If you have one person in the country who is, like, the best at explaining calculus, that person maybe should teach every calculus class in the country.

It’d be helpful if we could work through the idea that good teaching is *just* good explaining and vice versa. Someone here at Twitter Math Camp mentioned she conducts a math night for parents at the start of school. “I wish I had learned math like this as a kid,” they tell her. That realization, that there is and *should* be a difference between how math was taught then and now, is a giant first step.

**Featured Comments**

This shows the idea that children’s minds are empty vessels that need to be filled with knowledge and teachers are the keepers of that knowledge, whose sole job is to effectively pour said knowledge into the vessel. And if their minds didn’t get filled with our knowledge the fault must lie with our explanations.

This flies in the face of what we know about teaching and learning.

None of these reforms about math education can happen in a vacuum. There’s always a political side to what happens to people’s children, and if the way you help children learn math is important then the way you communicate with parents is also important.

## 28 Comments

## Tom Harrington

July 25, 2014 - 5:14 pmThat we should be teaching math differently seems obvious. In the past, humans did the calculations. Today, machines can do it. However, the idea that the person “best at explaining math” should teach every class is, at best, silly. People are different. Think differently. Reason differently. Construct knowledge and understanding differently. As a high schooler I discovered that a math explanation that made perfect sense to my best friend, Gordan, made no sense to me at all. I went on to teach math for 30 years, all the time remembering that what made perfect sense to me made no sense to at least 30% of my students.

## Kenneth Tilton

July 25, 2014 - 6:04 pmWho did Rand copy that one from? Can we not quote a serial plagiarist in an education blog?

For one, he gets it wrong: it will be five or fifty excellent calculus teachers, because learners vary. But what do plagiarists know? They just crib over the smart kid’s shoulder.

There is a good conversation to be had on the power of ed tech to leverage the best educators, but there is a reason (left as an exercise) we do not mention politicians in the workplace, military ward rooms, and bars.

## Kenneth Tilton

July 25, 2014 - 6:20 pm@TomHarrington Word up! I just met someone who flopped with math teacher A, “got it” with math teacher B, then blew away math teacher A when she got him again the third year.

Wait. We’re not programmable silicon devices?!

(a) Thank goodness.

(b) So how do we teach these infinitely variable organic systems?

Answering (b) should be a lot of fun, but we need to be honest with our students that we have no idea how to do it.

I guess that means we must invite them into the experiment as co-equal investigators.

Any one know a school board out there with the, um, nerve to implement *that*?

-k

## @cheesemonkeysf

July 26, 2014 - 4:36 amI realize that Senator Paul hasn’t gotten this far into the logical reasoning process, but… there’s no community in America that is going to accept a 1:2,000,000 teacher-student ratio for any course, much less for calculus.

Love,

me

(@cheesemonkeysf)

## Erica

July 26, 2014 - 5:49 amI don’t think there would be such a person, because different people find different explanations easy to understand. It depends a lot on the student’s background experience and knowledge.

## Susan from Clarkston

July 26, 2014 - 6:11 amIf Rand Paul’s idea is correct, then, Khan Academy should hire him to teach it for all the world to learn it (calculus).

## katenerdypoo

July 26, 2014 - 6:41 amThis shows the idea that children’s minds are empty vessels that need to be filled with knowledge and teachers are the keepers of that knowledge, whose sole job is to effectively pour said knowledge into the vessel. And if their minds didn’t get filled with our knowledge the fault must lie with our explanations.

This flies in the face of what we know about teaching and learning (perfect to pair this with that NY Times article or Eric Mazur’s confessions of a reformed lecturer).

A major problem is that because everyone has attended school and had teachers, they all think they have knowledge and understanding of teaching and how it works and/or should be done. This is why it is crucially important that we as educators continue to raise our voices and demand a seat at the policy table.

## TonyB

July 26, 2014 - 7:05 amSilly Rand Paul. As if there is “one person” who is “the best” at teaching calculus. No such animal. As a reputedly excellent calculus instructor myself, I cannot help but be aware that some students still don’t “get” calculus when I teach it, perhaps in part because they don’t “get” me. There is no one ideal math professor who is all things to all people. Different students require different styles.

## Steven Stowers

July 26, 2014 - 8:44 amI have a revolutionary new idea! If there’s one person in the country who is the best at explaining calculus (or anything else, for that matter), why don’t we get him/her to write a book about it so that the rest of us can learn from it?

