## [LOA] Hypothesis #1: Be Explicit

I’m going to lay out five hypotheses over the next five days that will be the current tally of my writing, reading, thinking about the ladder of abstraction this summer. These should all be tested, contested, and generally kicked around.

We represent towns with coordinates when our question concerns their location.

We represent data with tables because it keeps the data organized and sometimes reveals patterns.

We give points one capital letter and line segments two because it make them easier to talk about.

We turn real-world phenomena like trees and their shadows into right triangles when the tree-ness of the tree and the shadow-ness of the shadow don’t matter, when their height and length and and included angle are all we care about.

We climb the ladder of abstraction all the time. We teachers are good at that climb. We aren’t often explicit about the motivations and methods for making that climb.

We turn trees into line segments and cities into coordinates without so much as a word about that weird, violent stripping away of context. All of those implicit, elided abstractions in someone’s teenage years contribute to her adult sense that math is hopelessly abstract. We need to make these motivations and methods explicit.

“Let’s talk about these cities here. All we really care about is their location. Coordinates are a useful way of representing locations. Let’s lay down a grid so we can put numbers to those coordinates.”

Does it matter where you set the origin? Ask them. Then talk about it. I realize these kids are in ninth grade and should be totally adept at that kind of abstraction but let’s not assume that about them. Particularly when it just cost you an extra minute to have that conversation and make the abstraction explicit.

2012 Sep 18. Great line here from Frorer, et al, (1997):

And yet while abstraction in mathematics has some additional qualities or meaning, we rarely find it explicitly discussed let alone defined. You can pick up a book entitled Abstract Algebra and not find a real discussion of abstraction as a process, or of abstractions as objects.

Featured Comment

Erik:

I do not think use of the ladder metaphor is an admission that there is only a single way to get to certain understanding. I picture the ladder to mean that there is a path that I cognitively take to move along the spectrum of abstraction. This would allow room to climb a particular ladder to higher levels of abstraction and climb down another. The fact that there are multiple ladders to reach the same point does not invalidate the use of the ladder.

On another note, I think being explicit is enormously important especially if your goal as a teacher is to eventually make yourself useless to the students. Without revealing the undercurrents of your decision-making and assumptions, I think that you do not fully prepare them for life without you.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### David Stewart

September 17, 2012 - 6:36 am -

I’m in science education where the context is important at the beginning of the problem and again at the end. But we still have to climb the ladder of abstraction. If we don’t make it clear to novices why we abstracted in the way we did, they can’t follow us back down the ladder when we apply the answer back to the context.

In my research notebook I make notes at every step to explain why I have used the particular tool for the specific task. If another researcher, with the same training as me, didn’t have those notes, the rest of my data becomes almost meaningless to them.

In fact the choices available at each stage mean that the “ladder” of abstraction may not be the best metaphor. I’m not sure what metaphor woks better, a web?, but it should be clear the we made a choice about how we abstract away from context. There are often many ways that will work, not just a single linear path.

2. #### David Wees

September 17, 2012 - 7:12 am -

Just a small bit of push back – I don’t know that we have to be explicit all of the time about what abstraction we use. In fact, sometimes being explicit is just being confusing.

I’ve often noticed that most of my students can follow small abstractions that we develop together, but I’m not explicitly labelling the abstraction; we are using it as a tool during exploration of a problem. I do notice that with some of my students being more explicit helps, but I hesitate to do too much of the work for my students when it is unnecessary.

3. #### Jason Dyer

September 17, 2012 - 7:34 am -

@David Wees: I’ve also been wincing over the “ladder” metaphor but I’m not sure what the replacement would be. Not only is there more than one way to go up the ladder, but more than one way to go down. Where mathematics gets very interesting for me is abstracting a particular problem to, say, an inverse relation, then sliding down a different way to what appears to be a different problem but one that is mathematically the same.

Having the metaphor be a “ladder” feels like it undercuts the ability of mathematics to connect a whole network of problems.

