So here is a task that occupies maybe two rungs on the ladder of abstraction:

You bought some 40-cent pencils and some 70-cent pens. There were 10 of them in total for 460 cents. How many pencils and ballpoint pens did you buy?

At this altitude, it’s difficult to apply your intuition to a solution. The important information has already been identified. You’ll decide how to represent that information (symbolically, perhaps, with variables and two simultaneous linear equations) and what to do with that representation (manipulate those equations, solving for both unknown variables). But that’s the extent of the task ladder.

So watch and marvel, as I have, at video of a skilled Japanese teacher reconstructing that task ladder to include more rungs above and below what few currently exist. I have watched it five-or-so times since Steve Leinwand introduced it in his April talk at NCTM. I encourage you to do the same. For the sake of our conversation here, though, I have created a smaller excerpt of just the teacher’s *questions*:

Here are those questions and some commentary. The questions are listed in chronological order, like a script, but I’m sure you understand we’re starting at the bottom of the ladder:

- This is what we’re going to study today. What is this?
- What’s this?
- How much are they? Guess.
- So how much is a ballpoint pen?

I prefer tasks that make more intentional use of guesses, prediction, and intuition. But I don’t know what else he could do with this task except ask his students if they know what a ballpoint pen is and check to see if they have enough sense of that context to guess at its cost.

- You bought some 40 yen pencils and some 70 yen ballpoint pens. There were 10 of them in total for 460 yen. How many pencils and ballpoint pens did you buy?

The task is introduced.

- A table?
- Do you think the table would help you?
- Do you understand what the table is?
- Rio, do you think the table would be useful?
- Well then, if we’re going to make a table, we need some labels, don’t we?
- What kind of table?
- Do we need a label like this?
- Do you understand this?
- What else do we need as a label?
- Do we still need something else?
- How about this? “The total price.”
- What do you mean 0-10?
- You mean no pencils?
- And how many ballpoint pens?

This is one of my favorite moves in the lesson. A similar task in the United States might explicitly direct students to create a table. It might *provide* a table with several rows pre-filled. In both cases, *the problem would be abstracting itself*. It would be answering the question, “How do I represent the information that’s important?” *for* the student. It would be locking the process of abstraction inside a black box, far from the student’s view. And then twenty years later we’ll marvel when that student, now grown, complains that “algebra always seemed too abstract.”

But the Japanese teacher isn’t sending his students off to *discover* the abstraction. Yes, the *students* are making most of the crucial choices here â€” from the selection of a table to the labels of the table to the domain of the table â€” but the teacher steps in after each of those choices and makes them totally explicit.

The choice to include “0 pencils” in the domain of the table, for one example, is far from obvious. So when a student suggests it, the teacher underscores that suggestion *over and over again*, doing everything but sending a skywriter up in the air to write “YOU MEAN *NO* PENCILS?!” above the school in smoke.

- Well there are many combinations of the numbers right?
- Seika, can you tell me any examples of the combination?
- 3 pencils and 7 pens. Did you make the same combination?
- Is it correct?
- Are there any mistakes on the board, please tell me?
- But before we see that, there already is an answer, isn’t there?
- What is the answer?
- This pair?
- Is it similar with your notebook?

The teacher asks his students to calculate the table by hand. In another bright move, he passes out the table’s 11 columns to different students who calculate and return them. Now those columns are pinned to the board, arranged in no particular order.

He notes that they have solved the problem. One of the columns contains 460 yen. That’s the combination of pens and pencils they’re after. The task seems complete.

But the ladder continues upward.

- Can someone arrange the cards and make the table easier to see?
- Why did you arrange the cards like this?
- But why?
- It’s easier to see?
- The order of numbers?
- For example, if I change the cards like this, is it wrong?
- This is the answer, is that all?

*He’s still picking at the abstraction!*

Two girls come up and arrange all those haphazard columns into ascending order in a table. And he asks them, “Why did you arrange the cards like this?” He doesn’t let up. He takes one column and sets it out of order. “Is it wrong?”

No, it’s just less useful. A student says, “It’s easier to see.”

*What’s* easier to see?

The student explains that when we arrange the table in order, the change in total price is hard to miss. When they’re out of order that change is hard to see.

- How does our method match the Easy / Fast / Accurate rule?
- Is it fast?
- It is easy?
- Is it accurate?
- How can we solve the problem more quickly?

Now we’re abstracting over new variables: speed, ease, and accuracy. The students decide unanimously that their method is accurate. They’re mixed on ease and no one thinks it’s fast. So now we’re ready for a faster, easier method for a new problem.

- You bought some 60 yen colored pencils and some 80 yen markers. There were 12 of them all together. The price they cost was 820 yen. How many colored pencils and markers did you buy?
- What is the title of today’s lesson?

