Ok #MTBoS. What's your favorite first day of school activity? For high school. Something that helps me get to know my students??

— Ali Grace (@AGEiland) June 9, 2016

My contributions:

- Personality Coordinates Icebreaker.
- Who I Am. (Return these at the end of the year for more fun.)
- The Collaborative Icosahedron.
- Stacking Cups.

Help the rest of us out in the comments. What do you do on the first day of school?

**2016 Jul 27**. A Collection of First Week Activities.

To say that the community repository model has done wonders for open source software is a massive understatement. To what extent that success translates to curriculum I’m obviously unsure, but I have randomly-ordered reasons to suspect it’s appreciable.

I attended EdFoo earlier this year, an education conference at Google’s campus attended by lots of technologists. Speakers posed problems about education in their sessions and the solutions were often techno-utopian, or techno-optimistic at the very least.

One speaker wondered why teachers spend massive amounts of time creating lessons plans that don’t differ *all that much* from plans developed by another teacher several states away or several doors down the hall. Why don’t they just build it once, share it, and let the community modify it? Why isn’t there a GitHub for lesson plans?

I’m not here to say that’s a bad idea in theory, just to say that the idea very clearly hasn’t caught on in practice.

Exhibit A: BetterLesson, which pivoted from its original community lesson repository model to a lesson repository *stocked by master teachers* and now to professional development. (Its lesson repository is currently a blink-and-you’ll-miss-it link in the footer of their homepage.) The idea has failed to catch on with secondary educators to such a degree that it’s worth asking them why they don’t seem to want it.

Our room at EdFoo was notably absent of practicing secondary teachers so I went on Twitter to ask a few thousand of them, “Why don’t you use lesson download sites?” (I asked the same question two years ago as well.) Here are helpful responses from actual, really real current and former secondary teachers:

Using someone else’s lesson plan is like wearing a friend’s underwear. It may do the job but ultimately doesn’t fit quite right.

Their wheels aren’t the right size for my car.

Linux works because code compiles. Syllabi don’t compile. If I add a block/lesson, I never know who it helps.

I don’t require a script, just decent ideas now and then.

I’m not sure they solve for the problems they think they’re trying to solve. It takes time to read / internalize / modify others’ plans.

It’s challenging to sequence, connect, plan, and enact someone else’s lesson.

The plan itself is the least important element. The planning is what’s critical.

**2016 Jun 11**. Dwight Eisenhower:

In sum: “Small differences between lessons plans are enormously important, enormously time-consuming to account for and fix, and whatever I already have is probably good enough.” It turns out that even if two lesson plans don’t differ *all that much* they already differ *too much*.

Any lesson sharing site will have to account for that belief before it can offer teachers even a fraction of GitHub’s value to programmers.

**2016 Jun 8**. Check out Bob Lochel’s tweet above and Julie Reulbach’s tweet below. Both express a particular sentiment that the nuts and bolts of a lesson plan are less important than the chassis. (I don’t know a thing about cars.)

I was chatting with EdSurge’s Betsy Corcoran about that idea at EdFoo and she likened it to “the head” in jazz music. (I don’t know a thing about jazz music.) The head contains crucial information about a piece of music – the key, the tempo, the chord changes. Jazz musicians memorize the head but they’ll build and develop the performance off of it. The same head may result in *several* different performances. What I want – along with Bob and Julie and many others – is a jazz musician’s fake book – a repository of creative premises I can easily riff off of.

(Of course, it’s worth noting here that many people believe that teachers should be less like jazz musicians and more like player pianos.)

**Featured Tweets**

@ddmeyer @rawrdimus yes. We need ideas. Not lessons. That's our speciality.

— Julie (@jreulbach) June 8, 2016

@ddmeyer GitHub for lessons doesn’t appeal to mea, but GitHub for *planning* does.