4. #### Brian

September 17, 2012 - 8:56 am -

This reminds me of the Standards of Mathematical Practice in the Common Core standards, specifically #2 – Reason abstractly and quantitatively. Students are expected to decontextualize and contextualize throughout the problem solving process. If all goes well, this should be a skill developed throughout a student’s entire educational career starting in Kindergarten. Depending on the needs of your students, you might need to do some prompting when presenting a problem or you might need to do some prompting with individual students as they are working on their own. They key is to foster this skill within the students as opposed to making it something that the teacher does for them.

5. #### Erik

September 17, 2012 - 12:02 pm -

I do not think use of the ladder metaphor is an admission that there is only a single way to get to certain understanding. I picture the ladder to mean that there is a path that I cognitively take to move along the spectrum of abstraction. This would allow room to climb a particular ladder to higher levels of abstraction and climb down another. The fact that there are multiple ladders to reach the same point does not invalidate the use of the ladder.

On another note, I think being explicit is enormously important especially if your goal as a teacher is to eventually make yourself useless to the students. Without revealing the undercurrents of your decision-making and assumptions, I think that you do not fully prepare them for life without you.

6. #### Kelly Holman

September 17, 2012 - 12:42 pm -

If you want to make them independent, sometimes tell them explicitly why you make the choices you do. Sometimes ask them why. Sometimes ask them to make the choice. Have them do all the things you do.

By ninth grade, they should be adept at this, but they’re not because most teachers, IME, never talk about it.

Y’know that dome on a playground constructed of triangles? That’s what it is instead of a ladder. But Ladder of Abstraction is a lot easier to say.

7. #### David Cox

September 17, 2012 - 1:04 pm -

Jason:

Having the metaphor be a “ladder” feels like it undercuts the ability of mathematics to connect a whole network of problems.

I think that’s an important point, but the metaphor seems to work as long as folks still see Bloom’s as a hierarchy.

8. #### Joe

September 17, 2012 - 7:14 pm -

@Kelly: Love the image of the playground dome.

There’s something interesting about this problem, where it appears (let’s say) midway up the ladder. I can imagine a scenario where this problem works as-is, with follow up.

The problem makes a number of assumptions, the discussion of which would bog down the typical algebra class. Cities as geometric points. Signal strength as a function of distance. In “the real deal” problem, people don’t all live inside geometric points, and the cell phone company probably can’t build wherever it wants. But that discussion probably misses the point that the writers of this problem are trying to make.

So work the problem as-is, but follow up: can we solve this problem in a realistic situation? Give the class a map with three cities. What abstractions are needed so that we can use our previous strategies? This is where the explicit abstraction pays off, and students may be less confused by the questions since they have an idea of how the abstractions make the problem easier to solve. Did the fact that students decided to place an origin at different points change the final solution?

I don’t think we’re doing the work of our students by helping make the abstractions explicit—developing “the eye” for abstraction is not natural. We’re only doing the work if we never ask them to identify abstractions by themselves.

9. #### Santosh

September 18, 2012 - 4:18 am -

It’s fine to keep reminding students of the abstraction until they roll their eyes.

In Science, modeling is a really key skill. Teachers have to keep reminding students of the idealizations and simplifications needed to convert a real-life problem into a tractable one.

I guess one’s position on this is in line with which way one leans on math being elegant and beautiful or practical.

10. #### Elaine Watson

September 18, 2012 - 7:37 am -

How about the “chutes and ladders” of abstraction? …the “geodesic dome” of abstraction?…the “web” of abstraction?…the “long and winding road” of abstraction?

Whatever metaphor is used, I do think it is important to be explicit about why we are abstracting. As Dan said in his post, we abstract for different reasons…for making things easier to talk about, for stripping away any details that don’t get to the essence of what we care about in a particular problem. When we are using similar triangles and shadows, does it matter that the tree is green and has a conical shape or that it is red with a globular shape? The essence that we care about at that moment is its vertical height.

