Percy Jackson's Math Class

[I haven't written this post yet, and I know it's going to be more rambly than usual. Fair warning.]

My eldest child has been a big fan of Percy Jackson. Me, too, to be honest. I'm reading The Blood of Olympus now. The first Percy Jackson series is one of the few non-graphic novels my son read by choice. So this was a natural click for me: The Percy Jackson Problem at the New Yorker by Rebecca Mead.

It begins with a quote from Neil Gaiman (another family favorite):

“I don’t think there is such a thing as a bad book for children,” he argued, adding that it was “snobbery and … foolishness” to suggest that a certain author or particular genre might be a baleful influence upon young reading minds—be it comic books or the works of R. L. Stine. Fiction is a “gateway drug” to reading, Gaiman said. “Every child is different. They can find the stories they need to, and they bring themselves to stories. A hackneyed, worn-out idea isn’t hackneyed and worn out to them.” Well-meaning adults, he continued, can easily kill a child’s love of reading: “Stop them reading what they enjoy, or give them worthy-but-dull books that you like, the 21^st-century equivalents of Victorian ‘improving’ literature. You’ll wind up with a generation convinced that reading is uncool and worse, unpleasant.”

If you're a math teacher, how can you read this and not connect? We've been feeding students, as a profession, "worthy-but-dull" math for ages. (Worthy when it was good, that is. When it was bad, we're talking Tartarus.)

The author's argument is the counter to this idea that all reading is good.

Riordan’s books prompt an uneasy interrogation of the premise underlying the “so long as they’re reading” side of the debate—at least among those of us who want to share Neil Gaiman’s optimistic view that all reading is good reading, and yet find ourselves by disposition closer to the Tim Parks end of the spectrum, worried that those books on our children’s shelves that offer easy gratification are crowding out the different pleasures that may be offered by less grabby volumes.

I don't like this argument for reading. But I have made similar arguments in math. After a steady diet of exercises, students have no interest in problems.

But I think what I mean is that students have no experience with problems. The engagement that comes from finding a really meaty one. The question is whether reading Percy Jackson is really reading. I would argue that spending time on Tumblr is not reading (a current teenager discussion), and wonder if graphic novels are reading. (Aforementioned comic-obsessed son.)

I think this is an issue K-12. In elementary, there is a danger that teachers don't believe that students can do real problems. In high school, a desire to have the students do the basics first. Working with preservice and inservice teachers I try to stress and give experience with contexts that are problematic, but accessible. If it's not a problem, it's not doing math, no matter how many numbers and operations are involved.

Just being letters and words doesn't make it reading?

The author isn't so concerned with the Percy Jackson books, as with the forthcoming book of Greek myths as told by Riordan, writing in Percy's voice.

While the D’Aulaires wrote that “Persephone grew up on Olympus and her gay laughter rang through the brilliant halls,” Percy’s introduction to the story of Demeter’s daughter reads, “I have to be honest. I never understood what made Persephone such a big deal. I mean, for a girl who almost destroyed the universe, she seems kind of meh.”

It seems to me that this is some of what the common core struggle is about. Parents don't recognize newer curricula as math. (Which, of course, really has nothing to do with the common core in most cases; the Common Core gave them something to be against.)

The author closes with this concern:

What if instead of urging them on to more challenging adventures on other, potentially perilous literary shores, it makes young readers hungry only for more of the palatable same? There’s a myth that could serve as an illustration here. I’m sure my son can remind me which one.

Ooh, clever. I'm sure she knows which myth. What if after doing Desmos and Three Acts investigations, students don't want to do hackneyed word problems from the end of the chapter? That's probably not fair. Will they not be interested in the real problems of calculus, geometry, analysis and algebra? I think if we had a Percy Jackson parallel in math, the greater numbers of young people interested in math would mean a boom in STEM fields. The Percy Jackson problem? We should all have such problems.

This post started when I discovered no way to comment on the article. Because I wanted to share my daughter's response. And I want to think of this in terms of math, too. Here's Ysabela:

If they were arguing against, like, Twilight, where the language use is bad because the author has no writing experience, AND the plot and ideas are unoriginal/problematic, then I would agree with them. When Twilight becomes people's standard for literature, they start accepting total crap without a second glance, which is bad.

But Riordan understands language? And his plots (at least in the original series) were good? I'm not saying it was Harry Potter, but Percy Jackson was quality, and saying it wasn't just because the language is accessible to people who aren't scholars is just... really elitist. Like I know a lot of people who find reading really, really hard, but were able to enjoy Percy Jackson because it actually made sense to them. Plus, the series was narrated by a teenage boy, it was realistic.