## Michael Paul Goldenberg

July 26, 2014 - 9:02 amSomething comforting in the consistency of Paul’s predictably shallow “analysis” of yet another issue about which he clearly knows nothing. But therein lies his appeal to many: the ability to articulate perfectly the wrong ideas that they, too, assume are “common sense.” In a nation obsessed with rank-ordering everyone and everything (“Top Ten Hemorrhoids of All Time”; “The One Hundred Cleanest Fast-Food Rest Rooms You Must Visit Before You Die!”), the notion that there would be a “best” calculus teacher whom we could, of course, readily identify and agree upon, is perfectly sensible. And then, naturally, everyone should be taught by her. Because human interaction among teachers and students, students and students, etc., is in no way essential to learning mathematics (or anything else). Rugged individualism means never having to share or struggle with your own or other folks’ speculations and inklings about challenging ideas.

## Kenneth Tilton

July 26, 2014 - 9:06 amOK, people, but the flip side of “one best teacher” is students being subjected to teachers randomly positioned along a normal curve of quality.

All I see here is people knocking down the idea of one teacher. Anybody care to address how we use tech to leverage the best teachers so no one gets crippled because they happen to land one of the worst?

Perhaps the one thing we all agree on is that the single most important factor in the classroom is the teacher. So, again, how do we leverage the best among us?

btw, even the mediocre teachers will welcome a shift in which they become facilitators helping kids learn from the best, because in the end we all wake up in the morning hoping to do a good job.

-kt

## Michael Paul Goldenberg

July 26, 2014 - 9:39 am@Kenneth Tilton wrote in part: “Perhaps the one thing we all agree on is that the single most important factor in the classroom is the teacher.”

Where was that agreed upon? And in what is the teacher “the single most important factor” exactly? That claim gets bandied about a great deal, most frequently as a misstatement about closing ethnic learning/achievement gaps. And it simply isn’t so that either teachers or schools are the biggest factors in that regard. I’ll grant that I’ll take a solid teacher with sound content knowledge, basic pedagogical skills, and deep pedagogical content knowledge over lots of tech, a “great” textbook, etc., as I’ve seen the former work wonders with bad books and other disadvantages, while I’ve seen some teachers that could screw up any and all other positive resources available to them.

But that’s about as far as I’ll go, lest someone quote me out of context to support the notion that “great” classroom teachers are the key to curing all our social ills, particularly those so obviously grounded in economic and social inequity. I have no doubt that Mr. Paul would be comfortable with that notion, however.

That said, there are actually an increasing number of free or relatively-affordable online resources, including some damned good videos (not from Khan Academy) that really do get at some meaty K12 math issues intriguingly. If you’ve not looked at the free video minicourses James Tanton has up on gdaymath.com for solving quadratic equations and dealing with “combinations and permutations” (a distinction he neatly rejects and obviates), they’re one place to start. No, they’re not revolutionary in terms of “how to teach mathematics in a classroom,” since they are not teacher-student or student-student interactive, for starters. But they beat all other presentation of the ideas I’ve encountered (your mileage, of course, may vary on that score).

While I know from corresponding with Prof. Tanton that he plans to cover the full range of K-12 math in this courses, at least through calculus, I believe), building on his THINKING MATH textbook series among other sources, I can’t claim that this “solves” the issue of how to guarantee that all kids have access IN A CLASSROOM to a great math teacher. No one can realistically guarantee that, and many of those who use that rhetoric in the education deform movement are alarmingly hypocritical, in that they would never approve the costs involved in trying to make that happen.

If I were king of the universe, AND if I actually had at my disposal the necessary cadre of fine mathematics teachers to stock our K12 classrooms (not to mention our university classrooms), I’d wave my hands and do it without hesitation. But that cadre doesn’t exist, isn’t on the horizon, is actively undermined by parents, politicians, pundits, publishers, professors, pupils, and other pedagogues, not to mention the likes of Rand Paul and his daddy. As America by and large doesn’t know what mathematics is or what quality mathematics teaching looks like, it fights against many instances of such teaching and attempts to nurture more of it. Green’s article in the NYT Magazine gets at some this pretty accurately.