I know that I have been guilty of expecting that students “get” the abstraction automatically, perhaps because I never dissected the problem solving process in the way that Dan is.

11. #### Dan Meyer

September 18, 2012 - 8:00 am -

What Erik said.

Some of you guys are taking the ladder to be more prescriptive than it is. There are different questions to ask about a context, different ways to represent the information that’s relevant to that question, and different operations to perform on those representations, but once you’ve abstracted your way to an answer, you’ll be staring down from the top of a ladder — your own ladder, which may be different than mine.

Also, I anticipated “Be Explicit” would set off the alarm with progressive educators but I didn’t hedge well enough against that event. “Be explicit” doesn’t mean “talk at your students about abstraction” but it does mean “make sure no one walks away confused about abstraction.” Definitely ask questions. Definitely invite them to try out different abstraction and compare their effect later. But don’t let the class walk away without an explicit understanding of the abstractions we used today and why we used them.

12. #### Jessica

September 18, 2012 - 8:27 am -

In so much of what we do with students, we need to ‘crack open our brains’ and reveal our processes to students so those processes can then be transferred to and adopted by them. Whether it be abstraction or self-evaluation, we often hide our internal dialogue from those that we most want to posses it!

13. #### Elaine Watson

September 18, 2012 - 8:38 am -

Jessica,

Your last line is great: “Whether it be abstraction or self-evaluation, we often hide our internal dialogue from those that we most want to possess it!” I’ve been guilty of not being explicit enough with students, thinking “what is there to GET?…just THINK!” It is our job as teachers to scaffold HOW to think…and then slowly fade away as they learn to think on their own.

14. #### Max

September 18, 2012 - 9:15 am -

In addition to being explicit, I think there’s a really important unspoken piece here:

“We represent towns with coordinates when our question concerns their location…. We give points one capital letter and line segments two because it make them easier to talk about.”

Abstractions are used in contexts, to solve certain problems or make things easier. Without context, it’s harder to understand the abstractions. If you said to me “help, I need to fly to Key West, how do I find it?” I don’t know whether EYW or 24.5561° N, 81.7594° W or 3491 South Roosevelt Boulevard, Key West, FL 33040-5260 is more useful until I know whether you’re trying to type it into Orbitz, fly there, or put it into a GPS. And if I answered the wrong question, you’d think I wasn’t just wrong, you’d think I was nuts!

A big part of the job of making things explicit is making sure the students have experience in the contexts in which the abstractions are useful. Starting with abstraction (whether explicit or not) before context makes it possible that you’re talking about a feature of the landscape that’s not salient to your students, and they’re hanging your abstractions on the wrong things. E.g. we use coordinates when we care about cost, or something like that.

15. #### Bowen Kerins

September 18, 2012 - 2:50 pm -

What’s the meaning of “Can any other towns on the map benefit from the new cell phone tower”? What would be a meaningful answer to this question? Do only towns at the same distance to the tower benefit? Because really that makes no freakin’ sense to me.

Also, they’re lucky that the third town needed was Westfield instead of Seabury, because then the cell tower location (the point equidistant from the three cities) would be ridiculously inconvenient. So the telephone company doesn’t want the location equidistant from each city, they want the location with the smallest total distance to all cities. (A harder, more interesting problem.)

16. #### kmorrowleong

September 19, 2012 - 7:40 am -

Ladder. Geodesic dome. Each model of abstraction implies a maximum/greatest/best/highest characterization. In contrast, there is the lowest: lowest level, rungs, etc. If you aren’t operating at the highest level, you can not be the best. Malarkey!
To belabor the analogy, if you are standing at the top of the dome and can’t get down, you are useful to very few people. I don’t care if you are a student or teaching standing up there.
The CCSS has put the idea of generalization in our minds, and it is really a form of abstraction. CCSS has neglected to turn it around and prescribe the inverse: specifiying and applying generalizations to particular circumstances.
If that cell phone tower problem had been stripped of its textbook support structure of the coordinate plane, I would not have used that particular structure to solve the problem. I might have used templates of a circle, testing radii until I found an approximate best radius. An abstracted mathematical structure might have occurred to me then, but actively modeling the cell signal’s reach is my intellectual birthright.
If those of you at the top of the dome don’t understand my model, think it is inefficient or too concrete to be valuable mathematics, you are just as lost in my world as I am in yours.
Abstraction has an inverse, and it is just as valuable.

