## [Pseudocontext Saturday] Mazes

This Week’s Installment

Poll

What mathematical skill is the textbook trying to teach with this image?

Pseudocontext Saturday #2

• Using properties of symmetry (33%, 134 Votes)
• Completing the square (20%, 80 Votes)
• Identifying rational and irrational numbers (19%, 76 Votes)

Total Voters: 404

(If you’re reading via email or RSS, you’ll need to click through to vote.)

Rules

Every Saturday, I post an image from a math textbook. It’s an image that implicitly or explicitly claims that “this is how we use math in the world!”

I post the image without its mathematical connection and offer four possibilities for that connection. One of them is the textbook’s. Three of them are decoys. You guess which connection is real.

After 24 hours, I update the post with the answer. If a plurality of the commenters picks the textbook’s connection, one point goes to Team Commenters. If a plurality picks one of my decoys, one point goes to Team Me. If you submit a mathematical question in the comments about the image that isn’t pseudocontext, collect a personal point.

(See the rationale for this exercise.)

Current Scoreboard

Team Me: 1
Team Commenters: 0

The judges rule this pseudocontext because, given that awesome square maze, it’s very unlikely that anyone would wonder about the side length of the maze and unlikelier still that anyone would wonder if the side length was rational or irrational. An exhaustive search for a 1,225 ft2 square maze in Dallas, TX, produced no results, exacerbating the judges’ sense that the textbook is exploiting the world for the sake of math. That’s pseudocontext.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### John Golden

October 22, 2016 - 11:57 am -

Was hoping for space-filling curve or the left hand rule. Why can you finish HS math without knowing how to navigate a labyrinth?

2. #### cheesemonkeysf (Elizabeth)

October 22, 2016 - 12:05 pm -

*buries face in hands and softly weeps*

– Elizabeth (@cheesemonkeysf)

3. #### Mr K

October 22, 2016 - 1:24 pm -

You sure it’s not supposed to be identifying rational and irrational textbook problems?

4. #### Dennis Rankin

October 22, 2016 - 2:10 pm -

If this were the floor plan for a new high school, what would be a just amount of passing time for students to transfer between classes?

5. #### David Wees

October 22, 2016 - 2:37 pm -

My attempt at a non-pseudocontext:
How long would it take you to get through the maze by randomly guessing left or right at each turn?

(We actually visited a corn maze a couple of weekends ago with a four year old “leading” us through the maze, so this is an actual question I had.)

6. #### Paul Hartzer

October 22, 2016 - 3:03 pm -

I’m guessing symmetry even though mazes usually don’t have symmetry because textbooks are inane like that.

An interesting problem: Given a quadrilateral matrix of arbitrary points (that is, counting like a Go board, not a chess board), how many distinct mazes can be made with unique solutions? Is there a recursive function?

• #### Paul Hartzer

October 22, 2016 - 3:06 pm -

I meant to add: Assume that all maze walls connect neighboring points only. That is, all walls are either 1 unit long (horizontal or vertical) or sqrt(2) units long (diagonal). If no pattern exists, what if walls may only be 1 unit long.

7. #### Maureen

October 22, 2016 - 3:22 pm -

Logic and reasoning?
Or is it political? With that red and blue maze, this picture reminds me of a voter trying to get out of the “maze” of political crap we keep hearing from both candidates. Since we only see a way in, and no way out…it must be about our election.

8. #### Stacie

October 22, 2016 - 6:31 pm -

How about a probability question since a grid seems to be super-imposed above it? Find the probability that a person is in a red portion of the maze versus a blue portion?

9. #### Gaute

October 23, 2016 - 4:02 am -

I was hoping for one of the choices being connected to “something binary”. Could one say that 2^(n^2) is an upper limit for the number of choices you may have to make in an n-by-n maze? (I am sure this upper bound can be refined (c;

10. #### Laura

October 23, 2016 - 8:41 am -

I’m thinking about probability here in a different way…what is the probability you’d finish the maze if you just tured left/right/alternated between the two/some patterned combination of lefts and rights?

11. #### Michael Paul Goldenberg

October 23, 2016 - 9:40 am -

Was hoping this was some sort of graph theory, Euler circuit/path problem. Disappointed to not see that in the choices. :(

12. #### Xavier Bordoy

October 23, 2016 - 9:45 am -

None. Combinatory: how many ways are there? How many are for getting the goal?

