This is also why I think it’s a mistake to place children in charge of the speed of their learning, particularly during the early years of their education. If left to decide for themselves, many kids — and particularly those from at-risk backgrounds — will choose a relatively slow velocity of learning (again, because thinking is hard). The slow pace will lead to large knowledge deficits compared to their peers, which will cause them to slow down further, until eventually they “switch off” from school. The only way to prevent this slow downward spiral for these students is to push them harder and faster. But they need to be pushed, which means we should not cede to them control of the pace of their learning.

My own argument against personalized learning is that – in Audrey Watters’ fine formulation – it “circumscribes pedagogical possibilities.” Which is to say, a lot of fun learning in math class – argument, discussion, and debate chief among them – is ~~impossible~~ very difficult when you aren’t learning it synchronously with a group. Riley’s argument adds new dimensions to those concerns.

**BTW**. I left my own version of Riley’s second argument on Will Richardson’s blog, a forum where the value of student-personalized curriculum is, IMO, too often assumed to be utterly obvious and questioned only by cowards and cranks. Rather than spending his time tangling with anonymous Internet commenters, I’d like to know how a thoughtful technologist like Richardson would engage a critic like Riley.

**2014 Jun 24**. Mike Caulfield:

I often warn about overgeneralizing across disciplines but let me overgeneralize across disciplines here: if there is one thing that almost all disciplines benefit from, it’s structured discussion. It gets us out of our own head, pushes us to understand ideas better.

It teaches us to talk like geologists, or mathematicians, or philosophers; over time that leads to usStructured discussion is how we externalize thought so that we can tinker with it and refactor it.thinkinglike geologists, mathematicians, and philosophers.

**2014 Jun 25**. Alex Hernandez writes a thoughtful rebuttal.

It takes a certain amount of spatial skill to answer the question, “How high will the creamer be in the upside-down container? Will it be higher than the original? Lower? The same?” (I mean the *volume* is the same, after all.)

So I create a new PearDeck presentation and send the link out on Twitter asking just those questions. PearDeck then lets me capture the feedback of these students in realtime.

The teacher interface expands to let me know whose answers are close or not that close.

This sets me up for anything from an explanation of how to calculate solids of revolution in Calculus or a debate about covariation in Algebra.

If you’re an educational technologist and you think this is interesting, please notice that this is the *opposite* of individualized instruction. It’s socialized instruction. PearDeck would be much less interesting if you were the only person estimating, or if you were answering the question “Will it be higher or lower or the same?” alone.

Sometimes learning is less fun when you’re learning at your own pace.

(The answer.)

]]>That’s a very helpful comment from a recent workshop participant. Textbooks don’t have that same luxury.

Here’s an example. Watch how Connected Mathematics treats the classic Painted Cube problem:

Here are elements the textbook has already added:

**A central question**. (“How many faces of the little cubes have been painted?”)**A strategy**. (“Look at smaller versions of the cube.” It also tells you by omission that it’s impossible to find more than three faces painted.)**A table**. (For organizing your data.)**Table column headings**. (Edge length, total cubes, total cubes of each kind.)

If you *subtract* those elements and add them in *later*, you get to ask interesting questions and host interesting conversations with your students. Like this:

. (“What questions do you have about Leon and his cube?” And later: “Guess how many cubes don’t have any paint on them at all?”)**A central question**. (“What are all the possibilities for the number of faces that could have paint on them? Could five faces have paint on them? Can I tell you how mathematicians work on big problems? They look at smaller versions of the big problem. What would that look like here?”)**A strategy**. (“All of the numbers from our smaller versions are getting out of control. How can we organize all these loose numbers?”)**A table**. (“What kind of data should we look at? What about these cubes seems important enough to keep track of?”)**Table column headings**

You can always *add* those elements into the problem – the questions, the information, the mathematical structures, the strategy – as your students struggle and need help. But you can’t *subtract* them.

Once your students see the table, you can’t ask, “What tool could we use to organize ourselves?” The answer has been given. Once they see the table headings, you can’t ask, “What quantities seem important to keep track of in the table?” They know now. Once you add the strategy (“Look at smaller versions.”) their answers to the question “What strategies could we use?” won’t be as interesting.

In sum, much of the problem has been *pre-formulated*, which is a pity, seeing as how mathematicians and cognitive psychologists and education researchers agree that *formulating* the problem leads to success and interest in *solving* the problem.