17. #### Dan Meyer

September 19, 2012 - 9:50 am -

Kim:

Ladder. Geodesic dome. Each model of abstraction implies a maximum/greatest/best/highest characterization. In contrast, there is the lowest: lowest level, rungs, etc. If you aren’t operating at the highest level, you can not be the best. Malarkey!

I think this misunderstands the ladder analogy quite a bit.

The point isn’t to imply the superiority of one rung over another. (Repeatedly throughout this series, we’ve put primacy on being able to traverse all rungs of the ladder.) The point is that different tasks are possible at each rung, from calculation at higher rungs, to intuition and task definition at lower rungs. That’s a matter of fact, not a value judgment: it’s impossible to define a task that’s already been defined; it’s impossible to calculate an answer if there’s nothing there to calculate.

What we do with the metaphor is another matter.

18. #### K morrowleong

September 19, 2012 - 10:59 pm -

I don’t think it’s so much misunderstanding the ladder analogy as it is questioning the implicit hierarchy in the analogy. Ladders go up and they go down. While you may not intend this in your conceptual framework, “up” the ladder of abstraction is a value judgement that remains. For example, the elementary CCSS document allows for alternate strategies for division, but also clearly states that learning should culminate with the abstracted “standard algorithm.” That standard implies a hierarchy.

Philosophically I am interested in the unveiling of individuals’ hidden presuppositions, and in this case I mean their abstractions. As Max responded to the following:
Dan: “What matters isn’t the rung itself but how deftly you can move between all the rungs above and below you on the ladder of abstraction.”
Max: “It’s hard to see some of the abstractions I use to solve problems, since they have become so familiar as to feel like they contain all information in its original form…”
Max’s awareness of his own process of abstraction is what we want to see, but it is unclear whether he is making a value judgment. Neither do you communicate the idea that there is a hierarchy. But Elaine shared the thoughts I have caught myself thinking: “‘I’ve been guilty of not being explicit enough with students, thinking “what is there to GET?…just THINK!’” I worry about the teachers who have never had the guilt, only the thought. I would guess that they have a hierarchy.

I prefer the network analogy.

19. #### Jason Dyer

September 20, 2012 - 6:35 am -

What we do with the metaphor is another matter.

@Dan: I’m also thinking from the perspective of the (often math-hating, or least math-only-tolerating) K-5 teachers I’ve trained, who will have all the confusions pointed out above and more. If a bunch of math educators have strange ideas about the metaphor, what will teachers who have no predisposition towards math think?

20. #### Dan Meyer

September 20, 2012 - 6:50 am -

Jason:

I’m also thinking from the perspective of the (often math-hating, or least math-only-tolerating) K-5 teachers I’ve trained, who will have all the confusions pointed out above and more. If a bunch of math educators have strange ideas about the metaphor, what will teachers who have no predisposition towards math think?

I don’t personally feel unsettled about this metaphor, even given all the parsing in this thread. Regardless, a metaphor about abstraction is going to be most applicable in grades where they do the most abstraction. Less so in primary.

21. #### mr bombastic

September 20, 2012 - 5:00 pm -

It seems to me that abstraction is just as big an issue with grade school topics as high school topics.

22. #### Dan Meyer

September 20, 2012 - 6:14 pm -

Isn’t this just Piaget’s formal operations we’re talking about? Happens in adolescence, generally, right? I didn’t think this would be controversial but I don’t know Piaget as well as I should.

