And the other part of the problem is the kid’s maturity and independence. If you have a teen who is enthusiastic about math, that’s fine, they’ll do their schoolwork and then noodle around on Numberphile or Desmos. But younger children frequently lack the ability to direct their own mathematical lives. They want an adult to lead.

If you put those two pieces together — number-centricity and the need for adult guidance — it makes it very hard to give a young child what they want without accelerating them through the K-5 math curriculum.

I maintain that it’s worth trying to avoid as much of the school curriculum at home as you can. If you can get beyond the fun of “hey my six-year old is doing *ten-year old math*” you’re at risk of making it harder for schools to keep your kid happy. And while I’m not opposed to straight-up acceleration or grade-skipping in some cases, in general I think it’s best to avoid these with very young children, who however advanced they are in math often have totally age-typical emotional needs. (This is why, for all my love of Beast Academy, I don’t typically recommend it to parents.)

I’m trying to speak generally here, but without being super-weird about it I’ve encountered this both at school and at home. My oldest son is very enthusiastic about numbers, and I think it’s in his best interests to not race ahead too far of his classmates. And at school I have tried to express the perspective above to parents of my 3rd and 4th Graders, to mixed success. (Thankfully, most parents get it.)

OK, but all this leads to a question: what do you *do *with a kid who wants to talk about numbers during breakfast? and on the subway? and over dinner? and after dinner? and at many other times, especially when he’s bored and moody?

It’s in that spirit that I share some of the things that we’ve been talking about in my own home lately. Some of these I proposed to my oldest. Others he came up with on his own. I’ll add that in my teaching life I sometimes encounter accelerated students who totally lack the ability to ask questions independently. That doesn’t make sense to me, since often as kids become more independent they can get more out of their grade-level classes.

Anyway, here are some of the number-centric ideas my son and I have been talking about lately.

**Switcheroo Numbers**: Take a two-digit number, and swap its digits. 35 and 53. 72 and 27. 19 and 91. Then take the difference of those numbers. You will find that all those numbers are _________________. Why is that? Does it work with three-digit numbers? Four-digit numbers? Consider representing these numbers as sums — 30 + 5 and 50 + 3 — to play with some of these ideas.

**Writing a Scratch Program for Counting Factors: **My son is most eager to talk about factors, so we often end up drifting towards factor math. Together we made a Scratch program that finds all the factors of a number. I let him tinker around with it and he was able to show some indpenedence there, which I was gratified to see — coding is something unrelated to K-5 math that he can do outside of school. Together we added a factor counter to this program, so he could try to “set records” and find numbers with the most number of factors.

**Factor/Number Fraction: **Conversations about which numbers had a lot of factors led to discussions about the relationship of the number to its factors. My son loved talking about perfect, abundant, and deficient numbers. I, on the other hand, find this stuff dull as dirt, so I proposed a different question. 3 has two factors, so its factor to number fraction is 2 out of 3, or 2/3. 4 has three factors, so its fraction is 3/4. What numbers have the largest of these sorts of fractions?

**Building Bigger Factors: **This one also came from me, I thought he would like seeing how you could use multiplication to find factors of bigger numbers. For instance, the factors of 10 are 1, 2, 5, and 10. The factors of 9 are 1, 3 and 9. What are the factors of 9 x 10 = 90? How many factors does it have? And (if this is new territory for you) you might discover that there is a pattern that seems to hold for some, but not all numbers. (For instance, compare the factors of 9 x 10 and 5 x 10.)

**Triangular Square Numbers: **I don’t know if my boy got into triangular numbers from Marilyn Burns’ books or from Numberblocks, the TV show. But it might have been Numberblocks, with their annoyingly catchy “step squad” chant. You know the one I’m talking about.

Anyway, are any triangular numbers also square? That made for a few days of fun.

**The Difference Between Nearby Multiplications: **I don’t know why, a lot of my son’s math has to do with finding the difference between slightly different things. For instance, he came into the kitchen the other day eager to share a “trick.” Start with two numbers, like 2 and 7. Make 2 x 8 and 3 x 7. Find the difference between them. He thought there was a pattern but it took him a few false before he figured out what was going on, and then it was interesting to try to explain what was happening and why.

**The Sum of Consecutive Powers: **This one came from him, I think. Or maybe it was a book he was reading? I’m not sure. Either way, add consecutive powers of 2, like 4 + 8 = 12 or 32 + 64 = 96. His claim was that these numbers are always divisible by 3. Really cool, right? Try it with other bases besides 2. Why is this happening? Fun times.

**The “One and a Half” Sequence: **This one is very much kid-math, but I think it’s interesting and it occupied my son for a while. He spent a few days (or weeks? what is time?) looking for numbers that were “one and a halfs.” For example, for him 3 is a one and a half (because it is 2 and half of 2) but 4 is not. The full sequence of “one and a half” numbers for him was: 3, 6, 9, … right! He noticed that these were multiples of 3. And then what about “one and a third” numbers? “One and a fourth”? And so forth. An interesting regularity for him to notice and a challenging one for him to try to explain.

These are just the little number investigations that have landed at home. There are other things to recommend: Prime Climb, Math for Smarty Pants, The I Hate Mathematics Book are the biggest winners in our house at the moment. And of course it partly depends on your child’s taste.

But for parents of enthusiastic young children, I think the most important work at home is to broaden those tastes. Because even if what we love most is calculation and numbers, mathematics is big. Kids, especially those self-professed lovers of math, deserve the chance to explore it.

]]>Hello youcubians,

Three years ago we reached out to everyone who had taken Teaching Mindset Mathematics, a workshop we offer to share the curriculum, ideas and teaching practices from our 2015 summer mathematics camp for middle school students. We asked if they were planning to host a youcubed-inspired summer camp in 2019, and if so, whether they would be willing to participate in a study of student outcomes and teacher change. We heard from many people who were interested in engaging with us and ultimately enrolled 10 different camps and their school districts into our study, sponsored by the Gates Foundation.

