Thursday, December 18, 2014

Math problems of the week: Common Core-inspired algebra problems

From a "High School Algebra Core Curriculum Math Test: Math Common Core Sampler Test" at Math Worksheets Land.

Extra Credit: Is it possible to have mastered algebra without knowing:

a. the meaning of the phrase "solve using elimination" in 12 (as opposed to ...? don't all algebraic strategies for solving systems of equations involve eliminating one of the variables?)
b. what is meant by "the variable that is undefined" in 13 (as opposed to "the variable that is defined"?)
c. what matrices and disriminants are

Addendum: The Common Core on Matrices

Common Core advocates repeatedly state the "the Common Core is not a curriculum." Yet, for 6th grade all the way through twelfth grade, one of the topics is a topic that is not covered in many algebra curricula: matrices:

CCSS.Math.Content.HSN.VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

CCSS.Math.Content.HSN.VM.C.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

CCSS.Math.Content.HSN.VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.

CCSS.Math.Content.HSN.VM.C.9 (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

CCSS.Math.Content.HSN.VM.C.10 (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

CCSS.Math.Content.HSN.VM.C.11 (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

CCSS.Math.Content.HSN.VM.C.12 (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Tuesday, December 16, 2014

Autism Diaries: Strange Stories

Jill wanted to buy a kitten, so she went to see Mrs. Smith, who had lots of kittens she didn't want. Now Mrs. Smith loved the kittens, and she wouldn't do anything to harm them. When Jill visited she wasn't sure she wanted one of Mrs. Smith's kittens, since they were all males and she wanted a female. But Mrs. Smith said, "If no one buys the kittens, I'll just have to drown the kittens."
Question: Why does Mrs. Smith say this to Jill?
A "Strange Story" from Francesca Happe.
In her article "Understanding Minds and Metaphors: Insights from the Study of Figurative Language in Autism," Francesca Happe notes that even the highest functioning of her autistic subjects, those with normal IQs and relatively high Theory of Mind skills, made glaring errors in answering "Strange Story" questions like the one above about Mrs. Smith saying she would drown the kittens. A high functioning 17-year-old with autism, for example, said that Mrs. Smith's utterance was just a joke.

I haven't yet presented this story to J, who consistently rates as moderately autistic--and most decidedly not mildly autistic. But a recent exchange in the kitchen makes me want to (stay tuned).

"Why are you in the kitchen?" he asked.

Because I'm a neurotypical person with Theory of Mind skills in the normal range (for example, I know why Mrs. Smith told Jill she would drown the kittens), I can deduce J's subtext. He wants me to leave the kitchen so he can eat his supper in peace, without me suggesting that he use his knife and fork properly or informing him that it's late December so we don't need the fan on fast.

"Why are you in the kitchen?" he repeats.

"Because I like to be with you," I say.

"You're just joking."

"Why would I joke about that? I'm your mother and I like being with you."

"Why are you here?"

Discretely rubbing my arms against the brisk fan air, I try a new answer. "I'm enjoying the breeze in here."

Brief pause. Because I'm a neurotypical person with Theory of Mind skills in the normal range, I know he's stuck. Questioning my claim raises the uncomfortable question about why the fan is on.

But then, in an arch tone I've never heard him use before, followed by a victorious grin: "You're just trying to get me to turn the fan off!"

Sunday, December 14, 2014

Yet another baffler

In a move likely to cause political and academic stress in many states, a consortium that is designing assessments for the Common Core State Standards released data Monday projecting that more than half of students will fall short of the marks that connote grade-level skills on its tests of English/language arts and mathematics. 
The Smarter Balanced Assessment Consortium test has four achievement categories. Students must score at Level 3 or higher to be considered proficient in the skills and knowledge for their grades. According to cut scores approved Friday night by the 22-state consortium, 41 percent of 11th graders will show proficiency in English/language arts, and 33 percent will do so in math. In elementary and middle school, 38 percent to 44 percent will meet the proficiency mark in English/language arts, and 32 percent to 39 percent will do so in math. 
If the achievement projections hold true for the first operational test next spring, state officials will be faced with a daunting public relations task: convincing policymakers and parents that the results are a painful but temporary result of asking students to dig deeper intellectually so they will be better prepared for college or good jobs.
(From an article in last week's Education Week; emphasis added)

Question: How is it that this result will be temporary?

A. Teachers will rise to the challenge and alter their classroom instruction, causing student performance to improve substantially.

B. Teachers, parents, and the general public will react to the new proficiency standards in the same way that they react to high-level, high-stakes standards in general, and the resulting pressure will push the proficiency standards back down to previous levels.

Friday, December 12, 2014

Math problem of the week: Common Core-inspired first grade arithmetic

From an article in last week's Education Week, entitled Parents Get schooled on New Math Standards.

Here's more:
Jennifer Bonds, a parent at Old Orchard Elementary School in north-central Toledo, said she was watching her 3rd grader do math problems last year, "and I was like, wait a minute, I don't understand what you're doing!" The boy was calculating multi-digit addition and subtraction starting from the tens or hundreds place, and working down to the ones.

"I said, 'You can't do math from left to right,' " Ms. Bonds said.

