After over a month talking about policy, it’s fun to write about math classes for a change.

I’m teaching geometry, Algebra II, and Math For Kids Who Haven’t Passed The State Graduation Test Yet.

I’ve given this algebra readiness test to all my students for the past three years. I got it from a senior teacher at my last job, and it’s an excellent assessment of a students’ basic numeracy and first semester algebra skills. Can they substitute? work with negatives? multiply binomials? factor a quadratic? I don’t much care about second semester algebra (graphing parabolas, quadratic formula); my geometry students won’t need it, and my algebra II students will be reviewing the material again.

I know what some of you are thinking. “Why the heck are you giving your geometry and algebra II students a test in pre-algebra and first semester algebra? They already know that material, don’t they?”

This is me laughing at you naive folks. Ha ha!

Or, I could show you a graph of the results.

So the algebra II kids have taken a full year more of math than the geometry kids, and both groups have passed algebra. But the algebra II class is usually taken by kids who made it this far by their toenails, got low scores in both algebra and geometry. Geometry 9 is ninth graders who passed algebra the first time but chose not to take the honors course—and who aren’t in A2/Trig.

My Algebra II class is substantially stronger on average than last year’s class, which averaged around 20 wrong. I’ve got about 8 kids that got 0-2 wrong after finishing the test in ten minutes and should be taking A2/Trig, or even Honors. The Geometry 9 class is slightly stronger on average than my class from last year, which averaged around 12-13 wrong. My geometry classes last year were the strongest I’ve ever taught, but had a much weaker bottom than this class does. The strongest students in the Math Support class got 13-15 wrong, which is impressive.

For the uninitiated, there are two big pieces of info in this graph:

In all three classes but particularly the A2 and Geo, the range of scores on what should be an easy test is huge. The weakest students in both geometry and A2 got barely half right. I’m used to this—handling wide ranges in ability is probably my greatest strength as a teacher, although administrators don’t value it much. However, really think about that range and what it represents in terms of the ability gap within one classroom. And remember–this is a school that provides honors courses, so it tracks much more than my last two schools. The gap this year is considerably less than the scores from last year (which I can’t find, so you’ll have to take my word for it).

The other news is, of course, that the average score for the geometry class should be 5-6 wrong, and the algebra II students should, in a world where we worry more about what kids learn than what their transcript says, knock the test out of the park.

About half the kids in both geometry and algebra II classes should not be taking formal college prep courses, but rather an interesting math applications course, in which they continue to apply what they’ve already learned, rather than pile on new stuff.

Oh, well. That’s what we get for pretending ability doesn’t matter.

I don’t want to sound cynical or discouraged. I’m pumped. They’re a great group of kids and are stronger on average than my last year’s kids, with lots of high achievers. But what’s “normal” for me is clearly not anything that reformers understand or anticipate when they talk about high expectations or proficiency for all.

September 10th, 2012 at 8:57 pm

Seems there is a typo in the available answers for question 29. Or am I in need of some ‘math support?’

September 10th, 2012 at 9:25 pm

No, for some reason the PDF and the print out resolved incorrectly, while it’s correct (-2,3) on the Word file. I put the corrected graph on the board. I need to redo that before next year; it’s the second year that’s bit me.

September 11th, 2012 at 9:30 pm

BTW, such poor manners of me to be casting stones without first saying, great blog! I’ve loved it since first seeing it via Steve S.

I have a neighbor that’s a semi-retired (it’s Portlandia, we all “retire” for a stint at least once before 40) middle school teacher at a school primarily of -1 sigma types. She laughed out loud when I shared a couple of your more pithy excerpts. Among them, “I teach summer school because compared to regular school it’s like free money, ” and “algebra ‘needed skills’ list… integer operations and fractions! Damn. Why didn’t I think of that?”

September 11th, 2012 at 9:54 pm

Wow, I’m glad you like the blog! No problem on pointing out the error; I always tell my kids that there almost certainly will be a typo, the only question is how many. But that particular test, I’ve used three years running so they are mostly worked out. I can’t get that damn graph to resolve correctly.

Glad you like those lines. They’re among my favorite, as well. (To be sure, the second one isn’t mine! It was a colleague’s. I just report the facts.)

September 11th, 2012 at 2:59 am

Clearly there are a number of kids in your class who got through elementary school and a lot of middle school without having gained basic number sense.

But when you write about kids needing math ability or math interest to succeed in advanced math, I think Trig and pre-calc.

Knowing that z can stand for 19 and y can stand for 25 and that if you add the two you get 44 seems to be a kind of fundamental issue that everyone ought to be able to handle. Maybe I’m still hopelessly naive?

An experienced math instructor at a well-regarded private school here in Gotham once told me that her kids often have reading comprehension issues – they just can’t make sense of the math question ’cause they don;t know how to parse sentences and think about what is being asked. They can compute fine, they just can’t read, as it were. Do you see that issue?

It would be interesting to know the top 10 questions answered incorrectly; does it suggest something about the kids ability, or prior instruction?