So again I have to remind myself to be less helpful and be more thoughtful instead.

**BTW**. Of course I’m partial to Nicole Paris’ setup of the task:

**Great Help From The Comments**

I’m reprinting Bryan Meyer’s entire comment:

I don’t know that I have anything terribly insightful to add, but this seems like a fun conversation.

I don’t really see too much that is wrong with the problem/puzzle itself, which (to me) is something like:

I have this cube (show picture/tangible) made up of smaller cubes. If I dipped the whole thing in paint, how many of the smaller cubes would have paint on them? Is there a rule or shortcut we can create that would allow us to answer that question for ANY sized cube?

To me, the issue seems to be that the version we see in your blog post attempts to steer the direction of student thinking and leaves little room for play and divergent thinking/approaches. It “scaffolds” away all of the rich mathematical thinking and play in an attempt to cover standards. In particular, the unspoken assumption in the way it has been printed is that writing and graphing linear, quadratic, and exponential functions is the real “Math” in the task (things we can easily point to as belonging to the discipline/standards).

But, at it’s core and without all the mechanical scaffolding (as re-posed here), the question allows room for many mathematical strengths and habits of mind to be valued and sends different messages about what the real “math” might be: taking things apart and putting them back together, creating systems of organization, assigning variables, making generalizations, posing extension questions, etc. In addition, because it doesn’t dictate how to proceed, it encourages students to trust their own thinking and allows them to “see themselves” in the work that develops. The work of the teacher becomes to follow the student, looking for mathematically ripe opportunities in their work and thinking.

**2014 Jun 2**. Christopher Danielson brings his perspective to the task as writer of CMP.

Painted Cubes is a classic task, canonized right alongside the Pool Border task and Barbie Bungee, but that doesn’t mean it’s beyond help, or that everyone treats it exactly the same way.

Here’s a treatment from Connected Mathematics. What would you do with this and why would you do it? (Click for larger.)

]]>**Christina Tondevold** teaches her first three-act math task. There’s a lovely and surprising result at the end, when her students realize that with modeling the *calculated* answer doesn’t always match the *world’s* answer exactly.

After they all had taken the 24 bags x 26 in each bag, every kid in that room was so confident and proud that they had gotten the answer of 624, however … the answer is

not624! Why do you think our answers might be off?

**John Golden** creates and implements a math game around decimal addition called Burger Time with some fifth graders:

Roll five dice to get ingredients for your burger. The numbers correspond to how many mm tall each part is.

**Matthew Jones** creates a Would You Rather? activity and one of his “defiant” students makes an effective justification of a counterintuitive choice:

He blurted out “I want to paint Choice C.” I told them at the beginning that there was no right/wrong answer, they just had to justify it. I was lucky enough that he thought of the reason why you’d want to paint the larger one. The only reason you’d want to paint the largest wall is because you are paid by the hour. It was really interesting watching him come to that conclusion. And to take pride and ownership in the way that he did.

**Jennifer Wilson** describes one of NCSM’s “Great Tasks” and shows how she gathers and sorts student work with a TI-NSpire.

]]>Students are asked to create a fair method for cutting any triangular pizza into 3 equal-sized pieces of pizzas. I asked students to work alone for a few minutes before they started sharing what they were doing with others on their team. I walked around and watched.

Futurists and math educators talk past each other. If I could jump into any futurist’s head and encode any particular understanding there to make dialog easier, it would be this:

Adaptive learning is like an iPod with infinite capacity and infinite capability to play any song ever recorded or sung, provided those songs were written by Neil Diamond.

If all you’ve ever heard in your life is Neil Diamond’s music, you might think we’ve invented something quite amazing there. Your iPod contains the entire *universe* of music. If you’ve heard *any* other music at all, you might still be impressed by this infinite iPod. Neil wrote a lot of music after all, some of it good. But you’ll know we’re missing out on quite a lot also.

So it is with the futurists, many of whom have never been in a class where math was anything but watching someone lecture about a procedure and then replicating that procedure twenty times on a piece of paper. That entire universe fits neatly within a computer-adaptive model of learning.

