"When a student asks for help on a problem, I..."

[Click here for the bigger version of the pic.]

What's missing from this chart? What do you do to keep kids working on a difficult problem?

Two caveats to my chart:

1. I'm not saying that "promotes/eliminates struggle" are the only relevant factors when deciding how to intervene with a student. But one of the things I'm worrying about is how to make my kids more comfortable struggling with tricky problems. I'm interested in what I can say to a kid that will make him/her more likely to keep on trying something difficult.
2. There are obviously things that need to happen beyond the student-teacher interactions in order to promote the kind of environment that I'm looking for. Some of the stress my students experience when they get frustrated is intrinsic to the math problem. But some of the stress is social, a result of kids feeling like they're dumb for having to work so hard. We need to do things in class that normalize struggle and effort. (I think public presentation of interesting dead-end approaches and whiteboarding might be part of the solution.) So good interactions in the heat of the moment is just one aspect of what has to go on.

Links:

• "I ask for wrong answers to start the thinking process."  (Looking for Wrong Answers, Dave)
• YOUR POST HERE

A bunch of times that Sal Khan asks kids to memorize stuff

"You will never hear in any Khan Academy video 'memorize this formula.' " - Sal Khan

Well, that should be easy to check. And he's wrong.

1. "What we are going to do in this video is to give you just a ton of more practice and start you on your memorization of the multiplication tables." (Multiplication Tables, 0:13)
2. "If you don't get it, maybe you just want to memorize this." (Squeeze Theorem, 0:37)
3. "And it's good to memorize this." (Basic Ratios, 0:41)
4. "I think this is a good time to just memorize the sides of a 30, 60, 90 triangle, because that's something that one needs to know in life. It's surprisingly useful, especially once you start taking standardized tests or do trigonometry." (Circle Area, 1:59)
5. "You really just need to memorize that the hypotenuse is twice the shortest side." (Same video, 2:46)
6.  "You also should just need to memorize PV is equal to NRT." (Thermodynamics iv, 9:47)
7. "All you have to really memorize is this." (Harmonic Motion, 6:52)
8. "You can just memorize this... and you might want to just memorize it." (Trig Identities, 1:21 and 5:02)
9. "You should memorize this." (Law of Cosines, 8:46)
10. "So you should, to some degree, memorize these." (Quotient Rule, 3:38)
11. "And now I'm going to give you one of the few things in math that's probably a good idea to memorize." (Quadratic Formula, 5:15)
12. "And it's useful to memorize to some degree what this means." (Binomial Expansion, 1:33)
13. "It's something good to memorize," (Conservation of Energy, 2:53)
14. "This is something that you really should just memorize." (Trig Identities - Secant, 8:19)
15. "It's probably not a bad idea to memorize some form of this formula." (Projectile Motion, 4:55)
16. "It doesn't hurt to memorize this." (Radians and Degrees, 5:06)
17. "I'd like to help you memorize this." (Normal Distribution, 7:57)
18. "You probably won't be using this in your everyday life five or ten years from now, so it's OK if you don't memorize it, but temporarily, put this in your medium-term memory." (Subspace, 2:31)
19. "In future videos I will give you a little bit more intuition for why this works, or I'll actually show you how this came about. But for now, it's almost better just to memorize the steps." (Inverse Matrices, 6:24)
20. "And then this formula...just memorize. This "P e^rt' will make a lot of sense to you, and you will have a permanent neuron for it the rest of your life." (Compound Interest, 7:51)

There are a lot, lot, lot more times that Sal Khan asks students to memorize stuff. The way to get access to this is through the Khan Academy interactive transcripts. So just use your favorite search engine to look for the word "memorize" through khanacademy.org, and then click on the "Interactive Transcript" button on the video page. Then use your browser to search through the page for the word "memorize."


Anyway, happy searching.

