Is that a fork in your pocket, or are you ....

at Charts and Graphs

(photo thanks to Barry G; punchline thanks to Matthew T)

help desk - 'distributing the negative'

I'm working with a boy in a neighboring town who can solve equations with positive values like the following:

3 + 2(x + y) = 9

He is having difficulty solving equations that require him to distribute a negative:

3 - 2(x + y) = -3

I remember C. having trouble distributing a negative, and I remember stumbling over minus signs myself when I was a kid. At some point, I solved my problems by deciding to treat minus signs as either a -1 or the addition of a negative, depending on the expression I was dealing with.

Thus -x became (-1)(x) and x-8 became x + (-8).

I don't think anyone ever told me to translate expressions in this manner. Quite the contrary; I have vague memories of reasoning it out for myself on more than one occasion.


Here's the way a sheet I have from Glencoe says to teach distribution of the negative:

Use the Distributive Property to write each expression as an equivalent algebraic
expression.
a. 3(w – 7)
= 3[w + (-7)] Rewrite w – 7 as w + (-7).
= 3w + 3(-7) Distributive Property
= 3w + (-21) Simplify.
= 3w – 21 Definition of subtraction

Unfortunately, this sequence doesn't solve the problem. My student can simplify 3(w-7); what he can't do is simplify 3–2(x+y).

Today I tried having him draw huge brackets around 2(x+y), then simplify the 2(x+y), and then simplify the remaining expression:

3–2(x+y)
3–[2(x+y)]
3–[2x+2y]
3-2x-2y

In effect, I was turning the problem into two distributions: first the 2, then the negative sign.

This approach always worked for me, but the logic of it wasn't obvious to my student.

One more thing: this student probably had Everyday Math in elementary school, and his current class seems to be intensely procedural. The only textbook his teachers are using seems to be a NY state test prep book.

I'm eager to hear any thoughts you have both about procedural teaching (including mnemonics) and about how I might help this student make some sense of the math he's learning. Moreover, and I hate to say this, but if I'm going to help him make some sense of it, I have to do it on the fly. Our time together is extremely limited.

If anyone knows of a good set of "instructional worksheets," that would be fantastic. I'm combing through my own collection.

Last but not least, what do you think of this video?

Perceptual Learning

There was a story in the New York Times yesterday about perceptual learning, and it brought up two things for me:

a) I was surprised (relieved?) to hear that I wasn't the only one who has trouble with fractions.

b) I had a breakthrough moment a few weeks ago when Catherine told me to break out the number line.

(Cross posted on Perfect Score Project)

equation

from Science Daily:

Texas A and M University researchers ... have found that not fully understanding the "equal sign" in a math problem could be a key to why U.S. students underperform their peers from other countries in math.

"About 70 percent of middle grades students in the United States exhibit misconceptions, but nearly none of the international students in Korea and China have a misunderstanding about the equal sign, and Turkish students exhibited far less incidence of the misconception than the U.S. students," note Robert M. Capraro and Mary Capraro of the Department of Teaching, Learning, and Culture at Texas A&M.

[snip]

The problem is students memorize procedures without fully understanding the mathematics, he notes.

"Students who have learned to memorize symbols and who have a limited understanding of the equal sign will tend to solve problems such as 4+3+2=( )+2 by adding the numbers on the left, and placing it in the parentheses, then add those terms and create another equal sign with the new answer," he explains. "So the work would look like 4+3+2=(9)+2=11.

"This response has been called a running equal sign...

[snip]

The Texas A&M researchers examined textbooks in China and the United States and found "Chinese textbooks provided the best examples for students and that even the best U.S. textbooks, those sponsored by the National Science Foundation, were lacking relational examples about the equal sign."

Students' Understanding of the Equal Sign Not Equal, Professor Says
August 11, 2010

They had me until that last line.

How do Everyday Math and Terc teach the equal sign?

Why Connecticut Schools are Lagging

Throwing Curves has a new post on Why Connecticut Schools are Lagging. For those of us that follow CT education policy, this comes as no surprise. Still, it is nice to see new news sources picking up on what we've know so long.

up the down staircase

Sara (not sure who Sara is!) sends a link to this Everyday Math problem posted on the Well-Trained Mind forum.

