The speedometer in this video is broken.

Can you (or your students!?) fix it? Be careful: there are a couple of interesting ways to get this one wrong.

Also: what would the graph of speed v. time and position v. time look like here?

Let us know how you’re thinking about it in the comments.

**2015 Oct 17**. Updated to include the answer video and answer graph. You can also download these files at 101questions.

**The Judges**

- Malke Rosenfeld, who uses dance and choreography to explore mathematical thinking.
- George Hart, a research mathematician who also sculpts using geometry as his medium.
- Michael Serra, author of
*Discovering Geometry*, a geometry textbook infused from the front cover to the back with Michael’s love for math and art.

**Five Finalists**

Autumn, from Angela Ensminger’s class:

Theo, from Alice Hsiao’s class:

Trish Kreb’s seventh grade student:

John Grade & his daughter:

Maddie Bordelon and her math art team, “Right Up Left Down”:

[**BTW**. In an early draft of this post, I reversed the second and third prize winners. Mistakes were made. Apologies have been issued.]

**Third Prize**

Third prize, which is a medium-intensity high five delivered if we ever meet, and one copy of Weltman’s book, goes to Maddie Bordelon and her math art team, “Right Up Left Down.”

**Second Prize**

Second prize, which is sustained applause in a crowded, quiet room, and five copies of Weltman’s book, goes to Theo from Alice Hsiao’s class:

One judge wrote:

[E] completely holds my attention. The coloring choices pull me in and highlight the patterns and structure in a way that fascinates me. The long bands of white, blue and grey make a fantastic contrast to the brighter colors closer to the middle, which are also the shorter segments in the design. And, the bold outlines pull out the structure even more. I don’t know if it was intentional, but the overall effect of hand-coloring plus scanning the image made for a lovely final effect.

**First Prize**

First prize, which is 40 copies of Anna Weltman’s awesome book, goes to John Grade & his daughter.

[**2015 Oct 12**. John Grade is graciously passing his first prize down to the second prize winner.]

Our judges wrote about John Grade’s loop-de-loop:

It is very well constructed, brilliant use of color, and the number pattern chosen is pretty special.

A nice experiment to try Pi and see if a visible pattern emerges.

Congratulations, everybody.

**Honorable Mention**

I loved seeing students conjecturing mathematically about loop-de-loops, asking each other which ones converge and diverge, trying to predict the patterns they’d find in different strings of numbers. (See: Denise Gaskin’s comment for one example.)

Also, The Nerdery really sank its teeth into this assignment. This blog’s collection of programmer-types produced some great loop-de-loop visualizations:

- Josh G. used Scratch to let you manipulate every loop-de-loop of length three. (See also Scott Farrar with Geogebra; Jacob Klein with Desmos.)
- Dan Anderson used Processing to draw every loop-de-loop of length five.
- Joshua Green used PencilCode to let you draw
*non-rectangular*loop-de-loops. - Finally, Chris Lusto dazzled us with his “loop laboratory.” Great instructional design. No restrictions on the length of your loop-de-loop. Make sure you click through to screen 10.

Draw a shape on squared paper and plot a point to show its perimeter and area. Which points on the grid represent squares, rectangles, etc? Draw a shape that may be represented by the point (4, 12) or (12, 4). Find all the “impossible” points.

We could talk about adding a context here, but a change of that magnitude would prevent a precise conversation about pedagogy. It’d be like comparing tigers to penguins. We’d learn some high-level differences between quadripeds and bipeds, but comparing tigers to lions, jaguars, and cheetahs gets us down into the details. That’s where I’d like to be with this discussion.

So look at these four representations of the task. What features of the math do they reveal and conceal? What are their advantages and disadvantages?

**Paper & Pencil**

You’ve met.

**Dan Anderson’s Processing Animation**

Hit run on this sketch and watch random rectangles graph themselves.

**Scott Farrar’s Geogebra Applet**

Students click and drag the corner of a rectangle in this applet and the corresponding point traces on the screen.

**Desmos’ Activity**

277 people on Twitter responded to my prompt:

Draw three rectangles on paper or imagine them. Choose at least one that you think that no one else will think of. Drag one point onto the graph for each rectangle so that the x-coordinate represents its perimeter and the y-coordinate represents its area.

