The aim of Shell Centre Publications has always been to ensure that a number of seminal works in the field of mathematical education remained available. We have now reached the point where our most popular items are out of stock, and have come to the decision that it is time to stop storing and selling physical books. Digital distribution is the best way to keep these works available, so in the coming months, we will be making many of the publications on our list available, for free, as PDF downloads.

These books are just great. The Language of Functions and Graphs, in particular, has a couple of career's worth of great activities, lesson plans, and essays on teaching functions. Highly recommended.

[via Michael Pershan]

]]>I’ve been teaching for over 25 years and this is the best way to document what it is that happens in classrooms. A friend looked at my blog and then said to me, “Now I get what it is you really do.” Of course we can never actually capture all the moments, both large, small and in between, but I think all of our blogs together can do that.

And in the climate we find ourselves in today I think that is very important.

Joe's blog is one of my favorite recent subscriptions. You should subscribe.

It's really hard to find dedicated elementary math teacher bloggers for a lot of really good reasons. In particular, they generally teach *everything* so why would they specialize their blogging. All of this makes Joe's blog invaluable.

Two good starters:

]]>@ddmeyer Could you have written a list of 5 reasons you blog 5 years ago? And a list of 5 reasons you blog now? Lists match? What’s changed?

— Michael Fenton (@mjfenton) February 27, 2014

I'm reposting Michael Fenton's question here, less because I'm interested in you seeing *my* answers and more because I'm interested in seeing *yours*. Ignore his five-year qualification. If your motivations for blogging have changed over any stretch of time at all, let us know why.

In 2009, I blogged because:

**I wanted a record**of what I taught and believed about teaching that I could reflect on and laugh at later in my career.**I needed a community.**I taught in a rural district with five other math teachers (two of them married). Fine educators, but they were in different stages of their career and had answered a lot of questions I was just starting to ask. I needed people.

In 2014, I blog because:

**I want more interesting questions.**In 2009, I was asking questions about worksheet design, PowerPoint slides, and classroom management. By articulating my questions and noticing which of them created vibrant discussions and which of them fell with a thud on the bottom of an empty comments page, over five years I have moved on to some questions that make my work a joy to wake up to every day. eg. What do computers buy us in curriculum design? What does good online professional development look like? What does it mean for students to think like mathematicians and how do we scaffold that development? What is the "real world" anyway and what does it buy us in math class?**I need to stay connected to classroom teachers.**I'm fast approaching the date where I'll have been*out*of the classroom for longer than I was*in*it. Which scares the hell out of me and keeps me asking for advice from real life classroom teachers on this blog and reading, like, five hundred thousand teacher blogs every day.

A readership is more essential to my goals now than it was then. If you guys aren't tuning in and pushing back at my ideas and offering your own, those ideas get a lot dumber. (In 2009, by contrast, I had 120 students to let me know when my ideas were dumb.)

So a lot of what I do in my blogging lately is try to send you signals that I read and value and act on your responses. (See: the recent confabs; featured comments; putting the word out on Twitter that there's an interesting conversation brewing, etc.)

Perfect encapsulation of all of the above: this week's circle-square confab, which featured 62 comments from a pack of great teachers, creative task designers, and math education researchers.

That's why I blog now. Why do you?

**Featured Comments**

I started blogging way back in 2012 because I needed a way to reflect. I was in a very, um, rough patch in my teaching career and needed a way to get some thoughts out there. I was reading all kinds of other blogs and seeing what others were doing, stealing material from people left and right. Therefore, my blog was a way to thank people for giving me cool stuff.

Fast forward to 2014 and things have changed. I’m still reading blogs and stealing left and right, but I’m also trying to give back a little. As information kept pouring in, I started to get some ideas of my own. Sure, some of them are awful, but I’m proud of Barbie Zipline and some others. At this point, it’s still a 70/30 take/receive deal, but I’m all for it.

Bree:

I feel like I’m currently struggling to answer this question – which is probably why my blogging rate has been downwards of around once a month these days.

So many of the good teacher moves are invisible, and as I begin to blog I aim to capture some of the techniques that I have used to engage students. I often pass along worksheets and activities for teachers to use, but sometimes what I really want to pass along are the questioning techniques used throughout the lesson, along with a structure to ensure that students are discussing mathematics instead of working in isolation. Blogging allows for this extra commentary.

When I started blogging, I desperately needed to validate my experiences. I was teaching in big ol’ NYC, but at a private school with just one other math teacher. I needed to know: Was my teaching weird? Was I actually figuring things out about teaching, or just headed down my own idiosyncratic path? I wanted to say things that made sense to other people, so that I could be really sure that they made sense to me.

