- Eight new blog subscriptions from November & December.
- Five essential 2015 posts from this blog.
- Three bloggers I envy.
- Seventeen Great Classroom Action posts I never got around to posting.

**Blogging**

- We successfully goaded Brett Gilland into tweeting and blogging. His writing features art, wit, and insight for days. Best follow of my fall quarter.
- Jason D’Arcangelo is an elementary math coach, making him rare company online.
- Kendra Lomax does interesting work in elementary math education also, most recently with the University of Washington’s Teacher Education by Design project.
- Damian Watson just came off a two-year blogging hiatus with a post featuring Malcolm Swan, Andrew Stadel, and cognitive conflict, which pushes all three of my buttons.
- Meryl Polak likewise came off a maternity leave to post about her experience designing and implementing a 3 Act Math task.
- Geoff Wake was one of my colleagues at the Shell Centre when I set up a tent in their offices several years ago. Great guy. Interesting thinker. I’m excited to see him maintaining a blog.
- Jenn Vadnais does consistently interesting work with the Desmos Activity Builder. I’m tuned in, hoping to learn how she works.
- Glen Lewis blogs thoughtfully about technology, learning, and engagement in math education.

These blogs are each low volume, producing maybe one post per month. There is zero risk of getting overwhelmed here. Just toss them in Feedly or some other RSS reader and enjoy their insight whenever they find the time to share it.

**Honorable Mentions**

I don’t have a lot of envy in me for other Internet math ed types – their followers, retweets, subscribers, etc. Just keep working. What *does* turn me green, what I *do* covet, though, is another blogger’s ability to stir up conversation, to mobilize and collect the intellect of his or her readers. In 2015, that was Dylan Kane, the blogger whose posts invariably had me clicking through to the comments to see what he managed to provoke from his readers, then scratching my head trying to figure out how he did it.

If your heart belongs to *elementary* math education, the best moderators I have found there are Tracy Zager and Joe Schwartz.

**My Year in Review**

If you’ve come to this blog only recently, here are five posts that received a lot of traffic and commentary this year:

- The Math I Learned After I Thought Had Already Learned Math
- The Math Problem That 1,000 Math Teachers Couldn’t Solve
- WTF Math Problems
- Understanding Math v. Explaining Answers
- If Math Is The Aspirin, Then How Do You Create The Headache?

Looking for favorites from the wider online math education community? Check out the #MTBOS2015 hashtag. If I had to award my own MVP, it’d be Elizabeth Statmore’s “How People Learn” and how people learn where she turns essential research into manageable practice.

**Great Classroom Action**

And now, shamefully presented without commentary, seventeen posts I read in 2015 that had me check myself and think, “That classroom action is *great*!” I haven’t shared these yet and it’s time to clean the cabinet.

- Perplexity and Figuring It Out,
*Evan Weinberg* - Kicking Some Serious Triangle Booty,
*Kate Nowak* - Extra Time,
*Jonathan Claydon* - Statistics Arts and Crafts,
*Bob Lochel* - Trying out “I Have…Who Has?” for the First Time,
*Sarah Hagan* - Water Parks, Periodic Functions, and Mathematical Modeling,
*John Pelesko* - Dice bias. A statistics activity,
*Scott Hills* - Breaking Bad…Definitions,
*Jessica Murk* - Rumor has it…,
*Kaleb Allinson* - Day 160: Parentheses,
*Dan Burf* - “The Best Discussion I’ve Ever Had”,
*Thom Gibson* - Z-Score & Parent Ages,
*Dianna Hazelton* - The Math Modeling Cycle in Action,
*EPSmith* - Intro to Statistics (Unit 1),
*Pam Rawson* - Intro to Statistics (Unit 2),
*Pam Rawson* - odds + ends,
*Rachel Kernodle* - Collaborative learning with Canvas discussions,
*Mary Dooms*

The New York Times looks at the dismal testimony of an “accident reconstructionist”:

The “expert witness” in this case would not answer questions without his “formula sheets,” which were computer models used to reconstruct accidents. When asked to back up his work with basic calculations, he deflected, repeatedly derailing the proceedings.

Watch the video. It’s well worth your time and I promise you’ll see it in somebody’s professional development or conference session soon. It offers *so much to so many*.