— Ben Samuels-Kalow (@bensk) June 8, 2016

@ddmeyer you save time and learn new often better approaches by using someone else's plan, AND it still requires a lot of your own planning

— Paul Edelman (@TpTFounder) June 9, 2016

**Featured Comments**

There seems to be a general distrust of “other people’s lessons.” Which I get. But nothing about this model would change the extent to which you do or do not teach other people’s lessons, or the fidelity with which you do it. Again, the whole thing that got me thinking in this vein was the problem of managing, in some kind of coherent way, all the changes that teachers already make as a matter of course. If you’re starting with an existing curriculum, then you’re using other people’s stuff to some extent. And once you alter that extent, it might be nice to track it, for all sorts of reasons. Maybe classroom teachers don’t find that interesting, but somebody in the chain between publisher and implementer certainly does. Not totally sure who the best target audience might be.

Jo:

As an elementary math coach I don’t want a repository of lesson plans either but my teachers long for one. However, when given pre-written lesson plans they’re not happy with them–for all the reasons listed above.

The hardest thing about elementary math is that most elementary teachers go into teaching because they love reading and they want to share that. They rarely feel that way about math. So, they want a guided lesson that will teach the requisite skills. Unfortunately, it doesn’t work for them any better than it works for secondary teachers.Even in elementary it’s the process of planning that’s important. My brain needs to go through the work of planning–what leads to what, what is going to confuse the kids, what mistakes are they likely to make, what false paths are they likely to follow. The only way to deeply understand the material and how to present it is to plan it. The only way to truly understand the standard is to wrestle with what it really means.

Planning is the work; teaching is just the performance.

I get a lot out of reading other lesson plans/approaches to teaching/ideas, and steal activities fairly regularly, but my actual lesson plans aren’t copies of others’. It’s more like they’re inspired by what other people do. This is where the artistry of teaching comes in.

I get it – we don’t want a repository of lessons, but what happens once those lessons get downloaded and re-worked? Right now there isn’t a way to see derivatives of those lessons, which could be very important.

Brandon, I love that idea. Recipe websites do this — what can be substituted for what.. how can different teachers with different ingredients, different tools and in different places.. these are good parallels for the teaching world.

**2016 Jun 13**. Mike Caulfield offers an illustration of the value of *planning* relative to *plans*.

Danny Brown has expressed an interest in teaching mathematics that is relevant to students, relevant in important, sociological ways especially. This puts him in a particular bind with mathematics like Thales’ Theorem, which seems neither important nor relevant.

Here is Thales’ theorem. Every student in the UK must learn this theorem as part of the Maths GCSE. You are explaining Thales’ theorem, when one of the students in your class asks, “When will we ever need this in real life?” How might you respond?

He proceeds to offer several possible responses and then, with admirable empathy for teenagers, rebut them. Brown finds none of our best posters for math particularly compelling. You know the ones.

- Math is everywhere.
- Math develops problem solving skills.
- Math is beautiful.
- Etc.

So instead of fixing our posters, let’s fix the product itself.

Brown’s premise is that students are listening to him “explaining Thales’ theorem.” Let’s question that premise for a moment. Is that the only or best way to introduce students to that proof? [**2016 Jun 3**. Brown has informed me that explanation is not his preferred pedagogy around proof and I have no reason not to take him at his word. So feel free to swap out “Brown” in the rest of this post with your recollection of nearly every university math professor you’ve ever had.]

Among other purposes, every proof is the answer to a question. Every proof is the rejection of doubt. It isn’t clear to me that Brown has developed the question or planted the doubt such that the answer and the explanation seem *necessary* to students.

So instead of starting with the explanation of an answer, let’s develop the question instead.

Let’s ask students to *create* three right triangles, each with the same hypotenuse. Thales knows what our students might not: that a circle will pass through all of those vertices.

Let’s ask them to *predict* what they think it will look like when we lay all of our triangles on top of each other.

Let’s reveal what several hundred people’s triangles look like and ask students to *wonder* about them.

My hypothesis is that we’ll have provoked students to *wonder* more here than if we simply ask students to listen to our explanation of *why it works*.

**“Methods”**

To test that hypothesis, I ran an experiment that uses Twitter and the Desmos Activity Builder and is pretty shot through with methodological flaws, but which is *suggestive* nonetheless, and which is also way more than you oughtta expect from a quickie blog post.