Resulting in this activity on the overlay:

Again: what features of the math do they reveal and conceal? What are their advantages and disadvantages?

]]>**New Blog Subscriptions**

**David Sladkey**created a slick lesson in Desmos asking students to make their way around a maze using functions. Looking forward to more of that.**Bill Hinkley**taught with Zan Armstrong, one of Desmos’ crackerjack new hires. He seems to post once every couple of months with insightful, illustrated dispatches from his classroom.**Kelly Zinck**and**Erick Lee**co-author a blog (+1 fun). They’re a couple of math consultant co-workers (+1 fun) out of Halifax which is in*Canada*(+1 fun) who share both wacky and well-worn activities for math students (+1 fun). That’s four fun points. Easy call.- I don’t know who writes the blog Algeotrigcal but they had me at “Algeotrigcal.”
**Ruth Eichholtz**blogs thoughtfully about math education and technology. She has a particular post I*must*share with you later. Loved it. Best use of Classkick I’ve encountered so far. Wouldn’t dream of spoiling it.**Matthew Oldridge**tangles with the same crowd of traditionalists on Twitter that I tangle with only he does it with more patience and good humor. I was surprised I hadn’t already subscribed, frankly.**Selling B2E**. Gotta take care of business now.

**New Twitter Follows**

**Dylan Wiliam**. Perhaps you have heard of him.

**Commenters I Wish Had A Blog / Twitter Account / Zine / Etc.**

**Bret Giliand**, who gave me a good razzing last month. Somebody hook him up!

You should buy Anna Weltman’s new math book, *This is not a Math Book*.

You should buy several, probably, for all the little people in your life who are deciding *right now* what they think about math and what math thinks about them. If they’re taking their cues on that decision from someone who dislikes math or who dislikes little people, consider using *This is not a Math Book* for counterweight.

You’ll find dozens of pages of math art, math sketches, math reasoning, and math whimsy. I read it in one sitting outside a coffee shop one afternoon, big dumb smile on my face the whole time. Actually *finishing* the book, fully participating in Weltman’s assignments of creativity and invention, will take many more afternoons.

I’d like to send one of you a class set of Weltman’s book. Here is how you get it:

- I love Weltman’s Loop-de-Loop assignment. It lends itself to some of my favorite mental mathematical acts around prediction, sequencing, transformation, and questions like “what if?” So you or your students or all of the above should make an awesome Loop-de-Loop. (Here is Weltman’s instruction page and her student work page, but any piece of graph paper will work.)
- Scan and send it to ddmeyer+loop@gmail.com.
- I’ll pick my five favorites and ask some of my favorite math artist friends to pick the winner from those five. Winner takes all, which is to say 40 copies of
*This is not a Math Book*, from me to you. - Contest ends 10/6 at 11:59 PM Pacific Time.

Drawings, color, character work, mixed media, it’s all fair game. I can’t wait.

**BTW**. Over the next several days, Weltman is blogging interesting questions to ask your students about Loop-de-Loops.

**Featured Tweets**

@ddmeyer can this be outsourced to a class and we send in our favorites?

— Jonathan (@rawrdimus) September 29, 2015

Yes, do that!

Students, we need to do this! See me if you're interested! (I already have the book — you need it in your life!) https://t.co/wOucB0Iq51

— Holly Werra (@MrsWerra) September 29, 2015

Yes, do that!

@ddmeyer @tchmathculture @AnnaWeltman How much do I love that a class set is 40 books? Thanks for the nod to the reality some of us face. :)

— Becca Phillips (@RPhillipsMath) September 29, 2015

Got your back.

]]>I spent a year working on Dandy Candies with around 1,000 educators.

In my workshops, once I stop learning through a particular problem, either learning about mathematics or mathematics education, I move on to a new problem. In my year with Dandy Candies, there was one question that none of us solved, even in a crowd that included mathematics professors and Presidential teaching awardees. So now I’ll put that question to you.

Here is the setup.

At the start of the task, I ask teachers to eyeball the following four packages. I ask them to decide which package uses the least packaging.

With the problem in this state, without dimensions or other information, lots of questions are available to us that numbers and dimensions will eventually foreclose. Teachers estimate and predict. They wonder how many unit cubes are contained in the packages. They wonder about descriptors like “least” and “packaging.”