And then help us all understand what went wrong here. What’s your theory? Does your theory *explain* this catastrophe? Does it recommend a course of action? If you could go back in time and drop down next to this expert as he was learning how to make and analyze scale drawings, how would you intervene?

My own answer starts off the comments.

**BTW**. Can anyone help us understand how the expert came to the incorrect answer of 68 feet?

**BTW**. Hot fire:

The motorcyclist’s lawyer filed a counter-motion to refuse payment to the expert witness. It contained the math standards for Wichita middle schools.

[via Christopher D. Long]

**2016 Jan 2**. The post hit the top of Hacker News overnight.

**2016 Jan 2**. One of the Hacker News commenters notes that the actual deposition video is available on YouTube.

**Featured Comments**:

gasstationwithoutpumps offers one explanation of the error:

3 3/8″ at a 240:1 scale gives 67.5′ which rounds to 68′

It is easy to mix up 3/8″ and 3/16″, which is one reason I prefer doing measurements in metric units.

katenerdypoo offers another:

It’s quite possible he accidentally keyed in 6/16, which when multiplied by 20 gives 7.5, therefore giving 68 feet. This is also a reasonable error, since the 6 is directly above the 3 on the calculator.

Jo illustrates a fourth grader’s process of solving the scale problem.

Robert Kaplinsky chalks this up to pride:

Lastly, it’s worth noting that eventually the heated conversation shifts from the actual math to whether or not he will do it or can do it. At that point it seems to become a pride issue.

Alex blames those awful office calculators:

The reconstructionist is given an office calculator, which doesn’t even have brackets. He needs to enter a counter-intuitive sequence of “3/16+3” to even get the starting point. When I was at school I remember being aware that most people wouldn’t be able to handle that kind of mental contortion. They’d never been asked to.

So what’s the problem, and how might we solve it? Well, the man’s been given the wrong tool for the job. He’s never been asked to use the wrong tool before & so this throws him. This makes him defensive and he latches onto an excuse about formula sheets.

The motorcyclist’s lawyer is the unrelenting classroom didactic whose motivation is based on making his student look and feel stupid. I was waiting for Act 2 where the lawyer would jump up, grab his felt marker, and demonstrate just how easy he can show the procedure.

Anna:

]]>Interesting note: my grade 7 math class is in the middle of our unit on fractions, decimals, and percents, so I showed them this video so we could work on the problem. I thought they’d get a chuckle out of it and feel good about solving a problem that the expert on TV couldn’t solve.

Their reaction was unanimous. They identified with the guy and wanted them to give him his formula sheets. Some of them were pretty riled up about it!

They’re quite accustomed to me showing them videos and doing activities that are designed to build up their understanding that everyone approaches things differently, and we’ll all get there even if we take different paths. This guy wasn’t allowed to follow his path and do it his own way, and they were unfairly putting him on the spot and forcing him to do it their way.

It’s a rich problem, so I’ll use it again, but I think I’ll set it up and frame it a little differently next time!

When we’ve done analyses of the results of [our professional development efforts], we’ve found that teachers often move from a transmission approach where they tell the class everything and the students have been fairly passive, they’ve usually moved in two directions.

One is retrograde. They’ve moved towards individual discovery. They say “I’ve been saying everything to these students for so long. What I’ll do now is withdraw and let them play with the ideas. I’ve been saying too much. I’ll withdraw and let them discover stuff.” That’s worse than the place where they started.

The other place is where they move in and they challenge students and work with them on their knowledge together. That’s a better place. That’ smore effective.

And so in professional development, people take a path. Over time they might move from transmission to discovery to collaborative connectionist. So they might actually get worse before they get better.

That’s one of the problems with evaluating whether its been successful by looking at student outcomes. People take awhile to learn new things.

Earlier in the talk, he describes counterproductive designs for professional development:

Most of the time [in teacher professional development] we inform people of something and then we say “go and do it.” That’s not the way people learn. Usually they learn by doing something and then reflecting upon it.

So when you start with a professional development, you say, “Try this out in your classroom. It doesn’t matter if it doesn’t work. Then observe your students and then as a result you might change your beliefs and attitudes.”