She soon found out that, actually, he could. By attending parent math days held last year and this year at the school, Ms. Bonds learned that the common standards encourage students to add and subtract in a variety of ways other than the vertical carry-and-borrow methods she was taught, including by separating the tens and the ones.[Emphasis added.]  
Common Core advocates generally don't mention this, but alternative methods of adding have been around for a very long time. Nor are they incompatible with the standard vertical carry-and-borrow algorithms. One way to add a long column of digits in the ones (or tens, or hundreds, etc.) place, after all, is to make tens out of pairs and triplets of those numbers (and then to carry those tens over to the next place). I did that all the time as a kid.

What's changed is that, ever since the dawn of Reform Math in the early 1990s, these alternative methods have been crowding out rather than supplementing the standard algorithms.

What hasn't changed, meanwhile, is the fact that the standard algorithms, for arbitrary, arbitrarily large numbers, are by far the most efficient and the least prone to user errors. They are also, arguably, the best way to explore the subtleties of place value.

Finally, they are often easier to carry out than the alternative algorithms. Here's Ms Bond's take on the alternative method of adding: "It makes it look more difficult than it actually is."

Perhaps that's why American Reform math falls so far behind Singapore math and other more traditional curricula--even by the end of first grade.

Wednesday, December 10, 2014

Bonus Baffler

The curiously large gap between Student X's verbal and math SAT scores (see below) is mirrored by the curiously large gap between his reading vs. math scores on those mandatory NCLB/CCSS state tests. Year after year he earns a "Below Basic" score in reading and an advanced score, in the 99th percentile, in math.

He is fully included in a science and engineering magnet, with accommodations in English and social studies classes. His un-accommodated classes include an elite BC calculus class, discrete math, and AP computer programming. His unweighted GPA is approximately 3.5, and he earned a 4 last year on the AB Calculus AP exam (where it is assumed, based on his performance on practice tests, that he lost points for not showing sufficient work on the "free response" questions).

But his official state reading scores remain significantly "Below Basic."

Soon, for all but the most severely disabled 1-2 % of students, scoring proficient on the mandatory state tests will be mandatory for high school graduation. Given that these tests are supposed to measure 21st Century Skills, Higher-Level Thinking, and whether students are "College and Career -Ready," should students like this one be allowed to graduate from high school?

Or should they remain in high school for as many years as it takes for their reading scores to inch out of "Below Basic," past "Basic," and into "Proficient"--on the assumption that there will still be something left of the 21st Century by the time that happens?

Monday, December 8, 2014

December's baffler

The SAT Critical Reading score of Applicant X is 3/8 of his SAT Math score. There is a 500 point difference between the two scores.

1. Assuming that Applicant X grew up among native English speakers, what is his likely diagnosis?

2. Should you admit him to your engineering school?

Saturday, December 6, 2014

Puzzle Math, II: abstract pattern recognition

I’ve already blogged about the abstract patterns in math that are vanishing from today’s algebra classes. But my daughter just completed a particularly elaborate set of problems in her 1920s algebra book (Wentworth's "New School Algebra") for which abstract pattern recognition is absolutely essential:

To solve these equations, it’s massively helpful to recognize that each has the form of a quadratic—i.e., of ay2 + by +c.

For example, the first problem is of the form a(x2)2 + b(x2) + c, so if you let y = (x2), then you have y2 - 5y + 4 = 0. And if you’ve had enough practice with factoring, you immediately recognize the resulting pattern of coefficients and see what the roots of y are.

Similarly, in 8, the underlying quadratic structure is evident if (after adding 20 to both sides and putting the terms in descending order of their exponents) you take y = x1/4. The result: 2y2 - 3y + 20 = 0.

For 9, let y = xn, yielding 5y2 + 3y – 6 ¾ = 0.

For 11, let y = sqrt(x2 - x + 1), yielding 2y2 - y - 1 = 0, which, in turn, can be factored in a snap.

For 17, you see that you almost have a quadratic pattern. So, after adding 3 to both sides, express the 3 on the left hand side as 9 + -6. Then you have, with y = sqrt(2x2 + 3x + 9), y2 - 5y - 6 = 0—which (assuming you recognize the pattern of coefficients) is blissfully easy to factor.

Finally, for 20, which might at first seem despairingly removed from ay2 + by +c form, you notice that the coefficients of the x and x2 terms outside the radical are twice those of the corresponding terms under the radical. So you don’t despair, but subtract (4x + 9) from both sides of the equation. Then, as in 17, you split the constant term, expressing -9 and -6 + -3. Then you have 2x2 - 4x - 6 + sqrt(x2 -2x - 3) – 3 = 0. Let y = sqrt(x2 -2x - 3), and you get 2y2 + y - 3 = 0—which solves itself in a snap by factoring--assuming, once again, that you recognize the friendly pattern of coefficients.

Naturally, many American “math education experts” would look at this and, especially if they don’t actually try to do the problems (wouldn’t that be fun to watch!), write it all off as mindless manipulations of meaningless symbols. But what we have here, among other things, are wonderully abstract, multi-layered patterns—of the sort that are completely absent from Reform Math and Common Core-inspired math problems.

So many people wax poetic about how mathematics is all about patterns, and how doing math is all about pattern recognition. But how many of them realize what that really means? How many self-proclaimed math enthusiasts realize that the patterns of concrete quantities and 2-3 dimensional shapes so often used to illustrate mathematical beauty and mathematical depth are only the beginning, and that the most powerful patterns in math extend far below these shiny surfaces?

And how many self-proclaimed math experts realize, in the spirit of Puzzle Math, just how much fun it can be to unearth or sculpt out the underlying patterns and use them to disentangle what at first glance seems hopelessly disordered, unaesthetic, inelegant, and, in short, unpatterned?