## Observations on the PARCC sample Geometry exam

Part 1: Observations on the PARCC sample Algebra I exam
Part 2: Observations on the PARCC sample Algebra II exam
Part 3: Observations on the PARCC sample Geometry exam

Calculator part: 18 of 25

Use the information provided in the animation to answer the questions about the geometric construction.

To pause the animation, select the animation window.

The students are supposed to watch a video of a construction and then say things about the proof enacted through the constructions. This is a very specific skill that needs to be practiced. Daniel Schneider kindly sent me a link to a website with a large number of construction animations (along with proofs) in case you need more to use in class.

However, there’s a serious interface problem. Here’s what the video looks like when paused, as well as a question to go with it:

Point “C” is completely covered. Whoops.

Non-calculator part: 6 of 7

This is one of those simple-looking questions which has enough of a trick to it I’m not sure how many students will get it right.

Part A requires students to work a double-completing-the-square manipulation, hopefully not getting sidetracked by the presence of b on the right hand side:

$x^2 + y^2 - 4x + 2y = b$
$x^2 - 4x + y^2 + 2y = b$
$x^2 - 4x + 4 + y^2 + 2y + 1 = b + 4 + 1$
$(x-2)^2 + (y+1)^2 = b + 5$

Part B requires noticing that a radius of 7 means the right hand side will be 49, so $b + 5 = 49$ and thus $b = 44$.

In principle this problem is solvable, but the lack of partial credit on a problem with a “trick” that I worry a student who can normally complete the square would still get no points due to the indirection.

Calculator part: 13 of 25

This problem’s rough for three reasons:

a.) Even with the phrase “the pipe is open at both ends” placed in there, this is something of a background knowledge problem; the students need to know the “outer surface” excludes the circles on the top and bottom.

b.) There are volume formulas on the formula sheet but not surface area formulas. Thus the students need to have memorized $\pi d h$ or be able to extrapolate it, and know enough to exclude the circles.

c.) If $\pi$ is set to be 3.14, there answer comes out to be 1356.48. If $\pi$ is set to be 3.14159 or something with more digits (not unusual since graphing calculators have a “pi” button) the answer comes out to be roughly 1357.168. Rounding to the nearest integer thus can give either 1356 or 1357 as an answer.

Non-calculator part: 7 of 7

Out of all the problems on the PARCC final exam for geometry, 28% are related to transformations.

I can understand a transformational emphasis in general: it leads to a function transformation understanding of graphs (which is far more powerful and useful than looking at each kind of graph individually). However, why do so many of the dilation and rotation problem — 4 out of the 9 — involve centers not at the origin? This is not rhetorical; I really want to know where the utility is.

Non-calculator part: 20 of 25

This is one of the easier problems on the test, but assumes background the students don’t necessarily have. I can assume what a “collar” means here (even though I’ve never heard the word used in this context) but my ELL students are more likely interpret it as gibberish.

Calculator part: 10 of 25

This is very similar to the other problem in relying somewhat on background knowledge. Technically speaking one can ignore all the external stuff about merchant vessels and probes and focus on the math, but the brain of the ELL student doesn’t have an easy time removing the context.

Also note the weirdness of the rounding; in problem 20 the rounding needed to be done to the nearest tenth, while in this problem the rounding needs to be done to the nearest integer off the list.

Calculator part: 3 of 25

I’m noting this one because nothing in my current textbook (Carnegie Learning, written for Common Core) has anything resembling this kind of problem. Anyone have a source with problems that are similar?

Calculator part: 23 of 25

I don’t think I’ve ever give this much emphasis to the vocabulary of proof. Getting my students to keep the reflexive, symmetric, and transitive properties of congruence straight is going to be a nightmare and a half.

Ok, one last problem, from HS Sample Math Items, 7 of 10 (so not the final exam, but the open response part):

Here the angle bisector video returns (complete with unhelpful play button covering the diagram when paused) but the student is supposed to free-write a proof.

Here is how you type the first line as given:

1.) Pick “geometry” on the side and pick the short line in the upper right; that’s a “line segment” and will give you a blank box under a line segment so you can type letters.

2.) Type the letters you want under the line segment. If you accidentally type more than two letters any extra keypresses will be ignored.

3.) Go to “relations” above “geometry” and find the congruence symbol. Pick that. This will give the congruence symbol and a blank box.

4.) Pick the “line segment” and it will take the blank box that just appeared and put a line segment over it.

5.) Type the letters you need for the other line segment.

Now you have one step of the proof, now you just need to give a reason and then do four more steps.

(What would constitute a valid reason here, by the way? The mathopenref site I linked to early in this post just states “They were both drawn with the same compass width” — would this be considered valid by graders?)

## Robot maze puzzle

The puzzle above has a robot (marked with an arrow pointing up, or “north”) that you can control with a set of command cards that either move the robot forwards a set number of spaces (the + numbers) or backwards (the – numbers). After using a command card, the card is used up and can’t be used again.