You don’t set out by changing beliefs and attitudes. People only change themselves as they reflect on their own experiences.

And then productive ones:

If you’re designing a course, we usually start by recognizing and valuing the context the teacher is working in and trying to get them to explain and explore their existing values, beliefs, and practices.

Then we will provide them with something vividly challenging. It might be through video or it might be through reading something. And this is really different to what they currently do.

And through this challenge we ask them to suspend their belief and try and act in new ways as if they believed differently.

And as they do this we offer support and mentoring as they go back into the classroom to try something out.

And then they come back together again and it’s taken over then by the teachers who reflect on the experiences they’ve had, the implications that come out of their experiences, and recognize and talk about where they’ve changed in their understandings, beliefs, and practices.

What’s great about Malcolm Swan and the Shell Centre is their designs for teacher learning line up exactly with their designs for student learning. It all coheres.

]]>But here you have collected zero stars. No success.

That’s because your students need to set up parabolic, linear, exponential, sinusoidal, or rational functions to send the marbles on a trip through those stars.

Success!

That’s Marbleslides and you and your students should play it this week and let us all know how it goes. If you want a preview, head to student.desmos.com LINK and type “eht8”.

If you want to set up your own class, head to the Marbleslides activities listing, choose a function family, and get a classcode of your own.

Here are some quick, below-the-fold notes about what we’re trying to do here and why we’re trying to do it.

**Delight**. Whenever possible we want students to experience the same sense of delight about math that all of us at Desmos feel. Students can experience that delight both in pure and applied contexts and Marbleslides is that latter experience. Seriously, try not to grin.

**Purposeful Practice**. Picture two students, both graphing dozens of rational functions. One finds the experience dreary and the other finds it purposeful. The difference is the wrapper *around* that graphing task. If the wrapper is no more purposeful than a worksheet of graphing tasks, your student may fatigue after the first few graphs. In our Marbleslides classroom tests, we watched students transform the same function dozens of times – stretching it, shrinking it, nudging it up, down, left, and right by tiny amounts. That’s the Marbleslides wrapper. Students have a goal. Their pursuit of that goal will put you in a position to have some interesting conversations about these functions and their transformations.

**BTW**. Here’s the announcement post on the Desblog.

**Scott Keltner** sent a drone up in the sky while his students plotted *themselves* down below. I emailed Scott and asked for his lesson plan and he sent back something involving playing cards and, frankly, none of it made any sense to me and I seriously don’t understand how the downside of cost, time, and effort could possibly outweigh the upside of drones (!) but I’m *so curious*. Bug Scott via Twitter to write up what he did here.

**Julie Reulbach** used zombies to create a need for logarithms. Zombies are, obviously, catnip for some students, but that isn’t what caught my eye here. Julie understood that logarithms are a *shortcut* for inverting an exponential equation. And if you’d like to create a need for a *shortcut* it’s helpful for students to experience the *longcut*, however briefly. Watch her work.

**Ollie Lovell** used one of my unposted tasks with a group of students in Myanmar who spoke very limited English and whose classroom had no electricity. Imagine how your favorite lessons would *have* to change under those constraints and then read how Ollie changed his. I learned a lot.

**Sarah Hagan** shares a game from S T called Greed, which helped turn her students’ perception of box-and-whisker plots from useless to useful. Crucially, the game exploits the *need* for box-and-whisker plots, which is *comparison between multiple sets*. Creating a box-and-whisker plot for a single set of data *will* feel pointless, same as teaching someone to use a carrot peeler by using it to paint a house. That’s not what it’s *for*!

**NCTM is obviously interested in recruiting new members, along with all of their new ideas.**

Two years ago there was a panel discussion dedicated to technology in math education which featured a bunch of math Twitter-types. The following year saw an entire strand dedicated to ideas from those math Twitter-types. Then the math Twitter-types occupied the opening keynote at this year’s Nashville regional conference, immediately after which Robert Kaplinsky took my favorite photo from that conference.

These @NCTM confs are about the connections you make. cc: @dbriars @TracyZager @mslailanur @mathycathy @jreulbach pic.twitter.com/JhMaklOdSZ

— Robert Kaplinsky (@robertkaplinsky) November 19, 2015

Mark it, friends, or correct me if I’m wrong: that’s the first appearance of a current NCTM President at what the Twitter-types call a “tweetup.”