After those questions have their moment, I tell the teachers there are 24 unit cubes inside each package. Eventually, teachers calculate that package B has the least surface area, with dimensions 6 x 2 x 2.

We then extend the problem. Is there an even *better* way to package 24 unit cubes in a rectangular solid than the four I have selected? It turns out there is: 4 x 3 x 2. When pressed to justify why this package is best and none better will be found, many math teachers claim that this package is the “closest to a cube” we can form given integer factors of 24.

**The Problem We Never Solved**

We then generalize the problem further to *any* number of candies. I tell them that as the CEO of Dandy Candies (DANdy Candies … get it?!) I want to take *any* number of candies – 15, 19, 100, 120, 1,000,000, whatever – and use an easy, efficient algorithm to determine the package that uses the least materials.

Two solutions we reject fairly quickly:

- Take your number.
- Write down all the sets of dimensions that multiply to that number.
- Calculate the packaging for that set of dimensions.
- Write down the set that uses the least packaging.

And:

- Take your number.
- Have a computer do the previous work.

I need a rule of thumb. A series of *steps* that are intuitive and quick and that reveal the best package. And we never found one.

Here was an early suggestion.

- Take your number.
- Write down all of its prime factors from least to greatest.
- If there are three or fewer prime factors, your dimensions are pretty easy to figure out.
- If there are four or more factors, replace the two smallest factors with their product.
- Repeat step four until you have just three factors.

This returned 4 x 3 x 2 for 24 unit candies, which is correct. It returned 4 x 5 x 5 for 100 unit candies, which is also correct. For 1,000 unit candies, though, 10 x 10 x 10 is clearly the most cube-like, but this algorithm returned 5 x 8 x 25.

One might think this was pretty dispiriting for workshop attendees. In point of fact it connected all of these attendees to each other across time and location and it connected them to the mathematical practice of “constructing viable arguments” (as the CCSS calls it) and “hypothesis wrecking” (as David Cox calls it).

These teachers would test their algorithms using known information (like 24, 100, and 1,000 above) and once they felt confident, they’d submit their algorithm to the group for critique. The group would critique the algorithm, as would I, and invariably, one algorithm would resist all of our criticism.

That group would *name* their algorithm (eg. “The Snowball Method” above, soon replaced by “The Rainbow Method”) and I’d take down the email address of one of the group’s members. Then I’d ask the attendees in *every other workshop* to critique that algorithm.

Once someone successfully critiqued the algorithm – and *every single algorithm has been successfully critiqued* – we emailed the author and alerted her. Subject line: RIP Your Algorithm.

So now I invite the readers of this blog to do what I and all the teachers I met last year couldn’t do. Write an algorithm and show us how it would work on 24 or another number. Then check out other people’s algorithms and try to wreck them.

**Featured Comment**

Big ups to Addison for proposing an algorithm and then, several comments later, wrecking it.

**2015 Sep 25**

We’re all witnessing incredible invention in this thread. To help you test the algorithm you’re about to propose, let me summarize the different counterexamples to different rules found so far.

20 should return 5 x 2 x 2

26 should return 13 x 2 x 1

28 should return 2 x 2 x 7

68 should return 2 x 2 x 17

222 should return 37 x 3 x 2

544 should return 4 x 8 x 17

720 should return 8 x 9 x 10

747 should return 3 x 3 x 83

16,807 should give 49 x 49 x 7

54,432 should return 36 x 36 x 42

74,634 should give 6 x 7 x 1777

Explicit instruction (teaching by telling?) is appropriate, even necessary, when the knowledge is based in a social convention. Then I feel that I need to “cover” the curriculum. We celebrate Thanksgiving on a Thursday, and that knowledge isn’t something a person would have access to through reasoning without external input―from another person or a media source. There’s no logic in the knowledge. But when we want students to develop understanding of mathematical relationships, then I feel I need to “uncover” the curriculum.

Chester Draws responds and asks a question which I’ll extend to anyone who takes a similar view of direct instruction:

You expect student to “uncover” calculus?

Do constructivist teachers quietly just directly instruct such topics? Do they teach them, but pretend the students found them out for themselves? I can’t even begin to imagine how I could teach the derivative through constructivist techniques.

