It is odd that many people who deride the teaching of the traditional algorithms cite the availability of calculators as a reason not to learn the algorithm. I find it useful to regularly (every 6 months or so) have my kids (homeschooled) go over the operations of it, with the idea of helping them not only have confidence that what they are doing makes sense but also preparing them for them to understand the concept of algorithm by familiarity with a few specific examples (also on the list: standard stacking algorithms for multiplying, adding, subtracting).

A good idea is to practice occasionally with something like money: show 734 as 7 dollars, 3 dimes, 4 pennies. If you divide by, say 3, you start with 3 piles, each with 2 dollars; the left-over dollar is exchanged for dimes; etc. It is also useful to do the same problem on paper based on writing out the expansion 734 = 700 + 30 + 4 and then successively dividing each place value with remainder, and throwing the remainder downhill:

734 = 7x100 + 3x10 + 4

= 3x(2x100) + 1x100 + 30 + 4

= 3x(2x100) + 13x10 + 4

= 3x(2x100) + 3x(4x10) + 14

= 3x(2x100) + 3x(4x10) + 3x4 + 2

= 3x(2x100 + 4x10 + 4) + 2

= 3x244 + 2

This is "doing it by hand". The algorithm is a formalization. The algorithm is based on repeated division-with-remainder; you never really need to know what place value you are working with, or which side of the decimal point -- just divide and throw the remainder on the next lower place value. If kids understand this for integers, then dividing in the presence of a decimal point is just as easy. Note also how distribution is vital to long division, and since distribution is difficult for grade school kids this also gives practice recognizing and using that vital aspect of arithmetic.

Some critics say that the LD algorithm doesn't teach place value. Of course, it's not suppose to: it's supposed to divide numbers. But looking at it as an algorithm does indeed reinforce the concept of place value. Long Division is a keeper.

## Tuesday, December 15, 2009

### Favorite comments of '09: lgm on schools’ demands on parents

I have so many favorite comments that, to fit them in before the year turns, I must start posting them now. I'll do so in chronological order.

Here's the first, from lgm:

(re: parenting the the 21st century)

When I was a child, my school clearly defined the parent's teaching responsibility. It was to send a note in with the undone homework and notify the teacher if the child was struggling. The teacher would do the actual teaching. My district (Dept of Defense Europe) still has this policy.

In my child's public school district in NY, I am to make up whatever the class didn't get to. The penalty for not doing so is ineligibility for college prep courses. If I want my child to have the education I did I must tutor on the side and pay the community college for the senior year courses as it would be elitist to offer Calculus at the high school. I went to small schools. My kids have over 600 in their grade and so many specialists and paras that the principal has to develop a parking lot schedule to go along with the staff schedule. Academics was not the focus until NCLB came along.

## Monday, December 14, 2009

### 7th grade holiday math project: Just Because I Care About You

A Just - Because - I - Care - About - You MATH PROJECT!

You have been given $2,000 to buy gifts for ten different people in your life. You must decide who you want to give a gift to, what you want to buy them, and why you want to buy them this particular item. You must find a picture of this item with the price. Every item you select has a discount. You must find the discount for each item, calculate how much you will save, and how much the item will finally cost you.

Each student must complete a booklet consisting of 13 pages

Page one is your title page. This must include your name, and title of this project.

Pages 2 - 11 will display:

* A picture of a gift

* The original price

* The discount

* The final price with calculated sales tax ***

* Your math work

* Who the gift is for and why you chose this item for this person

Page 12 will show the price you spent for each item, how much money you spent all together, and how much you have left.

On page 13 you will donate the remaining money to a charity of your choice and explain why you chose this charity.

DISCOUNTS

20% off all major appliances (refrigerator, washer)

...

50% off all jewelry and clothing

***Please remember, there is a 8% sales tax on everything but clothing.

...Your project will be judged on creativity, accuracy, and neatness.

Sample

[Picture of Lamp]

A lamp for my friend Nancy.