Also:

My blog is for figuring things out, so that someday I’ll be able to help teachers and kids out in a real way.

]]>I used to blog because I felt like I was coming up with some innovative lessons and I was learning some new approaches.

Recently I haven’t blogged because I’ve been handicapped into traditional direct instruction lessons (through resources and student culture). Maybe when I’m not in a different school every year (or when I’m excited about the school where I teach) I’ll start blogging again.

Every student's initial graph was wrong. No one got it exactly right the first time. But Function Carnival doesn't display a percent score or hint tokens or some kind of Bayesian probability they'll get the next graph right. It just shows students what *their* graph means for *that* ride. Then it lets them revise.

David Cox screen-recorded the teacher view of all his students' graphs. This is the result. I love it.

**BTW**. I'm hardly unbiased here, having played a supporting role in the development of Function Carnival.

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.

Here's why I'm obsessed. In the first place, the task involves a lot of important mathematics:

- making sense of precise mathematical language,
- connecting the verbal representation to a geometric representation,
- reasoning quantitatively by estimating a guess at the answer,
- reasoning abstractly by assigning a variable to a changing quantity in the problem,
- constructing an algebraic model using that variable and the formulas for the area of a square and a circle,
- performing operations on that model to find a solution,
- validating that solution, ensuring that it doesn't conflict with your estimation from #3.

Great math. But here's the interesting part. Students won't do *any* of it if they can't get past #1. If the language knocks them down (and we know how often it does) we'll never know if they could perform the other tasks.

**What can you do with this? How can you improve the task?**

I'm going to update this post periodically over the next few days with the following:

- your thoughts,
- two resources I've created that may be helpful,
- commentary from some very smart math educators on the original problem and those resources.

Help us out. Come check back in.

**Previous Confab**

The Desmos team asked you what other Function Carnival rides you'd like to see. You suggested a bunch, and the Desmos team came through.

Man did you guys came to play. Loads of commentary. I've read it all and tried to summarize, condense, and respond. Here are your big questions as I've read them:

- Is learning to translate mathematical language the goal here? Or can we exclude that goal?
- What role can an animation play here? Do we want students to
*create*an animation? - What kinds of scaffolds can make this task accessible without making it a mindless walk from step to step? On the other end, how can we extend this task meaningfully?

There was an important disagreement on our mission here, also:

**Mr. K** takes one side:

It took me about 3-4 minutes to solve – the math isn’t the hard part. The hard part is making it accessible to students.

**Gerry Rising** takes the other …

If we want students to solve challenging exercises, we should not seek out ways to make the exercises easier; rather, we should seek ways to encourage the students to come up with their own means of addressing them in their pristine form.

… along with **Garth**:

Put it to the kids to make it interesting.

I'll point out that making a task "accessible" (Mr. K's word) is different than making it "easier" (Gerry's). Indeed, some of the proposed revisions make the task harder and more accessible *simultaneously*.

I'll ask **Gerry** and **Garth** also to consider that their philosophy of task design gives teachers license to throw any task at students, however lousy, and expect them to find some way to enjoy it. This seems to me like it's letting teachers take the easy way out.

Lots of you jumped straight to creating a Geogebra / Desmos / Sketchpad / Etoys animation. (Looking at **Diana Bonney, John Golden, Dan Anderson, Stephen Thomas, Angelo L., Dave, Max Ray** here.) I've done the same. But very few of these appleteers have articulated how those interactives should be used in the *classroom*, though. Do you just give it to your students on computers? To what end? Do you have them *create* the applet?

**Stephen Thomas** asks two important questions here:

- How easy is using [Geogebra, Desmos, Etoys, Scratch] for kids to construct their own models?
- When would you want (and
*not*want) the kids to construct their own models?

My own Geogebra applet required lots of knowledge of Geogebra that may be useful in general but which certainly wasn't germane to the solution of the original task. It adds "constructions with a straightedge and compass" to the list of prerequisites also, which doesn't strike me as an *obviously* good decision.

Lots of people have changed the wording of the problem, replacing the mathematical abstractions of points and line segments to rope (**Eddi, Angelo L**) and ribbon (**Lisa Lunney Borden**) and fencing (**Howard Phillips**).

This makes the context *less* abstract, yes, but the student's *work* remains largely the same: students assign variables to a changing quantity on the line segment, then construct an algebraic model, and then solve it. The same is true for some suggestions (though not all) of giving the students *actual* rope or ribbon or wire.