The demonstration camp that youcubed hosted in 2015 had resulted in significant improvements in student mathematics understanding and mindsets, and the purpose of the new study was to learn whether this could be replicated by teachers across the US in a variety of settings. Today, we are very excited to announce that the results of this research have been published in the journal Frontiers in Education. I would encourage everyone to read the full study, which can be found here.

So, a published study funded by the Gates Foundation. I clicked and opened up the paper.

Look, you know what’s coming at this point, right? We’ve been down this road before. I’m going to show you that the paper is a mess. And who cares? Nobody. People who care about research, including a great number of math education professors, do not have respect for Boaler’s output. YouCubed’s popularity stems I think from consultants, PD leaders, and from teachers themselves, not from the math education research community. Frankly, I think YouCubed would be just as popular if they never performed any research whatsoever. It’s totally inessential. As am I. As is this post. What am I doing with myself? How am I choosing to spend my life? That’s how I always end up feeling when I write about YouCubed.

Anyway, back to the paper, this is what they did:

A matched comparison analysis was employed to assess the effect of the approach on students’ achievement. School districts provided a variety of achievement measures of both participant and non-participant students (GPA and MARS scores, before and after camp participation; and a baseline math standardized test score), and a battery of control variables (race, ethnicity, gender, free and reduced-price lunch status, English learner status, and special education status).

OK basically the research team couldn’t run an experiment where they randomly place kids in these camps, so they did a very reasonable next-best thing: they created a control group, of sorts, by matching each child in their camp with a child with similar stats who didn’t go to the camp. So if Michael is a camper and has a GPA of 2.5 and a MARS score of whatever, etc., they tried to find a student Jackson who was not a camper but with the same GPA and MARS score. If Michael (and all the other campers) suddenly are performing much better after the camp than the “control” group then maybe the camp is responsible for it. Or maybe it’s some subtle selection bias.

This is all totally reasonable but in the end totally irrelevant for the issues I’m going to point out. I’m including it because, why not?

By the way, the details of this matching and a quantitative comparison of the groups are not detailed in the paper. Normally a paper would do that, but again that’s not so important in light of what I’m about to say.

OK, so we get to the part of the paper where we’re seeing how campers did compared to their control group. First issue is that in this table it looks like the “non-participants” in the matched group had a higher GPA after the camp than the campers.

The body of the paper says, “This analysis showed that at the end of the first term or semester back at school, the students who attended the youcubed summer camp achieved a significantly higher mathematics GPA (*p* < 0.01, n = 2,417).” Now this is the exact opposite of what the table says, which is a problem, because either the table is wrong or the sentence is false. And, honestly, I assume that the table is wrong and the entire thing is mislabeled. OK.

The paper goes on: “On average, students who attended camp had a math GPA that was 0.16 points higher than similar non-attendees (i.e., students from the same district and grade and who had a similar baseline math GPA and test score) (Table 4).”

This is a problem because the table also does not say this. The difference between a GPA of 2.679 and 2.622 is 0.057 points. That is not 0.16 points. What am I doing, providing this level of scrutiny to this paper, i.e. basic scrutiny? Shouldn’t I be writing a book? Doing math? Reading something? Was I put on this Earth to fact check these papers, providing infinitely more care than the authors themselves apparently did?

Anyway, presumably the issue is that the body of the paper misinterpreted its own table. The 0.16 is the effect size, provided (and clearly labeled) on the right side of the table, not the GPA.

And now a bit more from the paper: “In addition, compared to control students, camp participants were 6 percentage points more likely to receive a grade of B or higher, and 5 percentage points less likely to receive a grade of D or lower (Table 4).”

This is not right, again I think they must be rounding up the effect sizes and reporting them as a percentage change. Silly stuff, quite honestly, but none of it matters at all, I don’t know.

By the way, there is no supplemental information posted, despite this statement.

So, I don’t know. I wrote to corresponding author a month ago but haven’t heard back, I’m sure he’s busy and I’m not sure what he would say.

It’s absurd that this was published as is, and it obviously doesn’t speak well for YouCubed or Boaler that this was pushed through. I…I am unsure what any of this means. Like, yes, it’s bad. But in a way it’s so bad that it doesn’t really touch her core areas of advocacy. I *wish *the research she was attempting was better because then it would actually *be about *something. Instead it’s about nothing besides YouCubed and Boaler herself, who has managed to become *the *popular voice for math education research in this country while showing complete disdain for the work of research itself.

I need to find a better hobby.

]]>I finished writing this book in summer 2020, and it came out in April 2021.

I wrote the sort of book about teaching that I wanted to read — short, personal, aimed more at the working teacher than the teacher-in-training. I thought there were some original perspectives in there, though I think it’s fair to say it’s mostly a synthesis of research ideas.

I wrote a few posts this year that expanded on the techniques from the book:

What is “fading out a worked example” and why would it work and also when?

Good explanations connect particulars to principles

Start class with the good stuff

I was hoping to get started with a second book at some point in 2021, but it never happened. Pedagogically, this was a year of grappling with a million little things, rather than one big one.

Is there a book to be written about whole-class discussion, growth mindset, the equals sign, and math fact memorization? That is the book that (apparently) I am ready to write about teaching, but it just feels like a potpourri to me at this stage.

What actually even is a growth mindset?

The Change-Resistance Explanation for Why Kids Struggle So Much with Algebra

A growth mindset is a tendency to explain things in a certain way

These included some of my most popular posts this year, for what that’s worth.

Some questions and answers about whole-class discussions

What People Get Wrong About Memorizing Math Facts

On to the math! Honestly, I didn’t learn very much math this year, though the last few months of the year have been a bit better in that department.

People Actually Really Like Mathematics

Honestly, I know so little about policy. You don’t come to michaelpershan dot com to learn about politics. The best you can say about me is that I’m a dabbler and that I try to react thoughtfully to whatever happens to be causing a fuss at the moment.

Was it a mistake to call KIPP the top recommended charity in the United States?

Teachers are afraid of hybrid learning (picked up here)

What if we ban private schools?

Some basic questions about college admissions

If you trust hitcounts, these pieces were among my most popular. One of them is a would-be takedown of an incredibly popular education figure, another a joke, the last a very brief statement of purpose. I enjoyed writing each of these things but they don’t make it easier to figure out exactly what I’m doing here.