What amazing work from such meager beginnings.

A few final notes before we dive into your commentary:

**Titles.**What’s with titling the lesson at the end of the day? Anyone know what’s going on there?**Technology.**Blackboard. Chalk. Teacher voice. Student voice. Paper. Pencil. It’s multimedia in the most literal sense of the word. (Tentative hypothesis: it’s very difficult to work on the ladder of abstraction if your tasks are limited to one medium.)**Tables.**A table is a representation of a context’s most important information. It’s an element of an abstraction with which math teachers are*extremely*comfortable. I think we find it easy to assume, particularly at the secondary level, that what’s obvious to us about these representations is also obvious to our students. I love how every single aspect of the table in the video is up for negotiation and debate â€” its labels, the inclusion of zero values, the fact that you probably oughtta put*something*in ascending order.*Nothing*is taken for granted.**Time.**At the end of class, the teacher notes that, wow, that took a lot of time. It turns out a deep understanding of abstraction doesn’t come for free.

**BTW**: Michael Pershan offers an excellent analysis of Khan Academy as it relates to high-performing Japanese classrooms like this one. He closes with an alternate vision for Khan Academy that’s provocative. You’ll especially enjoy it if you find this #mtt2k business too mean-spirited. Commentary about that video belongs *on that video’s YouTube page*, though. All comments about it here will be shot on sight.

**Featured Comment**

Totally at the effectiveness of leaving the title to the end. The students not only see it because the content of the lesson is played out if front of them, but thereâ€™s a pretty clear agreement that the title of the lesson fits. One of the students even says, â€œCool.â€ How often do you get a student saying cool when you *start* a lesson by writing the title on the board.

## 41 Comments

## Tom Hoffman

August 2, 2012 - 8:17 am -Somebody send this guy a copy of Teach Like a Champion! These kids aren’t even SLANT-ing.

## Steve G

August 2, 2012 - 8:50 am -Saving the title of the lesson until the end of the day is a way to “check for understanding”. As in, Did you get what this was all about? Also, it’s another move up the LOA, generalizing the two questions that the students worked with into an idea about dealing with a whole phylum of questions.

But, Dan: “Technology: Blackboard. Chalk. Teacher Voice.” By your logic you could stop there and still have a multimedia, literally. (Replace blackboard with white board, Smart Board, PowerPoint; it makes little difference for my argument.) I found this example for your hypothesis underwhelming, even undermining. What would be a monomedia presentation–chalkboard with no chalk, teacher silent? That’s a pretty Zen lesson there, but I don’t know that you could call it math.

(Sure, you could have a lesson with teacher voice alone, but in my experience as a student and teacher that doesn’t really happen in *math* classrooms. You’ve got to have some representations of quantity somewhere, don’t you?)

## hillby

August 2, 2012 - 9:11 am -I agree with Steve G that the lesson title becomes a check for understanding, but it’s also a spoiler. Perhaps if they knew they were going to be making equations, the students would be less patient with the intermediate process of fussing over the table.

I also want to point out that in the last two years, every math classroom I’ve walked into looked more like this kind of back and forth, rather than the lecture strawman people keep propping up.

## Kelly Collins

August 2, 2012 - 9:21 am -I love how purposeful this teacher is in every single thing he does, not just in the questions he asks, but how his materials and setup relate to those questions. The table being cut up into pieces is *essential* to his question about the ordering of the table.

The flow of the lesson is also just so much better than in many math classrooms here because of his level of preparation. He didn’t waste time writing out the questions. He didn’t waste time drawing a table. He already had those things prepared. How much longer (and how much more boring) would it have been if he hadn’t already done those things? It seems like a small thing, but from where I’m sitting, it beats the hell out of just writing everything on the chalkboard/smartboard/whiteboard as you go. The lesson takes a while, but he spends his time wisely, on things that really matter. Just out of curiosity, how long was the full lesson?

After watching this video, I’m going to go ahead and prepare my own table cards! Laminated and with magnetic tape on the back so that I can use dry erase markers on them and reuse.

Really great post.

## Jim P

August 2, 2012 - 11:23 am -Breaking my response up to little pieces to help myself organize my thoughts.

Title:

I agree with Steve G and hillby on the idea of saving the title till the end. There are many aspects that are useful for saving the title until the end: checking for understanding, and allowing students freedom to freely think of ideas without being prescribed a method. It’s actually a very interesting approach that I might try with some topics. It has been iterated to me on many occasions that it is good to have the objectives of the lesson always up on the board somewhere. But this always bothered me: the fact that it does not allow for students to freely imagine and explore the mathematics. I am still a bit conflicted on this point. In the past I’ve tried putting the “title” of the lesson until after we’ve done act 1 (sometimes not until I’m well into act 2). I think it would be interesting to throw in some lessons where I save the title until the end, though. After all, students could be learning much more beyond just what the title says.