If the robot hits either the border of the grid or one of the black spaces, the robot stops moving and any remaining steps on the command card being used are ignored.

Landing on one of the spaces marked with circles causes your robot to turn 90 degrees. (That is, if the robot faces north it turns east, if east it turns south, if south it turns west, and if west it turns north.) The robot starts facing north. Can you get the robot to the star?

## Observations on the PARCC sample Algebra I exam

Similar to my breakdown of the PARCC Algebra II, here’s some question-by-question comment on the sample PARCC Algebra I Final Exam. I’m not too discouraged by the actual items on this one, but the interface needs work.

This is the second problem of the test. You click points to set a line, then click “solution set” to shade in one side. Note (just like the Algebra II test) the points snap to half-grid points, not to grid points, a circumstance I find hazardous. Also, if you click on solution set to color a side, then realize your line was wrong, you have to click back on “line” again but the color goes away, so you have to add the color back again after the line is fixed. This is true even if switching the line from solid to dotted.

I should add this is one case where I see the superiority of open response to multiple choice. Here’s an inequality graphing problem from our old state test:

The lack of choices makes the problem a dotted-or-solid / above-or-below question where the actual shape of the graph is given away.

The question here is fine, but what if a student drags in the wrong number and wants to fix their answer? Removing a number only works if you drag the number back to the original number boxes, just “tossing” the number to a random position outside the answer box doesn’t work.

One common technique in the PARCC interface is for students to fill in sentences with a drag-down menu. By my eye, though, the interface doesn’t look much like a sentence, and I could imagine a student not understanding they are placing words between f(2) and g(2) and so forth to produce something that is meant to be read from left to right.

I guess 4 answer boxes — clear overkill — is better than the situation with 2 answer boxes where the suggestion seems strong to fill both of them even if one of the answers turns out to be extraneous.

I’m pretty sure logarithms aren’t supposed to be on the Algebra I test? Also, the graph is drawn automatically through the points, unless it can’t like in the example above. It took me a bit of deciphering to realize there’s an asymptote on there (right on the y-axis) and the asymptote can be slid around, so the reason the graph wasn’t showing up is the points were on opposite sides of the asymptote.

Do the blanks really have to be so large? I admit to getting confused because the symbols spread out in a single function looked to me like function-break-really small expression-break-random parenthesis and I had to do a double-take before I realized what was going on.

I hope students have their window large enough to realize (or least deduce from there being a “Part A”) that there is a “Part B” to the question.

There’s a truly weird option to change colors of things. Sometimes I can get it to trigger but I’m not sure how. The upper right inequality in pink shows what things look like after you’ve messed with the color.

There’s even an interface for systems of linear inequalities. Notice how there’s still a snap-to-half-grid feature even when the y-axis goes up by 5.

Why does one “find the zeros” question have a drag-and-drop interface, while this one gives a list?

## Mathematics classes do not teach mathematics literacy

Both the phrases “mathematics classes” and “mathematics literacy” can be ambiguous, so I am meaning–

“Mathematics classes”: The high school curriculum delivered to a typical student in the United States. This usually excludes classes like “Statistics” or “Financial Math”.

“Mathematics literacy”: The kind of “good citizen” math that people refer to in articles like Headlines from a Mathematically Literate World. The word can also mean “ability to problem solve”, but that’s not what I mean here.

Taking the Headlines article and the classes of a typical high school mathematics student, how many of the headlines would a a student understand?

At the very least, understanding the entire list requires knowing about: correlation vs. causation, inflation, experimental replication, estimation of large numbers, incompatibility of comparisons with different conditions, understanding how tax brackets work, meaninglessness of predictions within a margin of error, statistically unlikely events, and reversion to the mean.

None of these will ever occur in an traditional math class. In other words, in the list of supposed math literacies, the typical math student in the US receives zero of them. (Some might possibly show up in a class labelled “Economics” or “Free Enterprise”, but those don’t get called Math Classes).

It’d be fair to argue I’m being highly specific in my starting definitions, but I often see the “good citizen” argument used during a general “why are we teaching math” type discussion which assumes a traditional math class track. That sort of argument only works if people are prepared to also overhaul the curriculum (by putting, for example, statistics before calculus as Arthur Benjamin discusses at TED).

## Why is a negative divided by a negative a positive?

So there’s a whole lot of posts, including one from this very blog, which give intuitive explanations of why a negative times a negative is a positive.

I haven’t seen nearly as much material for a negative divided by a negative. One can certainly appeal to the inverse — since $1 \times -1 = -1$, $\frac{-1}{-1} = 1$. Google searching leads to answers like that, but I’ve found nothing like the multiplication picture above.