YouCubed is More Than Just Sloppy About Research

All the numbers from 1 to 500, ranked from worst to best by a middleschooler

Where My Cynicism About Education Ends

If you’re reading this, it’s likely because at some point or another I wrote a worksheet that you liked. Or I shared someone else’s worksheet, and you liked it. Or you liked something I wrote about algebra, or whatever. But I spent a lot of time this year writing sorta-funny fiction (or straight up humor) and some of it got published in places.

Is any of this good? I couldn’t tell you. But for completeness, I include it below:

*The American Bystander*

“I Am Baby, Persecuted in City of New York”

“About the Murderer”

*Two Fifty One*

**Slackjaw**

“I Am A Mad Scientist, And I Am Begging You To Just Trust Scientists”

For the first summer in years, I didn’t teach. During some of that quiet time I sat down to really try to do service to this essay, something that I’ve been thinking about writing for nearly a decade in one form or another. My teen years were dotted with weird Jewish things that I’m spending my adult life trying to make sense of. Adult religion is so *normal*. Mostly for better.

“A Time To Die” in Tablet

I honestly don’t know! Looking back, it’s a lot. But do I regret any of it? Not really. I do wish that I was working on another book. Having a big project is good for me. Having a book is also quite honestly a nice thing for the ol’ ego. Blogging has never really been a “respectable” activity, in the sense that you could talk to family members about it without suffering a paroxysm of shame.

Then again, come to think about it, one of the few times this year that I told someone that I wrote a book, their reaction was to ask how anyone would ever hear about the book, let alone buy it? And then I had to once again shamefully admit that I’m a social media user. So there’s really no way out of the shame of writing. It’s an inherently shameful exercise, and maybe I’ll just try to own that embarrassment in the coming year.

]]>This is very much how progress with whole-group discussions has gone for me. I’ve never had a big “aha!” moment that has fundamentally changed how I’ve gone about running a discussion with a group of kids. Just one problem after another that I’ve slowly amassed answers to.

I think that there is value in whole-group discussions. I think part of the assumed classroom contract is that students will be able to share their thoughts and that teachers will care, and I think that’s good. Students probably need practice sharing their thoughts, and over and above any cognitive impact, the social value of doing this is immense — not just that they can divide fractions, but that they’re the *sort of person *who can divide fractions.

OK, but there are problems with discussion. They all cluster around questions of inclusion, status, participation. So here is a list of questions, along with how I currently deal with them.

**What do you do if a kid had a hand raised and was excited to share, but then put their hand down after someone else answered? And then you call on them, and they say “oh I was going to say what [other kid] said?”**

I say, “how were you going to say it?” Or “why do you agree?” Or “how would you have put it?” And if they tell me they were going to say the *exact* same thing then I say, “no two people put things in the *exact *same way, with the exact same words.”

Basically I don’t want to create a classroom economy where all we value is being first to say something.

**What if a kid says something and it’s complicated and you don’t think others heard them?**

I’ll ask them to repeat what they said, but I’ll warn the class that I’m looking for someone who will be able to restate the idea — their first words will have to be “[Student] said,” and the goal is to get as close to what the student actually said as possible. I’ll correct them if they start “umm basically what I think.” No! You’re restating. I learned this from the New Visions Mathematics Curriculum and their instructional routines.

**What if there are a ton of hands and you don’t know who to call on? **

Then I might ask the entire class to express an answer at the same time. This only works for short questions (obviously), but if it’s a question like “and what would you do next to solve the equation 2x = x – 10” and I’m looking at a ton of hands, I’ll say: “let’s all say what we think at the same time,” and then the class will.

**Really? How do you make sure they all speak at the same time? What can you possibly learn from that? **

I usually find that a quick countdown and a very visible arm motion gets a group to say things all at once. “OK, let’s all say it together, 1…, 2…,” and then I gesture and they speak.

What’s cool about this is that everyone can participate. If pretty much everyone has the same (correct) answer then I get to congratulate the class, it’s a nice social bonding moment. Hey, we all did it! Nice!

It’s surprisingly easy to tell when there’s dissent in the room when we’re all trying to respond as one. Which gives me a nice follow-up move: “alright, there was *definitely *some disagreement when we all just called out, let’s talk about that.”

**How do you make sure that you call on students in a fair way? In particular, how do you call on boys and girls equally?**

I always try to switch genders when I’m calling on students to speak. If I’ve called on a girl, I’ll call on a boy next. And so on.

Some of the most cringe-worthy teaching I’ve witnessed is a teacher saying to a group of girls, “hey, girls are a bit quiet back there, what do you have to say?” Ugh. Gross. Ugggggh.

**What if only boys are raising their hands?**

What, like half the class is raising their hands and it’s all boys?

**Yeah.**

In that case I might ask everyone to put their hands down, without any explanation, and I’ll switch to cold-calling.

**Cold-calling? Doesn’t that make students feel uncomfortable?**

Yes, sometimes, which is why it’s important to do it right.

If I’m worried that students won’t have anything to say when I cold-call them, I stop the class and say: “Hold on everybody, I think this question requires some thinking, and I want to give you a moment to think about it.” And after I write the question on the board or whatever, I’ll assign partners and ask students to share ideas about the question with those partners. While that’s happening, I’ll walk around and see how those discussions are going. Often I’ll hear a kid say something and decide to cold-call on them when we return to the whole-group.

If a student is particularly shy or not confident, I’ll interrupt their conversation and say: “I just was overhearing, what you said was really smart and interesting. I’d like to call on you to explain that to the class. Can I do that?”

But this all represents a significant time investment, and sometimes our whole-group discussions are much shorter than this. In those situations, I’ll find myself cold-calling even though I’m not sure what a student is going to say.

**What if you cold-call on a kid and they say something wrong? Isn’t that embarrassing? **

Potentially!