Technology:

I think if you limit the task to one medium (for example, only lecturing from the teacher), it limits student opportunities to fully explore and understand the concepts.

Table:

Like you, I loved the fact that each aspect of the table is being engaged. A lot of the feedback that the teacher was giving was noting a pause in the students thinking. “Does it make sense that we are doing this?”

Time:

Well invested time, I felt.

In general:

The “HAKASE” concept is interesting. I would like find more background on this at some point. I attempted to google “hakase” just now, and had no success. It seems to be a phrase that students were taught to memorize. Is there a greater philosophical discussion on whether good mathematics is always searching for “Easy / Fast / Accurate?” It struck me as an interesting topic. If I think about it on the level of discovering mathematical processes, then sure, that may be our ultimate goal. But if we are strictly talking solutions, then I think there is value in thinking about even the “difficult, slow, inaccurate” solutions.

but like I said… I want more background… which appear to be difficult to find using that phrase.

## Chris

August 2, 2012 - 2:13 pm -Totally at the effectiveness of leaving the title to the end. The students not only see it because the content of the lesson is played out if front of them, but there’s a pretty clear agreement that the title of the lesson fits. One of the students even says, “Cool.” How often do you get a student saying cool when you *start* a lesson by writing the title on the board.

Additionally, this lesson works on a low-tech level. He needs the students to contribute on paper so he can move things around. It’d be pretty tough to do this in a more tech way. The only way I might change this is to have my questions typed up on slides. On the other hand, he purposefully places what he’s written in certain spots. He doesn’t necessarily want them in digital form.

Very cool.

## Cindy Whitehead

August 2, 2012 - 2:18 pm -What about having students “guess” a combination? (Dan’s strategy of guessing “too high” and guessing “too low” – as in, a cost that’s too high and a cost that’s too low.) I think for this problem, students could “guess” the correct solution quite quickly since there are such limited options. Change up the total number of items (and cost) and you start to see the value in looking at the patterns in the table, as well as the “efficiency” of using equations. (This video skipped over that part. Is it on the original?)

Give students “table strips” to write a “possibility” on and put them up all over the board? Serves the same “messy to organized” prompting, and might also allow for missing values that should/could be filled in.

HA-KA-SE: I like evaluating a strategy this way. (I have used “Is it efficient? Is it accurate? We have even placed methods on a coordinate plan based on this.) It is important to differentiate that one method might be the most HA-KA-SE for one problem, but for an even slightly different problem, another method might take the prize. In addition, not all students share the same perspective about whether it IS fast, easy, and/or accurate.

## Steve G

August 2, 2012 - 3:40 pm -@Chris:

How often do you get a student saying cool when you *start*

a lesson by writing the title on the board?

Seriously! Spot on, Chris.

## Peter Price

August 3, 2012 - 5:26 am -Great post, Dan

Can I just say, I really enjoyed watching students enjoying the process of thinking about thinking about mathematics?

And like others above, I loved the idea of putting a title on the lesson, at the end. Sort of like saying “So, what have you learned through the process today? How has your thinking developed as you have followed today’s journey?”

Is this possible in the west (I teach in Australia)? I wonder how much of the beauty of this lesson comes from the pre-existing culture not just of this classroom, but of a general Japanese style of teaching math?

## Belinda Thompson

August 3, 2012 - 7:12 am -An important thing to note about this lesson, and something that I’ve learned from studying other Japanese lessons is that this teachers (most likely together with other teachers) have thought about and anticipated everything these students might say and do. If he finds that he missed something, it will be added to the plan for next time, and he will share this info with others. He knows which questions he’s going to ask and he knows the range of answers he’s going to get. His moves are impressive, but they are certainly not by the seat of his pants. In my coaching work, I implore teachers to solve every problem they present to students and to anticipate the good, the bad, and the ugly that will arise when students work on the problem. This is a tough sell in the west (we don’t want to presume or constrain), but is a critical aspect of lesson planning in Japan and China.

Classroom lessons from Japan and other countries in the TIMSS Video Study are available for free at timssvideo.com.

## Chris

August 3, 2012 - 7:28 am -I think you’re absolutely right, Belinda. I’d love to be able to do the same but that’s a tall order when you’re preparing numeracy, literacy and six other subjects everyday. Even more challenging if you’re teaching a combined grade. However, this is a strategy I hope to try to implement more often in the coming school year. Anticipating student misconceptions is key to steering the lesson.

## James Key

August 3, 2012 - 9:55 am -@Dan: you wrote “I prefer tasks that make more intentional use of guesses, prediction, and intuition. But I don’t know what else he could do with this task except ask his students if they know what a ballpoint pen is and check to see if they have enough sense of that context to guess at its cost.”