So I'm interested now in suggestions that change the students' *work*.

**Kenneth Tilton** proposes a "stack" of scaffolding questions:

- If the length of AB is 1, what is the length of AP?
- What is the ratio of AP to PB?
- Given Ps, the perimeter of a square, what is the area of the square?
- Given Pc, the perimeter of a circle, what is the area of a circle?
- How would you express “the two areas are equal” algebraically?

The trouble with scaffolds arises when a) they do important thinking *for* students, and b) when they morsel the task to such a degree it becomes tasteless. **Tilton** may have dodged both of those troubles. I don't know.

**David Taub** lets students *choose* a point to start with. Choosing is new work.

**Mr K** asks students to start by correcting a *wrong* answer. Correcting is new work.

It seems to me that a simple model of the problem, (picture of a string) with a failed attempt (string cut into two equal parts) should be enough to pique the kids “I can do better” mode. Providing actual string with only one chance to cut raises the stakes above it being a guessing game.

I think more important would be to start with some “random” points and some concrete numbers and see what happens.

**Max Ray** builds fluency in mathematical language into the *end* of the problem:

So I would have my students solve the problem as a rope-cutting problem. Then I would invent or find a mathematical pen-pal and have them try to pose the rope problem to them.

If our mathematical language is as efficient and precise as we like to believe, its appeal should be more evident to the students at the *end* of the task than if we put it on them at the *start* of the task.

**Gerry Rising** offers us an extension question, which we could call "Circle-Triangle." I'd propose "Circle-Circle," also, and more generally "Circle-Polygon." What happens to the ratio on the line as the number of sides of the regular polygon increases? (h/t **David Taub**.)

On their own blogs:

**Justin Lanier**offers a redesign that starts with a general case and then becomes more precise. I'm curious about his rationale for that move.**Jim Doherty**runs the task with his Calculus BC students and reports the results.**Mike Lawler**gives us video of his son working through the problem.

**2014 Feb 26.** *Some of my own resources.*

Here's one way this problem could begin:

- Show this video. Ask students to tell each other what's happening. What's controlling how the square and circle change?
- Then show this video. Ask students to write down and share their best guess where they are equal.

The problem could then proceed with students calculating whether or not they were right, formulating an algebraic model, solving it, checking their answer against their guess, generalizing their solution, and communicating the original problem in formal mathematical language.

Mr. K has already anticipated my redesign and raised some concerns, all fair. My intent here is more to *provoke* and less to settle anything.

I'm going to link up this video also without commentary.

**2014 Feb 27.** *Other smart people.*

I asked some people to weigh in on this redesign. I showed the following people the original task and the videos I created later.

- Jason Dyer, math teacher and author of the great math education blog Number Warrior.
- Keith Devlin, mathematician at Stanford University.
- Two sharp curriculum designers on the ISDDE mailing list, whose comments I'm reproducing with permission.

Here's video of a conversation I had with Jason where he processes and redesigns the original version of the task in realtime. It's long, but worth your time.

Keith Devlin had the following to say about the original task:

I immediately drew a simple sketch – divide the interval, fold a square from one segment, wrap a circle from the other, and then dive straight into the algebraic formulas for the areas to yield the quadratic. I was hoping that the quadratic or its solution (by the formula) would give me a clue about some neat geometric solution, but both looked a mess. No reason to assume there is a neat solution. The square has a rational area, the circle irrational, relative to the break point.

So in the end I just computed. I got an answer but no insight. I guess that reveals something of a mathematician’s meta cognitive arsenal. You can compute without insight, so when you don’t have initial insight, do the computation and see if that leads to any insight.

In the case of the obviously similar golden ratio construction, the analogous initial computation

doeslead to insight, because the equation is so simple, and you see the wonderful relationship between the rootsSo in one case, computation just gives you a number, in the other it yields deep understanding.

Off the ISDDE mailing list, Freudenthal Institute curriculum designer Peter Boon had some useful comments on the use of interactives and videos:

I would like to investigate the possibility of giving students tools that enable them to create those videos or something similar themselves. As a designer of technology-rich materials I often betray myself by keeping the nice math (necessary for constructing these interactive animations) for myself and leaving student with only the play button or sliders. I can imagine logo-like tools that enable students to create something like this and by doing so play with the concept variable as tools (and actually create a need for these tools).

