First, there is irony to be found in SRI's reporting of *usage* rather than *efficacy*. The Gates Foundation underwrote the SRI report and while Gates endorses value-added models of quality for *teachers* it doesn't extend the same scrutiny towards its portfolio company here. After reading SRI's report, though, I'm convinced this exploratory study was the *right* study to run. SRI found enormous variation in Khan Academy use across the nine sites. We gain a great deal of insight through their study of that variation and we'd be much poorer had they chosen to study one model exclusively.

SRI found some results that are favorable to the work of Khan Academy. Other results are unfavorable and other results seem to contradict each other. You can find many of the favorable results summarized at Khan Academy's blog. I intend to summarize, instead, the concerns and questions the SRI report raises.

**It isn't clear which students benefit from Khan Academy.**

Over the two years of the study, 74% of teachers (63 teachers in SY 2011-12 and 60 teachers in SY 2012-13) said Khan Academy was "very effective" at meeting the learning needs of "students whose academic work is ahead of most students their age." Meanwhile, only 25% of teachers gave Khan Academy the same rating for students who are *behind* most students their age.

One teacher reports that "the same students who struggled in her classroom before the introduction of Khan Academy also struggled to make progress in Khan Academy." She continues to state that those students "were less engaged and less productive with their time on Khan Academy [than their peers]."

Participating teachers don't seem to have a great deal of hope that Khan Academy can close an achievement gap directly, though they seem to think it enhances the learning opportunities of advanced learners.

But that hypothesis is contradicted by the surveys from Site 1, a site which SRI states "had some of the highest test scores in the state [of California], even when compared with other advantaged districts." In question after question regarding Khan Academy's impact on student learning, Site 1 teachers issued a lower rating than the other less-advantaged sites in the study. For example, 21% of Site 1 teachers reported that Khan Academy had "no impact" on "students' learning and understanding of the material." 0% of the teachers from the less-advantaged sites shared that rating.

SRI writes: “Whatever the reason, teachers in sites other than Site 1 clearly found greater value in their use of Khan Academy to support their overall instruction.” SRI is strangely incurious about that reason. Until further revelation there, we should file this report alongside notices of Udacity's struggles in serving the needs of lower-achieving students in their pilot course with San Jose State University in 2013. Their struggles likely relate.

**Khan Academy use is negatively associated with math interest.**

I'm going to jump quickly to clarify that a) Khan Academy use was *positively* associated with anxiety reduction, self-concept, and self-efficacy, b) all of these non-achievement measures are measures of correlation, not causation, and c) the negative association with interest isn't statistically significant.

But I'm calling out this statistically-insignificant, non-causal negative association between Khan Academy and *interest in math* because that measure matters enormously to me (as someone who has a lot of interest in math) and its direction downward should concern us all. It's very possible to get very good at something while simultaneously wishing to have nothing to do with that thing ever again. We need to protect against that possibility.

**Teachers don't use the videos.**

While Khan Academy's videos get lots of views *outside* of formal school environments, "more than half the teachers in SY 2011-12 and nearly three-quarters in SY 2012-13 reported on the survey that they rarely or never used Khan Academy videos to support their instruction."

One teacher explains: "Kids like to get the interaction with me. Sal is great at explaining things, but you can’t stop and ask questions, which is something these kids thrive on."

Khan Academy seems to understand this and has recently tried to shift focus from its videos to its exercises. In a recent interview with EdSurge, Sal Khan explains this shift as a return to roots. "The original platform was a focus on interactive exercises," he says, "and the videos were a complement to that."

Elizabeth Slavitt, Khan Academy's math content lead, shifts focus in a similar direction. "For us, our goal isn’t necessarily that Khan introduces new concepts to students. We want to give practice."

Khan Academy is shifting its goal posts here, but we should all welcome that shift. In his speech for TED and his interview with *60 Minutes* and my own experiences working with their implementation team, Khan Academy's expressed intent was for students to learn new concepts by watching the video lectures first. Only 10% of the teachers in SY 2012-13 agreed and said that "Khan Academy played a role in introducing new concepts." Khan Academy seems to have received this signal and has aligned their rhetoric to reflect reality.

**The exercises are Khan Academy's core classroom feature, but teachers don't check to see how well students perform them.**

73% of teachers in SY 2012-13 said "Khan Academy played its greatest role by providing students with practice opportunities." Over both years of the study, SRI found that 85% of all the time students spent on Khan Academy was spent on exercises.

Given this endorsement of exercises, SRI's strangest finding is that 59% of SY 2012-13 teachers checked Khan Academy *reports* on those exercises "once a month or less or not at all." If teachers find the exercises valuable but don't check to see how well students are *performing* them, what's their value? Students have a word for work their teachers assign and don't check. Are Khan Academy's exercises more than busywork?

SRI quotes one teacher who says the exercises are valuable as a self-assessment tool for students. Another teacher cites the immediate feedback students receive from the exercises as the "most important benefit of using Khan Academy." But at Site 2, SRI found "the teachers did not use the Khan Academy reports to monitor progress," electing instead to use their own assessments of student achievement.