But for math educators who have experienced math as a social process where students conjecture and argue with each other about their conjectures, where one student’s messy handwritten work offers another student a revelation about her own work, a process which by definition *can’t* be individualized or self-paced, computer-adaptive mathematics starts to seem rather limited.

Lectures and procedural fluency are an important aspect of a student’s mathematics education but they are to the universe of math experiences as Neil Diamond is to all the other amazing artists who *aren’t* Neil Diamond.

If I could somehow convince the futurists to see math the same way, I imagine our conversations would become a lot more productive.

**BTW.** While I’m here, Justin Reich wrote an extremely thoughtful series of posts on adaptive learning last month that I can’t recommend enough:

- Blended Learning, But The Data Are Useless
- Nudging, Priming, and Motivating in Blended Learning
- Computers Can Assess What Computers Do Best

**Featured Comments**:

]]>Can I offer another analogy for these technologists? Adaptive learning is like a guitar teacher who teaches you how to play harder and harder pieces of music but never teaches you how to improvise. So you can play a piece of music that is placed in front of you, but you’ll never be able to pick up a guitar and just play with a couple of friends. I would contend that the improvisor is better prepared to understand and even make music. I’ll bet Neil Diamond can pick up a guitar and jam.

What I love is that math lets you take some things you know and just by moving symbols around on a piece of paper find out something you didn’t know that’s very surprising. I have a lot of stupid questions and I love that math gives the power to answer them sometimes.

If you want to understand the Common Core’s fourth math practice standard, “Model with Mathematics,” you could do a lot worse than studying the mental feats Munroe performs in every single post of his What If? blog.

**Featured Comment**

I think the bit immediately prior is worth quoting as well, even if it’s a bit harsher:

And I love calculating these kinds of things, and it’s not that I love doing the math. I do a lot of math, but I don’t really like math for its own sake.

]]>Like on Mythbusters, Monroe is rarely content to stop with answering the question as stated: he generally keeps going bigger, faster, taller, or hotter until something explodes.

Before this last week’s OAME conference in Toronto, I’d only seen one use of Educreations: students record themselves teaching through a lesson or a problem as a kind of summative assessment. This assignment has been recommended to me in 100% of the tablets-in-education sessions I’ve attended. (Chris Hunter called the students “Khanabees,” which is clever.)

In her session at OAME, Marian Small used Educreations to show student thinking in its raw, unrehearsed form, full of loops and self-references, which for some purposes is more interesting than the polished Khanabees presentations.

The premise of her talk (PDF of her slides) was that the job of teaching comprises two very different, very difficult tasks:

- promoting student thinking through interesting questions,
- responding to that thinking in productive ways.

So her session was simple, but engrossing.

- She had students talk through and work out on an iPad interesting questions that they were seeing for the first time. (Here’s an example.) The iPad recorded their real-time sketches and markings and paired it to the their voice.
- She asked us what we’d say next and analyzed and critiqued our responses, highlighting their differences and categorizing them as either “scaffolding,” “redirecting,” “probing,” or “extending.”

An hour flew by.

This approach to representing student work has advantages and disadvantages relative to both scans of student work and videos that *show* the student and teacher. Rather than outline those differences myself, I’d rather take your thoughts in the comments.

If you’re into Talking Math With Your Kids or Math Mistakes, this approach to student work is worth investigating.

**Featured Comments**

Kate, referring to Small’s two bullet points above:

That is the most beautiful job description I’ve ever read.

I am not a teacher, but as a technologist/researcher, it strikes me we can, at least in the short run, have a lot more pedagogical insight with humans looking at a few these than machines crunching context-poor big data (even though I’ve argued the case of the latter before on this blog).

Wade Roberts, co-founder of Educreations:

Android is the next logical platform for us to support, but we don’t yet see sufficient demand to justify the cost of development. We’re monitoring Android tablet growth within schools, but iPad is still over 90% of school market. It is already possible to replay our videos on Android, however.

And:

]]>We’re incredibly excited about this use-case. People are often surprised when I tell them that just over half of the 5 million videos on our platform have been created by students.

Many, many thanks to everyone who helped me sort out some thoughts on this lesson in our previous confab post, including but not limited to Fawn Nguyen, Robert Kaplinsky, Bowen Kerins, Dan Anderson, and Max Ray, and the many participants I pestered at OAME2014 this last week.

Here’s the download link at 101questions.

**Act One**

Show this video.