Can anyone explain directly, at an intuitive level, why a negative divided by a negative is a positive? Or is the only way to do it to refer to multiplication?

## The evolution of mathematical exposition

More rambly and unsubstantiated than usual, apologies –

[Source.]

Theory: Mathematical exposition has evolved just like fiction writing has. However, tradition has held stronger in mathematics (likely due to a need for precision) and it means that clarity in writing is if not actively discouraged at least passively devalued.

Theory: We are not anywhere near the threshold of simplest and clearest explanations in the exposition of mathematical subjects.

Still, what used to be difficult is now considered easier. Various subjects have shifted their supposed level. For instance, not long ago College Algebra was the prestige class at the top of the high school level.

Furthermore, our raw definitions of what each class is has shuffled the actual content of subjects; Algebra I from the 1940s is not the Algebra I of today.

Theory: It would be possible to take a “hard” subject like group theory or transcendental number theory and make it comprehensible at a lower level. However, as there is no requirement to do so there is little motivation to make the subject easier. When a curriculum shift happens to move topics to a lower level, mathematical exposition evolves to catch up.

## Book update (and call for testers)

Don’t get too excited yet — the big one (Why Algebra Works, you’re best off reading this post for an idea of how it is being written) isn’t done.

However, in the midst of work I realized I was assuming the readers would remember how integer operations went, and it was quite possible they forgot, so I worked on a appendix. The appendix ballooned into a full fledged … short story? novella? … and got to the point that I even separated addition/subtraction from multiplication/division.

So the first part (addition/subtraction of integers) is close to ready, and it does follow my percepts, namely–

1.) that it should have a smooth writing style modeled after popular math articles (like Steven Strogatz or Martin Gardner) rather than textbooks

2.) that there are no “problems” but rather “puzzles”, roughly defined as anything that wouldn’t be out of place alongside a Sudoku book or in the middle of a Professor Layton game

3.) that there is a strong emphasis on meta-thinking; that is, having readers examine whatever mental model they are using in a particular part of mathematics and diagnosing where misconceptions may come about at the internal level.

As an example of #3, I start by asking the reader to add 2 + 2 (really), examining the possible ways of visualizing it and which ways might be more or less helpful.

In any case, everything is so unlike the textbook approach that I need some beta testers. In particular, while I would like some people who are adept in mathematics, I would also like some people who think that are not good at math or even actively dislike it. I’m guessing the latter don’t read this blog I’m going to need some help — if you know someone who might be a good candidate, could you send the word along? I’ll get back to everyone in a few weeks.

You can either post here or toss a line to my email over at my About Page.

## Impossible Learning: watch someone trying to learn calculus from scratch

David Wees retweeted this from Jared Cosulich:

It turns out Impossible Learning is a just-started-last-month blog where Jared is trying to learn Calculus and post about his struggles. It’s terrific and you should read it. See him ask the perennial question When Will I Use This?

And I immediately found myself saying “come on, when am I ever going to have to find the limit of this random equation”.

I felt like I was back in High School again.

But seriously, why is this one of the first things I’m directed to learn when I want to know more about Calculus? Why is it so hard for me to find some practical applications of this material? I know there is value in understanding the abstract math, but I’d like to balance that with at least some understanding of how this works practically…

I don’t think this actually counts as a proof, but it definitely made the “Power Rule” click for me a bit more. Basically it’s saying that the derivative of a square (x²) is two lines (2x) and the derivative of a cube (x³) is three squares (3x²).

So for a square to get a tiny bit bigger you need to add on two lines (one to the top and one to a side). Similarly for a cube to get a tiny bit bigger you need to add a square to three sides (e.g. top, right, and front).

## Counting puzzle from the first US puzzle championship

My students had fun with this one today. Part d is what showed up in the actual championship.

One night, in an attempt to combat insomnia, you begin counting (1, 2, 3, …), but you decide to do it digit by digit. As you go along, for example, the 15th digit you count is the 2 of the number 12.
1 2 3 4 5 6 7 8 9 10 11 12

a.) What’s the 50th digit you count?
b.) What’s the 100th digit you count?
c.) What’s the 1000th digit you count?
d.) What’s the 1,000,000th digit you count?

## Commentary on “Wrong Answer: The Case Against Algebra II”

There’s a new article by the author Nicholson Baker that is not raising as much a fuss as “Is Algebra Necessary?” from The New York Times last year, probably because it’s at Harper’s behind a paywall. Also, as I write this children are fleeing from algebra all over magazine stands:

The title strikes me as odd, given that the general argument is aimed at all algebra, and the text constantly references algebra as a whole rather than the quite US-specific version of Algebra II. (As far as I’m aware no other country has such a thing; most of them integrate all forms of mathematics together.) What is Algebra II anyway, and where is the cutoff point where Nicholson Baker considers algebra to be too hard to handle? He hints at what he’d like as a replacement, which strikes me as a math appreciation analogous to music appreciation courses:

We should, I think, create a new, one-year teaser course for ninth graders, which would briefly cover a few techniques of algebraic manipulation, some mindstretching geometric proofs, some nifty things about parabolas and conic sections, and even perhaps a soft-core hint of the innitesimal, change-explaining powers of calculus. Throw in some scatter plots and data analysis, a touch of mathematical logic, and several representative topics in math history and math appreciation.