SRI's report is remarkably incurious about this difference between the value teachers perceive of a) the exercises and b) the reports on the exercises, leaving me to speculate:

Students are working on individualized material, exercises that aren't above their level of expertise. They find out immediately how well they're doing so they get stuck less often on those exercises. This makes for less challenging classroom management for teachers. That's valuable. But in the same way that teachers prefer their own lectures to Khan's videos, they prefer their own assessments to Khan's reports.

One hypothesis here is that teachers are simply clinging to their tenured positions, refusing to yield way to the obvious superiority of computers. My alternative hypothesis is that teachers simply know better, that computers aren't a natural medium for lots of math, that teacher lectures and assessments have lots of advantages over Khan Academy's lectures and assessments. In particular, *handwritten student work* reveals much about student learning that Khan Academy's structured inputs and colored boxes conceal.

My hypothesis that teachers don't trust Khan Academy's assessment of student mastery is, of course, extremely easy to test. Just ask all the participating teachers something like, "When Khan Academy indicates a student has attained mastery on a given concept, how does your assessment of the student's mastery typically compare?"

Which it turns out SRI already did.

Unfortunately, SRI didn't report those results. At the time of this posting SRI hasn't returned my request for comment.

**Conclusion**

It isn't surprising to me that teachers would prefer their own lectures to Khan Academy's. Their lectures can be more conversational, more timely, and better tailored to their students' specific questions. I'm happy those videos exist for the sake of students who lack access to capable math teachers but that doesn't describe the majority of students in formal school environments.

I'm relieved, then, to read Elizabeth Slavitt's claim that Khan Academy doesn't intend any longer for its video lectures to *introduce* new concepts to students. Slavitt's statement dials down my anxiety about Khan Academy considerably.

SRI minimizes Khan Academy's maximal claims to a "world-class education," but Khan Academy clearly has a lot of potential as self-paced math practice software. It's troubling that so many teachers don't bother to check that software's results, but Khan Academy is well-resourced and lately they've expanded their pool of collaborators to include more math teachers, along with the Illustrative Mathematics team. Some of the resulting Common Core exercises are quite effective and I expect more fruits from that partnership in the future.

But math practice software is a crowded field and, for totally subjective reasons, not one that interests me all that much. I wish Khan Academy well but going forward I suspect I'll have as much to say about them as I do about Cognitive Tutor, TenMarks, ALEKS, ST Math, and others, which is to say, not all that much.

**BTW**. Read other smart takes on SRI's report:

————————–

Here are some closing words about "real world" math, mostly distilled from your comments on the last post. As with previous investigations, I am indebted to the folks who stop by this blog to comment and make me smarter.

**Real-World Math Is Hard To Define**

What other conclusion can we draw from the dozen-or-so definitions of "real-world math" I found here and on Twitter?

- It depends on whether we're talking about the world of procedures, concepts, or applications. [Karim Ani]
- A problem simulating a project / job / task being performed by someone performing their normal job duties, such as the example of the contractor building a pool to meet municipal code. [Ben Rimes]
- A problem involving objects or tasks that would be considered an experience students are likely to have in the "real world." [Ben Rimes]
- If it came from a math teacher it’s basically not real-world, unless it was a math teacher doing something in the outside world leading to an interesting problem in mathematics. [Bowen Kerins]
- An observation / question that would be interesting to humans outside a math class. [Kate Nowak]
- Something is “real” to a student if it’s concrete, attainable, comprehensible. [Michael Pershan]
- Something is “real” to a student if it has a non-mathematical purpose. [Michael Pershan]

**Real-World Math Doesn't Guarantee Interest**

David Taub argued the whole question confused "interest" with "real world." M Ruppel listed other criteria for judging the value of a question, one of which was "kids want to solve it."

Their arguments may seem obvious to you. They aren't obvious to the three people who emailed me here or these presenters at the California STEM Symposium or Conrad Wolfram or the New York Times editorial board.

As Liz said, "Real world is just a means to an end. The goal is interest." We should reject simple explanations of student interest.

**So Which Version Is More Interesting?**

Asking which of these three versions (candy, geometry, text) of the same math problem is more "real-world" was a pointless task since basically everyone has a different definition of "real-world" and it doesn't guarantee interest *anyway*

So let's ask about interest instead.

I used Mechanical Turk (and Evan Weinberg's invaluable Internet skills) to show a random version of the problem to 99 people. I asked them if they had a question or not.

Of the three treatments, only the *geometry* treatment was statistically better than a coin flip at generating questions. (Here's the experiment and the data.)

Then I showed another 80 people the same three treatments and asked how interested they were in the equal area question as measured on a Likert scale from -3 to 3, including 0. (This measured interest another way. Perhaps a question didn't occur to you impulsively but *once you heard it* you were interested in it.)

Here again only the geometry treatment had an interest rating that was significantly different from "neutral." (Experiment and data.)