I used to handle wrong answers in a bad way that emphasized (inadvertently) the student’s failure. Here’s how it went for me:

*Me: What’s 3 + 8? Charlie?*

*Charlie: 10.*

*Me: Yes, super interesting. Anyone else have something to say? Yes, Susan?*

This violates several principles of whole-group discussions that I have. One of those principles is:** If you call on a kid, don’t leave until they’ve said something smart.**

Another of those principles is: **Don’t set up a kid to get knocked down by some other kid.**

It’s never fun to say wrong things in front of other people, but a willingness to do so can accelerate learning. So I’m trying to reduce the social costs of wrongness, what people sometimes call “normalizing” error.

The way I do this (and in line with my two stated principles) is in two ways. First, I correct the child myself. Second, I give them a chance to revise. Here is what that above conversation might look like for me now:

*Me: What’s 3 + 8? Charlie?*

*Charlie: 10.*

*Me: So *[looking at Charlie] *it’s actually not 10, it’s 11. But how can you prove that it’s 11? *

Or maybe “how could you know it’s not 10” or “why is it 11” or “now given that it’s 11, what is 3 + 9” or whatever it is that you’re trying to teach.

But we’re going to stick with Charlie until he says something that everyone in the room agrees is smart, and then we’re going to give him credit for it, and then we’re going to move on.

**What if Charlie never says anything right?**

You gotta pick that follow-up question until Charlie has learned something or revised in some way. Hopefully it doesn’t take forever, and a few times a year it does and it can be a bit painful. I try not to give up, but you know, no absolutes in teaching.

**Why is this all about managing student responses? Isn’t there more to whole-group work than asking students questions and asking them to answer them? **

This is very important: no. I try to make as much of my whole-group time about asking questions and students answering them.

Do I give explanations? Absolutely. But I try to embed explanations in questions. Class begins with a question: “What fraction is this shaded piece of the whole?”

And then, after students respond, I say: *“Notice that you can write this as 1/2 divided by 2, and that it equals 1/4. And this is true in general. For example, what is 1/3 divided by 2?”*

And I think that’s a big part of how whole-group discussions can be engaging instead of a drag. A lot of boring discussions are light on questions for students and heavy on asides and reflections from the explainer. Questions, questions, questions. Let’s actually enshrine that as another of my Principles: **Almost everything in whole-group should either be a question or a response to a question.**

So, in summary:

- If you have a ton of volunteers and the question is short, ask everyone to say it at once.
- The goal is everyone participates.
- Give yourself rules to make sure you’re calling on volunteers equitably.
- Use cold-call so you don’t have to rely on volunteers.
- To prepare for cold-call, ask individuals to think and then give time to discuss with partners.
- If you call on a kid, don’t leave until they’ve said something smart.
- Don’t set up a kid to get knocked down by some other kid.
- Almost all time spent in a whole-group should be spent by students responding to a question.

Thwack, thwack, thwack, thwack.

]]>

I am very cynical about the impact of wealth on education, and especially on what that means for social mobility. I don’t think education is the great equalizer in American society. I think maybe at one point it was, but now? No. There is competition for whatever is educationally valuable, and those with the resources gobble up those things fairly efficiently. Thanks to widening inequality the bottom 25% of Americans have fewer chances to rise above through schooling just as the costs of not rising are as high as they’ve ever been. It’s not good, very bad, and it impacts education at every level.

I am somewhat cynical about the school curriculum, in particular the school math curriculum. Math is big, and the school curriculum is narrow, and a dozen mathematicians would end up with a dozen different curricula. Sure, every kid should know something about numbers and algebra. But what exactly are we going for in high school? What’s with the onslaught of function types and finicky expressions? But it seems hard at this point to change things in a major way — the stakes are too high, see everything above about widening inequality, we are apparently stuck with the math that we have, more or less.

I am a bit cynical about the ability of math class to keep people from killing each other, as Deborah Ball put it. The gap between some of these social aspirations and the reality of the classroom is just vast. I don’t seriously hope that my teaching helps create informed citizens, or sparks social change, or inspires people to act more kindly with each other…

…but that last one actually gets pretty close to the limits of my cynicism.

Because I’m not at all cynical about schooling. I think that schooling is essential. I think that learning is valuable, really any learning, even if what we’re learning is somewhat arbitrarily chosen. I think it’s good to spend your childhood learning. I think it’s good to spend your adulthood learning. I just like learning. I think a large part of being a kind and good person is being able to continue learning about others and their needs. Learning is good, I am not cynical about it.

And school is a good place for most kids to learn. Highly imperfect, but despite the imperfections I’m not at all cynical about the big picture. Get a bunch of kids together, get adults whose job is to be kind and nurturing towards these kids? I’m not cynical at all about that, this is my job, this is what I love doing. And because I’m not cynical about any of this, I’m pretty intolerant of adults who aren’t kind to children.

And even if I am not very optimistic about “teaching so that people stop killing each other,” I’m pretty committed to being kind to people in the classroom. I think that’s valuable. I think kids just need to be around people who are kind to them. Maybe because it’ll impact them in some way, or maybe just because it’s nice? Not in the future, but right now.

In other words, the kids are here, in your classroom, and wouldn’t it be better if they had a good time instead of a bad one? That’s what my teaching is about.

So, tallying it all up, I’m pretty cynical about the curriculum, teaching for social impact, or social mobility. What that leaves is the classroom, in particular my classroom, where every day I get another chance to do it right or not.

