The Red Cross is asking for certified teachers to help out in the shelters. What would you do if you were asked to help? What kind of activities could be done in a shelter with children and a minimal amount of supplies?

What would you recommend?

Given those constraints, I’d probably use the print-based portions of Jo Boaler’s Week of Inspirational Math along with a bunch of the Tiny Math Games we collected several years ago.

I’d probably cry a bunch, too, I don’t know, Jesus man what a horrible situation.

]]>*Increasing compensation and benefits*, though not without asking for extraordinary concessions from teachers in some cases. (Related:*The teacher pay gap is wider than ever*.)*Lowering standards*, as in Utah, where you can now become a teacher without any additional training beyond content certification, or in Pennsylvania, where they’re relaxing the content certification requirements themselves.*Abusing the H1-B program*by hiring teachers from the Philippines, Spain, and Mexico.*Increasing recruitment efforts*by reaching out to college sophomores and juniors more aggressively, by reaching out to former district graduates who might want to teach at their alma mater.*Increasing new teacher retention through mentorship programs*, especially residency programs, which often pair new and veteran teachers, offer stipends, and require multi-year commitments.*Sending administrators back into the classroom*, in one case.*Doing not really much of anything that I can tell*, in another case.

This is to say nothing of macroeconomic trends that may require *federal* action. For example, the 2008 recession resulted in the layoff of newer teachers (hi!) in favor of veterans who are now retiring and it turns out those newer teachers didn’t stick around.

I don’t claim any kind of policy insight. I just find all of this interesting and a little terrifying. If the states are laboratories of democracy, we’re about to watch a number of lab experiments performed on our nation’s kids.

**Increasing Compensation and Benefits**

There’s a bill to raise teacher pay by 3 percent, and another proposal for 4 percent. The leader of the Senate, President Pro Tempore Del Marsh, R-Anniston, is working on a bill that could offer some teachers a higher rate of pay if they elect not to enter the teacher tenure system.

This year, lawmakers made some progress addressing the problem. They sent the governor a bill that would provide $500,000 for incentives to drive teachers to rural Colorado. The legislation would: Create education training programs through coordinators at colleges in rural parts of the state. Provide stipends to offset tuition costs for student teachers who agree to teach in rural schools. Establish programs in rural areas to identify high school students interested in teaching. Provide money to teachers in rural districts who pursue national certifications.

In Pinellas County, teachers can get up to $25,000 in incentive pay to work at low performing schools, which are currently under investigation by the U.S. Department of Education. But to get the full amount, they must have a master’s degree, be at the top of the pay scale, work 90 extra minutes a day, plus some Saturdays — and attend summer sessions. “There’s an ambulance waiting for them at the end of the rainbow there, because it’s almost undoable,” Gandolfo said.

The Blue Ribbon Commission on the Recruitment and Retention of Excellent Educators – a panel of 49 state office holders and educators – last year issued a report with suggested solutions for state legislators to consider in the 2016 session, however, only one related bill passed. It sets up the New Generation Hoosier Educators Scholarship, which offers up to $7,500 per year for students who commit to teaching in Indiana for at least five years.

Iowa:

Iowa College Aid has announced that 126 teachers were offered awards as part of the Iowa Teacher Shortage Loan Forgiveness Program. The average repayment was $5,493, for a total of $692,171.

The Legislature has supported the 2013 recommendations from Gov. Butch Otter’s education task force, which were created to provide Idaho with an unprecedented five-year plan to improve K-12 education and teacher satisfaction. One of those recommendations — the estimated $125 million dollar career ladder — increases teacher pay. Another piece — the state’s leadership premium package — designates additional money for hard-to-fill positions.

Moore said that his district has offered candidates for positions such benefits as a four-day school week, health benefits and a competitive salary, and candidates are still not willing to accept positions in rural areas.

The school district is currently in negotiations with the tribe in Owyhee to see if they will be able to provide housing for teachers. However, Zander acknowledge that finding housing in areas like West Wendover where there isn’t any teacher specific housing available may still be a challenge.

“When trying to find in-demand teachers, districts can negotiate certain options such as initial placement on the salary guide, where the Superintendent does have the ability to recommend that a teacher be place on a higher level,” said Belluscio. “And there’s also the concept of signing bonuses for teachers who would be in a difficult to recruit area.”