September 11th, 2012 at 6:15 am

“Clearly there are a number of kids in your class who got through elementary school and a lot of middle school without having gained basic number sense.”

And high school. Most of my algebra II students are juniors and seniors, which means they took algebra and geometry in high school as well.

No, I don’t see language ever being an issue, particularly not reading.

“It would be interesting to know the top 10 questions answered incorrectly; does it suggest something about the kids ability, or prior instruction?”

It says something about the kids’ ability. I wrote abouot the myth of “they’ve never been taught” here: https://educationrealist.wordpress.com/2012/07/01/the-myth-of-they-werent-ever-taught/

September 11th, 2012 at 9:01 am

Dear Educationrealist !

1. Great mathematician V. I. Arnold (1937-2010) insists that formula

(1+x)^alpha =

=1+

+alpha*x+

+[alpha*(alpha-1) / 2!]*x^2+

+[alpha*(alpha-1)*(alpha-2) / 3!]*x^3+

+[alpha*(alpha-1)*(alpha-2)*(alpha-3) / 4!]*x^4+

+ …,

{for integer positive alpha it is a finite sum}

and, as such, was known to Vieta

(i.e. befrore Newton.)

It was the genius of Isaac Newton

(and here I agree with Arnold)

— to understand (and to claim) that this is

a valid (but infinite) series

for any alpha — be it fractional, irrational, negative, or even complex.

E.g. for alpha=(-1) this is a series

for the sum of infinite geometric progression.

2. Here is how I taught it for alpha=(1/2) to my 3 kids

(not simultaneously, but with account of age.)

Sqrt(a) is such a number b,

that b^2 = a.

Sqrt(1) is evidently 1.

If “x” is small, then probably Sqrt(1+x) is close to 1.

So, denote Sqrt(1+x) as (1+y)

Then from definition of what Sqrt is, (1+x) = (1+y)^2 (exactly.)

You draw simple picture of a square with a side (1+y)

to show, that its area equals

1^2, + areas of two rectangular stripes 1*y, plus area y^2,

i.e. (1+y)^2 = 1 + 2*y + y^2.

If, by assumption, “y” is small in comparison with 1,

then y^2 is especially small,

and in the first approximation it can be ignored.

So, we got 1+x =(approx)=1+2*y,

Sqrt(1+x) =(approx)= 1 + (1/2)*x,

i.e. which is the first term of Newton’s

(not Vieta’s, but _Newton’s_ !!! ) binomial formula.

With respectful greetings, F.r.

September 11th, 2012 at 11:42 am

“I’ve got about 8 kids that got 0-2 wrong after finishing the test in ten minutes and should be taking A2/Trig, or even Honors.”

Why aren’t they taking those higher classes?

September 11th, 2012 at 1:14 pm

I asked them. The answers are mostly about the counsellor’s recommendation or their fear of A2/Trig. One of them couldn’t get it in his schedule, but his math teacher allowed him to test out of A2/Trig, so he’ll be gone.

September 11th, 2012 at 3:53 pm

To illustrate the point about Newton’s binomial formula,

you may suggest to your students to use

“ MicrosoftWindows.Programs.Accessories.Calculator. (‘scientific view’) ”

to calculate {-1+Sqrt[1+10^(-200)]}.

What is nice about this example is the following.

Each (repeat, each) of the entries in this expression can be dialed

on Windows Calculator (‘scientific view’).

The correct result, 5*10^(-201),

also may be meaningfully displayed by the program.

But uncritical straightforward execution

of the expression presented above yields __zero__ as the result.

Meanwhile, the correct result is direct consequence of

Newton’s binomial formula for the power alpha=(1/2):

Sqrt(1+x) =(approx)= 1+(1/2)*x.

Sure, for mathematically-oriented students

there is another way to struggle with the use of

Windows Calculator (‘scientific view’).

Multiply this number by fraction,

which is identically equal to 1;

namely by 1 = = = {1+Sqrt[1+10^(-200)]} / {+1+Sqrt[1+10^(-200)]}.

Then the resulting numerator becomes 10^(-200),

while Windows Calculator (‘scientific view’)

yields the result =2 for the denominator, which is more than good enough.

With best wishes,

your Florida resident.

September 11th, 2012 at 4:07 pm

I am quite sure almost any student who ends up qualifying for pre-university (we call them junior colleges here) in Singapore can ace this test. They are slated for much crazier stuff (http://www.seab.gov.sg/aLevel/2013Syllabus/9740_2013.pdf) anyway. But that’s only the top 30% of the cohort.

What’s the comparative statistic for the US? And if there’s a difference, why is there a difference? Teaching methods? Motivation? Human biodiversity?

September 11th, 2012 at 8:52 pm

Yeah, I’m teaching kids from the 30th to 60th percentile, overall. Anyone higher than 60th percentile is probably in another class.

November 9th, 2012 at 10:36 pm

50 questions, 4 multiple choice answer options each. And yet the mean score for Alg II and Geometry is < 12.5. What the hell?

November 9th, 2012 at 10:37 pm

Oh wow, those are wrong answers, not right answers. My mistake.

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