Brian Lawler responds that, yes, it’s possible to teach calculus without direct instruction, and offers up his Interactive Mathematics Program Year 3 unit “Small World” as evidence. Pulling out my copy of IMP, though, I find pages in the Small World unit that directly instruct students in the calculation of slope, the calculation of average rate of change, and the definition of the derivative. This appears to answer Chester’s question, “Do constructivist teachers quietly just directly instruct for such topics?”

So I hope Marilyn and anybody else with similar ideas about direct instruction will take up Chester’s question with force. It’s an important one, and mandates like “uncover the curriculum” seem more descriptive of philosophy than practice.

It’s worth pointing out in closing that this direct instruction in IMP is preceded in each case by activities through which students develop informal and intuitive understandings of the formal ideas. This is in the neighborhood of pedagogy endorsed by *How People Learn*, which, again, you should all read. It just isn’t the example of direct instruction-less calculus Lawler seems to think it is.

**BTW**. Clarifying, because I’m frequently misinterpreted: I don’t think learning calculus without direct instruction is logistically possible over anything close to a school year, or that it’s *philosophically desirable* even if it were possible.

**BTW**. Elizabeth Statmore offers an excellent summary of the pedagogical recommendations in *How People Learn*.

**BTW**. Chester references “constructivist teachers.” Anybody who sniffs back at him that “constructivism is *actually* a theory of learning, not teaching” gets week-old sushi in their mailbox from me. I think his meaning is clear.

**Featured Comment**

I consider CPM pretty strongly constructivist, and I am currently LOVING that calculus program. My kids are generating much better insights and dialogues around calculus than I have seen with other programs and what I am seeing suggests much better expected results on the AP exam. Don’t know if it is sufficiently pure to pass the “no instruction evar!” test that you seem to be using.

Also, it is probably worth noting that the VAST majority of AP workshops are centered around making instruction more constructivist, because the typical textbook presentation is so MASSIVELY DI. That makes this whole conversation feel like it is taking place in some bizarro universe. The question I tend to find myself asking is this:

“You expect students to understand calculus when taught using only DI?

Do traditionalist instructors just quietly slip in investigations, but pretend the students figured things out from their amazing lectures? I can’t even imagine how I would teach derivatives without heavy exploration of finite differences, secant and tangent lines, and distance/time vs velocity/time graphs.”

]]>No matter what anyone says, “uncovering curriculum” is just discovery learning & concept formation in a different cloak. I started teaching back in the 1970’s when this method was in full bloom. Once we’d set the stage for ‘aha moments’ of understanding to occur, we let the struggle ensue. We were limited to asking a learner nudging questions, redirecting his/her efforts in a more fruitful direction when the chosen path was a dead end, and simplifying the problem so he/she’d trip over the gem. As one after another student ‘got it’, we imagined we could hear a series of little light bulbs popping on over their heads until the light of understanding in the room was blinding!

The problem was that it’s impossible to be at every student’s side to ask just the right question at just the right moment all at the same time in a class of 25 or 30. First discoverers would end up telling their more baffled peers the secrets (hidden direct instruction). Some of the really lost got direct instruction from older siblings or parents.

The awful thing was that students who didn’t get it couldn’t turn to the teacher for help, because they’d only get more questions & more ‘lead up’ to the place where the leap of understanding had to be made. Direct questions were not to be met with direct answers. The teacher didn’t believe in telling.

And that’s the sort of dishonest thing about this method. The teacher has all the secrets. Everyone knows this is the case. The students’s job becomes finding and digging up treasure. For a some this is an act of learning. For many it’s like being in an elaborate guessing game with the prize of enlightenment denied those who are not capable players. The teacher had all the secrets but never tells.

But what many teachers don’t get is that taking students through a process of discovering Uncovering those secrets is not the same as constructing personal understanding. And they also don’t factor into the learning experience the fact that direct instruction is everywhere. If you won’t share the secrets & homework has to be done, kids will ‘uncover’ up what they need online. Kids can circumvent your process with the tap of a finger if it will get them what they (or their parents) believe they need.

Although deepening understanding is the goal, getting a toehold on being able to do some stuff can be a great place to start. How many of us who drive have a deep understanding of the physics, chemistry, and engineering that makes up our vehicles? Yet we can use them skillfully to solve problems. The same goes for cooking. If you can follow a recipe you can feed your family — which is a pretty fantastic result achieved without understanding how & why the recipe works. There’s nothing wrong with passing on knowledge and then getting on with helping kids learn how to apply it confidently and in a broad range of circumstances.