I just would like to note that this lesson was probably given on Pseduocontext Saturday (they go to school 6 days a week in Japan, right?). Exactly when do I find myself knowing the prices of the individual items, as well as the total cost of the order, but having no idea how many of each I bought? Probably never. But this video underscores this point: you don’t need “realistic, real-world problems” to have a *great math lesson.* Someone recently pointed out that what students need is to perceive the *relevance* of a problem, which is another way of saying they need to care about solving it. Realistic contexts can be dull; contrived contexts can be exciting. (That being said, I love all the real-world stuff I get from this blog!)

Titling. Wow — this was so cool! Another way of saying, “So we solved a problem today — congratulations. Can you describe the general class of problems like this one? Can you concisely articulate the strategy we used to solve our problem?” Great move.

Dress code: guess I need to step up my game and order me some suits! And were those tennis shoes?

Provisioning: As others have noted, I was struck by the prepared elements — he had the “labels” ready to go. The use of magnetized blocks was awesome.

Media: the simplicity of this lesson was great. I have to admit, I’m a little geeky about technology. I use it a lot, but sometimes I think the return on my {time spent preparing tech lesson}:{instructional value} ratio is poor. I’m just finishing up a two-week summer session, where I teach for 3.5 hours straight every day. I don’t even try to plan everything down to the minute like I would for a 45-minute class period, and it’s interesting to me that I find I do some of my best teaching in these circumstances. I just have a couple of whiteboards, some markers, and a lot of generic teaching tools at my disposal (varying the arrangement of desks, varying the grouping/pairing arrangments, calling strategies, etc.). And “planning” consists of making a list of a half-dozen topics to treat with the students. I take about 5 minutes to plan for the day, but of course I’m drawing on 9 years of experience as well.

Math observations. For what it’s worth, here was my thought process on this problem. 10 pens would cost $7.00 — that’s too much. 10 pencils would cost $4.00 — that’s too little, but now we’re getting close. I started thinking about what would happen if I swapped out one pen for one pencil, and that’s when it hit me: the pens cost $0.30 more than the pencils, and I need to close a $0.60 gap. So I need two pens! Hence I need 8 pencils, 2 pens.

I think this kind of thought process is another important element in the “geodome of abstraction” (someone rightly pointed out that a ladder is too restrictive). I never wrote anything down, I used no equations, made no table. Just two guesses, followed by a flash of insight.

Since we’re on systems of equations, and since I’m on “using intuition rather than formal algebra,” I would just like to share a strategy I use for teaching systems. Here is my contrived pair of examples:

1. Dave buys 2 hot dogs and 2 sodas for $13.00.

2. Tom buys 2 hot dogs and 3 sodas for $15.50.

3. How much does each item cost?

Can you solve this problem without writing anything down? How did you do it? I give this to my students and then show them this problem:

1. Sarah buys 1 box of Junior Mints and 2 tubs of popcorn for $13.00.

2. Kristen buys 2 boxes of Junior Mints and 1 tub of popcorn for $11.

3. How much does each item cost?

Students should have a bit more difficulty solving this one, so I ask them to figure out what it was that made the first problem so easy. Answer: Dave and Tom bought the same number of hot dogs. Now comes the big moment: can they figure out that doubling one of the orders in second problem will produce the same situation?

All the above should be totally accessible, and without introducing formal algebra, you can motivate the idea of 1) using elimination to solve systems of equations and 2) mutliplying the terms of one equation by a scalar to prepare for elimination.

This comment was *a bit long* — guess it’s time to start my own teaching blog! :-)

## Robert Berkman

August 3, 2012 - 10:13 am -Why isn’t anybody commenting on the use of the blackboard in this lesson? If you notice, the work is well organized, the teacher is prepared with different pieces of the puzzle, and he almost never erases what is on the board. This is a much, much different presentation than you would see in any American classroom.

## Bridget

August 3, 2012 - 10:41 am -I agree with Cindy as I immediately thought of efficiency when I saw the fast-easy-accurate. Algebraic thinking is what we want to move kids towards-what they could do in the intermediate grades with guess and check, now can be done with methods that take into account patterns…thus a more efficient method. Older students should have multiple methods of approach to a problem and being able to choose the most efficient method, along with providing an argument to support that method, I believe is the most important thing a teacher can teach their students. The teacher in the video was making visible the thinking of this problem…a skill that is sometimes overlooked in the classroom b/c the questions seem so minute-that the classroom teacher most likely believes they are stating the obvious. Further, it is more proof that math teaching is about the process, not the product.

## lesanno

August 3, 2012 - 11:54 am -I love the title-at-the-end approach, but I have a niggling concern. I work solely with students who have learning disabilities, and while they would all benefit from this kind of information processing, some of them have such impaired working memories and language processing speeds that they would struggle to achieve it as it’s presented here.