Khan Academy has a really rich resource for teachers

Forget the videos for a moment. Khan Academy has, below every video, a collection of student questions on the material. To be fair, the questions are mostly irrelevant. But there a ton of gems buried in there, and Khan Academy is definitely a place that I'm going to stop when I'm trying to figure out where my students are going to trip up on a topic. A lot of these questions are going to make their way onto the Math Mistakes blog eventually, but for now, here's a sampling.

Sorry for the small picture size. Click on 'em though -- it'll only take a second.

The thing that gets me most excited are the questions on the primary school material. One of my weaknesses as a teacher is my inability to effectively help students who are struggling with negative numbers, fractions, multiplication and division of fractions, square roots and the whole array of stuff that kids are supposed to know coming into my classrooms but don't. And the blogging/tweeting community skews towards teaching older kids, so I have fewer folks to lean on.

Digging into what kids struggle with in earlier math material is proving to be super interesting (of COURSE kids would think that a negative plus a negative should be a positive) and I'm betting that it'll make me a better teacher.

A pretty wonky post about how different curricula handle the properties of arithmetic in Algebra 1

I'm working on my Algebra 1 curriculum, and I came across two really different approaches to teaching the properties of arithmetic. Because I find them interesting, I want to share them, record how I plan to use them, and then ask a question at the end.


Here's how CME introduces the properties of arithmetic to 9th graders:

Students are asked to complete the above table and find several patterns in the table. They do this with the multiplication and addition tables. Students are given a big, arduous, concrete task and asked to find shortcuts. Students are given hints directing them towards particularly interesting shortcuts. Then kids share their work, and then we call it a day.*

* This table is actually from Day 4 of their sequence. Day 1 observes patterns in the addition and multiplication tables, restricted to non-negatives. Day 2 reviews arithmetic with negative numbers. Day 3 extends us to the full addition table. 


It's not until the next day that we have any sort of general expression of the properties of integers, and at this stage we still don't use any algebraic notation at all. Instead we articulate the "Any order" and "Any grouping" properties of arithmetic. Then we extend these principles to cover rational numbers and then the reals. 


Finally, students look at various algorithms and try to explain why they're true. But that's really hard, using English, and so after students struggle to justify the reliability of these algorithms using our common language and vocabulary, the students are presented with variables. Variables are ways of talking about arithmetic with generality. And, boom, then comes the rest of the Algebra 1.


Now, here's how Park Math introduces the properties of arithmetic to their kids:

From there they move to finding counterexamples to claims about this rule, motivating the importance of mathematical justification. From there they quickly move to the introduction of new rules and, then comes this problem:

In a sentence, CME eases kids into variable expressions via the need to justify arithmetic shortcuts. Park Math assumes that kids understand variable expressions (and equations) and asks them to justify algebraic rules.

The personal puzzle is how to hack these two approaches apart and come up with something coherent and useful for my classroom. I'm assuming the pace of CME is too slow for my kids but the abstraction of Park Math is too much to start with. My current plan is to start with CME and move on to Park.

But the more interesting question is why these two curricula assume such different things about the students who are beginning Algebra. Does anyone know why that is? Is there something in the water in Massachusetts/Maryland?

Exeter versus Park Math

Within the big tent that is problem-solving, there are some really different approaches to teaching. In this post I'm going to draw out one of those differences, offer an interpretation of their significance, and then invite you to disagree with me in the comments. Deal? 

There are two free, complete, and well-regarded high school curricula available that I know of: the Park School's and Exeter's. (For some quick reading on these schools, click here, here, here, for Heaven's sake don't click here, and here.)

Let's start by looking at two problems, the first from Park Math, and the second from Exeter.

These problems look pretty similar, and I think that the generalization is fair: there isn't a great deal of difference between the types of questions that students are asked to solve in the two curricula. But these fairly similar problems are situated in entirely different contexts. In the Park Math curriculum this problem serves as a concrete hook to hang the rest of a series of linear functions problems on. In the Exeter problem set, this problem is preceded by a rates problem and followed by a problem that asks students to practice a guess-and-check technique.