I asked teachers to send their students to a link. That link randomly sends students to one of two activities. In the control activity, students click slide by slide through an explanation of Thales’ theorem. In the experimental activity, students create and predict like I’ve described above.

At the end of both treatments, I asked students “What questions do you have?” and I coded the resulting questions for any relevance to mathematics.

77 students responded to that final prompt in the experimental condition next to 47 students in the control condition. 47% of students in the experimental group asked a question next to 30% of students in the control group. (See the data.)

This *suggests* that interest in Thales’ theorem doesn’t depend strictly on its social relevance. (Both treatments lack social relevance.) Here we find that interest depends on what students *do* with that theorem, and in the experimental condition they had more interesting options than simply listening to us explain it.

So let’s invite students to stand in Thales’ shoes, however briefly, and experience similar questions that led Thales to sit down and wonder “*why*.” In doing so, we honor our students as sensemakers and we honor math as a discipline with a history and a purpose.

**BTW**. For another example of this pedagogical approach to proof, check out Sam Shah’s “blermions” lesson.

**BTW**. Okay, study limitations. (1) I have no idea who my participants are. Some are probably teachers. Luckily they were randomized between treatments. (2) I realize I’m testing the *converse* of Thales’ theorem and not Thales’ theorem itself. I figured that seeing a circle emerge from right triangles would be a bit more fascinating than seeing right triangles emerge from a circle. You can imagine a parallel study, though. (3) I tried to write the explanation of Thales’ theorem in conversational prose. If I wrote it as it appears in many textbooks, I’m not sure anybody would have completed the control condition. Some will still say that interest would improve enormously with the addition of call and response questions throughout, asking students to repeat steps in the proof, etc. Okay. Maybe.

**Featured Comments**

Danny Brown responds in the comments.

Michael Ruppel responds to the charge that Thales theorem isn’t important mathematics:

]]>As to the previous commenter, Thales’ theorem is not a particularly important piece of content in and of itself, but it’s one of my favorite proofs for students to build. It requires careful attention to definitions and previously-learned theorems as well as a bit of creativity. (Drawing that auxiliary line.) Personally, my favorite part of the proof is that students don’t solve for a or b, and in fact have no knowledge of what a and b are. but they prove that a+b=90. The proof is a different flavor than they are used to.

Are your students overgeneralizing their models? After working exclusively with proportional relationships for the last month, are they describing *every* new relationship as proportional?

This isn’t a task, or a lesson, or anything of that scope. It’s a resource, a provocation, one that gives students the chance to check their assumptions about *what’s going on*.

Play this video and pause it periodically, asking students to decide for themselves, and then tell a neighbor, what’s coming next.

10 marbles weigh 350 grams. So 20 marbles should weigh how much? I’m curious which students will say the answer is less than, exactly, or more than 700 grams. I’m curious which students will say it’s impossible to know.

Reveal the answer.

That will be surprising for some. Now invite them to speculate about 30 marbles. 40 marbles. And 0 marbles.

Let me end with three notes.

**First**, my thanks to Kevin Hall who had the fine idea for the video and encouraged me to make it. I’ve never met Kevin. That’s the kind of internet collaboration that makes my week.

**Second**, the stacking cups lesson offers a similar moment of dissonance. Can you find it?

**Third**, here’s Hans Freudenthal on technology in 1981:

What I seek is neither calculators and computers as educational technology nor as technological education but as a powerful to arouse and increase mathematical understanding.

**Featured Tweet**

I always like creating a proportional reasoning speed bump by giving these types of questions.

**Featured Comment**

]]>Hey! Nice idea for helping kids make the turn from proportional to linear relationships. There were two things I wanted to change:

• the discrete nature of the domain

• the way it’s not clear in the still images whether we are being shown the mass of just the marbles or the mass of the marbles + the glass together (the brief shot of the balance scale with the glass on it at the beginning of the video wasn’t doing it for me).So I made a video! Here! It was shot on my phone using a jar of cumin to stabilize, so it could certainly be professionalized.