Bracy said part of the state’s teacher shortage has to do with fewer college students enrolling in education majors. Another part has to do with low pay, he said. “It sometimes makes the teaching profession not as attractive as others,” Bracy said. Governor Pat McCrory recently signed the state budget, which would allow a 5 percent increase in pay for teachers. That means the average teacher could make close to $50,000.

The state’s teacher shortage loan forgiveness program has so many applicants that it can no longer fund all who apply, North Dakota University System financial aid director Brenda Zastoupil told the Education Standards and Practices Board on Thursday. This year, 628 teachers applied to receive up to $1,000 in forgiveness from their college loans. Last year, the number was 525 and the year before, 471. The program allows teachers to apply multiple years for a maximum of $3,000.

Voters will have a chance to help teachers out come November due to a proposed state question that will raise sales taxes by 1 percent statewide, which is expected to generate $615 million annually, with about 70 percent designated for a $5,000 pay raise for Oklahoma teachers and other funding for K-12 schools. Republican Gov. Mary Fallin also has discussed a possible special session that could restore some education funding.

Frances Welch, dean of the College of Charleston’s Teacher Education Department, said she has seen student enrollment in her department shrinking for years now. She said the state needs to expand student loan repayment programs, such as the Teaching Fellows scholarship, and minority recruitment programs, such as Call Me Mister.

The Office of the Superintendent of Public Instruction asked the Legislature this year to raise the salary for beginning teachers, and add signing bonuses and other incentives to make the profession more attractive.

Yet, the salary issue cannot be ignored. According to the Wisconsin Department of Public Instruction, the average public school teacher had 14 years of experience and made a salary of $49,908 in 2013-’14. By comparison, the U.S. Census Bureau calculated that the average salary for a bachelor’s degree holder nationwide in the same year was $62,048.

**Lowering Standards**

Along with the substitute regulation change, the state board has proposed to eliminate the state regulations requiring that teachers be “highly qualified.”

He’s one of the many Savannah-Chatham schools hired through the Alternative Pathways to Teaching program. It’s a way those with a Bachelor’s degree in any field can become a teacher in Georgia, through a work-as-you-go certification program that takes one to three years to complete.

The board discussed the possibility of busing students to Stanfield, or using curriculum the ODE has established for small schools where an adult supervises classroom learning but does not have to be a licensed math teacher.

The prospect is sufficiently worrisome to Pennsylvania Department of Education officials that they are eliminating obstacles to becoming a teacher – for instance, by easing math requirements on the basic skills assessments that teaching students take as sophomores and have struggled to pass.

Utah:

Utah has a severe teacher shortage, so it decided to do something about it. Under a new rule, schools can now hire people to teach who have no training in the profession. None whatsoever.

**Abusing the H1-B Program**

At the start of the summer, Goodsell hired a few recent graduates from out-of-state education colleges. Next, he began Skyping with high-school math and science teachers in the Philippines who had applied for work visas to teach in the United States. He selected 11.

Lewis said her district hopes to fill math, science and special education jobs with teachers from the Philippines. Capitalizing on connections of previously recruited Filipino teachers, officials conduct interviews via Skype and, if successful, navigate the long process of securing an H-1B visa.

Superintendents in Jackson Public Schools, Noxubee County, Holmes County, Meridian and Gulfport have all hired foreign teachers on temporary visas called H-1Bs, along with others. According to MDE, there are 451 teachers in the state with degrees from outside the country.

Dallas employs some 350 teachers from Puerto Rico, including some who have become administrators. The district’s H-1B visa program recently brought in 30 Mexican teachers who will stay for three years. Dallas ISD also takes part in the Texas-Spain Visiting Teacher Program, a J-1 visa initiative that each year brings about 75 Spanish teachers to the district for three years.

**Increasing Recruitment Efforts**

Wilson says with money tight, they’ve been all about hiring quality teachers instead of just filling their quote. He says that all starts with getting good student teachers from [Southern Illinois University].

Krysl says because of the hiring freeze, they looked for talent from colleges, towns, and states they hadn’t actively recruited from. That’s where they found teachers like Amy Shamp who moved here from Owatanna, Minnesota to teach at Curtis Middle School.

Murphy said the school system has been working to stay in contact with graduates interested in becoming teachers to return to Charles County. “We’re hoping to create that pipeline and connect with those students throughout the years,” Murphy said.