Instead, I would try giving a different/broader title at the beginning of class (Say, Solving two-piece problems. It doesn’t have to be pretty.), then repeat that title throughout the lesson so they have an anchor to which they can attach what they learn. Without a title for what they’re doing, one that is repeated and emphasized throughout the material, it is exceedingly difficult for (some of) them to keep the different moments, ideas, etc. connected in their mind. At the end I could ask them to re-title the lesson. Then, throughout subsequent projects, activities, etc., I would either link the two titles firmly together, or outright replace the first title with the second.

## Jim P

August 3, 2012 - 1:00 pm -@Belinda post#11

It would be interesting if they design the lesson together and places the emphasis on expecting potential responses — and then adding to that. I’m going to explore that TIMSS website more when I have some time, and make myself a list of things that I want to anticipate (student misconceptions, main concepts, …etc… just the categories)

@James Key post#13

I also noted the simplicity of it all… It made me wonder the same ratio problem that you addressed. Lately I’ve been so entrenched in trying to learn and incorporate new technology, that I haven’t really thought about the ratio of time invested vs instructional value. I still will pursue learning about new technology, I think, but it’s worth re-aligning myself and think about the truly important thing here.

@lesanno post#16

Good point about students with struggles without the title being there in the beginning. What you said addresses the small conflict I had with the title as well… for precisely the same reasons that you stated. Your suggestion of having a broader title is interesting, and I might give that a try!

## Doug Smith

August 4, 2012 - 7:55 am -@Belinda

Perhaps they incorporate Math Plays, which would definitely help produce this type of lesson. In a math play, teachers sketch out a lesson, script, and roles, and then act out the lesson.

One thing that I wonder about with the title at the end is what the students are told at the beginning of the class as to what they will be working on. Do the students have clear learning intentions from the start? Or are they just exploring a mathematical idea and seeing where it takes them?

## mr bombastic

August 4, 2012 - 9:06 am -Is this teacher a “skilled” teacher or an average teacher in a math education system that provides an environment where average teachers can be succesful?

The Japanese take a bottom up approach. Small details of each lesson are discussed, the lesson is observed multiple times in classrooms and adjusted, and the write up is kept at the ministry of eduation and readily available to teachers.

How useful would it be to regularly observe high quality lessons and to have easy access to a high quality sample lesson plans – for any topic – including expected student responses?

I never quite understood the importance of a title or objective or whatever. It should be obvious what you are trying to do in a lesson (figure out how many pens and pencils). If it isn’t, a title or objective won’t help. He did give a title at the end, but really didn’t place much emphasis on it, and the kids didn’t seem to either.

## James Key

August 4, 2012 - 5:36 pm -@19 mr bombastic: “I never quite understood the importance of a title or objective or whatever. It should be obvious what you are trying to do in a lesson (figure out how many pens and pencils).”

Hi. First, I might characterize the lesson a little differently. Was it really about figuring out how man pens and pencils were purchased? On a surface level, yes, but surely that is incidental to the broader purpose of the lesson: exploring linear equations by means of 1) tables and 2) “math sentences.”

Second, I tend to have the same mindset as you in regards to “it should be obvious what you’re trying to do.” For instance, if I give a lesson where students discover the distance formula by means of the Pythagorean theorem, and then practice it a few times on some specific pairs of points, I would expect everyone in the class to be able to say, “Well, today we figured out a formula that allows you to find the distance between a pair of points in the coordinate plane.” Maybe they should even be able to articulate the rationale for the distance formula (makes use of the PT). But it is always surprising to me *how difficult it is* for students to articulate the purpose of the lesson at its end. That is — I’m surprised that it is difficult at all. *But it is.*

I think this discussion fits in nicely with the broader conversation on this blog about abstraction — for students to say what the lesson is about, they have to *step above it.* At any given moment in time during your 45-minute lesson, they may know what’s going on. But to see how the pieces fit together to form the whole — and to understand the significance of it all — and to be able to articulate it to their classmates and their teacher — these are all high-level thinking skills that should not be taken for granted.

In fact, you just gave me an idea. Maybe I’ll ask my students to keep a math journal this year, and every day they get to answer the same question: What was the purpose of today’s lesson?

## Sam Andrews

August 5, 2012 - 1:09 pm -The reason the Japanese math teacher asked his students what the title of the lesson “should” be is to help clarify and cement the lesson. The teacher went from asking what math concept they used that day to what the lesson or skill they learned was. The ladder of abstraction might be something like [less to more abstract] solving word problems, using tables to solve word problems, turning word problems into tables to learn about them and find the solution.

The goal of the lesson was to learn how to use tables to solve word problems (a practical solution), and to learn how writing a word problem as a table can help you solve math problems (turning a sentence into an abstract table of numbers).