Leslie Dietiker (Boston University) describes how you can make an inaccessible task more accessible by giving students *more* work to do (more *interesting* work, that is) rather than less:

If the need for the task is not to generate a quadratic but rather challenge students to analyze a situation, quantify with variables, and apply geometric reasoning with given constraints, then I'm pretty certain that my students would appreciate a problem of cutting and reforming wire for the sake of doing exactly that …

**More Featured Comments**

I disagree with people who are saying that this problem as written is inherently bad or artificial. As an undergrad math major, a big part of the learning for me was figuring out that statements worded like this problem were very precise formulations of fundamental insights — insights that often had tangible models or visualizations at their core.

I remember lectures about knots, paper folding, determinants, and crazy algebras that the lecturers took the time to connect to interesting physical situations, or even silly but understandable situations about ants taking random walks on a picnic blanket. For a moment I even entertained the idea of graduate work in mathematics, because I realized that math was actually a pretty neat dance between thinking intuitively and thinking precisely.

Terrence Tao writes about that continuum here.

tl;dr version: Translating this problem from precise to intuitive and intuitive to precise, is part of the real work that research mathematicians (and their college students) do, and not something we should always keep from our students. It’s a skill we should help them hone.

**2014 Mar 4**. As usual Tim Erickson got here first.

I messed up her chocolate milk order a few years back. This is a new ratio task I first heard from Colin Foster at the Shell Centre last winter.

Here's the download link, which includes the third act.

**Featured Tweet**

Love all the strategies my Ss used with Nana's Paint Mixup from @ddmeyer. pic.twitter.com/HmCWOuAvyd

— Kate Fisher (@_K8Fisher) February 20, 2014

**Featured Comments**

Don:

I dunno, this one made me wish the follow-up showed him just throwing the paint away and starting over. There’s just not enough investment in materials & time to make me think past. Plus if 6 tablespoons was enough paint to do the job then 30 is just a waste of paint far in excess of throwing away 6.

@Don, what if he had used up all the white paint after putting in the 5 scoops of white? Then he’d have to figure out how to do it by just adding more red.

**2014 Mar 11**. Great extension for Algebra students from Paul in the comments:

]]>I used the task to set up this question in an Algebra class.

The students were very puzzled when their intuition about the solution did not match arithmetic or a demonstration with cubes.

“I have two cans of pink paint variations in the following ratios. Neither is perfect.

Nana’s Pink 5 Reds : 1 White

50/50 Pink 1 Red : 1 WhiteI think that a perfect pink will be 3 Reds : 1 White

Can I make it by mixing Nana Pink and 50/50 Pink?”

Students expected the solution to be one cup of each. How wonderful the sound of a perplexed group of students when their arithmetic did not match their intuition.

Sense making ensued for many minutes with pictures, cups with cubes and more arithmetic.

It's about the quickest and most concise illustration I can offer of Guershon Harel's necessity principle. The moment of need is brief, but really hard to miss. It sounds a lot like laughter.

**2014 Feb 19**. Christine Lenghaus adapts the interaction for naming angles:

]]>I drew a large triangle and then lots of various sized ones inside it and asked the students to pick an acute angle. I asked a student to describe the one they were thinking about and then another student to come up and mark it! This lead to discussion on how best to label so that we both agree on which angle we were talking about. Gold!

Here are three e-mails I received from three different people over the last three months. Spot the common theme.

November:

My co-teacher and I were puzzling over what kind of problem would create an intellectual need for systems. Do you have anything you could send, by chance?

December:

We are planning to launch a unit on systems of equations in early January (after December break) and wanted to try out your approach to create an intellectual "need."

January:

Showing two straight lines on a piece of graph paper and finding points of intersection has very little significance to most people. I'm looking for a real-world problem that has an answer that is not self-evident, but which requires a little thinking and finding the intersections and is infinitely more productive and satisfying and will stay with them for the rest of their lives. That is what I am looking for.

I receive these questions on Twitter also. I find them almost impossible to answer because *I don't know what your class worships*.

Here's what I'm talking about:

**Class #1**

You start class by asking your students to write down two numbers that add to ten. They do. Most likely a bunch of positive integers result.

Then you ask them to write down two numbers that *subtract* and get ten. They do.

Then you ask them to write down two numbers that do both at the exact. same. time. "Is that even possible?" you ask.

Many of them think that's totally impossible. You can't take the *same* two numbers and get the *same* output with two operations that are *natural enemies* of each other. They'd maybe never phrase it that way but the whole setup seems totally screwy and counterintuitive.

Then someone finds the pair and it seems obvious in hindsight to most students. We've been puzzled and now unpuzzled. Then you ask, "Is that the *only* pair that works?" knowing full well it is, and the class is puzzled again.