Give an algebraic function whose graph has one positive root, a negative y-intercept, and an asymptote at x = -5, if that’s possible. If it’s impossible, explain why you can’t.

Maybe the student *can* determine the function. At some point, an advanced algebra student *should* determine the function. But what do I learn from a student who *can’t* determine the function? What does a blank graph tell me?

The student might understand what roots, intercepts, and asymptotes are. She might understand every part of the task except how to form the function algebraically. I won’t know because I’m asking a very formal task.

This is why a lot of secondary math teachers ask a *less* formal question first. They ask for a *sketch*.

Sketch a function whose graph has one positive root, a negative y-intercept, and an asymptote at x = -5, if that’s possible. If it’s impossible, explain why you can’t.

Think of what I know about the student that I didn’t know before. Think of the feedback that’s available to me now that wasn’t before.

Desmos just added sketching into its Activity Builder. That was the result of months of collaboration between our design, engineering, and teaching teams. That was also the result of our conviction that informal mathematical understanding is underrepresented in math classes and *massively* underrepresented in computer-based mathematics classes. We want to help students express their mathematical ideas and get feedback on those ideas, especially the ones that are informal and under development. That’s why we built sketch before multiple choice, for example. I’m stating this commitment publicly, hoping that one or more of you will help us live up to it.

But the biggest advantage of a tutor is not that they personalize the task, it’s that they personalize the explanation. They look into the eyes of the other person and try to understand what material the student has locked in their head that could be leveraged into new understandings. When they see a spark of insight, they head further down that path. When they don’t, they try new routes.

EdSurge misreads Mike pretty drastically, I think:

What if technology can offer explanations based on a student’s experience or interest, such as indie rock music?

Mike is summarizing what great face-to-face tutors do. They figure out what the student already knows, then throw hooks into that knowledge using metaphors and analogies and questions. That’s a personalized tutor.

But in 2016 computers are completely inept at that kind of personalization. Worse than your average high school junior tutoring on the side for gas money. Way worse than your average high school *teacher*. I don’t think this is a controversial observation. In a follow-up post, Michael Feldstein writes, “For now and the foreseeable future, no robot tutor in the sky is going to be able to take Mike’s place in those conversations.”

So it’s interesting to see how quickly EdSurge pivots to a *different* definition of personalization, one that’s much more accommodating of the limits of computers. EdSurge’s version of personalization asks the student to choose her favorite noun (eg. “indie rock music”) and watch as the computer incorporates that noun into the same explanation every other student receives. Find and replace. In 2016 computers are great at find and replace.

This is just a PSA to say: technofriendlies, I see you moving the goalposts! At the very least, let’s keep them at “high school junior-level tutor.”

**BTW**. I don’t think find-and-replacing “indie rock music” will improve what a student *knows*, but maybe it will affect her *interest* in knowing it. I’ve hassled edtech over that premise before. In my head, I always call that find-and-replacing approach the “poochification” of education, but I never know if that reference will land for anybody who isn’t inside my head.

My biggest professional breakthrough this last year was to understand that every idea in mathematics can be appreciated, understood, and practiced both formally *and also informally*.

In this activity, students first use their informal home language to describe how the red point turns into the blue point. Then, more formally, I ask them to predict where I’ll find the blue point given an arbitrary red point. Finally, and *most* formally, I ask them to describe the rule in algebraic notation. Answer: (a, b) -> (a/2, b/2).

It’s always harder for me to locate the informal expression of a idea than the formal. That’s for a number of reasons. It’s because I learned the formal most recently. It’s because the formal is often easier to assess, and easier for *machines* to assess especially. It’s because the formal is often more powerful than the informal. Write the algebraic rule and a computer can instantly locate the blue point for any red point. Your home language can’t do that.

But the informal expressions of an idea are often more interesting to students, if for no other reason than because they diversify the work students do in math and, consequently, diversify the ways students can be *good* at math.