The title at the end of the lesson helped the students remember that they were learning about this second more abstract lesson as they solved the more mundane problem about pens and pencils.

## mr bombastic

August 6, 2012 - 9:41 am -@Sam & James, The teacher asked what the lesson should be titled and gave the students 10 seconds to answer (two call out answers). He then provided a title for the lesson: from tables to math sentences. This is more or less window dressing to wrap up the lesson, yes? I see the real attempts to get the students to solidify the ideas being made during the lesson – not at the end.

The distance formula is a pet peeve of mine â€“ who is this Ms. Distance that stole a formula and named it after herself? I would say your students used the Pythagorean Th. to find the distance between two points. They did not figure out a formula.

## Dan Meyer

August 7, 2012 - 7:38 am -Steve G:Fair critique. I’ll start amending that by saying, “The classroom media need to allow for progressive disclosure and the teacher needs to take advantage of it.” This might punt the problem of defining “abstraction” over onto “progressive disclosure” but I think the second term is far easier to define and implement. Just don’t show everything all at once. Then make smart decisions about what to show and when. That’s very difficult if a teacher sets out to fully define a task using a static medium like paper.

lessano:This concern interests me. I think every student (not just those with learning disabilities) needs to know a concise statement of their task â€” ie.

what we’re doing hereâ€” something that’s easily conceptualized even if it isn’t easily accomplished. That could be, “Let’s figure out how many pens and pencils are in the box.” Later it can be made clear to the students that we just learned something called “solving simultaneous linear equations.” That second prompt is harder to conceptualize, though. It’s functionally meaningless to the student before they’ve done the work of the lesson. So what I’m wondering is, whyopenwith that title?## mr bombastic

August 7, 2012 - 10:18 am -@Dan, when do you think it would be appropriate to tell these students they are solving a system of equations? To me, that day is nowhere in sight.

Obviously this is a systems problem, but their method makes it much less apparent that two equations, or even variables, are involved than the usual Algebra I methods.

I think the students would view it as one equation with three variables: Cost = 40(#pencils) + 70(#pens). I donâ€™t believe they are thinking of #pencil + #pen = 10 as an equation that they are using, just a fact that there are 10 writing utensils. I would not budge from the marginal difference approach and word problems at that age.

## louise

August 7, 2012 - 4:40 pm -I thought we were giving it a title because we are told we have to. Math may be hierarchical, and sequenced, but it’s really messier than “today we will have naming of parts.” Giving titles to lessons to compartmentalize the learning makes students think they don’t interweave.

It *is* an equation with three variables. Or five if you like. One of them is the dependent variable. Two are fixed values for this problem. We replace the dependent variable with the answers we know; we replace the constants for this scenario, and we are down to 2 equations with two unknowns.

## Erik V.

August 8, 2012 - 5:51 am -@louise, you said exactly what I was thinking.

Naming the lesson at end works to undermine the compartmentalization of mathematics for both the teacher and student. As an elementary school teacher, I see this overt, artificial separation into discrete skills greatly diminishing the students’ overall conceptual understanding. I see this every year as I watch students struggle with the presented notion that ratios/proportions differ substantively from fractions.

Naming the lesson at the end invites connections and drawing on previous knowledge to apply to new problems/situations. Also, this backwards design technique helps focus instruction and questioning for the teacher. Plus, naming the lesson forces the students to label the abstraction and by matter of circumstance to ascend the ladder.

## James Key

August 8, 2012 - 6:37 am -@mr bombastic, “The distance formula is a pet peeve of mine â€“ who is this Ms. Distance that stole a formula and named it after herself? I would say your students used the Pythagorean Th. to find the distance between two points. They did not figure out a formula.”

Hi. I’m not sure I understand the point you’re making. Is it that it bothers you to call it a *formula*, as opposed to calling it a simple “application of the PT?” Or are you suggesting that they did not *discover* the formula, in the sense that I wasn’t explicit about having them generalize a specific numerical example to a case that involves variables?

If your actual point is the first one (quit calling it a formula), then can’t we say the same thing regarding just about any formula?

## Mr Macx

August 8, 2012 - 8:32 am -This is fantastic. Something that I wonder, which is the same thing that I wonder when I watch Catherine Humphries videos in her and Jo Boaler’s book: What were the moves to set up classroom expectations that allowed this kind of participation from the class?

I could get my higher achieving classes to try this no problem, and then I could focus on being as good as this teacher. But my lower achieving students are coming from a ton of failure in math, and won’t stick there necks out for anything.

I’m working on ways to avoid the situations I found myself in last year, which was:

Question. Crickets. Question (modified). Crickets. Answering my own question, followed by next question from plan. Crickets. Then direct instruction on exactly what I want to see as I admit defeat.