Education Minnesota is proposing several solutions to attract and keep good teachers. The union wants to build a pipeline program to get students interested in teaching as early as high school. It also wants to change the licensure process and provide a stronger financial incentive. Once teachers get into their careers the union wants more collaboration between educators and administrators, stronger financial benefits and more professional development.

“Our students need, and our districts need, creative bright young people to enter the profession,” he said. “The way to encourage that is to maintain high standards but for our policymakers to show that they value teachers and they respect teachers and they listen to the voice of teachers when they make education policy.”

“The unfortunate reality is fewer people today are choosing to enter the education field and lead our classrooms,” said Rhode Island Teach for America Director Heather Tow-Yick. “There are a number of factors contributing to this: we’re seeing an increasingly polarized public conversation around education; the broader economy is improving and people who experienced the national recession during their college careers are showing more interest in what they see as financially sustainable professions; teacher satisfaction has dipped steeply in recent years; and education has been deprioritized by most Americans when they rank major issues facing our country.”

“I think it’s all about getting people to go into the teaching profession and that means talking to maybe junior and seniors in high school or freshman and sophomores in college,” he said.

**Increasing New Teacher Retention through Mentorship**

Ritter in 2012 founded the Arkansas Teacher Corps, which recruits the highest achievers from all majors and fields to teach in districts that struggle to hire and retain quality teachers. Corps teachers receive a $15,000 stipend, paid over three years, in addition to their regular teaching salary for working in such a district.

“More teachers are leaving, and fewer teachers are going into the profession,” said Corey Rosenlee, president of the Hawaii State Teachers Association, which represents 13,500 teachers. “We cannot find even emergency hires for these positions.” DOE officials contend that the hiring process is selective and that more focus on a mentoring program for new teachers will continue to boost retention rates.

In Richmond, a more intensive teacher residency is helping retain both new and veteran teachers, said Jan Tusing, a teacher leader-in-residence at VCU’s Center for Teacher Leadership, which runs the residency program. The residency program is an intensive multiyear process where new teachers are paired with a co-teacher. The program’s original goal was to provide extra help for new teachers, but it ended up helping retain the experienced teachers who served as mentors also, Tusing said. That’s a win-win for districts.

Ritter in 2012 founded the Arkansas Teacher Corps, which recruits the highest achievers from all majors and fields to teach in districts that struggle to hire and retain quality teachers. Corps teachers receive a $15,000 stipend, paid over three years, in addition to their regular teaching salary for working in such a district.

**Sending Administrators Back into the Classroom**

“You have so many more schools than you did 10 years ago,” Armenta said. “We’ll do what we did last year, and that is, in those classrooms where we haven’t found teachers yet, we will send in administrators from city center again to fill in.” That means administrators who have desk jobs now but are still licensed to teach will be in the classroom.

**Doing Not Really Much of Anything That I Can Tell**

“We’re looking at issues in terms of effort, in terms of teaching our teaching workforce overall, making sure our students are able to navigate properly with the numeracy aspects in the classroom, and making sure they can really solve the problems necessary. So there’s not just one silver bullet, there’s a lot of things in motion here.”

**Featured Comment**:

My only complaint is some of your pejorative language (e.g., “abusing work visas”; “experimenting on students”). You may have good reason to disagree with some or all of the responses, but it seems that this requires some argument rather than name-calling.

That these states are performing experiments on kids seems just empirically true to me.

I am, indeed, editorializing about the H1-B. That program was meant to staff hard-to-find individuals in specialty fields. While I agree that teaching is a specialty field, teachers aren’t hard to find so much as hard to compensate and treat in a way that’s fair. Teachers are, in my view, underpaid, undervalued, overworked, and deprofessionalized. The solution is to fix those working conditions, not exploit an immigration loophole to find people who will tolerate them.

**Featured Tweets**

1. Part of neoliberal ed reform has been to denigrate teacher prep. Enrollment down in most ed programs in the US. This cuts supply.

— Ilana Horn (@ilana_horn) August 16, 2016

2. Re: H1-B visas: as @LoraBartlett's research showed, many overseas trained teachers send remittances home & live in economic precarity.

— Ilana Horn (@ilana_horn) August 16, 2016

This makes them less able to advocate for themselves as workers.

— Ilana Horn (@ilana_horn) August 16, 2016

3. As @DanaGoldstein and others have talked about, shortages also a result of turnover. Best strategy on Dan's list: retention & pay.