The informal expressions aren’t just interesting work but they also make the formal expressions *easier to learn*. I suspect the evidence will be domain specific, but I look to Moschkovich’s work on the effect of home language on the development of mathematical language and Kasmer’s work on the effect of estimation on the development of mathematical models.

Therefore:

- Before I ask for a formal algebraic rule, I ask for an informal verbal rule.
- Before I ask for a graph, I ask for a sketch.
- Before I ask for a proof, I ask for a conjecture.
**David Wees**: Before I ask for conjectures, I ask for noticings.- Before I ask for a calculation, I ask for an estimate.
- Before I ask for a solution, I ask students to guess and check.
**Bridget Dunbar**: Before I ask for algebra, I ask for arithmetic.**Jamie Duncan**: Before I ask for formal definitions, I ask for informal descriptions.**Abe Hughes**: Before I ask for explanations, I ask for observations.**Maria Reverso**: Before I ask for standard algorithms, I ask for student-generated algorithms.**Maria Reverso**: Before I ask for standard units, I ask for non-standard units.**Kent Haines**: Before I ask for definitions, I ask for characteristics.**Andrew Knauft**: Before I ask for answers in print, I ask for answers in gesture.**Avery Pickford**: Before I ask for*complete*mathematical propositions, I ask for incomplete propositions.**Dan Finkel**: Before I ask for the general rule, I ask for a specific instance of the rule.**Dan Finkel**: Before I ask for the literal, I ask for an analogy.**Kristin Gray**: Before I ask for quadrants, I ask for directional language.**Jim Murray**: Before I ask for algorithms, I ask for patterns.**Nicola Vitale**: Before I ask for proofs, I ask for conjectures, questions, wonderings, and noticings.**Natalie Cogan**: Before I ask for an estimation, I ask for a really big and really small estimation.**Julie Conrad**: Before I ask for reasoning, I ask them to play/tinker.**Eileen Quinn Knight**: Before I ask for algorithms, I ask for shorthand.**Bill Thill**: Before I ask for definitions, I ask for examples and non-examples.**Larry Peterson**: Before I ask for symbols, I ask for words.**Andrew Gael**: Before I ask for “regrouping” and “borrowing,” I ask for grouping by tens and place value.

At this point, I could use your help in three ways:

- Offer more shades between informal and formal for the blue dot task. (I offered three.)
- Offer more SAT-style analogies. sketch : graph :: estimate : calculation :: [
*your turn*]. That work has begun on Twitter. - Or just do your usual thing where you talk amongst yourselves and let me eavesdrop on the best conversation on the Internet.

**BTW**. I’m grateful to Jennifer Wilson and her post which lodged the idea of a secret algebraic rule in my head.

**Featured Comment**

**Allison Krasnow** points us to Steve Phelp’s Guess My Rule activities.

David Wees reminds us that the van Hiele’s covered some of this ground already.

]]>Two examples from my recent past.

**Combining Like Terms**

Why did we invent the skill of combining like terms in an expression? Why not leave the terms *uncombined*? Maybe the terms are *fine*! Why disturb the terms?

One reason to combine like terms is that it’s easier to perform operations on the terms when they’re combined. So let’s put students in a place to *experience* that use:

Evaluate for x = -5:

3x + 5 + 2x

^{2}– 7 + 8x – 5x^{2}– 11x + 4 – 5x + 3x^{2}+ 4 + 3x – 6 + 2x + x^{2}

Put it on an opener. The expression simplifies to x^{2}, giving students an enormous incentive to learn to combine like terms before evaluating.

[I’m grateful to Annie Forest for bringing the example to mind. She also adds a context, if that’s what you’re into.]

**Parentheses**

When students first learn to graph points, the parentheses are the first convention they throw out the window. And it’s hard to blame them. If I told you to graph the point 2, 5, would you *need* the parentheses to know the point I’m talking about? No.

So why did mathematicians *invent* parentheses? What purpose do they serve, assuming that purpose isn’t “tormenting middle school students thousands of years in the future.”