## Julia Tsygan

August 8, 2012 - 1:48 pm -I wathed the video after reading the main post as well as all comments, and maybe that was a bad way to go about it because boy did it not live up to my expectations.

The teacher is organized, yes, but IMO that’s only to be expected of any teacher working in a half decent system which allows teachers time for preparation. The fact that this is a lesson study makes the preprepared materials less impressive. This lesson would have worked well on PowerPoint as well, and cost less time, craft supplies and storage space.

I was likewise not impressed by his questions to the students, or rather by the lack of time he allowed them to work out these questions in their minds. Above all, it worries me that students either volunteered answers out loud or raised their hands. With 27 students in class, how can the teacher know that even most of the got the big ideas? I’d rather he had picked kids randomly or use a clicker/red-green card system. In general it’s a problem working in dialogue with the class as one big group because students are rarely if ever, mentally, one big group. I often prefer to use smaller groups, or even a think-pair-share system.

I liked that he left the title to the end, thereby connecting the ideas into a more meaningful whole. But overall I much prefer some of the timss videos which highlight the complexity of the tasks given to young students and the progress through individual and group work through which the students master new problems.

Also, we should remember that studies of Japanese and in general East Asian classrooms show huge diversity in teaching practice. This is illustrated more clearly in the second timss video study (1999?) which shows that Hong Kong favors a heavily lecture-based approach, whereas in Japan the students do most of the talking. So let’s learn from the Japanese, but let’s not imagine that their way is necessarily the best way for us.

## John

August 8, 2012 - 9:38 pm -In video 4 of 6, at 7 minutes, teacher sends a student to the board to draw “the graph”.

At 8 minutes of that video, the teacher basically says: here try this, erases part of what the student writes, and then does a stacked bar chart.

What does this little episode accomplish for the whole lesson? Was he just honoring a students attempt to visualize?

## Pat

August 9, 2012 - 7:33 am -On top of all the other great comments and discussion, what I found intriguing–coming from a school that focused on a highly structured classroom environment–was that no one was raising their hand. Students were calling out ideas at will, which I think generated better discussion as well as the sense that everyone was participating, instead of the two or three that raise their hands most often. Clearly a great culture had been established, but this generally more loose structure I think would be frowned upon in many education circles, though it seemed advantageous here.

@mr bombastic: love the video-bank of quality lessons, sample lessons, and student responses idea. Thousands of expert educators are doing this work every day, yet every day I think folks continue to remake a perfectly good wheel, because they don’t have access to the resources they need, namely exposure to expert educators over and over again.

## Brian

August 9, 2012 - 10:52 am -@mr bombastic and Pat: I wholeheartedly agree with you! I often feel that teachers are torn between performing in many roles. One role that demands a lot of time, but often gets pushed to the side, is lesson design. Who has time to craft high quality lessons when you’re busy returning parent phone calls, reviewing classroom data, grading papers, putting up a bulletin board, etc, etc.? How great would it be to select from a pool of great, tested lessons rather than reinventing the wheel over and over again?

Even within my last school district there was little sharing and communication between schools. All the fourth grade teachers might get together over the summer at a math professional development workshop, but during the year we hardly spoke, nor were we given time to collaborate or share lesson plans/ideas.

Granted, putting together a bank of quality lessons won’t necessarily mean it gets used. Teachers would still need the time to browse the bank, watch the videos, judge the quality of what they’re seeing, and make adjustments as needed for their particular group of students. You would also have to build trust in the lessons. From my experience, teachers are often mistrusting of outside resources especially if the district is pushing the teachers to use them. Despite best intentions, things like this can backfire because teachers feel their professional judgment is being called into question.

Don’t get me wrong, I still love the idea. I would have loved a bank from which to pull great Texas history lessons when I taught fourth grade! I just know that it’s a complicated issue, sadly.

As for the topic of titling the lesson, I think it comes down to knowing your class. I agree that having students title the lesson at the end is a great way to check their ability to summarize their learning. It’s easier said than done, and it likely takes practice for the students to get good at it.

But I will agree that some students may benefit from giving a title at the start of the lesson. It’s not like “From Tables to Expressions” encapsulates the entire lesson and ruins it for the students in any way. Rather it can give organization to the lesson that a teacher can use to keep the students focused on the goals of the lesson.

Think about the fact that he points out the answer to the first problem just a few minutes into the lesson, but he continues to talk about making the whole table. At times when I want to focus on the process I have given away the answer almost immediately so that the students can focus on how to arrive at the answer. It annoys the kids who want to solve it fast and raise their hand ready to shout out the answer, but it helps students realize that the answer isn’t all that matters.

Giving away the title or the answer doesn’t necessarily take away from the learning, it just necessitates a different kind of discourse in the classroom. And again, it depends on your students. You need to base all of these decisions around them, their abilities, and their needs.