As teachers, it’s our job to pack our tool boxes with as broad a range of strategies as possible so we can help each kid forge the connection of skill development & growth of understanding. I urge colleagues not to become so enamored with one approach that they become ‘one trick ponies’. I fear it will not serve you or your kids well in the long run.

In an influential book for the National Academies Press in the US [

How People Learn], the constructivist position is explained in terms of the children’s bookFish is Fish. In the story, a frog visits the land, and then returns to the water to explain to his fish friend what the land is like. You can see the thought bubbles emanating from the fish as the frog talks. When the frog describes birds, the fish imagines fish with wings, and so on.The implication is that we cannot understand anything that we have not seen for ourselves; each individual has to discover the world anew.

Meanwhile, here is how the cognitive scientists who wrote *How People Learn* *actually* interpret *Fish is Fish* (pp. 10-11):

Fish Is Fishis relevant not only for young children, but for learners of all ages. For example, college students often have developed beliefs about physical and biological phenomena that fit their experiences but do not fit scientific accounts of these phenomena.These preconceptions must be addressed in order for them to change their beliefs(e.g., Confrey, 1990; Mestre, 1994; Minstrell, 1989; Redish, 1996).

To illustrate this phenomenon we need only look at Ashman’s essay itself. Ashman came to his essay with the common misconception that constructivists believe that “each individual has to discover the world anew.” Even though the *How People Learn* authors interpret *Fish is Fish* explicitly, that explicit interpretation wasn’t enough to dismantle Ashman’s misconception.

Perhaps the *HPL* authors should have taken their own advice, anticipated Ashman’s misconception, and addressed it explicitly. It turns out they did exactly that in the next paragraph:

A common misconception regarding “constructivist” theories of knowing (that existing knowledge is used to build new knowledge) is that teachers should never tell students anything directly but, instead, should always allow them to construct knowledge for themselves.

A book is nothing if not a medium for explicit instruction and Ashman illustrates the limits of that medium here. Explicit instruction is powerful, certainly, and I can’t think of any influential scholars, least of all the authors of *How People Learn*, who would deny it. But it often isn’t powerful enough on its own to remedy a student’s existing misconceptions. Luckily, *How People Learn* offers many more powerful prescriptions for teaching and it’s free.

**BTW**. Always relevant: The Two Lies of Teaching According to Tom Sallee.

**New Blog Subscriptions**

**Patty Stephens**is an instructional leader in Washington state. I’m hoping she writes more about her Teacher Fellows program, which attempts to build teaching capacity throughout the state. (Ditto Bryan Meyer about his Teacher Partnership Program, while I’m here.)**Bridget Dunbar**has been blogging and tweeting for years but left her first comment on my blog last week, pointing me to her exceptional post comparing the pros and cons of three representations of the same problem.**Ryan Muller**is a software developer who writes about education research at his Learnstream blog. He seems curiously unaffected by education research’s typical turf wars, just happy to read and write about what he reads. Refreshing.**Timothy McEvoy**writes thoughtfully and critically about math edtech, a genre of writing that is in short supply.**Tom Bennison**runs the #mathsjournalclub chat on Twitter – a discussion group for math education research, which is the kind of social unit I’m already missing from grad school.**Julie Reulbach**offers us all a daily photo and caption from her innovative algebra classes. Yes, please.**Kris Boulton**applies our headache metaphor to a question about slope. Watch how his subtle alterations to the same task make its mental controversy more acute. More like this, please.**John A. Pelesko**and**Michelle Cirillo**are a university-level mathematician and math education researcher, respectively, and have paired up for a blog dedicated*exclusively to mathematical modeling*! Their post “Is this mathematical modeling?”*gets it*.**Sam Shah’s**new group blog around good questioning strategies had me at “Sam Shah’s new group blog.”**Nathan Kontny**writes breezy narratives about entrepreneurship, at least one of which (on audience) is still rattling around my head one month later.

There are lots of good reasons to ask students for multiple representations of relationships. But I worry that a consistent regiment of turning tables into equations into graphs and back and forth can conceal the fact that each one of these representations were invented for a purpose. Graphs serve a purpose that tables do not. And the equation serves a purpose that stymies the graph.