23. #### Courtney Weitzer

September 21, 2012 - 8:45 am -

Does being able to add and multiply large numbers in your head make you a good mathematician? I think it’s part of it but does not necessarily mean anything. But unfortunately, there is a common misconception that mental math = genius. Probably because schooling is still flawed in teaching students mathematical concepts and problem solving skills instead of just focusing on formula and procedure: the illusion of “how” to solve a problem. In that sense, something as simple as one of the 4 operations becomes a mechanical process because of *stupid* terms like “carrying” and “borrowing” that emphasizes process over concepts as early as 1st or 2nd grade. What do you think about this? Can this all be fixed in the early grades with something as simple as the Japanese number system which explains regrouping, place value and base ten extremely well?

24. #### Kelly Holman

September 22, 2012 - 2:35 pm -

Regardless, a metaphor about abstraction is going to be most applicable in grades where they do the most abstraction. Less so in primary.

If primary school kids dealt more with realistic problems, not just mechanics, they wouldn’t have so much trouble with abstraction when they get older. To accomplish that, K-5 teachers would need to be more comfortable with the ladder/web/geodesic dome of abstraction.

25. #### Brian

September 23, 2012 - 7:46 am -

Kelly,

I’m not sure how often you visit K-5 classrooms, but I have known many teachers, myself included, who spend time with realistic problems and abstractions. The difference at the elementary age is the amount of scaffolding and support students need from the teacher and each other. There may be an issue of transfer, of maintaining a skill, or of consistency between grade levels with regards to students’ flexibility with moving between situations and abstractions, but it is not fair to generally characterize lower grade teachers as deficient in working on these skills.

26. #### Kelly Holman

September 23, 2012 - 5:34 pm -

Sorry, I didn’t mean to particularly criticize K-5 teachers or imply that all are neglecting realistic problems. In my experience as a student, teacher, and tutor, I’ve seen most teachers at all levels focusing on mechanics and theory. I’m not sure I’m all that great at teaching the way I aspire to teach, but this blog and others have been a breath of fresh air. I was overjoyed to find that my dissatisfaction was perfectly reasonable, that there really was a better way to teach.

27. #### Jason Dyer

September 24, 2012 - 7:06 am -

@Courtney Weitzer: The regrouping etc. in Common Core was cribbed pretty much directly off of Japanese curriculum. (Also, some of the folks in Singapore were apparently looking into cribbing off the Common Core approach to fractions. A lot of their curriculum started in the US. We’re just a big, chaotic country that has trouble coming to agreement.)

Isn’t this just Piaget’s formal operations we’re talking about?

@Dan: No.

Mathematical abstraction starts in pre-K with counting. There are all sorts of understandings that need to be reached before a child can be said to “count”, like —

* Knowing that the sequence is always the same.
* Knowing that the last number named is equivalent to the size of the set.
* Knowing that each element of the set is counted once and only once.
* Knowing that the number of items in a set is invariant if the set does not change.

Each one of these (except for arguably the first) reflects the child building an abstraction. Usually it is formed by the child encountering lots of instances until the idea is realized — for example, counting lots of “4” sets until the idea that the “four” at the end of the sequence somehow corresponds to the number of the things in the set.

Forming the first step up the ladder, here, is then not done off a single problem but a network of interconnected problems. According to Piaget the early grade level students will not be able to think about the abstraction without some concrete instance, but even if that is so (Piaget’s sort of fuzzy on number sense to begin with) an abstraction must be formed nonetheless for a student to know that “five” is five.

As a later example, the same sort of principle applies for models of multiplication. Here is where the idea of a ladder starts to be dangerous, because the grade 2+ teachers dealing with it often get the impression there is only one way up the ladder for each problem, and that multiplication is only conceivable abstractly in one way (for example, repeated addition). This causes issues in later grades with problems that ought to be parallel, but from the student’s perspective aren’t (for example, the gas problem quoted in the link above).