And don't even get me STARTED on the D'Aulaires, they're SO AWFUL. They watered down the myths so much they were almost unrecognizable, "for kids," and then wrote it at like a college reading level. Plus they organized it like total tools. I can't express in words how much I would have rather had a Riordan book of myths than the D'Aulaires when I was younger.

Hello Katie @ Society 6

As she steps out the door now, she's railing against having spent two weeks on factoring. Because the last day before the mini fall break they learned the quadratic formula. "And it always works!" Do you know why it works? "He showed us from \(ax^2+bx+c\). It's extra credit on the test." How much more dangerous is it that she believes math is that pointless and uninteresting?

So, I think I'll side with Neil Gaiman on this one.

What's a Problem?

We had a fun class in the elementary math course today. Introducing SMP1 - problem solving, we got to an interesting question: what's a problem?

Here's the story as told by the residue on my whiteboards.

Schema Activation: jot down about a time you solved a real problem in your life. You won't have to share with anyone if you don't want to - this can be private.

After a few minutes to write, I shared how one of the big justifications for teaching and prioritizing math in school, other than the jobs to which it gives access, is that it teaches problem solving.

 Actually more yes than I expected!

People asked if I meant did math class help with the problem that they journaled about or in general. I said we want to know about problem solving in general, but they could use their instance as a specific.







Next question: is it possible that math class could help teach problem solving? Short time to discuss at table.








These were definite yesses, with a lot of confidence.




One of the reasons I like teaching teaching math is teaching is so much like math to begin with. So rephrasing: our problem is how to go from the current situation to get what we want.

So the next prompt was to quickly brainstorm. Ideas for making this happen.

















I shared that I liked how many of these were things that were up to the teacher. And I paraphrased Marzano, about how there is bad news: a small percentage of factors affecting student learning are under the teachers control; good news: that percent still makes a big difference.

What did they like?

• The emphasis on real life. This brought out uniform hatred for unlikely impractical story problems.
• Logic problems: one of the students shared how engaging and powerful these were for here. I asked about the contrast with real world, but people were comfortable with the crazy logic problems because the emphasis was on how did you do it.
• Teachers can ask for multiple ways and have students compare them.

Okay. So if we're going to teach problem solving, we need problems. I asked a question, then asked them to think about if that was a problem or not. There was a tub of square tiles on each set of two tables.

After they all had answers, I asked them to think about the second part. After a short time to discuss, we went to +cheesemonkeysf 's talking points structure for the statement.

This is a problem.

We're still working on the structure, so some of my feedback was about that. The no comments idea is hard.

Pretty strong agreement. People were willing to share their reasoning with the whole class ...  even the small minority. Maybe especially the small minority?

I was really happy to see the "depends on what the teacher does" idea come up. And I added the "depends on students" too. This is not a problem for me. For first graders, a serious problem - there aren't enough tiles in the tub to cover! For them... well, they talked about methods, chose how to do it, discussed results; these are problem indicators to me.

Then we went back to the question. What answers did they find?

 There was shock at the diversity.

One question I ask a lot is: is this a question where different answers could all be right?

They discussed and...

There was concern about the largest answer being too big, but that table and an adjacent table figured out the first group's table is actually larger.

I shared how measurement is one of my favorite content areas, exactly for this reason that a diversity of answers is to be expected. It can be culture setting.

So, with all this, definitely time for a problem. One of my favorites. How many pentominoes are there?


We played with domino patterns last class, so that's a natural starting point. We agreed that there was only one way to make a domino with two squares. If they touch, they have to share an edge.

With three squares, the issue came up about putting them "in the middle." That's solved by the edge rule. What about the elbows in a different direction? No, the class agrees, if you can turn it to match, it's the same shape.

So what I want to know: is how many pentominoes?



People got to work with tiles and graph paper. No group found all the tetrominoes as a step, which I was trying to suggest. A couple of times I had tables report on how many they had and were they expecting more. 8 more, 9 more, 12 probably, etc. Next time, 12, 14, 15 and a mix of have them all and think there are more. When our time was up, they sent emissaries to the board to draw what they had:

Now the best part: math fight! Are flips the same or not?

They divided up into groups based on their answer: 14 flips matter, 9 flips are the same. They shared reasoning. Flips matter because these are flat things and a flip requires another dimension. And if you try to match them up they do not line up. Flips don't matter because you can get them to line up, and what's the difference between a flip and a turn, really?

I refuse to settle the issue, and ask them to make a complete set before Monday.

If you care to comment or tweet a response, I'll share your answers with them:

• How do you recognize what is a problem for your students?
• Are flipped pentominoes the same or different? Why?