Ask: “What would be a fair price for the Money Duck?”

You guys were right. In the end it makes more sense to pose the student as the seller. It’s more productive and more interesting even though its easier to empathize with the buyer initially.

**Act Two**

Ask: “What information would you need to decide on a fair price?”

Now we’re going to introduce the probability distribution model.

It’s unusual so we’re going to do several things in order:

- We’re going to ask for speculation about what it means. Then we’re going to tell them what it means. (A: It shows every possible event in the space of events along with their likelihoods.)
- We’re going to show more contrasting cases. (See: Schwartz, 2011.) Impossible cases and possible cases. We’ll ask them which are impossible and why. Then we’ll tell them which are impossible and why. (A: The probabilities have to add up to 1. Each rectangle here has to stack up and reach exactly the 1 line.)
- Now we’ll ask the students, “If you’re selling the Money Ducks for $5, why is each of these distributions bad for business.” (A: The first distribution means word gets out that you’re cheaping your customers and eventually no one will buy your ducks. The second distribution means you’re losing loads of money.)
- Show the students four distributions and ask them to make up a price for the distributions that would be fair to both buyer and seller, that wouldn’t result in money lost or gained.

After laying all this informal groundwork, we’re ready to transition from qualitative descriptions to numerical and define expected value.

- We ask them to calculate the expected value of the distributions from #4 and compare those values to their prices. If their intuitions were sound and their calculations correct, their intuition will support the validity of the expected value model.

**Act Three**

There’s no act three here. We don’t know the probability distribution of the Money Duck (I asked) so we can’t validate. That’s okay.

**Sequel**

Let’s show the students the actual price of the Money Duck and ask them to determine a probability distribution that would give them a $3 profit per Money Duck as a seller. Answers, happily, can vary.

**BTW**. The bummer-world version of this problem reads like this:

A carnival game is played as follows: You pay $2 to draw a card from an ordinary deck of 52 playing cards. If you draw an ace, you win $5. You win $3 if you draw a face card (Jack, Queen, King) and $10 if you draw the seven of spades. If you pick anything else, you lose your $2. On average, how much money can the operater expect to make per customer?

**2014 May 12**. You should definitely read Dan Anderson’s experience running this lesson with students.

**2014 May 19**. Also, Megan Schmidt had some interesting results with students particularly w/r/t the question, “Which distributions are impossible?”

My sense is this is an object with a lot of potential for a math teacher. I’d like to know how you’d harness that potential.

A particular question I’m wrestling with is whether or not to put the student in the position of:

- the buyer, asking “Is the Money Duck worth its price?”
- the seller, asking “How should I price the Money Duck?”

Over the next few days I’ll update this post with comments I’ve solicited in advance from some of my favorite curriculum designers. I’d love to add your thoughts to the pile.

Featured task designer **Fawn Nguyen**:

I’d be more interested in having students be sellers rather than buyers. Buyers’ incentives seem more qualitative: soap quality, residue, allergy, shape, smell.

Being sellers, students could do a lot with cost analysis given production constraints.

Possible task:

- Show the picture. It’d be cool to have time lapsed video of soap breaking down under a shower stream or, to not waste water, one could actually document soap usage over time.
- Ask students how much they’d be willing to buy it for.
- Decide on a central measure of their answers to use (mode or median?) or go with a small range of values. (I actually surveyed my algebra class: out of 36 kids, 13 said they’d be willing to pay between $4 and $6, while 15 were willing to pay $7 to $10 for a bar.)
- Students are now sellers of the soap. How much should they price it knowing what people are willing to pay?
- Give constraints: cost of raw material, time to make, number of workers, etc.
- If 5,000 bars were made, what is the distribution of the bills in them? What about 10
^{6}bars?

Featured task designer **Robert Kaplinsky**:

Thinking about it, I would position the student as the seller of the money duck for two main reasons. Ultimately, knowledge of the quantities of each denomination would be needed by both the seller and buyer. Unfortunately, the buyer would most likely never have access to that information (since I assume that it is not like a lottery where they divulge the odds of winning) and would have to guess whereas the seller could reasonably have that information.