So while we try to improve our practices and those of colleagues who are willing to share, take risks, think, reflect, and so forth, we need to recognize the limitations of getting good math teachers into many, let alone most or all, classrooms. We simply can’t wait for that to happen because if it does, it will take, at the very least, decades. I’ve been looking at mathematics teaching professionally for over 25 years. I turned 64 yesterday. While I’m thrilled by all the thoughtful, reflective, talented, insightful, and gifted young (and a few not so young) math teachers who are actively sharing their thinking and practices in various communities on the Internet and in other contexts, there just aren’t enough to go around or to counter all the dead wood. And the scary part is that there is plenty of NEW dead wood being produced and ushered into classrooms every year. Some of them are salvageable, but many will be as bad or worse than some of the horror shows most of us have experienced in one or more capacities – as students, colleagues, coaches, supervisors, parents, etc.

So as hypothetical King of the Universe, I might be able to “get rid of” the bottom 5, 10, 25, 50 or even 90% of the “bad” math teachers out there (and let’s pretend that we had some accurate and meaningful way to determine such things), thus satisfying the deformers’ fondest dreams. But then what?

## Michael Paul Goldenberg

July 26, 2014 - 9:45 amOne more thing regarding KT’s last post: I don’t believe that it’s as true as we might hope that all/most/many of the mediocre and worse than mediocre teachers want to improve. Part of why they aren’t already better is because they truly don’t get it. Lots of them think they’re excellent and that the problem is the kids or any other number of factors completely beyond their control either now or ever And I’m not talking about the realistic concerns about poverty and its ill effects, but simply the idea that there’s no reason to rethink what it means to do, know, teach, or learn mathematics because, after all, they already know the math themselves and teach it: the darned kiddos just don’t/can’t learn it.

## Joel Patterson

July 27, 2014 - 3:36 amDan, the point you made about a teacher doing Math Night to help parents understand is key. None of these reforms about math education can happen in a vacuum. There’s always a political side to what happens to people’s children, and if the way you help children learn math is important then the way you communicate with parents is also important.

So many bad educational ideas, like “value-added measures” and “cubicle warehouse with math software” get packaged & sold to parents–and that builds distrust toward teachers (deserved or not). Paying attention to the political facet of better math learning is just something you have to do if you want better math learning to propagate.

## Jon Michael Stanford

July 27, 2014 - 7:02 amI like that he is actually trying to think outside if the box here. It sounds implausible at first, but would be be very to implement in actuality, via relatively old technology. I could see 5-15 minutes of a class being “taught” by this “mega teacher,” but then the rest by the real life teacher who is able to tailor the lesson to the individual needs of the students.

## Dan Meyer

July 27, 2014 - 12:24 pmThanks for the commentary, team. I pushed responses from

KateandJoelup to the main post.Jon Michael Stanford:This is fairly inside the box thinking, really. Massive online learning, online learning, courses-by-TV, courses-by-mail – been done been done been done been done. All of these efforts have worked for certain students and not for others and this all depends on how we define “worked.”

## Jon Michael Stanford

July 27, 2014 - 1:16 pmDan,

I agree. It’s not new. I’ve tried using Khan Academy and found out it was great, for about 10% of the students. Most kids like to come to class and be engaged in real life, bouncing ideas off each other in a competive, uplifting environment.

I’m just saying, there are a lot of teachers out there who are ineffective. It seems easier and more efficient to make the effective teacher’s strategies more accessible.

Perhaps this is implausible in the end.

But, it is very good that Paul has a focus on effective teachers. This, I think, is a turning point.

If you watch his father, too, I think this is the “Paul” approach: make a point which stirs the pot, is not really effective in the end by itself, but opens up to more discussion to find a better answer than the “establishment.”

Sorry for all the “quotes.” ;)

## Joel Patterson

July 28, 2014 - 2:33 am“there are a lot of teachers out there who are ineffective. It seems easier and more efficient to make the effective teacher’s strategies more accessible…it is very good that Paul has a focus on effective teachers.”

I want to point out something about word choice, Jon. It’s the use of “effective teachers” or “ineffective teachers” which implies that the teacher is like replacing old guitar strings with new strings to get better music. I’m the same teacher that I was in 1995 when I started, and I started teaching with lots of great explanations for math & physics, but my effectiveness has improved with better methods of teaching/learning in the classroom (my students do more of the explaining than I do). Therefore that is why the Japanese lesson study forbids using personal terms in the critiques to keep the

lessonas the subject.We need to push back on politicians, editorial boards, pundits, and school administrators who talk about “effective teachers” because they lose focus on the one thing every teacher can use: methods of learning that build curiosity.

## Shaun Errichiello

July 28, 2014 - 10:05 amI try and imagine all the progress that would be lost throughout history if we applied Paul’s thinking to long ago.