Up is negative; down is positive.

That strikes me as a bad mnemonic to teach kids who are going to be encountering coordinate planes just a few years from now.

She got the answer wrong, too.


Up the Down Staircase

Branching Out

In a completely throwaway gag line, football writer (and high school math teacher) Mike Tanier makes a mocking reference to Everyday Math in his offseason column.

I could never be a documentary cameraman. At one point in Sunday's episode of Human Planet (a BBC/Discovery Channel production in the Planet Earth vein) a father takes his two children on a five-day hike along the frozen Zanskar River in Northern India so they can attend a boarding school. The river slowly melts during their journey. At one point, the 11-year-old daughter must crawl along a tiny, cracking ice ledge over the rushing, freezing waters.

...With the poor girl's luck, the school she risked her life to attend just adopted the Everyday Math curriculum, making the whole trip worthless.

Football Outsiders is largely a sabrmetrics site, so the readership is more numerate than average. Still, it warms my heart to see Everyday Math mocked so casually outside of the usual context. I had been planning a post linking Bill Walsh to practicing to mastery, so I guess this serves as a nice segue.

waiting for the teacher, part 2

part 1 is here

Allison writes:

Oh, one other tidbit: the activities in Everyday Math ALWAYS involved manipulatives being arranged in some way. That meant you HAD to stay at your table and wait for the teacher--you couldn't carry it up to the teacher's desk and show your work and ask where you were stuck because it was impossible to transport.

waiting for the teacher

Allison writes:

In the last few months, I visited over a dozen elementary schools. Mostly I visited kindergartens, but whenever possible, I visited the 1st, 2nd, 3rd and 4th grades as well.

Over and over I saw schools where "math class" was the same template: children doing activities from Everyday Math on their own in chaotic, loud classrooms where students didn't have individual desks but had to sit at group tables (sometimes putting up their books and folders to act as little cubicle walls) while they waited for a teacher or an aide to interact with them. Uniformly, I saw half a dozen kids doing nothing at all in those times; another half a dozen chatting or playing but obviously not doing anything, and a precious few trying to block out the stimulus. Some read cheap fiction books.

No one could have learned anything in such a room even before you find out that the task at hand is some bizarre manipulative task in Everyday Math that had no goal or explained purpose anyway.

The teacher didn't spend more time with those having trouble it seemed, either, because those having trouble weren't even bothering to do the activity.

Choosing Carnival Junk Food

It’s standardized test time in Connecticut and my child will be busy participating in this painfully drawn-out process over the next couple of weeks.  While I should clarify that I think testing *can* be useful for diagnostic purposes, I consider the following question from the eighth grade CMT Mathematical Applications section to be yet another example of why I find my state's manner of assessing students quite useless:

Sample Item 8-5 (Numerical):  Buying Tickets

The carnival offers you two different options for buying tickets.

                OPTION A:  $2.00 per person plus $0.75 per ride
                                                                OR
                OPTION B:  $5.00 per person plus $0.25 per ride

If your uncle gave you $10 for the carnival, which option – A or B – would you choose.  Show the mathematics you used to determine your answer.

OPTION CHOSEN: _______

Explanation:

This poor excuse for a word problem is just one example of why we started homeschooling.  While I’m assuming the objective of the question is for the student to show mathematically that option B is the better choice because you can go on 20 rides as opposed to only 10 rides with option A, the question does not indicate that the goal is to go on as many rides as possible.  I can easily imagine any of my children (including the 8-year old) coming up with alternate scenarios that could make either option the better one.  Unfortunately, I can just as easily imagine a scenario where the person responsible for grading 200 tests containing these strange open-ended mathematical responses before the end of their shift would mark their mathematically and numerically accurate answer WRONG.
 

Let's say the student were to choose OPTION A since he really likes to eat junk food at carnivals (just like his favorite uncle who spots him the $10) but hates the rides because they make him dizzy (thereby leaving him with $8 to spend on food instead of $5).  Would that be counted as a correct answer?  I would argue that either option could be the better choice depending on the objective—which was not made clear.  Mathematics is supposed to be clear, precise, and accurate. This question is just silly.