Just five years ago, these Twitter-types occupied the fringe. It’s so nice to see everybody making friends and learning from one another. This only bodes well.

**NCTM’s new conference website has promise.**

The history: Zak Champagne, Mike Flynn, and I ran Shadow Con as an experiment in extending the face-to-face conference experience. We offered speakers a more powerful platform on the web for interacting with attendees (live and virtual) than NCTM’s existing read-only conference program website.

We reported the results of that experiment to NCTM’s executive team and that was the last any of us heard from them until this year’s regional conferences when they tweeted out their new conference website. Look at it!

The featured speakers at the regional conferences each get their own page on a WordPress installation. On first glance those pages look *just* like a conference website. Title, description, and time. Just the facts. But speakers can add files, videos, and other resources. Then there is a comment box where attendees can get in touch before and after the session.

A colleague of mine remarked: “It’s a mixed bag.” Yeah, but *what a mix!*

Out of the 28 featured sessions across the three regional conferences, seven presenters don’t seem to have visited their page. That lack of attention has basically zero downside. Their pages look *just like* they would on any read-only conference program website. Title, description, and time. Just the facts.

And across the other 21 sessions, there is a *pile* of activity!

- David Wees asks Jon Wray to clarify his session description. Savvy shopping, David! Helpful replying, Jon!
- Christina adds a Yelp-style review to Graham Fletcher’s page telling people they should definitely come to his session. Great social proof, Christina!
- Donna Leak’s page has video of her presentation.
*Video!* - A presenter had to cancel her session but leaves behind an apology and her slides!
- David Barnes collects tweets from his session and adds them to the comments afterwards. Tweets!
- Math Twitter-types like Annie Fetter, Kate Nowak, and Max Ray trick out their pages with resources and interact with attendees like the web veterans they are!

All of this is *possible* without NCTM site’s but none of it is *easy to do* and none of it is *easy to find*.

So here is my hope for the future of NCTM conferences.

**Extend this website to cover all presenters from all NCTM conferences and offer it to affiliate organizations for their conferences as well.**

I want to click Annie Fetter’s name on *one page* and see all the talks she’s ever given, across geography and time, including five years ago at some random state affiliate conference I never knew existed.

I want to search for “Kate Nowak NCTM” in Google and find her past conference pages and also her upcoming talks.

Before I attend a conference, I want to locate presenters whose talks seem to provoke a lot of online discussion afterwards, and then attend *those*.

If NCTM makes this commitment, they’ll increase their value to current and prospective members several times over.

For current members, they increase the value of conference attendance and decrease the pressure on attendees to attend *every* session. (Expect the question “Will you be posting your resources to your page?” to float around Twitter in the weeks leading up to every conference.) The conference page will connect attendees and speakers in the twelve months between annual conferences.

Prospective members, the kind who wonder “Why NCTM?”, may start to land on conference pages more often than Pinterest boards when they search for resources. As those prospective members explore the resources on those conference pages, NCTM can recommend journals, articles, books, tasks, and *other conference pages* that may also be helpful. NCTM can point those visitors to upcoming conferences and sessions on those themes, converting non-members into members and members into stronger teachers.

Until future notice, I am a single-issue voter in all NCTM elections and this is my issue.

]]>Sometimes I see a worksheet online and I say to myself, “That should stay a worksheet. Paper is the right home for that math. Any possible benefit from moving that math to a computer is more than outweighed by the hassle of dragging out the laptop cart.”

Other times I see a worksheet and it seems clear to me that a different medium would add – you name it – breadth, depth, interest, collaboration, etc.

That’s the case with Joshua Bowman’s implicit differentiation worksheet, which he shared on Twitter. It’s great in worksheet form. But the Desmos Activity Builder can add a lot here while subtracting very little. Activity Builder is the right home for this math.

Here is the activity I built in Activity Builder:

And here are some differences, from small to large:

**Simplify Assignment Collection**

Bowman is asking his students to do their work in Desmos *anyway* and then copy and paste their calculator link into a Google Doc for feedback.