— Ilana Horn (@ilana_horn) August 16, 2016

@ddmeyer bill for separate, non tenure salary schedule in AL didn't pass last session. But this did…https://t.co/28cQgL528j

— Dr Amy Fowler Murphy (@amykfmurphy) August 16, 2016

Karim Ani, the founder of Mathalicious, hassles me because I design problems about water tanks while Mathalicious tackles issues of greater sociological importance. Traditionalists like Barry Garelick see my 3-Act Math project as superficial multimedia whizbangery and wonder why we don’t just stick with thirty spiraled practice problems every night when that’s worked pretty well for the world so far. Basically everybody I follow on Twitter cast a disapproving eye at posts trying to turn Pokémon Go into the future of education, posts which no one will admit to having written in three months, once Pokémon Go has fallen farther out of the public eye than Angry Birds.

So this 3-Act math task is *bound* to disappoint everybody above. It’s a trivial question about a piece of pop culture ephemera wrapped up in multimedia whizbangery.

But I had to *testify*. That’s what this has always been – a testimonial – where by “this” I mean this blog, these tasks, and my career in math education to date.

I don’t care about Pokémon Go. I don’t care about multimedia. I don’t care about the sociological importance of a question.

I care about math’s power to puzzle a person and then help that person unpuzzle herself. I want my work always to testify to that power.

So when I read this article about how people were tricking their smartphones into thinking they were walking (for the sake of achievements in Pokémon Go), I was *puzzled*. I was curious about *other* objects that spin, and then about ceiling fans, and then I wondered how long a ceiling fan would have to spin before it had “walked” a necessary number of kilometers. I couldn’t resist the question.

That doesn’t mean *you’ll* find the question irresistible, or that I think you *should*. But I feel an enormous burden to testify to *my* curiosity. That isn’t simple.

“Math *is* fun,” argues mathematics professor Robert Craigen. “It takes effort to make it otherwise.” But nothing is actually like that – intrinsically interesting or uninteresting. Every last thing – pure math, applied math, your favorite movie, *everything* – requires humans like ourselves to testify on its behalf.

In one kind of testimonial, I’d stand in front of a class and read the article word-for-word. Then I’d work out all of this math in front of students on the board. I would circle the answer and step back.

But everything I’ve read and experienced has taught me that this would be a lousy testimonial. My curiosity wouldn’t become anybody else’s.

Meanwhile, multimedia allows me to develop a question with students as I experienced it, to postpone helpful tools, information, and resources until they’re necessary, and to show the resolution of that question as it exists in the world itself.

I don’t care about the multimedia. I care about the testimonial. Curiosity is my project. Multimedia lets me testify on its behalf.

So why are you here? What is your project? I care much less about the *specifics* of your project than I care how you *testify* on its behalf.

I care about Talking Points much less than Elizabeth Statmore. I care about math mistakes much less than Michael Pershan. I care about elementary math education much less than Tracy Zager and Joe Schwartz. I care about equity much less than Danny Brown and identity much less than Ilana Horn. I care about pure mathematics much less than Sam Shah and Gordi Hamilton. I care about sociological importance much less than Mathalicious. I care about applications of math to art and creativity much less than Anna Weltman.

But I *love* how each one of them testifies on behalf of their project. When any of them takes the stand to testify, I’m *locked* in. They make their project my own.

Again:

Why are you here? What is your project? How do you testify on its behalf?

**Related**: How Do You Turn Something Interesting Into Something Challenging?

[Download the goods.]

]]>Here is a *very* valuable conjecture:

The spelling of every whole number shares at least one letter with the spelling of the

nextwhole number.

Which is to say that:

- “one” and “two” both share an “o”
- “two” and “three” both share a “t”
- etc.

Could that possibly be true for *every* whole number?

If I were starting a course on geometry or a unit on proof or an activity on deductive logic, I would introduce this conjecture very early in the process. Let me explain what I find so very valuable about this conjecture.

Deduction is hard. It’s an abstract mental act that *adults* find difficult. (See: the van Hiele’s and their levels.) Too often we rush students to that abstract act, rushing them past the lower van Hiele levels, and we ask them to argue deductively about objects that, to them, are *also* abstract.

I suspect that, to many students, those proof prompts read something like this:

Given that the base bangles are twice the tonnage of the circumwhoozle and the diagonalized matrox is invertible, prove that all altimeters cross the equation at Quito, Ecuador.

The word “prove” is weird. And, unfortunately, so is every other word in the sentence.