]]>499. 149

498. 373

497. 55

496. 137

495. 18

494. 59

493. 452

492. 40

491. 470

490. 107

489. 379

488. 37

487. 165

486. 97

485. 377

484. 236

483. 252

482. 275

481. 479

480. 366

479. 297

478. 122

477. 465

476. 49

475. 154

474. 187

473. 191

472. 139

471. 235

470. 365

469. 315

468. 392

467. 414

466. 299

465. 117

464. 415

463. 425

462. 475

461. 48

460. 440

459. 173

458. 193

457. 142

456. 199

455. 229

454. 249

453. 329

452. 412

451. 218

450. 492

449. 43

448. 408

447. 311

446. 248

445. 150

444. 96

443. 306

442. 459

441. 146

440. 141

439. 123

438. 376

437. 319

436. 182

435. 60

434. 206

433. 177

432. 240

431. 28

430. 476

429. 389

428. 372

427. 214

426. 382

425. 225

424. 125

423. 444

422. 51

421. 127

420. 447

419. 8

418. 53

417. 342

416. 473

415. 274

414. 65

413. 321

412. 56

411. 245

410. 287

409. 257

408. 496

407. 375

406. 347

405. 417

404. 190

403. 468

402. 24

401. 438

400. 5

399. 266

398. 353

397. 16

396. 327

395. 4

394. 323

393. 22

392. 352

391. 91

390. 396

389. 212

388. 131

387. 215

386. 317

385. 129

384. 20

383. 442

382. 436

381. 99

380. 367

379. 82

378. 75

377. 103

376. 144

375. 100

374. 197

373. 270

372. 209

371. 201

370. 114

369. 37

368. 394

367. 58

366. 395

365. 437

364. 159

363. 259

362. 54

361. 243

360. 264

359. 439

358. 261

357. 185

356. 232

355. 289

354. 482

353. 135

352. 330

351. 304

350. 455

349. 296

348. 81

347. 38

346. 357

345. 364

344. 241

343. 90

342. 308

341. 341

340. 361

339. 242

338. 356

337. 3

336. 94

335. 344

334. 36

333. 47

332. 461

331. 128

330. 124

329. 88

328. 1

327. 318

326. 359

325. 272

324. 489

323. 178

322. 31

321. 222

320. 217

319. 325

318. 499

317. 490

316. 14

315. 413

314. 349

313. 401

312. 410

311. 407

310. 466

309. 83

308. 474

307. 405

306. 160

305. 106

304. 157

303. 210

302. 431

301. 138

300. 383

299. 307

298. 443

297. 427

296. 11

295. 195

294. 380

293. 45

292. 340

291. 110

290. 397 & 332 (tied)

288. 402

287. 263

286. 220

285. 497

284. 186

283. 370

282. 181

281. 80

280. 203

279. 21

278. 108

277. 424

276. 345

275. 445

274. 430

273. 227

272. 116

271. 120

270. 233

269. 246

268. 133

267. 313

266. 111

265. 386

264. 115

263. 136

262. 416

261. 400

260. 487

259. 72

258. 130

257. 262

256. 155

255. 278

254. 258

253. 9

252. 388

251. 277

250. 104

249. 354

248. 140

247. 87

246. 19

245. 238

244. 265

243. 471

242. 419

241. 143

240. 89

239. 355

238. 310

237. 491

236. 467

235. 500

234. 483

233. 93

232. 303

231. 368

230. 171

229. 305

228. 333

227. 198

226. 221

225. 109

224. 10

223. 448

222. 219

221. 456

220. 433

219. 42

218. 267

217. 384

216. 387

215. 404

214. 284

213. 205

212. 71

211. 337

210. 46

209. 273

208. 312

207. 166

206. 35

205. 32

204. 67

203. 480

202. 70

201. 226

200. 163

199. 92

198. 294

197. 26

196. 41

195. 271

194. 231

193. 230

192. 486

191. 44

190. 134

189. 86

188. 324

187. 216

186. 423

185. 79

184. 282

183. 50

182. 288

181. 29

180. 348

179. 293

178. 30

177. 322

176. 291

175. 78

174. 290

173. 298

172. 68

171. 172

170. 334

169. 161

168. 283

167. 279

166. 255

165. 485

164. 228

163. 162

162. 343

161. 204

160. 350

159. 84

158. 421

157. 478

156. 477

155. 25

154. 145

153. 192

152. 446

151. 435

150. 180

149. 385

148. 339

147. 168

146. 113

145. 360

144. 390

143. 23

142. 295

141. 434

140. 256

139. 351

138. 454

137. 39

136. 184

135. 371

134. 411

133. 276

132. 381

131. 251

130. 253

129. 281

128. 406

127. 73

126. 247

125. 76

124. 280

123. 292

122. 62

121. 374

120. 27

119. 460

118. 362

117. 472

116. 77

115. 112

114. 176

113. 331

112. 2

111. 495

110. 309

109. 52

108. 418

107. 320

106. 183

105. 151

104. 57

103. 61

102. 391

101. 174

100. 7

99. 268

98. 148

97. 175

96. 152

95. 300

94. 101

93. 484

92. 462

91. 285

90. 481

89. 213

88. 102

87. 153

86. 239

85. 95

84. 224

83. 393

82. 338

81. 17

80. 358

79. 286

78. 223

77. 346

76. 326

75. 316

74. 105

73. 119

72. 33

71. 74

70. 302

69. 156

68. 34

67. 167

66. 118

65. 399

64. 498

63. 207

62. 336

61. 188

60. 66

59. 378

58. 208

57. 254

56. 12

55. 301

54. 398

53. 98

52. 488

51. 429

50. 132

49. 426

48. 85

47. 422

46. 260

45. 147

44. 244

43. 179

42. 189

41. 450

40. 158

39. 314

38. 409

37. 196

36. 202

35. 363

34. 457

33. 63

32. 64

31. 403

30. 463

29. 458

28. 451

27. 200

26. 211

25. 453

24. 164

23. 250

22. 194

21. 464

20. 13

19. 126

18. 449

17. 234

16. 335

15. 432

14. 121

13. 494

12. 493

11. 170

10. 441

9. 6

8. 428

7. 15

6. 469

5. 369

4. 269

3. 169

2. 420

1. 69

]]>But the bigger sloppiness was their interpretation of a study by Jason S. Moser. The study found that people improved their performance on some task after they made mistakes, the reason being that they noticed the mistakes and improved. In the hands of YouCubed, this Moser study was cited as saying “the brain sparks and grows when we make a mistake, even if we are not aware of it,” i.e. precisely the opposite of what it says.

(Robert Kaplinsky also looked into this, by the way, and never heard back from Moser.)

I don’t want to write the same thing over and over again, because that’s boring, but I do want to add that it’s not just about their interpretation of existing research. The research YouCubed itself releases is often extremely misleading, or so shoddy that it’s hard to even talk about it *as *research.