In terms of determining the selling price:

- The seller thinks: profit + cost of “real money” + production cost.
- The
buyerthinks: amount buyer would regularly pay for soap + amount of money buyer could win + amount buyer would pay for the novelty of having a money duck soap – minus whatever margin they are hoping to profit.I think that the seller’s situation is

muchmore manageable. We can have Act Two information to determine the profit and production cost. The math will come from determining the cost of the “real money.” As for the buyer, there are many assumptions that will not fall as conveniently into Act 2. There may be disagreement as to what a buyer would pay for soap, the novelty cost, and profit margins. Accordingly, I think it is much easier to do the seller situation.

My favorite question here would be to set a price for the duck and ask the seller’s question, what distribution should I use for the bills? But only after the kids have determined that an equal distribution is profitable for the buyer.

Mr K:

My thought is that the kids will gravitate to the “Should I buy it” question, and the real learning comes from shifting them around to the sellers side of the problem.

I’d set up many sets of two types of groups, producers (sellers) and consumers (buyers). The producers would determine what distribution of the bills (ha) go in the ducks and then set a price based on that info. They make up 20 “ducks” with their distribution. The consumers would go up to a producer and be given the odds of each type of bill and the price of the duck. It’d be up to them if they’d like to buy that groups ducks.

how about using the money duck to construct a survey. Survey students as to what odds (probability) of winning any prize would encourage them to muy multiple money ducks.

I presented the Money Duck to my grade 4/5 class this morning with the question: “What is the most that you would pay for a Money Duck bar of soap?” Group conversations were animated as was the class discussion.

Questions that students raised included: “How likely would it be for the package to have 10, 20 or 50 dollars?”, “Is the money planted in the middle of the soap? If so, would someone break the soap to get at the money?” [in which case the soap didn't matter at all in the pricing] and “What is the quality of the soap?” [If it was a good quality soap many were willing to pay more]. Some students started taking the seller’s point of view and gave ideas how they could increase profits.

The discussion lasted nearly half an hour. Incidentally, the average price the students were willing to pay was $5. (Figuring that the soap itself was worth 3 or 4 dollars, and then factoring in the minimum prize of 1 dollar).

Clearly this idea has potential at multiple grade levels.

How many money ducks would I have to buy to be guaranteed to score at least one duck with a $50 bill inside?

From what I’ve heard, nearly all of the ducks are $1 ducks. It’s like buying a lottery ticket — you expect to lose, but it can be fun for some people anyway.

Bit from an Amazon review of a different money soap:

“I used to work at a warehouse for online gag gifts until it went out of business I had four boxes of these prob 200 bars or so and never got more than a five.”

I think legally they only need to have one duck in the entire country that has a $50 in it to claim there could be a $50, so that’s likely your odds.

First, show this video.

Ask students to tell their neighbor how much they’d be willing to pay for the Money Duck. Find the high and low in the class.

Now there are a series of questions I’d like students to confront, including:

- What is a probability distribution and how do we represent it?
- What does an
*impossible*probability distribution look like? Why is it impossible? - If you’re a seller, what kind of probability distribution is bad for business and why?
- Rank these distributions in order of “I’d definitely buy that for $5!” to “I definitely
*wouldn’t*buy that for $5!” - What would a fair price be for each of these distributions so that over time you wouldn’t lose or win any money?”

I’m trying to progressively formalize a) this new, strange representation of probability and b) the calculation of expected value.

The first two questions assist (a), basically asking “What *is* this thing?” and “What *isn’t* this thing?”

The next three questions assist (b), applying progressively more nuance to the concept of expected value. First, the concept is either/or. (“Who gets screwed?”) Then it becomes ordinal. (“Rank ‘em.”) Then numerical. (“Put a price on ‘em.”)

Kids will struggle at different moments in this sequence, but that’s okay because the purpose of the sequence isn’t that they *discover* the concept of expected value. The purpose of the sequence is to make make my eventual formal explanation of expected value much more comprehensible. (See Schwartz on contrasting cases.)

So explain how mathematicians calculate the expected value of a distribution. Now let’s go back and calculate the exact expected value of the distributions in #5.

Here are those questions and screens packaged as a Keynote slideshow (also PowerPoint, if you must) and as a handout.

Now let’s show them the answer, what the Virginia Candle Company *actually* charges for the Money Duck.

Now every student should create a distribution that results in some profit at this price over time.

Fin.

Follow on with practice and assessment.

What would you change, add, or subtract from this sequence?

]]>