These types of  Everyday Math word problems (I'm being generous here by calling it a word problem) used to make me crazy when my child would come home with them in 4^th grade.  Now here we are in 8^thgrade running in circles all over again.

Meanwhile back in Singapore children are answering this:

Hooke's law for an elastic spring states that the distance a spring stretches is proportional to the force applied. If a force of 150 newtons stretches a certain spring 8 cm, how much will a force of 400 newtons stretch the spring? (New Elementary Math 2 Placement Test)

 *sigh*

déjà vu

Two parents discover Everyday Math:

As it turns out, our school district is using a controversial math curriculum called Everyday Mathematics, also known as "Reform Math." EM, as Everyday Mathematics is referred to by teachers, was developed by the University of Chicago, and according to their website, it is in use by about three million students nationwide. Here is one example of how simple addition "can" be performed using EM:

An example of what EM calls the "lattice method" for performing multiplication:

What becomes immediately clear is that several extra steps are now necessary to accomplish simple beeline computations. More steps will result in more errors -- only an idiot would claim otherwise. Eventually, EM students are taught four ways to add, five ways to subtract, four ways to multiply, and two ways to divide (traditional long division has been eschewed completely). Rote memorization is de-emphasized, and calculators (as well as estimating) are introduced in grade two.

Here is the basic rationale behind EM, directly from the University of Chicago website:

Research has shown that teaching the standard U.S. algorithms fails with large numbers of children, and that alternative algorithms are often easier for children to understand and learn. For this reason, Everyday Mathematics introduces children to a variety of alternative procedures in addition to the customary algorithms.

Links to or excerpts of said research are not provided -- we are to simply take these statements as fact. EM further claims to "make mathematics accessible to all students" by:

Incorporating individual, partner, and small group activities that make it possible for teachers to provide individualized feedback and assistance.

Encouraging risk-taking by establishing a learning environment that respects multiple problem solving strategies.

This couple is politically conservative, and as a result, the one thing they've got wrong is the idea that  liberal parents like this stuff when they don't. Not for the most part.

This passage took me aback:

What's worse, the methods purportedly being used to convince school boards to adopt EM reek suspiciously of Rules for Radicals: ^*

State that the traditional approach hasn't worked

Disparage testimony from those against the adoption as ideological and politically-motivated arguments

State that the success of any program depends on the teacher

Bring in teachers from affluent school districts as witnesses

Bring in a witness from a university

Never thought of these tactics in terms of Saul Alinsky.

Sheesh.


^* Barry's article!

Arguments against Corrective Math

Just came across three articles arguing against using DI's Corrective Math as remediation for students in Philadelphia's low-performing "Empowerment Schools."  The venue is The Notebook, an online journal that describes itself as an "independent voice for parents, educators, students, and friends of the Philadelphia Public Schools." The author is a lecturer at the University of Pennsylvania Graduate School of Education, and a consultant for two of these Empowerment Schools.  

Using familiar arguments and buzzwords, she argues that the solution instead is better implementation of the existing ("rigorous," "research-based") Everyday Math and Math in Context curricula.

If you have a moment, please comment on these articles (most of the existing comments support the author's views):

http://www.thenotebook.org/blog/102440/correcting-corrective-math-problem

http://www.thenotebook.org/blog/102440/correcting-corrective-math-problem-ii http://www.thenotebook.org/blog/102455/correcting-corrective-math-problem-iii

(I haven't yet commented myself because, as a Philadelphia parent and educator, I wanted to pitch a response article first).

the things money can't buy, part 2

From the Everyday Math response (pdf file) to the new CCSI standards:

It may be that some on the Mathematics Working Group, perhaps over-generalizing from their personal experience in school, underestimate the cost of bringing all children to mastery on, say, decimal long division with the traditional algorithm.

I have an idea.

How about the schools economize on curriculum materials by purchasing Primary Mathematics instead of Everyday Math, so we have enough money to teach everybody long division?

please explain

You really have to see it to believe it.

What happened to "Show your work"?


source: Math Problem of the Week

the final word

This may be the best comment Barry has ever written.