That's a huge difference between Park Math and Exeter. Actually, I think that it's two huge differences.

• In Park Math problems are organized by topic, as in a traditional curriculum. Linear functions is followed by geometry of lines, which is followed by coordinate geometry, etc. Things work entirely differently for Exeter. In Exeter's problem sets there are a cluster of densely related topics that drop in and out of the curriculum as the year goes on. There is no linear functions unit in Exeter; questions relating to linear functions appear on the second page and appear on nearly every page for a month or so. In contrast, there is a well-defined chapter on linear functions in Park Math.

• Because topics are not organized by topic in Exeter, the curriculum does not offer students much context at all for any of the problems that appear. In Park Math problems are very carefully scaffolded so as to allow students to discover solutions to more difficult problems without the help of a teacher. Questions continuously build, taking you deeper and deeper into a subject.

This difference signifies a big difference in the pedagogical assumptions of each school. The Exeter curriculum seems premised on the idea that deep learning happens when students make connections, and solving difficult problems divorced from context forces one to make connections. Students will learn best when forced to situate a new problem among the rest of mathematics. Categorizing problems for students as they're learning them is like organizing questions by topic on an exam. The context of similar problems makes retrieval way too easy on students, and they learn less from it. At the end of the day, it's all about making connections between different mathematical topics.

Park Math, on the other hand, seems premised on different assumptions. Their curriculum is consistent with a vision of learning that values narrative above making connections. Their curriculum is designed to offer concrete hooks and guiding problems for students to dive into. Then the problems develop and complicate the introductory problems and then, after the development, students are offered an array of difficult problems that use the mathematics that was developed. They're betting that it's sufficient to allow students to make connections in this section, after the story of (say) linear functions has been told.

So, that's the framework I'm offering for the differences between Park Math and Exeter's approaches. Park Math favors narrative in learning, whereas Exeter prefers making connections.

Every problem set that I've ever made follows the Park Math format. Exeter is the weird one here for me -- I've seen very little that resembles their approach. But, for a while now, I've been a bit worried that my scaffolded problem sets are offering too much support for my students. I'm thinking that I focus too much on narrative, and not enough on helping my students make connections.

Homework, for the comments:

1. Do you agree with Michael's analysis? Why or why not?
2. What other important differences between the two curricula do you see?
3. When designing a lesson, do you aim for narrative or connection-making? Does it depend on the lesson?
4. How could you add more connection-making to your classroom, without fully implementing the approach of Exeter Academy?

Drills for Teachers

What does it take to become great at teaching?

That guy looks pretty good. He's got a tie. His arms are crossed. He moves SUPER SLOWLY.

Anyway, if I polled 100 teachers about what it takes to be great, I think 75 would say "experience." Then 24 would say "reflecting on experience."

Part of the getting great definitely involves the classroom experience. Good teachers use the classroom to try new lesson structures, practice tricky management techniques and ask subtle questions of students. When classroom experience is combined with regular reflection, these teachers do get better.

Good teachers know how to use the classroom to improve. But I don't think that's enough to get great. I've got a theory about what it takes to become a great teacher. I think that great teachers find way to practice regularly, outside the classroom. But what does it mean for a teacher to practice?


"The Mistake Most Weak Pianists Make is Playing, Not Practicing."