You define systems of equations as "finding numbers that make statements true" and you spend the next week on statements that have only one solution, that have infinite solutions, and the disagreeable sort that don't have *any* solutions.

Students learn to identify the kind of scenario they're looking at and how to find its solutions quickly (if any exist) using strong new tools you offer them over the unit.

**Class #2**

The same lesson plays out but this time, after we've determined the pair of numbers that solve the system, a student pipes up and asks, "When will we ever use this in the *real world*."

**Worshipping the Real World**

David Foster Wallace wrote about worship – the secular kind, the kind that applies to everybody, not just the devout, the kind that applies especially to us teachers in here:

If you worship money and things — if they are where you tap real meaning in life — then you will never have enough. Never feel you have enough.

If your students worship grades, they won't complete assignments without knowing how many points it's worth. If they worship stickers and candy, they won't work without the promise of those prizes.

If you say a prayer to the "real world" every time you sit down to plan your math lessons, you and your students will never have enough real world, never feel you have enough connection to jobs and solar panels and trains leaving Chicago and things made of stuff.

If you instead say a prayer to the electric sensation of being *puzzled* and the catharsis that comes from being *unpuzzled*, you will never get enough of being puzzled and unpuzzled.

The first prayer limits me. The first prayer means my students will only be interested in something like The Slow Forty – a real world application of systems. The second prayer means my students will be interested in The Slow Forty (because it's puzzling) but *also* the puzzling moments that arise when we throw numbers, symbols, and shapes against each other in interesting ways.

The second prayer expands me. Interested people grow more interested. Silvia writes, "Interest is self-propelling. It motivates people to learn thereby giving them the knowledge needed to be interested" (2008, p. 59). Once you give your students the experience of becoming puzzled and unpuzzled by numbers, shapes, and variables, they're more likely to be puzzled by numbers, shapes, and variables later. That's fortunate! Because some territories in mathematics are populated *exclusively* by numbers, shapes, and variables, in which cases your first prayer will be in vain.

That's why I can't tell you what to teach on Monday. Your classroom culture will beat any curriculum I can recommend. I need to know what you and your students worship first.

**BTW**

- Review assignment: Which prescriptions from our earlier review of curiosity research are evident in Class #1 above?
- Michael Pershan draws a similar distinction between pedagogy and curriculum.

**References**

Silvia, PJ. (2008). Interest — the curious emotion. *Current Directions in Psychological Science*, 17(1), 57–60. doi:10.1111/j.1467-8721.2008.00548.x

- The Cannon Man's graph is piecewise quadratic and linear.
- The Bumper Car's graph is piecewise linear, which has thrown a bunch of students.
- The Ferris Wheel's graph is sinusoidal.

But we found ourselves wondering if there were other rides and other graphs and other great ideas we had missed. So we're kicking this out to you in this week's Curriculum Confab:

What would be a worthwhile ride to include in Function Carnival? What would you graph? Why is it important?

I'll post some great responses shortly.

**Previously**:

In the last confab, we looked at a math problem inspired by Waukee Community School District's decision to let their buses idle all night. Molly showed us how to make a good problem out of it, and a lousy problem also. Great confabbing, people.

]]>Fearing our buses wouldn't start due to cold, our district let them idle overnight. The first student question this morning: "How much did that cost?"

- That's kind of amazing.
- There's a local, personally relevant, real-world math problem somewhere in there for students to work on and learn from. But one of my theses with fake-world math is that
*relevance and the "real world" aren't necessary or sufficient*. They don't guarantee interest and they don't guarantee learning.

So tell me about an *effective* treatment of this situation in math class. (Draw on research on curiosity, abstraction, and the CCSS modeling framework if they're helpful.) Also tell me about an *ineffective* treatment of this situation in math class.

**BTW**. "Curriculum Confab" will be a recurring feature around here, similar to our early "What Can You Do With This?" days only with more design and theory attached.

**2014 Feb 02**. Molly helps out enormously with this confab:

Ineffective: If gas costs 3.38 per gallon, and the bus burns 1.1 gallons per hour idling, what is the cost of the fuel burned by 32 buses over a period of 13 hours?

Effective: 1. What questions do we need to ask in order to answer this question?

The first treatment offers no "information gap" of the kind that's generative of student curiosity. Moreover, curious or incurious, the first treatment doesn't have students *doing* modeling of the sort promoted by the CCSS, where students set themselves to " identifying variables in the situation and selecting those that represent essential features."

I'd only add one question to Molly's effective treatment: "How much would you *guess* it cost the district to keep the buses idling overnight?"