Practice Problems

I'm trying to shift towards standards based grading (SBG) this semester for the content grade in my geometry class, and so I'm paying a lot of attention to the kinds of problems I write. My tendency is to write big, open, sprawling problems that cross many standards, but I haven't yet worked out how to do that and SBG.  I know that I want problems where students can show understanding without necessarily getting to a correct answer.  Mathematicians are wrong a lot. What distinguishes us, I think, is that we often know whether we are wrong or right.

My students asked to have the time before the test to practice, instead of the book group I wanted to do. (110 min class and a 1 hour exam.) Very reasonable.  But that meant that I had to come up with practice problems! I'll post those now, then the test when all students have taken it.

Photo by scrappy annie @ Flickr

Do you give students practice?  What's the relationship between practice and the assessment problems?  This was a big topic in my student teacher observation this morning also.  

 322 Midterm Practice

The Standards

A. Analysis of characteristics and properties of
2-D geometric objects: number of edges; side length; angle size; parallel; perpendicular; convexity, etc.

1. concept: definition and recognition
2. application: use to sort and characterize, build or draw
3. combination: consider multiple properties in a single object
4. familiarity with examples: triangles, quadrilaterals, polygons

D. Distance, area, angle

1. concept: definition and key properties
2. formulas: use and derivation
3. connections among formulas

Try your choice of the following problems. Look for problems that allow you to problem solve and share your thinking.

Which standards could you demonstrate on which problems?


1) Connect each side property to an angle property and draw a different polygon to match each pair.

(at least) 3 congruent sides		no adjacent congruent angles
pair of perpendicular sides		(at least) 1 angle > 180 degrees
3 parallel sides		pair of adjacent congruent angles

Draw 3 different connections and try again!

2) Sometimes quadrilaterals are defined by their diagonals rather than by sides and angles. Determine which quadrilateral goes with which definition below, and make your argument. If no quadrilateral goes with a definition, state why.
a. Diagonals both bisect each other.
b. At least one diagonal bisects the other.
c. Diagonals are equal length.
d. Diagonals are perpendicular.
e. Diagonals do not intersect.
f. Diagonals are perpendicular bisectors.

3) On our Area on a Grid class workshop, find the areas of the shapes using formulas.

Area on a Grid


4) On graph paper, divide a square up into exactly 7 triangles with as many different triangle types as possible. Can you get all 7 types?

5) Area
a) Make an area formula for a trapezoid, or prove the one you know, using the formulas for rectangles and triangles.
b) Make an area formula for a kite.
c) Make an area formula for a chevron, using the diagonal lengths.

6) On graph paper, find squares with areas listed or argue why you can’t: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
a. Note: 5 is possible.
b. Find the edge length for each square you found using the Pythagorean Theorem. What do you notice?

7) Draw a 2 circle Venn diagram with the labels only in your mind (eg. a line of symmetry and at least one pair of parallel sides). Write in the quadrilateral types where they go, and give to a tablemate to figure out the labels.

Variable and a Problem

The Geogebra post on variable was gathering resources for a middle school inservice with math teachers and special education teachers.  (I've added the geogebra files for my sketches.)  Unfortunately, everything that worked smoothly the night before caused hangups on the day.  Sigh.

I'm working on this project with Esther Billings and Pam Wells, who both have a great grasp on teaching this subtle concept and are amazing teachers.  Pam is a wonder at working from student/participant work towards her goals, and Esther has this great integration of research understanding with what it means for teaching.

Objective:  TPW understand variables can represent changing quantities or represent an unknown; experience doing algebra with meaning and understanding of underlying concepts.

The day before they had spent a lot of time working on and watching student work for finding the patterns from pictoral sequences and expressing the relationship symbolically.  Including the very nice Modeling Middle School Math video of the Beams and Rods problem, from the Math in Context curriculum Building Formulas lesson (videos 8-13).  

We wanted to keep the element of having the teachers have a chance to do mathematics instead of just talk about it, and we also wanted it to connect to the Connected Mathematics Project Bags of Gold problem from the Moving Straight Ahead unit.  Swapping problems help develop the idea of equality, which is central to the idea of variable as unknown.  The bags of gold gets at it by putting the same amount in each bag, but you don't know how much that is.  So if 2 bags and 3 coins is the same as 15 coins...

I hate contrived situations, so my first thought about swapping was Magic cards (or Pokemon or Yu-Gi-Oh) but that seemed irrelevant to the audience.  Pam had the nice Stuart Murphy book Dinosaur Deals about that sort of thing.  My next inspiration was currency trading.  Found the current rates, jiggled it around, and realized it was entirely proportional.  Save it for later.  I wanted an exchange plus.  I thought about a classroom rewards situation (these were teachers) that a friend uses, and then complicated it.  So I wrote this problem:

Classroom Rewards

For small achievements or solid whole group work, students in Ms. Smittyck’s class earn a white chip (like presenting a problem at the board).  For more significant achievements  or an action that benefits others (like raising your grade in the class or helping another student meet a standard), a student can earn a red chip.  For very notable work or effort or action on behalf of another student (like figuring out a way to increase recycling in the classroom), students can earn a blue chip.  If you can get to 6 blue chips, you get a jolly rancher on Fridays for free.