13. #### Paul Hartzer

October 23, 2016 - 11:14 am -

Unlike the dart one (last week), now that I see the explanation, it does make some sense. If you’re *building* such a maze, you will need to know that the diagonal walls will have lengths that have to be estimated. Although, as with the dart puzzle, most people will simply measure a drawing rather than calculating out the lengths, at least I can see how some people might use a formula.

I get the point, but on the other hand, if we go too far the other direction with this, we run the risk of communicating that all higher math is pointless because people will just estimate everything they can’t do with plain arithmetic. During one of my education classes, I remember trying to develop a “real world” geometry application of calculating how much carpet is needed for a room. I was told by several people (including the professor) that they’d just take basic measurements and estimate, because there’s going to be waste anyway.

So how *do* we create mathematics problems that are real and that don’t rely on overly fake scenarios? Or do we stop trying to create real world mathematics problems entirely?

• #### Julian Gilbey

October 25, 2016 - 11:33 am -

These are really good points and questions. My sense is that creating ludicrously fictitious contexts is pointless, as it gives the message that mathematics is disconnected from the real world problems that people care about. On the other hand, I am struggling to think of a good example of a genuine real-world application of rational versus irrational numbers. Yet I think there is a really interesting story to tell nonetheless about the history of their discovery and the fear they were held in – it is far more engaging than this pseudocontext, and may help to explain the terminology “irrational” (which in English has quite negative connotations). There is also the potentially interesting discussion of how to represent these numbers – how can computers handle them, for example? (I’m not sure whether this counts as “application”, though.) But some of this gets quite sophisticated quite quickly.

There is also the point to make that much mathematics is interesting for its own sake: it is a subject of intrinsic beauty, and humans as a species like problem-solving. (Think of the prevalence of crossword puzzles, the sudoku craze and so on, which probably have no significant practical applications.) And math is about problem-solving, just because. Sometimes, the problems are practical, sometimes not. But it is the problem-solving aspect which is common to these, not the practical applications.

• #### Paul Hartzer

October 25, 2016 - 1:05 pm -

@Julian: “Pythagoras allegedly killed someone over the square root of two” does seem to get kids’ attention. ;)

• #### Dan Meyer

October 26, 2016 - 12:34 pm -

Julian:

Yet I think there is a really interesting story to tell nonetheless about the history of their discovery and the fear they were held in – it is far more engaging than this pseudocontext, and may help to explain the terminology “irrational” (which in English has quite negative connotations).

Agree. I’d like to know more about the genesis of “irrationality.” Somewhere in the inception of the idea is a conflict which we can adapt for the classroom.

14. #### Andrew Knauft

October 26, 2016 - 11:40 pm -

I am that unlikely individual: put me in front of a square maze and tell me it is 700 square feet, and I will quickly start to wonder how and why they built it that way. And yet, put that same maze in a textbook problem as one of four options to investigate for irrational side lengths and I no longer care. I suppose that is exactly what the “pseudo” in pseudocontext hits at: the problem *could* come up in the real world, and if it does it might very well be an interesting problem for that person in that moment; but divorced from that it becomes hollow and forced.

• #### Dan Meyer

October 27, 2016 - 3:58 am -

Is there any way for curriculum designers to notch a win with you, if the “designed” aspect of their work seems to disagree with you?

15. #### Simon Gregg

October 27, 2016 - 5:18 am -

The story that the irrationals were a problem for Pythagoras, though attention-grabbing, is, like the maze in Dallas, exploiting the world – or in this case history – for the sake of mathematics. See my post:
http://seekecho.blogspot.fr/2016/03/hippasus-wasnt-pushed.html
There is still the possibility of imagining how their first discovery must have come as a surprise. I mention a possible line of thought here:
http://seekecho.blogspot.fr/2013/08/not-quite-pythagoras.html

16. #### CateMD

October 28, 2016 - 4:57 pm -

I teach pre-algebra as well as algebra. I sometimes show these images to my students. They thought it might be about transformations (translate, rotation and reflection), although they didn’ t see where reflection might apply.