This seems like a truly random collection of topics, but it sounds like Mr. Baker is basing his list off the notion of using The Joy of x by Steven Strogatz as a possible text.

I have had some personal experience with using popular texts; when I team-taught statistics at the University of Arizona we used The Drunkard’s Walk, Fooled by Randomness, and Struck by Lightning: The Curious World of Probabilities. While it led to interesting discussions, there just wasn’t enough meat to have students do problem solving based solely on the text. The math from the books was purely a passive experience. (We still did just fine by augmenting with our own material.)

I still hold forward the absurd idea that students still solve math problems in a math class. If you’re designing a freshman mathematics-teaser course, I might humbly suggest Problem Solving Strategies: Crossing the River with Dogs, which has the virtue of steering away from algebra as the sole touchstone for problem solving.

Back to Mr. Baker’s attempt to define Algebra I:

Six weeks of factoring and solving simple equations is enough to give any student a rough idea of what the algebraic ars magna is really like, and whether he or she has any head for it.

Mr. Baker himself seems to have a confused idea of what algebra is like, but he’s not alone. (See: my prior post on what is algebra and why you might have the wrong idea.) Also of note: some countries don’t bother to factor quadratics. (I haven’t made a map comparing countries, but it seems to be continental Europe that ignores it and just says “use the quadratic formula”.)

I could see solving equations managed in six weeks, but in a turgid, just-the-rules style that would be the opposite of what we’d want in this kind of appreciation class in the first place. Just the concept of a variable can take some students a month to wrap their head around, making me disturbed by the notion that six weeks would be enough for a student to find “whether he or she has any head for it”. (This isn’t even touching the issue of just how much is internal to the student. I’ve heard dyscalculia estimates of up to 7% of all students, but excluding that group a great deal of the attitude seems culturally specific. Allegedly in Hungary [I don't have a hard research paper or anything, this is just from personal anecdotes] it is quite common for folks to say mathematics was their favorite subject in school and the level of disdain isn’t remotely comparable to the US.)

They are forced, repeatedly, to stare at hairy, squarerooted, polynomialed horseradish clumps of mute symbology that irritate them

The article’s invective along these lines makes me wonder again how much the visual aspect to mathematics is the source of hatred, as opposed to the mathematics itself.

This picture is from the Adventure Time episode “Slumber Party Panic” and is supposed to represent the ultimate in math difficulty. However, the math symbolism is strewn truly at random and there is no real problem here. I am guessing a good chunk of the audience couldn’t even tell the difference and for them, any difficult math problem looks like random symbolic gibberish.

This is related to another issue, that of bad writing. Here’s Mr. Baker quoting a textbook:

A rational function is a function that you can write in the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial functions. The domain of f(x) is all real numbers except those for which Q(x) = 0.

I claim the above definition is simply bad writing, and a cursory check of the Internet reveals several definitions I’d peg as clearer in a students-who-don’t-like-math-are-trying-to-read sense:

any function which can be written as the ratio of two polynomial functions.

a function that is the quotient of two polynomials

“Rational function” is the name given to a function which can be represented as the quotient of polynomials, just as a rational number is a number which can be expressed as a quotient of whole numbers.

Hardy’s A Course of Pure Mathematics, page 38:

the quotient of one polynomial by another

Godfrey Harold Hardy admittedly goes on from there, but his text is written for mathematicians. High school texts have no such excuse.

In a similar vein, the article later quotes a 7th grade Common Core standard:

solve word problems leading to equations of the form px + q = r and p(x+q)=r, where p, q, and r are specific rational numbers

which I suppose is meant to be horrifying, but in this case the standards are written for the teachers and aim to remove any ambiguity. Here’s problems that matches the standard:

1. You bought 3 sodas for 99 cents each and paid 10 cents in tax. How much money did you pay?
2. You bought 4 candies for 1.50 each and paid \$6.20. How much was tax?

The standards can’t simply produce many examples and gesture vaguely. They must be exact. Just because 4th grade specifies that students “Explain why a fraction a/b is equivalent to a fraction (n×a)/(n×b)” does not mean students are using variables to do so. (In case you’re curious what it does look like to the 4th graders, Illustrative Mathematics has tasks here and here matching the standard.)

There’s lots more to comment on, but let me leave off for the moment on this quote, because I’m curious…

Math-intensive education hasn’t done much for Russia, as it turns out.

…is this statement (in the last paragraph of the article) accurate?