There is a particularly clear example in a recent release, “Raising Expectations and Achievement: The Impact of Two Wide Scale De-Tracking Mathematics Reforms.” The authors, Jo Boaler and David Foster, want to show that eliminating tracking leads to better outcomes for kids.

The state of California used to allow 8th Graders to take a variety of courses: Pre-Algebra, 8th Grade Math, Algebra 1, Geometry, or Algebra 2. Students would take a test at the end of the year in whatever course they were in. Which is rather chaotic, sure, but this is California we’re talking about, my understanding is if 100 people sign a piece of paper they have to pick a name out of a hat and then that person has to be governor. California is a bit chaotic.

Anyway, California wanted to do something about this, so they made everybody take a class called “8th Grade Math,” and then they all took the same test.

Well, first of all, are you surprised that scores were better on average after they made everyone take the same test? Would you have predicted it?

I’m not surprised, though I wouldn’t have predicted it. That’s because the tests are totally different. I don’t have any clue what the Geometry test in California is like! Why would I have an opinion about whether students would do better on that or this 8th Grade test? And even if they did, what would that even show? Maybe the 8th Grade test is easier and kids who used to take Algebra 2 aced it.

Anyway, as it turns out, scores went up after this “de-tracking” and Boaler and Foster call this “Study 2” in support of de-tracking.

There are other publications that are less funny, but no less flawed. In particular, there is a MOOC paper where 100s of people dropped out of YouCubed’s intervention, and the study just ignores it. But when hundreds of people are dropping out of your intervention…if hundreds of people stop taking a drug in the middle of the trial, you need to ask some questions, questions like “why did they stop taking it?” and “did their hair grow back after they stopped?” and so on.

So, what can I say, YouCubed and Boaler are not producing legitimate research.

Which, as I often point out, wouldn’t be so concerning on its own. This is the world we live in. People misuse research, literally all the time.

But YouCubed’s *entire *shtick is that they are research-based. They have risen to influence by beating the drum of research with particular enthusiasm. Their mission mentions research not once, but three times: “Our main goal is to inspire, educate and empower teachers of mathematics, transforming the latest **research **on maths into accessible and practical forms. We know from **research **how to teach mathematics well and how to bring about high levels of student engagement and achievement but **research **has not previously been made accessible to teachers.” Research, research, research.

All this while Jo Boaler has become the closest math education in the US has to a celebrity. (You’ve seen #JoOnAStick, perhaps?) And of course, part of what makes her an important voice in education is that she is a researcher, who does research, and can speak for research.

So, what is going on?

I don’t know. It’s clearly all connected to the first big controversy in Boaler’s career surrounding the Railside study, but I don’t really understand the full trajectory. Clearly, Boaler now sees the world of math education in terms of conflict, frequently highlighting the fact that her agenda has opponents, signing every YouCubed email with “Viva La Revolution [*sic*].” Was this tendency towards conflict a result of the wars of Railside, or one of their causes?

Either way, it’s been four years since I started writing about YouCubed and I’m feeling ready to go further than “sloppy.” For whatever reason, YouCubed as an organization frequently produces or cites research in ways that don’t show what they claim to show. As a result, I wouldn’t trust the organization (or Boaler) to make any research-based claim.

And the issues will continue to grow until the mathematics education community decides that it isn’t OK to mislead people about research.

]]>When students believe they can get smarter, they understand that effort makes them stronger. Therefore they put in extra time and effort, and that leads to higher achievement.

This is fine, and I think I get it. But what does it mean for kids to “believe they can get smarter”? Not to get extremely pedantic, but I have some very pedantic questions to ask. What does “believe” mean? Does it mean that you sit a kid down over a nice glass of apple juice, start shooting the breeze about life, work, how busy we all are, and then raise the question: do you think you can get smarter?

And what does it mean to “get smarter”? Or to “understand” that effort makes one “stronger”? What if I think that working hard increases my knowledge but leaves me at the same smartness? What if I tell you over apple juice that effort makes me stronger, but then in the moment I curl up in a ball of learned helplessness?

Pedantry aside, after reading up and thinking about these things I don’t think any of these questions really matter. There’s a way of defining growth mindset that makes a lot more sense to me, and I think it’s true to research. Here goes:

When students have

a tendency to explain their successesor failuresin terms of their efforts, they tend to work harder. Therefore they put in extra time and effort, and that leads to higher achievement.

I think this has a few advantages over the “belief”-based definition. Here they are:

- If you have a “belief” and an “understanding” about intelligence then you presumably carry that around with you from room to room. But a tendency to explain your experiences in certain ways can quite clearly vary from room to room, even minute to minute. (Maybe this tendency is really just what we mean by “belief”?)
- It makes clear that mindset training does not require lying to people. Every success we have is the result of our innate abilities along with our efforts. Defined in terms of a tendency to explain, the only question is which of these factors we are paying attention to in the moment. If our attention is constantly drawn to innate abilities, that will be demotivating. If our attention is drawn to the
*equally valid*factors that are in our control, we will likely be more motivated. - It explains why some of those studies have such eye-popping results. Not because they’ve instantly changed some deeply held belief concerning the nature of intelligence with a few videos and lectures about brains. It’s because they have
*created a context in which students have are set up to explain successes/failures in terms of effort*. If you’re running an experiment and researchers make a big stink about effort, and then in the context of that same group activity you have a success/failure, you will of course be more likely to turn to that explanation.

So I’m very happy with all this.

One thing I wonder is whether it’s somehow important, for the purpose of all these materials and studies, to define mindset in the Mindset Works way. It makes less sense to *me*, but maybe it’s important. The interventions involve taking a thing that we think of as innate and telling people that it’s actually malleable. “Intelligence can grow”; “Smartness depends on effort”; “Your brain gets bigger”; “Talent takes hard work.” If this messaging is effective, maybe it works because people with a tendency towards innate reasons are reminded of it when they reach for those “fixed” explanations?

Probably. Probably it’s a bad messaging to say, “of course you have some innate limits, but life goes better if you don’t stew on that and instead think about the things you can control.” God knows that I’ve spent my fair share of time stewing about how life would be different if I had spectacular literary talent, a tremendous hook shot, God-level charisma and an inspiring childhood, the ability to create a search engine in 1998. Probably the innate-to-malleable messaging in its most direct form — “talent is a matter of effort” and so on — is perfectly appropriate.