My close friend, Nancy, just got married. At the Craft Show last month, she admired a lamp which bears a resemblance to this one. She said it was the perfect lamp for her foyer. I could not pass it up.

[Various calculations]

[Final price]

## Saturday, December 12, 2009

### How not to handicap students with Asperger's

Reading this article from yesterday's British newspaper, the Daily Mail, I was struck not just by the impressive accomplishments of its subject, but by the following points, which I've put in bold face:

A schoolboy is studying for a maths degree at the age of 12.My question is, how likely are similar children to experience similar recognition and accommodations in present day American schools?Cameron Thompson has been accepted by the Open University on its BSc Honours course and expects to graduate when he is 16.

The child prodigy already has A* grade GCSEs [a standardize British subject exam] in Maths and Additional Maths.

The youngster, who has a form of autism called Asperger syndrome, scored 100 per cent in all of those tests, so his teachers decided to put him in for the exam proper last May.

Cameron's father said: 'He is in the second year of the course and in the first unit last year he had a final score of 89 per cent.

'That unit usually starts in October and ends the following June - Cameron finished it a couple of weeks ago.

'The second unit starts in February and he says, quite seriously, that he is going to have letters after his name by next October.

'He also plans to have graduated with a BSc [Bachelor of Science undergraduate degree] honours degree by the age of 16 and he is on course for that.'

Mr Thompson, who works in IT, added: 'His abilities are remarkable but all this does have its challenges as we have thought for some time he has Asperger Syndrome.

'This means he has trouble dealing with other children and tends to lock himself away for days.

'He has never been officially diagnosed but we are thinking of having that done.

'However, Maelor School have been brilliant with him and have provided well for his special needs.'

And what does this mean for the future of American children with Asperger's Syndrome?

## Thursday, December 10, 2009

### Math problems of the week: 6th grade Everyday Math vs. Singapore Math

**1. The final arithmetic problem set in the 6th grade Everyday Mathematics Student Math Journal 1**, p. 161:

2. The final arithmetic problem set in 6th grade Singapore Math Primary Mathematics Workbook 6B, pp. 14-18: |

## Wednesday, December 9, 2009

### Autism Diaries XIV: More erudition

"Wilson! Wi-i-i-lsON! I will save you!" says J, smiling goofily as he chases the basketball he kicked off the court down the grassy slope.

Yes, a basketball; not a volleyball; and yes, the force of gravity; not an ocean current. But nonetheless, an endearing reference to, and partial reenactment of, a scene from one of the few adult he really enjoys and understands.

And one of the most non-mischievous, "neurotypical" jokes we've seen from him.

## Monday, December 7, 2009

### Feedback loops or vicious cycles?

Suppose you're in a position to design and disseminate K-12 academic curricula, and that you believe strongly that this curricula should teach students skill X. Suppose, furthermore, that data shows that students are deficient in skill X. Suppose, finally, that you believe that emphasizing A, B, and C will teach skill X. So you design and disseminate a k-12 academic curriculum that emphasizes A, B and C. New data then emerges that shows that students are still deficient in skill X; some of the data suggests that the problem is getting worse.

What do you do at this point?

1. Reform the curriculum so that it puts an even greater emphasis on A, B, and C?

2. Question your initial belief that emphasizing A, B, and C will teach skill X, and try a new strategy?

Hints:

X = "higher level thinking skills"

A = explaining how you solved problems

B = "reflecting" on your learning process

C = "inquiry" and "argumentation" over specific content

"You" = a member of the current education establishment

## Saturday, December 5, 2009

### Education and the needs of businesses

An article in this week's Education Week reports on concerns that "the push for ‘21st-century skills’" by the Partnership for 21st Century Skills, or P12, "is an attempt by technology companies to gain more influence over the classroom."

The article notes that, "for Ken Kay, the president of P21, such criticism amounts to a 'cheap shot' by those who don’t believe that the education system should be more responsive to business needs. "

This prompted me to post the following comment:

There are two sorts of "business need" that come to mind. One is the need of technology companies to sell their products to schools. The other is the need that companies in general have for a skilled workforce.