Study Hacks is one of my favorite blogs, written by Cal Newport, a professor of Computer Science at Georgetown. On the blog he published a letter he received from an accomplished piano player. The letter details the differences between the habits of the top piano players and those of the musicians just below the top. Here's an excerpt, with four strategies for practicing that separate the top musicians from the rest:

• Strategy #1: Avoid Flow. Do What Does Not Come Easy.
  “The mistake most weak pianists make is playing, not practicing. If you walk into a music hall at a local university, you’ll hear people ‘playing’ by running through their pieces. This is a huge mistake. Strong pianists drill the most difficult parts of their music, rarely, if ever playing through their pieces in entirety.”
• Strategy #2: To Master a Skill, Master Something Harder.
  “Strong pianists find clever ways to ‘complicate’ the difficult parts of their music. If we have problem playing something with clarity, we complicate by playing the passage with alternating accent patterns. If we have problems with speed, we confound the rhythms.”
• Strategy #3: Systematically Eliminate Weakness.
  “Strong pianists know our weaknesses and use them to create strength. I have sharp ears, but I am not as in touch with the physical component of piano playing. So, I practice on a mute keyboard.”
• Strategy #4: Create Beauty, Don’t Avoid Ugliness.
  “Weak pianists make music a reactive  task, not a creative task. They start, and react to their performance, fixing problems as they go along. Strong pianists, on the other hand, have an image of what a perfect performance should be like that includes all of the relevant senses. Before we sit down, we know what the piece needs to feel, sound, and even look like in excruciating detail. In performance, weak pianists try to reactively move away from mistakes, while strong pianists move towards a perfect mental image.”

The best musicians figure out ways to focus on the hardest parts of their skill. They don't just practice by performing complete pieces all the time and reflecting on the performance. As important as planning for their performances are, that's not all that they do.

When teachers try to get better using the classroom they're playing, not practicing. It's good, but there needs to be something else.

In the excerpt from the piano player above he advises focusing on the hard parts of the skill and systematically focusing on weaknesses. How many teachers think like this?


Drills for Teachers

Good drills are focused, purposeful and difficult. Reflecting on a recently delivered lesson is purposeful, but not focused. Planning a lesson is purposeful and focused, but not necessarily difficult. If you're intellectually drained after planning a lesson then it's probably a drill. If not, then planning that lesson is probably not making you much better.

Piano players get better by running scales, strengthening their fingers, training their ears, focusing on the hardest parts of pieces, and complicating tricky sections of the piece. What are some analogous drills for teachers?

This question is important and exciting to me. I am dying to know what you guys think. Here's what I've been able to come up with so far:

• Designing curriculum - This is a common drill, even if many teachers don't see it as such. When you design your own curriculum you need to think hard about multiple aspects of the lesson. You are forced to be creative, as opposed to reactive. 

• Mistake Analysis - This is why I made my MathMistakes site. One of the skills that great teachers have is the ability to quickly analyze student work and offer a response. The classroom is an inefficient way to improve at this, but fortunately there is a large supply of student work that we can practice on. This is focused on a particular aspect of teaching, it's purposeful and oh-my it can be hard.

• Analyzing the work of others - There a bunch of ways to do this. One way is to analyze video of other teachers. This is why I've curated a Classroom Footage page (see the link at the top). You could also focus on the curricular work of others, or on short Khan Academy videos. Via @Trianglemancsd a good question to ask is "What are the pedagogical assumptions of this video/curriculum/text?" Another way that I like to structure this drill is to list what's working and what could be improved.

Are there drills that you use, or have seen used? Can you think of any drills for classroom management? Comments are open for business.

Call for criticism and feedback

Hey smart people. I'm teaching a course in "How not to be an awful person" next year, and I've got a draft of the course plan. I'm hoping to get some criticism and help from you all in making this better.
Self and Society - Course Plan v1.9

Here's the criticism that I've heard so far from readers:

• It's unprofessional to discuss charity choice with students. Likewise for how much money they choose to spend. In general, teachers should not ask students to reflect on the details of their personal lives. (From a colleague)
• Kids will be more interested in the abstract, big ideas than in personal reflection.
• The trip to the food pantry/soup kitchen should be earlier in the year. 
• There is redundancy in the course plan.
• There is insufficient attention paid to dispositions, emotions, thoughts and too much on actions.

While I'd appreciate ALL of your thoughts and criticism, feedback concerning the above are especially appreciated.

Thanks in advance, readers. I owe you one.