By asking for all three representations time after time, my students may have gained a certain conceptual fluency promised us by researchers like Brenner et al. But I’m not sure that knowledge was ever effectively *conditionalized*. I’m not sure those students knew when they could pick up one of those representations and leave the others on the table, except when the problem told them.

Otherwise, it’s possible they thought each problem required each of them.

The same goes for representations of one-dimensional data. We can take the same set of numbers and represent its mean, median, minimum, maximum, deviation, bar graph, column graph, histogram, pie chart, etc.

So here is the exercise. Take one representation. Now take another. Why did we invent that other representation? Now how do you put your students in a place to experience the *limitations* of the first representation such that the other one seems *necessary*, like aspirin to a headache?

**Featured Comment**

Ok. First is bar chart, second is box plot.

All situations in statistics require some data, and the best data is that which students compile themselves. For this comparison a single set of data is best presented as a bar chart, but compare the data from five or more distinct groups of subjects, same measure, and the multiple strip bar chat is a bloody mess. Five box plots above the same numberline, and so much more is revealed, at a small cost of loss of detail.

I used to think that box plots were a waste of time until I saw the above usage.

The same is true of physical representations. I am thinking of many algebra growth problems that involve squares and growing patterns. It is valuable to ask students to go through the actions of adding squares to watch a pattern grow through the addition of tiles. This action can help them have the physical experience of a rate of change. But this representation also has its drawbacks. It is clearly cumbersome and not efficient.

I think of them all as connected to making predictions about data – certain representations lend themselves to different ways in which data is presented, and certain representations help make predictions about that data.

Tables are great when you need to generate data from a scenario – you have a situation that has been given to you and you need a place to start. Creating a table for some initial data helps you see the patterns in whats happening and helps you make littler predictions about where the data is going. If I want students to appreciate tables, I give them a visual pattern or a scenario problem with a starting condition and a rate of change, then ask them some questions about what will happen.

Graphs are great when you’re given several random data points that, even when arranged as a table, don’t indicate a clear pattern. Sometimes plotting these visually helps you predict what other points could be missing, or what other points exist as the pattern continues. This is especially true for situations whose solutions depend on two variables, such as only having 30 dollars to spend on item A that costs 2 dollars and item B that costs 1 dollar. When I want students to appreciate graphs, I give them one of these situations (which usually lends itself to standard form of an equation, but they don’t know that) or I give them several data points and ask them what’s missing in the pattern. This is easier to see when organized visually and you you a particular shape to your points rather than a random collection.

Equations are the most efficient way to make predictions about patterns – if you’re given an equation, there’s no reason to have any other representation. Equations are useful for predicting far into the future for your data – maybe you can figure out the first few terms of your pattern, but trying to generate the 100th term is inefficient. Using an equation is like being omnipotent with a set of data. When I want students to appreciate equations, I give them a scenario but ask for a data point in the absurd future where the table or graph necessary to find the point would be too large and unwieldy to use.

The order I’ve presented these in this comment is also my typical order for presenting these representations to students: tables are useful at the beginning to generate data; graphs are useful once you have lots of it that may or may not be organized and may be missing some points, and equations are good for predicting the future.

A curious consequence might be: it’s not particular situations that necessitate one representation versus the other; rather, its what data you choose to give them at the beginning and what you ask them to do with it that makes one representation more valuable than another.

Jodi:

So very true. This skill seems to be neglected in our classrooms. Computers can take one representation and switch to others over and over again, much faster than humans can. If switching back and forth is your only skill, I can easily replace you with a $100 calculator from Target. And the calculator will be faster and more accurate.

But if I’m training students to be problem solvers who are smarter than computers, the “which representation is needed here” is a much more important question. I’m not aware of a computer that can answer that question.

Draw a simple line on a graph.

Now what is the value at x = 1.37?

Now they see that the equation is quicker and more accurate than the graph — even when inside the graphed region.

Or draw two lines that do not meet at integer values. Where do they meet, exactly? Hence that simultaneous equations are better in some situations than graphs.

But again, we can draw y = log x crossing y = x

^{2}quicker on our graphics calculator than we can solve it.(Of course y = x

^{2}doesn’t cross y = log x, but they only know that if they graph it!)

**BTW**: Essential reading from Bridget Dunbar also: Effective v.Efficient.