Re: Success Adds Up for D.C. Schools' Math Program by Bill Turque

The myth that teaching algorithms and procedures deprives students of conceptual understanding is as prevalent as the myth that saltpeter quells sexual appetite and is put into prisoner's food. Procedural fluency leads to conceptual understanding. Procedures are not taught in isolation and even a sidelong glance at math textbooks used in the 50's and 60's (an era that was supposedly dominated by rote learning) will illustrate the fact that explanations for procedures were given, and that such books contained many word problems to ensure that students could apply procedures and concepts to problems.

For a description of Everyday Math that is not quite as flattering as this Post piece, see my description of how I countered the effects of EM with my daughter and her friend:

More information on how Clifford Janey managed to get EM adopted in Washington DC

Seriously, what more is there to be said on the subject of Success Adds Up!!!!!

Everyday Math author defends his program against Katharine Beals

In today's Philadelphia Inquirer Letters to the Editor, excerpted here:

Katharine Beals' article on the use of "reform math" with students with autism contains many misperceptions about Everyday Mathematics that, as the program's coauthor, I want to clarify ("The 'reform math' problem," last Monday).

Everyday Mathematics was designed for general education students, but it has been effective in special education, including with students with autism.

Beals' claim that students spend large chunks of time working in unsupervised groups is untrue. A teacher supervises student group work at all times. While some assignments are "open-ended and language-intensive," many are not. A balanced curriculum needs simple exercises to build basic skills, as well as more difficult problems.

Beals writes that students "lose points for failing to cooperate in groups, explain their answers, and comprehend language-intensive problems." While decisions about how to grade students are made at the local level, many people believe it's reasonable to require students to work cooperatively, explain their work, and understand word problems.

Everyday Mathematics is not just a "sequence of themes," but a carefully organized sequence of lessons resulting in mastery of a specific set of goals. Its approach is well supported by research, the authors' experience, and decades of classroom experience.

Naturally, accommodations for teaching children with autism must be made, and that's what professionals always do. As with any tool, Everyday Mathematics must be used with professional judgment.

Andy Isaacs

Chicago

The Revolt Against Lousy Math Instruction May Just Go Viral

This example was just posted at BoingBoing, a large group blog, at this post.

Do You Understand My First-Grade Child's Homework?

The blogger asks

My six-year-old told me she doesn't understand her homework. After studying it for 15 minutes, I *think* I understand what she's supposed to do, but I'd like a second opinion.

Is it from Everyday Math?

Go add to the BoingBoing comment fun, if you like.







(I have another question -- homework for six-year-olds? I'm ok with requesting reading at home, but that's it. Period. The end.)

Speaking of the spiral...

...my problems of the week this week show the spiraling vs. the linear approach within chapters called "Addition and Subtraction" in the 2nd grade Everyday Math vs. Singapore Math curricula.

Op-Ed in the Philadelphia Inquirer on autistic spectrum students and Reform Math

Here!

For all the talking points that Reform Math proponents deploy in response to the general criticisms, I haven't yet seen any talking points that respond to concerns about children on the autistic spectrum. Has anyone else?

Since it's well-documented--and generally agreed--that AS children require structure, direct instruction, and discrete tasks, and that many of them have the potential to excel in math, and since the education establishment's purported missions include (1) mainstreaming and (2) catering to different learning needs, I believe this is a fruitful message to keep plugging.

the natives are restless

How much of this has to do with that asinine Everyday Math? I'm fighting every week with the teachers to keep a calculator out of my kid's hands until she can do arithmetic on her own, something she's eminently capable of doing someday (she's six, for crying out loud). Yes, including long division (not that it's part of the curriculum -- why think for yourself when you can use the outboard brain?) and maybe some introductory algebra. These teachers, who'd stare like deer in headlights if asked to calculate the future value of anything and come out with a number, give me this endless stream of talking-point baloney from the Everyday Math "how to reassure parents who are scared because it's not like the math they remember" book. They don't understand why I'm so stuck on the idea that my daughter should be able to do her own adding, subtracting, dividing, multiplying. AAARGH.

Amy from Iowa

You can vote to "Recommend" your favorite comments on the story. Amy got my vote.

Speaking of Everyday Math, remember this?