So I cherish opportunities to help students argue deductively with *concrete* objects, which is what we’re working with here, with the spelling of whole numbers. This conjecture also gives students several different angles on the proof act.

You can ask students to find a counterexample, for example, a useful strategy when first interrogating a conjecture.

Once students have tried several different numbers they may satisfy themselves that the conjecture is true. This is one of the naive proof schemes Harel & Sowder observed in the students they studied. When this proof scheme surfaces in conjectures about geometric shapes, it’s challenging to summon up one new shape after another to challenge the student’s proof by example. It’s trivial, by comparison, to summon up one new *number* after another and ask the student to check her hypothesis again.

At a certain point in this process, likely after you give several numbers in the millions, your students may transform in two ways:

- They’ll get tired of trying example after example. “Proof by examples means you have to try
*all*the examples,” you can say, giving you both a moment to reflect on the need for a more rigorous proof scheme, like deductive reasoning. - They’ll notice that every number in the millions shares an “n” with every
*other*number in the millions. And same for the billions. And same for the trillions. And … same for the*hundreds*. And*so on*.

And suddenly we’re on our way to a proof by *exhaustion*, which is much more rigorous than a proof by example. Nice.

This conjecture also leaves ample room for you and your students to pose *follow-up* conjectures. Like, “Does it work for all integers, or just whole numbers?”

I saw the conjecture and saw its value immediately. This is a very valuable kind of conjecture, I thought. But I don’t have many of them. Do you have another you can trade?

[via Futility Closet.]

**BTW**. You’re worse off in at least one way now than before you knew the conjecture was true. Now, when you ask your students, “Could that *possibly* be true?” you’re going to have to *pretend*.

**Featured Comments**:

Max:

What I like about game strategies is you go from “what seems to work,” to “will this always work,” to “here’s why this does/doesn’t work” pretty seamlessly.

Is it true for any other language with an alphabet? It fails for German (5/6 using umlauts, 7/8 otherwise), French (2/3), and Spanish (7/8).

What letter(s) of the alphabet do(es) not appear in the spelling of the first 999 whole numbers? Prove it.

]]>List all the factors of every number from 1-100. What do you notice? Which numbers have an even number of factors? An odd number? What do you notice about numbers with an odd number of factors? Can you prove which numbers beyond 100 will have an odd number of factors?

- Invite students to try a task that is intuitive, but inefficient or inaccurate.
- Help them understand some math.
- Invite them to
*re-try*the task and see that with math it’s more efficient and accurate.

That’s an instructional design pattern meant to help students see that the math they learn is power rather than punishment. Most instructional resources do a great job at #2, which they decorate with images of *other people* using that math in their lives. *Some* resources invite students to use the math themselves in #3. But without experiencing #1 the advantage of math may be unclear. “Why do I need to learn this stuff?” they may ask. “I could have done this by guesswork just as easily.”

We should show them the limits of guesswork.

Last week’s installment of Who Wore It Best looked at three textbooks each trying to exploit billiards as a context in geometry. None of the textbooks applied all three steps. I needed a resource that didn’t exist and I spent two days building it. Here is how it works.

**Inefficient & Inaccurate**

Play this video. Maybe twice.

Ask students to write down their estimates for all eight shots on this handout.

For instance:

@ddmeyer CBCAABAB Great discourse between hubby and I. I want a paper version so I can sketch and protractor. Probably all wrong.

— Adrianne Burns (@a_schindy) July 19, 2016

**Some Math**

Several of the textbooks simply *assert* the principle that the incoming angle of the pool ball is congruent to the outgoing angle. Based on Schwartz & Martin’s work on contrasting cases, I’ll offer students this page as preparation for future instruction.

What do you notice about the reals that isn’t true about the fakes?

**2016 Jul 31**. Edited to add this literature review, which elaborates the positive effect of contrasting cases (and building explanations on student solutions) in more detail.

**2016 Jul 31**. Also, in the spirit of “you can always add, you can’t subtract,” I’m sure that before I showed all four contrasting cases and the labels “real” and “fake,” I’d show the individual cases *without* those labels. Students can make predictions without the labels.

**Efficient & Accurate**

Now that they have an introduction to the principle that the incoming angle and the outgoing angle are congruent, ask them to apply it, now with analysis instead of intuition. Have them record those calculations next to their estimates.

Then show them the answer video.

Have the students tally up the difference between their correct calculations and their correct estimates. If that isn’t a positive number, we’re in trouble, and essentially forced to admit that the math we asked them to learn isn’t actually powerful.