## mr bombastic

August 9, 2012 - 11:29 am -@Julia, I am not overly impressed with this teacher either, and agree the student dialogue could have been managed better. I donâ€™t believe this lesson would be as good using PowerPoint. Very little is erased from the board as the lesson progresses, so if my mind wonders for a bit, or I forgot something, the information is still there in front of me. The Japanese teachers do a lot of purposeful circulation (looking at notebook work) which is one way to check for understanding as is the number of students calling out answers. How did the title connect the ideas into a more meaningful whole?

@John, the student was making a graph of #pens vs #pencils and did not appear to have a plan for approaching the original question. The only way that I can see to use the students graph would be to pursue the usual intersection method to solving a system â€“ way too much to take on at this point. Instead, the teacher drew in the bars to make it clear that the graph was just a more complicated way of showing that we have 10 writing utensils and got on with the lesson.

@Pat, I feel like I am remaking a perfectly mediocre wheel most of the time. It is very tough to do this when you have no one to critique and you have to wait a year to try again.

@Brian, I think a lot of the problems you speak of are do to the much more varied opinions in the US on what a good lesson looks like.

## Julia Tsygan

August 9, 2012 - 2:31 pm -@mr bombastic, I think I imagine using powerpoint in a different way than usual for this kind of task. Do one slide with the question, then actually make – as in get out of presentation mode and into writing mode – the table together with the students. That thing he did with shifting the pieces of paper/table around? Doable by cutting+pasting columns in the table in powerpoint. The graph would be trickier, maybe.

I’m not saying there’s anything wrong with chalk and paper and magnetic strips, it’s just that I’d never ever make time to even find out where I can get magnetic strips, or laminate paper, and if I did then once I’d made everything I’d immediately lose the materials in some cupboard.

Purposeful circulation (what else are you gonna do while students are working?) does show that students are filling in tables etc, but it doesn’t reveal whether the students are drawing any constructive conclusions from the activity – unless students are also in writing answering prompts such as “what patterns do you see?” but I didn’t observe such prompts in this video.

Regarding the idea of lesson banks etc – it sounds wonderful in theory, but I haven’t yet made it work in practice. I keep and freely share all my lesson plans with my colleagues, but when we attempt to use each other’s it’s just too hard to figure out what the other person intended, the detail of it all. I tried using betterlesson, but that didn’t work either. What HAS worked is to find and adapt specific activities, rather than complete lessons, that other teachers have come up with. I find that galore on the internet, especially on all the math blogs out there. In fact there is too much, too many choices! Sometimes it seems more work to try to choose among a dozen great ideas and I end up just designing something from scratch instead.

## Bridget

August 12, 2012 - 4:03 am -I’ve thought about the title at the end some more and realized it’s the equivalent to writing a summary after reading a story in ELA. That’s a check of a student’s comprehension and ability to synthesize information… Don’t we want our math students to do that?

Further, it made me reflect on why I don’t put objectives/ lesson titles up- it’s all math and I think I subconsciously felt that writing titles piecemealed the subject into bits–bits that students can’t bind back together on their own.

I think that I will try the “summary at the end of the story” approach.

## JJ

August 14, 2012 - 6:45 pm -With the title at the end it allows the students to think about and reflect upon what they have worked on, gives the students an opportunity to use mathematical language, it also makes the lesson more concrete and succinct for some of the learners. I would think that this would also allow a greater number of students to make connections to the lessons/learning that they have done and will do.

As for the white shoes, all teachers and students in Japan and Korea change into indoor shoes when they arrive at school (similar to taking off ones shoes when entering someone’s home). He would have had his dress shoes on when he left home and commuted to work.

## Christopher Danielson

August 17, 2012 - 8:58 am -Interesting stuff. Ton of fun. The video and your commentary bring up lots of ideas, but I’ll stick to one talking point.

Is work in classrooms ever really limited to one medium? I would concur that only one medium may be *valued* in a classroom or other teaching and learning setting. But even when everything takes place in an iBook setting, for instance, people are making notes on scrap paper.

And to your point, Dan, those notes on scrap paper need to be part of the lesson. They often represent important, student-generated rungs on the ladder. The relationship is a bit of a chicken-and-egg one, I suppose. Do we start with more forms of media and highlight more rungs on the ladder as a result, or do we start by valuing those rungs, and the media become necessary consequences of that decision?

## Dan Meyer

August 17, 2012 - 1:15 pm -Christopher:I’d argue “yeah.” When a student receives a task on paper â€”Â either a handout or a page out of a textbook â€” its author has likely tried to fully contain it on that paper. It’s no longer up for any kind of negotiation. You can’t ask the student to think about the information that’d be necessary for completing the task because the information is already on the paper. Etc.