I can't help wondering to what extent the P12 group has surveyed actual businesses. The last time I checked in, businesses were bemoaning the scarcity of those with basic numeracy and literacy skills.

The only relatively new skill that I can think of that schools should be teaching is computer programming--and by this I mean actual programming courses; not courses in Power Point, Photoshop, and Excel. Why do so few schools teach, for example, Basic, Pascal, C, and Java?

In defense of p12, a subsequent commenter wrote:

Included [in necessary 12st Century Skills] are team building and project management, skills that help workers in am [sic] ever expanding workplace with ever shrinking human interaction.Are businesses really crying out for K-12 schools to teach "project management" and "team building"? My suspicion is that they're more interested in "team players," in the sense of players who are skilled enough to get their part of the job done properly, without other team players having to do it over again for them.

## Thursday, December 3, 2009

### Math problems of the week: 6th grade Connected Math vs. Singapore Math

1. The final fractions calculations problems in 6th grade Connected Mathematics 2, Bits and Pieces I: Understanding Fractions, Decimals, and Percents, p. 52:

Extensions:

For exercises 55-60, find an estimate if you cannot find an exact answer. You may find that drawing a number line, a hundredths grid, or some other diagram is useful in solving the problem. Explain your reasoning.

55. What is 1/4 of 12?

56. What is 2/9 of 18?

57. What is 1/4 of 3?

58. What is 3/4 of 8?

59. What is 2/9 of 3?

60. What is 3/4 of 3?

2. The final fractions calculations problems in 6th grade Singapore Math Primary Mathematics 6B Workbook, p. 13:

Find the value of each of the following:

(a) 1/2 + 1/2 × 1/4 - 3/8

(b) 2/5 × (5 - 3) ÷ 7/10

(c) 2/3 ÷ 4 × 3/4

(d) 2 ÷ (1/2 + 1/4) × 3/8

(e) (1 - 3/8) ÷ (1/2 + 1/3)

(f) 1/6 + 5/6 ÷ 5/6 - 2/3

3. Extra Credit

Which problem set involves more mathematical "extensions"?

Find an estimate if you cannot find an exact answer. You may find that drawing a number line, a hundredths grid, or some other diagram is useful in answering this question. Explain your reasoning.

## Tuesday, December 1, 2009

### Digesting my Op-Ed responses

The emails reacting to my November 9th Op-Ed have stopped coming in, and so it's time to sum up the 40+ responses.

The biggest contingent were parents and educators of autistic spectrum children who agreed with me that Reform Math is shortchanging their children.

Nearly as numerous were those who agreed with my points, but felt that Reform Math shortchanges a much larger group of kids. (I agree. My focus on autistic spectrum children was partly a matter of news topicality, and partly because I think that these are among the most vulnerable of affected children.)

Less than five people wrote in defense of Reform Math. Those most angered by my piece were a couple of teachers who have autistic spectrum students in their classrooms and claim that Reform Math works just fine for them. ("Please check your facts before writing articles," wrote one.)

This got me thinking about how it is that some teachers can come to believe, based on actual classroom experience rather than ed school indoctrination, that Reform Math is better than traditional math for their most vulnerable students.

What I think is going on here is that people are confusing success with Reform Math with success in math. As should be apparent from my Problems of the Week posts, Reform Math, in comparison with non-Reform Math, offers a much smaller number of easy problems and leaves out all sorts of difficult concepts and procedures. Issues of speed, accuracy, and mathematical challenge do not arise very much.

Therefore, if you are a teacher working with students who struggle with actual math--as many lower-functioning children on the autistic spectrum do, for all the strengths that higher functioning autistic children have in math--and if you confuse Reform Math with actual math, you may come to believe that Reform Math serves your students much better than non-Reform Math does.

That's why we desperately need measures of actual math achievement, instead of all those state tests that, designed (as one follow-up letter to the editor points out) in lockstep with Reform Math, simply measure Reform Math achievement.