I’ll wager your class average is positive, though, and on the last three shots, which bank off of multiple cushions, *very* positive.

Because math is power, not punishment.

[Download the goods.]

**2016 Jul 26**. I have changed a pretty significant aspect of the problem setup after receiving feedback from Scott Farrar and Riley Eynon-Lynch. Thanks, team.

**2016 Jul 26**. I’ll be changing the name of this activity shortly, on request from a Chicago educator who thinks his students will read violence into the title. That makes sense to me.

**2016 Jul 28**. Changed to “Pool Bounce.” I am amazing at titles.

**Featured Comment**

]]>I love this partly because the fake ones

lookfake, and students have to think about why and are given materials to test their hypothesis. You’re making students refine their intuition to include mathematical precision, which they can then use to solve the rest. I feel like this honors and builds on the knowledge they already have in a way that’s far more motivational than throwing out some big-words statement about angles of incidence and reflection.

**What You Said**

In the preview post, commenters called out the following turn-offs in different versions.

- “It jumps to the math notation too quickly.”
- “There is a ton of language in these problems.”
- “Two of the books just state that the angle of incidence and angle of reflection are the same and the other just expects students to
*know*that.” - “I feel like if I sat down and solved the problem that follows their explanation, I’d be copying their steps rather than really thinking it out for myself in a way that would make sense of it.”

On Twitter, Rose Roberts urges us to *be careful here* as, “Problems involving pool and mini-golf were *the reason* I decided I hated geometry in 8th grade. The sole reason.”

I’ll try to summarize the critiques using language that’s common to this blog without putting *too* many words in my commenters mouths. These textbook treatments rush to a formal level of abstraction too quickly. They don’t do a sufficient job developing the question for which “angle of incidence = angle of reflection” is the answer, or helping students develop an *intuition* about that answer.

In *Discovering Geometry*, for example, the formal equivalence statement is given and then the text asks students to apply it with their protractor.

A number of my commenters offer variations on, “Just take ’em to the pool hall!” This idea *sounds* great and will scan to many as suitably progressive, inquiry-based, student-centered, etc. But I’m unsatisfied. Mr. Bishop took us to the pool hall when I was a high school student and let us watch a local pro knock down a rack. I think he let us shoot a bit ourselves. I remember enjoying myself. I don’t remember learning more math than I did in his classroom lesson.

Pro pool players don’t use protractors.

For one reason, they’ve internalized that mathematics through practice. For another, the player can’t measure the angle of the ball in real time. The ball moves too quickly and the pool player’s eye-level view of the pool table is unlike the bird’s-eye view that would allow her to measure that angle.

This is a problem.

**What I Need**

Here is the resource I need. I’d like students to experience mathematical analysis as *power*, rather than *punishment*.

So let’s start with a tool that comes easily to students: their intuition. Let’s invite them to use their intuition in the context of a pool table. And let’s establish the context so that their intuition *fails* them, or at most earns a C-.

Then, let’s help students learn how to analyze the path of the pool ball *mathematically*. We’ll repeat the previous exercise and point at the end to the superior results that accrue when students analyze the pool table m*mathematically* instead of *intuitively*. (If superior results *don’t* accrue, we should either re-design the context to better highlight math’s power on a pool table or admit to ourselves we were wrong about math’s power.)

John Golden gets us *close* to that resource, inviting teachers to pull out still frames from this video of billiard shots for student analysis. But that analysis is much more complex than the level of the textbooks we’re critiquing today. Billiards ricochet off of other *billiards* in that video.

The resource I need doesn’t seem to exist yet, so I’ll try to build it. I’ll start with this game. Stay tuned.

]]>Here are eight different yet interacting moving parts that I believe has to go into any reform aimed at creating a high-achieving school using technology to prepare children and youth to enter a career or complete college (or both).

Notably, none of them are explicitly about technology.

]]>“These are the dimensions of the rectangle that has the largest area given a fixed perimeter.”

“WHAT IS A SQUARE!” I yell out while my competitors are still thinking quietly. I have disqualified myself and ruined the round, but I don’t care. I start high-kicking around the set while security tries to wrangle me away and I *still* don’t care because I finally found some use for this fact that takes up a *significant chunk* of my brain’s random access memory.

