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?

Observations on the PARCC sample Algebra II exam

Fresh off the presses at

you can now find complete PARCC sample exams. Since it’s the one I most likely have to worry about next year, I tried out the Algebra II exam.

While I haven’t checked the other tests yet, the Algebra II has laid off the “show your work” type question. I’m going to put this in the positive column (I was deeply worried about open responses being graded by an army of interns; plus the turnaround needs to be very fast because this is technically the final for our Algebra II class. Imagine having to shave off an extra two months from the end to give time for the test to be graded). [ADD: According to this comment, this only applies to the final exam, and there are still "show your work" type questions on the other tests.]

Some question-by-question comments:


I admit I was baffled enough by this one to check Wolfram Alpha (I got m = -15/2 where m = 1 was extraneous), until I realized the key phrase “you may not need to use all of the answer boxes”. We’re going to have to specially train students for that trick, I think.


I tested this one on my students. The majority put the value for x which makes the statement true (1) not the extraneous solution (-3/4). I can understand the motivation here: forcing students to go through the factoring rather than just test numbers or even eyeball the thing, but putting the extraneous solution as an answer is rare and I can imagine a legion of students being confused by this.


I always considered the trick of rejiggering a number’s base (turning 4 into 2^2) to be something of a side trick; I’ve never had to use it outside a college algebra book.


Because saying “complete the square” would be too easy. In all seriousness, this is one of those problems I would have many students know how to do but get intimidated by the language of the question.


So the first 8 questions (which I was just quoting from) are in the non-calculator portion of the test. The remaining 26 are from the calculator portion. The students will have access to a TI-84 emulator.


I’ve got mixed feelings on this one. In a way it’s a neat bit of factoring (yank out the (x-y) terms, and what remains will be a difference of squares, tossing an extra (x-y) on the pile) but it also looks nothing like the kind of factoring from any textbook I’ve used; I’m fairly sure our Carnegie Learning book (which was written from scratch specifically for Common Core) has nothing like it.


The number of fancy widgets to enter answers has dropped considerably, but here’s one; you drag and drop in order.

Also noteworthy: there are two problems involving average rate of change. I think in our textbook there was enough material to squeeze out a single day. The PARCC writers must be putting rather more emphasis on it.


Here’s another widget: the dots are dragged around to form the graph. I was somewhat thrown by the dots not just snapping to the grid but also the halfway points. I believe this is a bad idea. A student could easily think they have a correct answer but leave the dot halfway rather than right on the appropriate point.


My students would be thrown off by the wording here. I think the intermediate step is actually helpful, but the phrase “Product of greatest common factor and binomial” looks so technical many of them would shut down.


I’m used to tests that make the statistics so easy students can answer them with no training whatsoever, but this is what I’d call a real statistics question.


This is a widget that lets you select an interval on the number line. In the process of trying to enter an answer the number line disappeared. Then going back a question and returning I was unable to get anything to appear at all.

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 –



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.


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