But that’s the messaging that for years left me unsure whether I should believe the results of growth mindset research. They might be good messaging, but they rang untrue. But if “teach people that practice grows your brain” is just a snappy way of saying “encourage people to explain their achievement in terms of effort,” I can absolutely live with it.

]]>One of the most popular curriculums in the US for elementary school is EngageNY/Eureka, and they have a version of this activity that they call “Sprints.” A “sprint” is 44 problems that you’re supposed to solve as quickly as you can. It’s a bunch of problems on a page.

Some people love these activities, but a lot of people in math education hate them. Many of them are haunted by memories of “Mad Minute” drills in school, where you’re supposed to answer as many math fact questions as you can in a minute — that’s mad. These opponents prefer non-stressful practice that doesn’t have a time constraint. In their classrooms the facts are maybe presented one at a time on the board, and students are asked to derive the solution any way they like. Then they talk about the strategies people used. It’s not problems on a page, it’s a problem on the board, and we’re talking about it.

A good example of a proponent of this view is Jo Boaler, whose YouCubed has a position paper titled “Fluency Without Fear.” Besides for number talks (talking about a problem on the board) she also calls for practice that is fun and engaging, stuff like dice games and puzzles. As usual, the essay talks a lot about “brain research” that supposedly proves that this stuff works. Sure it does, whatever. This is the other perspective.

And what I’d argue is that actually *both *of these approaches make the same mistake, and it’s a big one. Because what both approaches assume is that if kids solve a problem a bunch of times, they will commit it to memory. And the thing is that this is not true, at least not true enough.

What we need is a theory — an explanation, really — for when people remember something. Let’s not make it complicated, we can put it very simply: people remember something when they’ve successfully remembered it a bunch of times. I don’t think this is saying anything that retrieval practice advocates haven’t already said — the best practice for remembering something is practicing remembering it.

Suppose that I ask you to find the product of 12 and 5. What is going on in your head as you answer the question? Maybe you start counting by 5s. Or maybe you remember that 12 x 4 is 48, and you do 48 + 12. Or you grab a piece of paper and start adding 12s. None of this is retrieval practice — none of this is practicing pulling the fact out of memory.

(To be fair, maybe you did practice pulling 12 x 4 out of memory. And maybe, when we’re talking about this problem after you solve it, you’ll end up having to remember that your answer to 12 x 5 was 60. But neither of these are sure things, though they might help if you keep doing them for enough time.)

There is a very simple question we can ask to see if math fact practice is likely to help in the most direct way: Are kids practicing remembering? Or are they practicing something else?

Let’s apply this test to EngageNY’s Sprints. Will kids be practicing remembering when working on them? Here’s a basic point: *not unless they are successful*. If a kid is unsuccessful at pulling a math fact out of memory, what are they going to do? They are going to try to *derive *it, using some sort of strategy. Maybe they’ll try using the most basic sort of strategy, something like skip-counting for multiplication or counting on fingers for addition. But of course they’ll derive — what else are they supposed to do?

The “Sprint” aspect of this is encouraging kids to move quickly, which is really only possible if they have very efficient strategies or have many facts in memory. True, if kids are successful then they’ll get some good practice. But if kids are not successful, they will be forced to derive solutions, which will take more time and is *not *the sort of thing we were trying to help kids get better at. They will complete much fewer problems, especially if they thoughtfully arrive at solutions to the questions. It’s a trap. Not hard to believe that this stresses some kids out.

Which means that “problems on a page” is lousy practice that helps strong kids get stronger but leaves students who know fewer facts spending time practicing something else, a.k.a. Not Very Good News.

OK, but we have precisely the same problem with the “Fluency Without Fear” activities. What are kids thinking about during a Number Talk? They are thinking about — the whole point is to think about — the various strategies that we’ve used to derive some fact. That’s great if you’re trying to study strategies. But it’s not giving anyone a chance to practice remembering stuff.

(And I really do think it’s good to teach strategies. For some facts, like 9 + 8, a lot of successful people just use a very efficient strategy and never end up memorizing it. Plus, you can’t memorize every useful fact, at some point mental strategies come into play. But also because it’s easier to memorize 9 x 5 if you know that it’s going to end in a 5 or a 0, easier to remember 7 x 9 if you know the digits are going to sum to 9, easier to remember 9 + 8 if you know it’ll be in the teens. Strategies are useful, but practicing with strategies isn’t retrieval practice.)

In sum: the vast majority of math fact instruction doesn’t focus students on the thing that it’s purporting to teach. Good news, though, it’s far from impossible to engineer practice that does focus on memorization. But how?

The basic answer here is “flash cards,” which has two big advantages. First, if you fail you can turn over the card and stick the fact back in your memory, then try again. You aren’t left to derive the fact (though you can). Second, you can repeat problems frequently to practice the problems that you didn’t answer correctly.

There are more complicated things to say. Brian Stockus just wrote a great post showing how he is doing one-on-one flashcard work with his daughter. I’ve written about some of the ways I’ve used flashcards in my 3rd and 4th Grade classrooms. And of course there are a million computer programs that promise to help teach kids math facts…they all are basically flashcards, each and every one of them, combined with some sort of gamey practice. You want research on how to help kids with difficulty learning facts how to learn facts? You’re going to find a lot of flashcards.

Nothing is a sure thing, and I don’t mean to make this sound easy. There are no guarantees in teaching, especially when you’re working with a whole class. Follow some of the links above and you’ll find lots of practical advice on how to manage the difficulties. Given our focus on retrieval practice, it goes without saying that you should only introduce a few new facts at a time, and the goal needs to be for students to be *successful* at remembering them by the end of the practice session.

There’s no point bemoaning the state of discourse in education, it’s bad, everyone knows it’s bad. Stop me if it sounds like I’m bemoaning, but it seems to me that pretty much every discussion about math facts misses the point, viz. everything I said above. People don’t ask the right question, which is “how do kids remember stuff?” Or rather they do, but answer the question in clearly insufficient ways.