It’s a question you’ll find in every quadratics unit, every textbook, everywhere. I could have selected this week’s Who Wore It Best contestants from any print textbook, but instead I’d like to compare *digital* curricula. I have included links and attachments below to versions of the same task from GeoGebra, Desmos, and Texas Instruments, three thoughtful companies all doing interesting work in math edtech. (Disclosure: I work for Desmos, but don’t let that fact sweeten your remarks about the Desmos version or sour your remarks about the others. Just be thoughtful.)

So: who wore it best?

Click each image for the full version.

**Version #1 – GeoGebra**

**Version #2 – Desmos**

**Version #3 – Texas Instruments**

Steve Phelps suspects I stacked the deck in favor of Desmos here, taking full advantage of our platform while taking only *partial* advantage of GeoGebra and the Nspire. John Golden concurs, hypothesizing that “there would be a worksheet to go with the GeoGebra sketch.”

So a note on sampling: the GeoGebra example is the most viewed lesson on the subject I could find at their Materials site. The Texas Instruments lesson is the *only* lesson on the subject I could find at their Activities site. I told Steve, and I’ll tell you, that if anybody can come up with a better lesson on either platform. I’ll be happy to feature it. This isn’t much fun for me (or useful to Desmos) if I stack the deck.

Both Lisa Bejarano and John Golden call out the Desmos lesson as “too helpful” – they know how to make it sting – in the transition from screen 5 (“Collecting data!”) to screen 6 (“Here! We’ll represent the data as a graph for you.”).

I’l grant that it seems *abrupt*. I don’t think this kind of help is *necessarily* counterproductive, but it doesn’t seem as though we’ve developed the question well enough that the answer – “graph the data!” – is sensible. The Texas Instruments version has a solution to that problem I’ll attend to in a moment.

My concern with the GeoGebra applet is that the person who made the applet has done the most interesting mathematical thinking. I *love* creating Geogebra applets. I generally *don’t* have a good story for what students *do* with those applets, though. In this example, I suspect the student will drag the slider backwards and forwards, watching for when the numbers go from small to big and then small again, and then notice that the rectangle at that point is a square. The person who made the applet did much more interesting work.

Let me close with one item I prefer about the Desmos treatment and one item I prefer about the Texas Instruments treatment.

First, my understanding of Lisa Kasmer’s research into estimation and Paul Silvia’s research into interest led me to create this screen where I ask students “Which of these three fields has the biggest perimeter?” knowing full well they all have the *same* perimeter:

Still later, I ask students to estimate a rectangle they think will have the *greatest* area. That kind of informal cognitive work is largely absent from the TI version, which starts much more formally by comparison.

TI does have a technological advantage when they allow students to sample lots of rectangles and quickly capture data about those rectangles in a table.

Desmos is working on its own solution there, but for now, we punt and include prefabricated data, which I think both companies would agree is less interesting, less useful, and more abrupt, as I mentioned above.

That’s my analysis of these three computer-based approaches to the same problem. What’s your analysis? And it’s also worth asking, “Would a *non-*computer-based approach be even better?” Is the technology just getting in the way of student learning?

You can also pitch your thoughts in on next week’s installment: Pool Table Math.

**2016 Jul 8**. Steve Phelps has created a different GeoGebra applet, as has Scott Farrar.

**2016 Jul 9**. Harry O’Malley uploads another GeoGebra interpretation, one that strikes a very interesting balance between print and digital media.

- In what ways are they different?
- What do their differences say about their authors’ beliefs about students, learning, and math?
- Would you make changes? Which and why?

Every secondary teacher and secondary textbook author knows that parabolas are #realworld because they describe the path of projectiles subject to gravity. Forgive me. “Projectiles” are not #realworld. “Baseballs” are #realworld.

But let’s not relax simply because we’ve drawn a line between the math inside the classroom and the student’s world outside the classroom. Three different textbooks will treat that application three different ways.

Click each image for a larger version.

**Version #1**

**Version #2**

**Version #3**

Chris Hunter claims, “The similarities here overwhelm any differences.” That’s probably true. So let’s talk about some of those similarities and what we can do about them.

**My Least Favorite Phrase in Any Math Textbook**

They each include the phrase “is modeled by,” which is perhaps my least favorite phrase in any math textbook. Whenever you see that phrase, you know it is *preceded* by some kind of real world phenomenon and *proceeded* by some kind of algebraic representation of that phenomenon, a representation that’s often incomprehensible and likely a lie. eg. The quartic equation that models snowboarding participation. No.