Two white chips can be traded in for a jolly rancher.
Six white chips can be traded in for three red chips and a jolly rancher.
Five red chips can be traded in for two blue chips and a jolly rancher.

What would be a fair trade involving white chips and blue chips?  At the end of the year, what would be a fair trade for blue chips in terms of jolly ranchers?  What other questions does this raise?  

Connections:  Does the situation make sense?  Do you need more information?  What does an answer to the problem look like?

Focus:  How will you answer the question?  Have you ever solved a problem like this one?  Is there a representation that would be helpful?

Activity:  Solve the problem.  Try to record your thinking.
Extension:  If you were going to let the class trade in chips as a whole group for a pizza party or doughnut day, what would make a reasonable goal?  Why?

Reflection: How would you check your work?  Now that you know the answer would you solve it another way?

Teacher Work
This was my first time with the group, so I was worried:  was it too messy? Too easy?  Too contrived?  (I hate contrived problems - especially mine!)

But they were amazing.  They dived right into the problem, asking great making sense questions.  What is this?  How does that work?  Why wouldn't they just...?

People worked with equations (because that seemed more mathy), tables (because that seemed helpful) and making pictures with blocks (which we had put out on the table beforehand without comment).  The blocks group had the most rapid progress and worked things out in multiple ways.  Nobody's answer matched mine.  But that wasn't the point.  They did want to be told the answer, but were okay with 'later.'

The free Jolly Rancher caused the most problems.  People thought you were trading the 6 blue for 1 Jolly Rancher, or that it was in addition or a one time thing.  The other confusing thing was that the red and blue chips come out surprisingly (to people there) close in value.  Given that, I reworked the problem a bit.  This retains the messiness and elements of non-proportionality.  If you want to make the problem considerably cleaner, make it 11 white chips for three red chips and a Jolly Rancher


Classroom Rewards, v2

For small achievements or solid whole group work, students in Ms. Smittyck’s class earn a white chip (like presenting a problem at the board).  For more significant achievements  or an action that benefits others (like raising your grade in the class or helping another student meet a standard), a student can earn a red chip.  For very notable work or effort or action on behalf of another student (like figuring out a way to increase recycling in the classroom), students can earn a blue chip.  If you get into the 6 blue chips club, you get a jolly rancher every Friday without having to trade anything in.

Two white chips can be traded in for a jolly rancher.
Twelve white chips can be traded in for three red chips and a jolly rancher.
Ten red chips can be traded in for two blue chips and a jolly rancher.

What would be a fair trade involving white chips and blue chips?  At the end of the year, what would be a fair trade for blue chips in terms of jolly ranchers?  What other questions does this raise? 

Questions
How would you work on this problem?  How would students?  Is it too messy for students?  If you're interested, I'd be curious about your comments.

Money Problems

Excellent variation on the "how many ways to make change problem" at the New York Times today. The Freakonomics column is reporting the work of Patrick DeJarnette. I can see giving this problem from 4th grade to linear algebra!

Click on the money tag to see my previous money games. Click on the cartoon to go to the excellent Non Sequitur website.

Good Problems

Where do you get good problems for your students?

One source is that problem-of-the-day widget at the bottom of the blog. A couple times a week, I'm copying those, put them into a Word document, and then save them for a good opportunity.

But my all time favrite source is from the English (or British?) parallel to the NCTM: Nrich. Problems are sorted by content, tagged, by grade band (stage) and challenge level (number of stars). Some are unsolved, but accessible. Almost all are clever and/or interesting. Soooo nice! Give them a try. Here's an account of a teacher and how they use Nrich.

Here's one that I gave on a math for middle school final this semester:
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Take, for example, four consecutive negative numbers, say

−7, −6, −5, −4

Now place + and/or − signs between them. e.g.

−7+−6+−5+−4
−7− −6+−5− −4

There are other possibilities. Try to list all of them. Now work out the solutions to the various calculations. e.g.

−7+−6+−5+−4=−22
−7− −6+−5− −4=−2

Choose a different set of four consecutive negative numbers and repeat the process. Take a look at both sets of solutions. Notice anything? Can you explain any similarities? Can you predict some of the solutions you will get when you start with a different set of four consecutive negative numbers? Test out any conjectures you may have. Try to explain and justify your findings.