If we propped up one teacher and said, “this person is the best, let’s use their style everywhere,” there would be no room for innovation (certainly not for three act lessons).

What a sad world it would be!

## TonyB

July 28, 2014 - 10:27 amThe myth of “the one best teacher” is compounded with the “one ideal solution” illusion. As a result, education is unduly fad driven. Right now there’s a boomlet for “accelerated curriculum” such as Statway. You know what? At my college it works

great! But it’s not for every student who needs algebra and statistics. Ditto with our self-paced drop-in learning center. Some students need more structure. Years ago we introduced “splits” which take a one-semester class and slow it down into a two-semester sequence. Some slow-learners like it.I hope we’re finally taking seriously the problem of properly pacing students in the learning environment that best suits them, whether standard lecture classroom, accelerated or decelerated program, self-paced, on-line, etc., etc. Of course, we’re hobbled by limited resources from offering every learning mode for every course, but the emphasis on proper placement is one trend I heartily support.

## Eric I.

July 30, 2014 - 7:34 amIs this a future gem from Rand Paul?

“If you have one person in the country who is, like, the best at performing heart surgery, that person maybe should perform every heart surgery in the country.”

## John

July 30, 2014 - 10:19 amLess than 5% of American high school students graduate with calculus behind them, and my bet is that he’s one of them. So he’s not the idiot people on this forum want to make him out to be. Clearly he will never intend to take away math teaching jobs (if there ARE any in the US), but I’m 100% behind ONE calculus expert documenting what it’s really all about.

Our calculus text books contradict each other so much that even Americans who take calculus in high school really do NOT know which version is correct, much less how to explain it.

At least give Rand Paul that much credit, even if he is dead wrong (and contrary to his father) regarding illegal aliens.

## Kenneth Tilton

July 30, 2014 - 10:47 am@Michael, you are right. Many teachers blame the students for their bad teaching. Well, maybe if a world of better teaching grows around them they will quietly mend their ways.

@Joel, you are right, too. I should be talking about ineffective teaching. Those teaching that way are victims of our “cowboy” approach to teaching, in which every teacher is supposed to go out their and figure it out for themselves. What profession works that way? If we make teaching a true profession with clear best practices and books and software, /then/ we can start talking about individuals who fail to measure up to those standards. I doubt there will be many.

## Chris

August 8, 2014 - 9:03 pmIt bothers me when I hear parents saying they wish they’d been taught like that (especially based on a one-off lesson). I bet their teacher from back in the day wishes they’d always turned up as a motivated, focused student eager to learn?

## Joel Patterson

August 10, 2014 - 3:26 pmChris, I’ve heard that statement a few times, and it bothered me, too. Then I figured they were just trying to express hopefulness about their child making it further along than they did… which is pretty much every parent’s goal.

## Joel Patterson

August 10, 2014 - 3:47 pmJohn–um, when I read this:

“Our calculus text books contradict each other so much that even Americans who take calculus in high school really do NOT know which version is correct, much less how to explain it.”

I’m not sure ‘contradict’ the accurate word for this. Every Calculus book I’ve seen has the same definitions for the derivative, and same set of integral tables, same infinite series for sin(x) and cos(x). Some develop ideas in a different order, some use different proofs for the FTC, but they are all pretty much factually correct.

Why not let the different textbooks coexist and give teachers the time to compare the differences for themselves to decide what’s best for the students they have?

## Jason Dyer

September 26, 2014 - 8:02 amI’ve heard the “let’s all use the same Calculus textbook” argument before but:

1. Try to get a group of people to argue which textbook is the Master Textbook to Rule Them All. Things will fall apart. By popularity I would guess Stewart, but you’ll find a lot of Stewart haters out there.

2. Actually putting forth special effort to make the One Master Textbook isn’t any different than picking an already existing one. Nobody puts all the work in attempting to write a bad textbook. (And arguing that ‘it would be a team effort’ — well, plenty of textbooks are also team efforts. Often the solo efforts end up being more readable.)

3. Mathematical exposition has changed, historically, over time. This isn’t just manner of proof, but manner of writing and presentation and ordering and other important cognitive things. Assuming that a snapshot can perfect the presentation for all time is bizarre. The closest that’s ever been reached is by Euclid (and there are places that still use original Euclid) but it is very hard to argue there is no better way to write a geometry textbook.

4. Context matters: someone gunning to be a mathematician and someone aiming to be an engineer will value different things in their learning (and, indeed, there are sometimes different classes with different textbooks).