Chris Hunter notes that the equations “come from nowhere” and seem like “magic.” True.

@dmcimato and John Rowe point out that what *normal people* wonder about baseball and what *these curriculum authors* wonder about baseball are not at all the same thing.

That isn’t *necessarily* a problem. Maybe we think we should ask the authors’ questions anyway. As John Mason wrote in a comment on this very blog on the day that I now refer to around the house as John Mason Wrote a Comment on My Blog Day:

Schools as institutions are responsible for bringing students into contact with ideas, ways of thinking, perceiving etc. that they might not encounter if left to their own devices.

But these questions are *really* strange and feel exploitative. If we’re going to *use*, rather than *exploit*, baseball as a context for parabolic motion, let’s ask a question like: “Will the ball go over the fence?”

And let’s acknowledge that *during the game* no baseball player will perform *any* of those calculations. This is not *job-world math*. So the pitch I’d like to make to students (heh) is that, yes, your intuition will serve you pretty well when it comes to answering both of those questions above, but *calculations* will serve you even better.

Ethan Weker suggests using a video, or some other visual. I think this is wise, not because “kids like YouTubes,” but because it’s easier to access our intuition when we see a ball sailing through the air than when we see an equation describing the same motion.

Here’s what I mean. Guess which of these baseballs clears the fence:

Now guess which of *these* baseballs clears the fence:

They’re different representations of the *same* baseballs – equations and visuals – but your intuition is more accessible with the visuals.

We can ask students to solve by graphing or, if we’d like them to use the equations, we can crop out the fence. If we’d like students to work with time instead of position, we can add an outfielder and ask, “Will the outfielder catch the ball before it hits the ground?”

This has turned into more of a Makeover Monday than a Who Wore It Best Wednesday and I shall try in the future to select examples of problems that differ in more significant ways than these. Regardless, I love how our existing curricula offer us so many interesting insights into mathematics, learning, and curriculum design.

**Featured Comment**

]]>I’ll throw ours into the ring: In which MLB park is it hardest to hit a home run?

In the report, “Equations and Inequalities: Making Mathematics Accessible to All,” published on June 20, 2016, researchers looked at math instruction in 64 countries and regions around the world, and found that the difference between the math scores of 15-year-old students who were the most exposed to pure math tasks and those who were least exposed was the equivalent of almost two years of education.

The people you’d imagine would crow about these findings are, indeed, crowing about them. If I were the sort of person inclined to ignore differences between correlation and causation, I might take from this study that “applied math is bad for children.” A less partisan reading would notice that OECD didn’t attempt to control the pure math group for *exposure to applied math*. We’d expect students who have had exposure to *both* to have a better shot at transferring their skills to new problems on PISA. Students who have only learned skills in one concrete context often don’t recognize when new concrete contexts ask for those exact same skills.

If you wanted to conclude that “applied math is worse for children than pure math” you’d need a study where participants were assigned to groups where they *only* received those kinds of instruction. That isn’t the study we have.

The OECD’s own interpretations are much more modest and will surprise very few onlookers:

- “This suggests that simply including some references to the real-world in mathematics instruction does not automatically transform a routine task into a good problem” (p. 14).
- “Grounding mathematics using concrete contexts can thus potentially limit its applicability to similar situations in which just the surface details are changed, particularly for low-performers” (p. 58).

**BTW**. I was asked about the report on Twitter, probably because I’m seen as someone who is super enthusiastic about applied math. I *am* that, but I’m also super enthusiastic about *pure* math, and I responded that I don’t tend to find categories like “pure” and “applied” math all that helpful. I try to wonder instead, what kind of cognitive and social work are students *doing* in those contexts?

**BTW**. Awhile back I wrote that, “At a time when everybody seems to have an opinion or a comment [about mathematics education], it’s really hard for me to locate NCTM’s opinion or comment.” So credit where it’s due: it was nice to see NCTM Past President Diane Briars pop up in the article for an extended response.

**Featured Comment**:

]]>What is often overlooked in these kind of studies is the students who are enrolled in the various courses. The correlation between pure math courses and higher level math exists because higher achieving students are placed in the pure math classes, while lower performing students are placed in applied math.

Same thing is true for studies that claim that students who take calculus are the most likely to succeed in college. No duh! That is because those who are most likely to succeed in college take calculus.

The course work does not cause the discrepancy, the discrepancy determines the course work.