Activity Builder simplifies that collection process. Students do their work in the Desmos activity. Desmos sends you all of their graphs, quickly clickable.

**Ask More Questions**

When students see worksheets with seventeen questions running (a) through (q), they lose their mind. Let’s lighten their cognitive load and keep question (q) out of their visual space while they’re considering question (a).

This isn’t necessarily an improvement, especially if my new questions just ask students to repeat the same dreary work several hundred times. So:

**Ask More Interesting Questions**

I added six more questions to Bowman’s worksheet, and they share particular features.

First, they ask students to work at several different levels, from informal to formal. For example, I wanted to ask questions about:

- a blank graph – “What do you think the shape of the graph will be?”
- the graph – “Add up all the intercepts. What is that sum?”
- the graph
*and*some tangent lines – “Multiply their slopes. What is the product?”

These questions move productively from informal understandings to formal understandings, but *they don’t live well together on the same piece of paper*. You can’t ask students, “What do you think the shape of the graph will be?” when *the graph is farther down the page*.

Another example:

Bowman’s worksheet asks students to find the equation of the tangent lines to the intercepts of the graph. Some students may use sliders, other students may differentiate implicitly.

I can quickly figure out which group is which by asking them to multiply their slopes together and enter the product in a new question. Which students differentiated and which students experimented?

Long before I ask students to *calculate* that product, I ask them to simply estimate its sign. Envision the tangent lines in your head. Without knowing their exact slopes, what will their product be? That’s an informal understanding that assists later, formal understandings.

So again:

- Simplify assignment collection.
- Ask more questions.
- Ask more
*interesting*questions.

Best of all, this Desmosification took minutes. Start somewhere. The tools are all free forever. Thanks, Joshua, for sharing your worksheet and letting us take a crack at it.

**Featured Comments**

I also like how the overlay view of your student answers could help lead to new questions, like seeing trends for student mis/understanding.

This is great…but I need more. I want a way to be able to provide feedback to my students as they work through these activities.

]]>Where @ddmeyer convinces me it's time I learn how to use the @Desmos activity builder. https://t.co/PBJbfB66YT

— Joshua Bowman (@Thalesdisciple) November 18, 2015

Here is my speaking calendar for 2016. Some of these sessions are private, others have open registration pages (see the links), and others have waiting lists. Feel free to send an e-mail to dan@mrmeyer.com with inquiries about any of them. It’d be a treat to see you at a workshop or a conference.

**BTW**. After my keynote address at Nebraska’s state conference on September 9, 2016, I’ll have worked with teachers in every U.S. state. It’s been such a privilege getting to know so many interesting people doing so much interesting work. If you have attended any of my sessions, you’ve heard me express how indebted I am to participants from *other* sessions for the questions they ask about my ideas and the ideas they share themselves.

I haven’t been able to shake one particular exchange, though.

Halfway through the comments, two people who disagree with each other as completely as anyone could each made a precise and articulate case for their diametrically opposite theories of learning.

Ze’ev Wurman, a longtime advocate of traditional math instruction:

I thought the purpose of a problem in a classroom is to check whether a student knows sufficient math to solve it, rather than learn bout the nature of human thinking processes. If it is the latter, Dan is completely right, except it belongs to cognitive science experiment rather than a classroom.

Brett Gilland, an infrequent blog commenter who should comment more frequently:

I can not disagree with this enough. The purpose of a problem in my classroom is almost always to understand the nature of that human’s thinking processes. This allows for amplification, further investigation into how the student is able to navigate similar problems with subtle variations and complications, and attempts to draw student mental models into internal conflict to create pressure for remediation and revision of said mental models.

I suspect that Gilland’s employer, and certainly the parents of his students, would also disagree. Some quite strongly. The primary purpose of school is to educate the kids at hand, not to train the teacher. This doesn’t mean that teachers do not learn from experience, but if gaining experience and insight is the primary reason for what the teacher does, he’d better get approval from an IRB and a waiver from each individual student or parent that attend his class.