Why the geometry treatment? I don't know. It's more abstract than the candy treatment, which features objects from outside the math classroom. 88% of the people I surveyed in the first experiment answered "within the last year" to the question "When did you last use math to solve a problem in life, work, or school?" That's a math-friendly crowd. It's possible that a class of elementary schoolers would find the candy treatment more interesting and that a coffee klatch of research mathematicians would tend towards the *text* treatment.

I don't know. I'm just speculating here that *real world* is a pretty porous category. And for the sake of *interesting* your students in mathematics, it's more important to know *their world.*

**2014 Mar 26**. Fawn Nguyen asked her eighth grade geometry students which version they preferred.

Are hexagons less real-world to an eighth-grader than health insurance, for example? Certainly most eighth graders have spent more time thinking about hexagons than they have about health insurance. On the other hand, you're more likely to encounter health insurance *outside* the walls of a classroom than inside them. Does that make health insurance more real?

I don't know of anyone more qualified to answer these questions than our colleagues at Mathalicious who produce "real-world lessons" that are loved by educators I love.

I'm sure they can help me here. Here are three versions of the same question.

**Version A**

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.*

**Version B**

*When do the circle and the square have equal area?*

**Version C**

*Where do the circle and square have the same number of candies?*

**Version D**

[suggested by commenter **Jeff P**]

You and your friend will get candy but only if you find the spot where there’s the same number of candies in the square and the circle. Where should you cut the line?

**Version E**

[suggested by commenter **Mr. Ixta**]

Imagine you were a contractor and your building swimming pools for a hotel. Given that you only have a certain amount of area to work with, your client has asked you to build one square and one round swimming pool and in order to make the pools as large as possible (without violating certain municipal codes or whatever), you need to determine how these two swimming pools can have the same area.

**Version F**

[suggested by commenter **Emily**]

Farmer John has AB length of fencing and wants to create two pens for his animals, but is unsure if he wants to make them circular or square. To test relative dimensions (and have his farmhands compare the benefits of each), he cuts the fencing at point P such that the area of the circle = area of the square, and AP is the perimeter of the square pen and PB is the circumference of the circular pen.

**Dear Mathalicious**

Which of these is a "real world" math problem? Or is *none* of them a real-world math problem?

If anybody else has a strong conviction either way, you're welcome to chip in also, of course.

**Featured Mathalicians**:

- Kate Nowak says "real world" means "interesting to humans outside a math class."
- Ginny Stuckey asks if I'm trolling.
- Karim Ani is on vacay and promises a response when he gets back to business. [
**2013 Mar 22**: He delivers.] - Matt Lane asks, instead, if the task imbues people with joy for mathematics.
- Chris Lusto asks, "Is the question self-referential?"

**Featured Comments**

These are the two ideas that seem to be confused here are just "real world" and "interesting". There seems to be an inherent assumption that they are somehow related when I doubt they even should be. Sometimes they will overlap, and sometimes not, but it is a bit random and quite personal in my opinion.

I caught the maintenance staff at my high school using "real-world math."

The problem probably lies in the many ways that one could define "real world".

What qualifies as "real world"?

- A problem simulating a project/job/task being performed by someone performing their normal job duties, such as the example of the contractor building a pool to meet municipal code?
- Is a problem involving objects or tasks that would be considered an experience students are likely to have "real world"(dividing up the M&Ms to be equal size)?
- Actual evidence of math in the "real world "(video or otherwise) being applied as a part of someone's job (the maintenance staff example above from @Kevin Polke) could qualify, or perhaps the application of math to a problem that is foreseeable in that person's job duties.
I think it's best to take the more pluralistic viewpoint on this one, as it would be quite a task to attempt to statically define the exact nature of "real world" math, as there are always countless more examples lining up to disprove whatever narrow definition you choose.

Liz:

"Real world" is just a means to an end. The goal is interest.

What matters for a ‘good task’ is not whether it’s real, it’s whether

1) its meaning is clear right away

2) kids want to solve it

3) have the mathematical tools to solve it (even if not very sophisticated tools)

I’ll say again though: I don’t really care if a problem is real-world. There are so many great problems that aren’t, and so many terrible problems that are. I don’t think it carries huge added value. Everyone decides what’s “real” to them (as Jeff P said). Right now for my kid, 7-5 and 5-7 being related to one another is plenty real, even though there is no connection yet to physical objects or money or any of that.

**Featured Tweets**

@ddmeyer Intriguing matters to my students -Wonder if 'real world' imperative comes from assumption math is horrible http://t.co/WkgCG7lAmr

— Cathy Bruce (@drcathybruce) March 11, 2014

@ddmeyer a missing aspect in the "Real-world math debate? "Nothing ever becomes real 'til it is experienced." – John Keats

— Micah Hoyt (@MicahHoyt) March 11, 2014

**2014 Mar 26**. Fawn Nguyen asked her eighth grade geometry students which version they preferred.

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!