biased test questions - integrated algebra

A lot of people have been talking, and will be talking for a while, about the Integrated Algebra curriculum/exam in New York this year. For now, consider the following question (#31) from this brand-spanking-new exam:

"Tom drove 290 miles from his college to home and used 23.2 gallons of gasoline. His sister, Ann, drove 225 miles from her college to home and used 15 gallons of gasoline. Whose vehicle had better gas mileage? Justify your answer."

Two questions:
(1) Do you think this question is harder for students whose parents do not own cars? Or perhaps, have not spent considerable time in cars?

(2) If yes, does this make the test "unfair"?

Upon seeing the question, the eyebrows went up, the blood boiled as I thought about my kids from upper Manhattan who, in general, do not spend time in cars. Buses, subways, yes; cars, not nearly as much. They don't gas them up, don't drive them around, don't have a sense of what gas mileage really is. I don't think this is even about being economically disadvantaged; people in the five boroughs do not drive as much as people outside of the five boroughs. I don't own a car. Most people I know don't either, even if they could afford one.

My kids know ratios and rates. Miles per hour; feet per minute; words per minute, etc., etc. "Gas mileage"? Can't say I used that phrase, ever, in class.

In the end it's yes and yes. The question is far harder if you've never filled a car with gas and driven it around. Let alone if you don't even know what the phrase "gas mileage" means. My best students struggled with this question. Put the phrase in the performance indicators or write a fairer question, New York.

I've always thought it would be interesting to write an exam that had a bias the other way, where my kids would "get" all the questions and everyone else would struggle with vocabulary, context, meaning. Is there a way to "invert" this question?

when do children follow rules?

A lot has been made recently of complex/"strict" rule systems and other infrastructure at charter schools and B.O.E. schools around the country. My own two cents? Children meet teacher/school expectations when...

(1) they can "do it" (that is, they possess mental/physical/social/emotional capabilities for meeting expectations)

(2) they know/understand how to "do it" (that is, they know exactly what the expectations are)

(3) they want to/have to "do it" (that is, they feel compelled extrinsically, intrinsically, or morally)

More about these underlying philosophies to come. They will show up everywhere. I don't think they're really any different from the reasons why adults do the things they do. But effective strategies have to involve a mix of these elements.

i'm going to miss this place

Just a couple snapshots near school.

it's just that simple

Ad for NYC Teaching Fellows near school:


In case you can't read the fine print: "I show kids that I'm willing to do whatever it takes for them to succeed."

Oh man. Wish I had thought of that! It's that simple. Just show 'em you care. The rest will take care of itself. Trust me.

real life math

Friend and I stop for a beer in the afternoon, drawn in by the following deal on a chalkboard:

"Cana Two For One".

It turns out that a cana is priced at $2.50 and a regular old pint is $5.00. The special deal puts the Cana at a quarter of the price of a pint, provided you drink at least two. In fact, you can get no less than four canas for the price of one pint. Excellent? Certainly seems that way, until you consider the size of the cana versus the pint. It gets a little dicier. The cana is freaking tiny. Exhibit A:

(That would be pint on the left, cana on the right.)

So what is the better deal? Four canas or one pint?

Comparing volumes is usually an estimation killer, since, in my experience, the brain is usually pretty bad at estimating volume. This is probably because quadratic relationships are somewhat "unnatural", at least when compared with linear ones.

But we push through. We call each glass a cylinder (imperfect, but it'll have to do). The pint's diameter registers at one of my pointer fingers, almost exactly. Conveniently, my finger is divided neatly (intelligently designed?) into three pretty equal sections; the cana's diameter is almost exactly two-thirds of the same finger. So we can call the radius of the cana two-thirds of the radius of the pint. Similarly, the height of the cana works out to be about two-thirds the height of the pint.

We're good to go. Let's call the pint volume V, the pint radius R, and the pint height H. Similarly, the cana volume will be v, the cana radius r, and the cana height h.