People do *not *necessarily remember the things they derive. Repeatedly deriving something in a way is practice *avoiding *retrieval from memory, which is (I admit!) a very mathematical thing to do. Mathematicians love talking about formulas that they derive every time and can never seem to remember. These theorems or formulas aren’t anything but upper-level math facts.

So we should remember that this is a real phenomenon, and that it’s true for little kids as well. If you want people to remember something, it’s often not enough to get them to derive it, whether on a big page of problems or as part of a number talk. As usual in education, the people with the strongest opinions have missed the point, and apologies for just a bit of bemoaning.

]]>- solving problems like 2 + 8 = ___ + 3 (it’ s not 10 and it’s not 13)
- knowing how to define the equals sign as “is the same as” (not “the answer is”)
- remembering equations like 2 + 8 = ___ + 3 after seeing them (“encoding in memory”) even if they aren’t in the most common a + b = c format (often kids reconstruct the uncommon ones incorrectly)

Taken together, these can predict a certain amount of a kid’s future success in learning algebra. And this prediction goes beyond overall math ability, IQ, or many other things that you might want to control for.

Here’s a fantastic new paper from leading researchers on all this. The intro and discussion at the end contain tons of readable, thoughtful exposition on all these things:

Looking at these relationships between early equivalence knowledge and later algebra success leads inevitably to a conclusion: it’s really important to help kids understand how equations and equality work in their early years of school. If you can improve knowledge of equivalence, more kids will learn algebra.

OK, but why does this stuff help? A lot of theories don’t add up, but Nicole McNeil writes about a “change-resistance” hypothesis that makes a lot of sense to me.

The hypothesis goes like this: It’s harder to learn a second language as you grow older. Your knowledge of the first language is so strong that you lose flexibility. Your understanding of language is highly structured by your deep and thorough experience with the first language, and it is really hard to change how you think. You may never be 100% successful, you will never sound like a native speaker, you will never feel entirely comfortable with your non-native tongue. Not because of what you *haven’t* learned, but because of what you already *have*.

Students usually encounter equations for the first time at school, and when they do it’s often a heavy dose of equations that look pretty much the same: NUMBER SOMETHING NUMBER EQUALS BLANK. Four plus three equals blank. Five minus one equals blank. Sure, sometimes you get a question mark or a box instead of a blank. Yes, eventually multiplication and division make an appearance. Either way, there is this very rigid format to the equations kids experience in their early years.

The change-hypothesis account says, this changes kids. This is their native language.

It explains why kids can’t solve equations like 10 + 2 = ___ + 3, instead answering 12 or 15. Isn’t that how equations always work? It explains why kids define the equals sign in a narrow way as “here’s the answer” — that’s how it’s being used in all the equations they’ve experienced! And it explains why their memories have a hard time holding on to the nontraditional equations, as memory has been structured around the a + b = c format.

Now, here comes a subtlety, because we haven’t explained why this impacts later algebra success. A clean story would be that these mathematical equivalence skills are lacking for algebra students. They’re clearly prerequisite for success with algebra. If you think that equations are always telling you to perform some operation with a numerical result, yeah, algebra is going to be tough. If you can’t solve equations like 3 + 10 = __ + 5, why would you expect to be able to solve 2x – 3 = 5 + x? If you never learn mathematical equivalence, of course algebra is going to be tough for you.

Here’s the thing, though:

- The best predictor of later success is solving those problems (3 + 10 = __ + 5), following by encoding, and having a good definition of the equal sign doesn’t predict much at all
- Kids pretty much learn how to solve those types of problems as they get older (in one study that we’ll get to in a moment, undergrads solved 91.8% of these problems correctly when untimed)

But McNeil and others have an explanation for all this, which is that it’s not just about the learning. Go back to the language analogy — maybe you taught yourself to conjugate correctly in French, even though it’s not your native language. Maybe you studied really hard and practiced a great deal. But what happens in stressful moments, when you aren’t able to explicitly think through the situation? What happens when you’re negotiating over the phone and trying to remember the correct suffix for the verb? Or what happens when you’re trying to read an especially tricky French text?

The change-resistance explanation says that the initial, narrow way of thinking about equations never goes away, and it impacts your ability to learn more advanced material later.

I love some of the predictions and studies McNeil has used to test this hypothesis. My favorite are when she takes adults who — as I mentioned a moment ago — can pretty much solve the 3 + 10 = __ + 5 equations when you give them enough time, and she shows that their native language is lurking beneath the surface. There are two ways that she does this:

- Rushing them with a time constraint, and showing that when you rush a competent adult they start to make the same mistakes that 2nd and 3rd Graders make — and eye-tracking data shows that they don’t look across both sides of the equals sign when analyzing the equations, consistent with the left-to-right way of reading basic traditional equations
- Asking people to solve an arithmetic problem (like 8 + 4)
*reduces*their ability to solve equations like 3 + 10 = __ + 5 under time pressure, compared to a control condition where participants had to add colors instead of numbers (blue and green makes ____)

According to this view, what ends up making it harder to learn algebra is this strong bias towards a + b = c equations. It’s this tendency to see equations of this type more easily. It’s possible to learn algebra even if you have this “native language” but it requires a certain amount of mindful redirection of your attention. This saps your available cognitive resources — a little or a lot, depending on the strength of the a + b = c paradigm — and makes it harder for you to learn algebra.

It also explains why interventions that simply expose students to nontraditional problem solving formats (such as 4 = 2 + 2) can make a difference — you’re really trying to disrupt the strength of the a + b = c paradigm in its formative years. McNeil’s current approach though is more holistic, focusing not just on nontraditional equation formats. I suppose this makes sense — you need to give kids a way to avoid reforming that strong bias towards a + b = c even if they continue to see problems in that format after the intervention.

If the change-resistance story is right, though, I’d think that the ultimate solution to the problem would be teachers and curricula using a variety of equation formats. There’s no real reason why equations have to all look like NUMBER OPERATION NUMBER EQUALS BLANK. I don’t think anyone says changing the way equations look would magically help everyone become great at algebra, but I think there’s a very plausible explanation for why it could really help.

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