Funny thing, that. My employer, my parents, my students, my district, the state evaluator for my school, etc. all support my teaching. Most quite strongly. This might be due to the fact that when most people hear “I work really hard to understand your child’s thought processes so that I can better guide their thinking and draw out subtleties and conflicting mental models,” they don’t think “Oh my God, that man is performing experiments on my child to improve his educational practice.” Instead, they think “Oh my God, that man really cares about what is going on inside my child’s head and is attempting to tailor instruction to what he finds there. Thank goodness he isn’t stuck with a teacher who believes that teaching is just lectures interspersed with quizzes to determine if my child gets it or needs to be droned at more with another utterly useless generic explanation that takes no account of what my particular child is thinking!”

I’m sure that everyone walked away feeling like their side won, but one side is wrong about that.

**Featured Comments**

“Every school should be organized so that the teachers are just as much learners as the students are.” (Adding It Up, 2001, pg. 13)

]]>Alas, I feel that Mathematics is reaching a junction – in which the traditionalists and the progressives must come to a head and work together to forge a stronger future for our young mathematicians! Whilst today’s world demands an ability to think and to use available resources to find new meaning, we must not forget those who generated those resources in the first place. a fine craftsman needs to learn the tools of his trade before he or she can produce the creative thinking in his head. A computer programmer must use efficient logic before we can play those game or use those apps to be progressive learners. To what degree should thinking, reasoning, and problem solving come before skills acquisition , or vice versa? Sound like the chicken and the egg to me.

My biggest fear for us as teachers is that we are robbing our young people of the beauty and passion for maths through repetitive, applied drill, just so that they can demonstrate a high level skill skill that they will never use. or, we are not providing enough technical skills to enough of our students to ensure quality craftsmanship I am saddened every single time that my best mathematics students tell me they want to study medicine or dentistry instead of using their mathematical ability to grow our field.

Let us first and foremost provide our students with mathematical challenge- that requires both the creative solution and advanced skills acquisition. Whether the challenge be abstract or modelling a real situation need not matter. What is important is bringing back the passion for mathematics that we, as mathematics educators share, passion- through intrinsic motivational challenge and drive.

- Is there a problem that could be completely explained using symbolic notation alone?
- Is there a problem that symbolic notation cannot sufficiently explain?

I vote yes for both and have added my examples in the comments.

**BTW.**

I realized that *the headline* from the Garelick & Beals article doesn’t match their *argument*.

The headline: “Explaining Your Math: Unnecessary at Best, Encumbering at Worst”

Their article: “At best, verbal explanations beyond ‘showing the work’ may be superfluous; at worst, they shortchange certain students and encumber the mathematics for everyone.”

I can see why *The Atlantic* would want to sharpen their writing for the headline. They qualify themselves *twice* in the article (“at best” and “may be”) *barely* making a claim.

So if they think symbols are *always* sufficient explanations, let’s offer questions in the comments for which they *aren’t*. If they think verbal explanations are *sometimes* necessary, let’s let them articulate *when*.

**Featured Comment**:

]]>There’s a place in instruction (somewhere between ages 3 and 8) where each of the symbols “3” and “+” and “4” and “=” and “7” each need explanation, which might look like

… + …. = … …. = …….I am pretty sure that Common Core haters dislike the notion that any of that ought to be explained, that they would prefer that this just be one of the 55 addition facts that ought to be memorized, and let’s move on.

At the same time, requiring 3rd or 4th graders to explain why 3 + 4 = 7 seems ridiculous, EVEN THOUGH SOME 3RD GRADERS HAVEN’T MASTERED ADDITION.

I would ask my 8th graders to explain why 3x + 4 != 7x, but I wouldn’t ask for this in a Calculus class, EVEN THOUGH IT HAS NOT BEEN MASTERED BY ALL.

My point is that we use symbols for efficiency, to avoid explanation. Symbols are NEVER enough explanation at the beginning of instruction.

A student’s presence in a given class presumes a level of previous mastery and efficiency WHICH IS NOT ALWAYS THERE, and instruction examples demonstrate the level of explanation that is expected.

To Chester Draws about the quadratic, I would hope for words like “Quadratic => 0, 1, or 2 solutions” in an explanation.

So a really good question is “what level of explanation should students be expected to demonstrate on a national test? (all right, multi-state, but I am in favor of a national curriculum).