The pint volume is given by:
The cana volume is given by:
We've established relationships between R and r and H and h:

Now we can substitute into the equation for v:

Simplifying the squared business:
Finally multiplying all fractions:
Which leads to final comparison of v and V:
This works out nicely; eight-twenty sevenths is pretty darn close to one-third (nine-twenty sevenths). So we are getting a deal (well, friend is, since I opted out), since four canas equate with the price of one pint, but three canas has already equated the volume in one pint.

Not bad for a Saturday afternoon. Facility with fractions and formulas. We hit quite a few Integrated Algebra performance indicators.

How hard is graphing y > 2x + 1?

Coming up this week (for my Integrated Algebra students) is that algebraic mother lode: graphing linear inequalities (A.G.6: Graph linear inequalities, for the standards-aligned). This has been saved for the year's end since it is such a killer. The problem is, from the state's point of view, it should be easy for a ninth grade student. It should be, based on the state math performance indicators.

What do you have to do if you want to graph y > 2x + 1? (If you are a college-educated adult and have no idea what I'm talking about, this raises some other juicy questions about math education/retention/cultural significance.) Well, for starters, you have to recognize that you are going to need to graph the line y = 2x + 1. This sort of graphing is taught in seventh grade (7.A.7: Draw the graphic representation of a pattern from an equation or from a table of data) and again in eighth grade (8.G.17: Graph a line from an equation in slope-intercept form (y=mx+b)). This will no doubt call on you to plot points in the coordinate plane, something begun in fifth grade (5.G.12: Identify and plot points in the first quadrant).

You've also got to recognize also that since this is a strict inequality (>) you need to graph using a dashed line. (Why? Because the line does not include points in the solution set of the inequality.)

Finally, choose a point that is not on the line, plug in the numbers in the inequality (5.A.3: Substitute assigned values into variable expressions and evaluate using order of operations) and then evaluate the truth of the resulting inequality (4.A.3: Find the value or values that will make an open sentence true, if it contains <>) . If it is true, shade in that side of the line; if false, shade the other side.

And that's it. We'll let alone the case where y is not isolated (a further algebraic challenge) or the case of the system of linear inequalities (yup, two or more at once).

But for most students I teach, culling all of these skills at once is a lot to ask. If a student is weak in just one of these skills, and many are weak in several, the whole operation fails. Graphing a linear inequality becomes a labor that can last half a period, and most often contains at least one error. It can cause such frustration even though it is really just a combination of things my students already, supposedly, know.

So what's the problem? What happened to these students in fourth, fifth, seventh, and eighth grade? Were their teachers checked out? Did they leave these standards out of their instructional plans? Were their lessons poorly planned? It's possible, but I doubt this is a leading reason.

I believe that there were days in 2006 when my students were expert graphers. I think there was a day or two in 2004 when students were evaluating expressions with the best of them. Their teachers led them along through all the proper steps and understandings, and the students could do it. But this is not good enough today, in ninth grade algebra.

Based on my limited experience, I'm tempted to say the problem is the kind of understanding my students come to high school bearing. While students may have, at one time, under the right circumstances, with the proper scaffolding, etc. etc. known how to graph, plot points, evaluate expressions, and determine the truth of inequalities, they never truly understood the meaning of these skills. If my students are reaching into a vat of collected, memorized skills, and pulling them out, one at a time to graph y > 2x + 1, then of course it's hard. They could pull the wrong one. They could pull out a skill that is dusty, rusty and incompletely memorized. They are constantly in fear of "doing the wrong thing" or doing it "the wrong way".

If, on the other hand, they see a coordinate plane and recognize it as a sea of points, each of which registers true or false in the inequality; if plotting points is "second nature"; if the coordinate plane is readily divided by linear equations with differing slopes; in short, if students have a thorough understanding of what these various elements represent, then graphing inequalities will happen quickly and easily, conforming to a body of knowledge already obtained. The prerequisites need to be understood deeply and richly.

How do we instill this in children? I don't know. I'd like to know. I'm working on it.

Have students learned if they forget later?

lunch scenes

From today: Gunshots that dispersed all the normally happy kids at the park out my classroom window during period 5. Also, I ate lunch.