So, again — for us we can handle the idea of the ladders having multiple starting points and multiple ending points, but K-5 teachers will (from what I have seen in practice) take such analogies more literally. Because of this I personally would feel uncomfortable using the ladder analogy as a teacher training tool.

Mind you, I’d *like* this thing to be hashed out. I’m not trying to rain on the party here, given I have attempted (and had trouble with) training in the exact thing being discussed here. I just think (if nothing else) you might want to test your analogy on some live subjects who are less math-inclined before you roll with it.

28. #### Max

September 24, 2012 - 7:22 am -

I’ve noticed that the “Ladder of Abstraction” metaphor is being considered in two really different ways. In one way, it seems to hang together really well (to me at least) and in another, it breaks down quite quickly.

Here’s the first:

Any individual, in solving a math problem or considering a mathematical situation, makes choices to move back and forth between making the situation more concrete and making it more abstract. She considers, for example, the real terrain, then a map of the terrain, then a lattitude/longitude grid, then makes a hypothesis about the grid which she tests on the map, then she studies the real terrain a bit more to add another layer to her grid, etc.

In that case, the ladder’s an apt metaphor (although abstractions that don’t turn out fruitful may occasionally lead to an imaginary “branched” ladder). It’s allowed people to have a conversation about the ways that abstraction and concretization (if that’s a word) are always available at any stage in problem-solving and that there are really important skills that aren’t about reaching the optimum rung but are instead about being facile climbers, I think the ladder draws attention nicely to the existence and availability of multiple stages and the requirement to get good at going up and down — for this thinker, on this problem.

The second way the ladder metaphor seems to be being used (and is not, I think, the way Dan or others who’ve used the ladder intended it) is as a way of organizing the whole domain of abstractions and concretizations available to math learners or that could be involved in thinking about a particular math situation.

It’s certainly not the case that multiplication is the top (or middle) rung of a single path of increasing abstractions from counting groups of beans. Or that Dan is advocating the answer to a math problem comes on some rung of a single ladder.

I think a geodesic dome or web or network may be a more apt latter for the space of all representations of mathematical situations, and they can vary in all sorts of dimensions.

But an individual’s path through that space while working in a single mathematical situation tends to move “up and down” along the abstraction axis, and can be (ha!) abstracted by squishing the other axes we aren’t paying attention to into a vertical path. A ladder, if you will.

And that abstraction is useful to me when thinking about, for example, how to represent the situation to the learner just thinking about abstraction for this task for my student — where on the ladder do I want them to start? What tools for concretization do I want them to have available? What tools for abstraction?

Maybe that will help people define more whether they have a problem with the ladder metaphor for an individual problem-solving path, abstracted to one dimension, or whether they were thinking of the whole collection of representations for big mathematical concepts.

29. #### Dan Meyer

September 24, 2012 - 9:15 am -

@Jason, thanks for relinking your post on the number sense activity and clarifying Piaget.

@Max, no complaints at all. I agree with your second-to-last graf, in particular, about the implications of the ladder metaphor and its uses in teaching, curriculum development, etc. The framing of abstraction as a geodesic dome or a web just isn’t useful to me. It leaves me twiddling my thumbs thinking, “So now what?”

30. #### kmorrowleong

September 24, 2012 - 8:20 pm -

“I just think (if nothing else) you might want to test your analogy on some live subjects who are less math-inclined before you roll with it.” – Jason
I concur with Jason in this respect. My work has been with elementary teachers who really wish that I, as a mathematics coach, had never darkened their doorstep. They have real fear, and this thread has helped me to identify the object of their fear: abstraction beyond their ability to connect it to something concrete. They don’t own the power to mathematize their own world.
Lesh has a network-like model for representing mathematical problems. Five nodes: real-life situations, pictures, verbal symbols, manipulatives, and written symbols (numbers/operators). Start with one node. Translate the math into the other four. Get all five? That’s when the learning happens.

R. Lesh and H. M. Doerr (Eds.), Beyond constructivism: A models & modelling perspective on mathematics problem solving, learning & teaching. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.