It turns out that, while it’s very easy to graph a single point with or without parentheses, graphing *lots* of points becomes very difficult without the parentheses. So let’s put students in a place to *experience* that need:

Graph the coordinates:

-2, 3, 5, -2, 8, 1, -4, 0, -10, 4, -7, -3, -2, 7, 2, -5, -3

You can’t even easily tell if there are an even number of numbers!

[My thanks to various workshop participants for helping me understand this.]

**Closer**

The need for combining like terms is Harel’s *need for computation* and the need for parentheses is Harel’s *need for communication*. I can’t recommend his paper enough in which he outlines five needs for all of mathematics.

My point isn’t that we should avoid real-world or job-world applications of mathematics. My point is that for some mathematics that is actually impossible. But that doesn’t mean the mathematics was invented arbitrarily or for no reason or for *malicious* reasons. There was a *need*.

Math sometimes feels purposeless to students, a bunch of rules invented by people who wanted to make children miserable thousands of years in the future. We can put students in a place to experience those purposes instead.

**Previously**

We explored these ideas in a summer series.

]]>Probably 40 years ago, I was an invited guest at a national summer conference whose purpose was to grade the AP Examinations in Calculus. When I arrived, I found myself in the middle of a debate occasioned by the need to evaluate a particular student’s solution of a problem. The problem was to find the volume of a particular solid which was inside a unit three-dimensional cube. The student had set up the relevant integrals correctly, but had made a computational error at the end and came up with an answer in the millions. (He multiplied instead of dividing by some power of 10.) The two sides of the debate had very different ideas about how to allocate the ten possible points. Side 1 argued, “He set everything up correctly, he knew what he was doing, he made a silly numerical error, let’s take off a point.” Side 2 argued, “He must have been sound asleep! How can a solid inside a unit cube have a volume in the millions?! It shows no judgment at all. Let’s give him a point.”

What a fantastic dilemma.

Pollak argues that the student’s error would merit a larger deduction in an applied context than in a pure context. In a real-world context, being wrong by a factor of one million means cities drown, atoms obliterate each other, and species go extinct. In a pure math context, that same error is a more trivial matter of miscomputation.

The trouble is that, to the math teachers in the room, a unit cube *is* a real-world object. They can hold a one-centimeter unit cube in their hands and, more importantly, they can hold it in their minds.

The AP graders aren’t arguing about grading. They’re trying to decide *what is real*.

What a fantastic dilemma.

**Featured Comment**

Whenever I wanted to give students the most amount of partial credit, my coop teacher would ask me the poignant question, “What exactly are you assessing?” I found this was a great question to continue asking myself. So, in the example you gave, are you assessing students’ ability to perform mathematical functions correctly or are you assessing their ability to connect those math functions to the real world?

Benjamin Dickman alerts us that Pollak’s piece is online, free, along with a number of his modeling tasks.

]]>**Annie Forest** gives you ten ideas for your last week of class:

Here is my criteria for what makes a good mathy activities for the end of the school year: no/low tech; still incorporate math or problem solving in some way; fun and engaging.

Hey I’ll pitch one in! Here’s an eight-year-old blog post of mine. Every student starts with a 2D paper circle and by the end they’ve collaborated to construct a 3D icosahedron!

**Marissa Walczak** started carrying around a whiteboard as she helps students with their classwork:

If I wanted to show something to students I would always have to ask if I could write on their paper (which I really don’t ever want to do), or I’d have to say “wait for one sec” and then I’d go grab a piece of scratch paper, or I’d draw something on the board and then it’s far away from the group and then everyone sees it even though I don’t want everyone to see it.

**Christine Redemske’s** class takes Popcorn Picker to the literal limit, making cylinders that are shorter and shorter and wider and wider.

**Tina Cardone** gets a lot of mileage out of a very simply-stated arithmetic problem:

]]>In the next question students needed to decide what half of 2^50 would look like. All around the room students wrote 2^25. But children! We just talked about that! And then I realized that 1) it’s far from intuitive, that’s why they included more questions in the book to solidify this idea and 2) the language changed.