*In a blog post a while back, I threw down the gauntlet to algebra teachers: identify four big ideas that could ground the course and make it more intellectually worthy. My new virtual friend Patrick Honner took me up on it here. The 4 ideas he identifies are:*

*1) Algebraic Structure *

*2) Binary Relations*

*3) The Cartesian Plane*

*4) Function*

*Here is my reply:*

Touché, Patrick! At first blush, these are fine candidates for big ideas. However, before we can say for sure, of course, we need to define a big idea – and warn about a common misconception as to what a big idea is.

Let’s start with the misconception. A big idea is not an overarching category that merely relates many pieces of content together. So, *equation* or *number* are not big ideas to me; they are just names for a vast category of entities. Similarly in English: *word* is not a big idea. In fact, even a little kid already brings that idea to the class.

What is a big idea? It is an idea that provides new insight, order, and usefulness for otherwise confusing, discrete-and-isolated-seeming elements. A big idea has intellectual power, in other words: it helps you make sense of things and enables transfer and new connections. The *periodic table* is a big idea. So is *axiomatic system*.

Another way to say it: *number* and *word* are foundational ideas but not “big” ideas. Once you get what they categorize, their power is over. *Force* is not a big idea but *F = ma* is one of the biggest ideas ever in intellectual history. The same is true of *democracy* in history. It establishes a category, a kind of governance, but beyond that you gain no intellectual power beyond the initial concept. *Inalienable rights*, however is a big idea because we keep using, refining, and questioning the idea to advance freedom.

I am inclined, therefore, to say that the *Cartesian plane* is a big idea. Indeed, it was a big idea historically, as you know. Suddenly, seemingly difficult to solve geometry problems could be solved more easily by analytical means, and soon thereafter math was no longer limited to equations to the 3^{rd} power.

But *Cartesian plane* as a phrase leaves me a bit cold. It doesn’t capture the power of the idea. I think we need a phrase like “we can map all possible 2-dimensional figures and relationships using a simple coordinate system” or some such thing (with less ugliness).

Similarly, *binary relations* is a powerful idea because once you get it, many seemingly disparate and random algebraic problem solving moves make sense. However, again, I am inclined to state the idea somewhat differently to stress the “power” angle. I think the more helpful way to state it is: *binary relations provide useful equivalences, based on complete reciprocity* because the equivalences are key to problem finding and problem solving.

I realize that you may mean all of my phrasings to be implied; I just want to stress the “bigness” aspect a bit more.

By this reasoning, I am less sure about the idea of *algebraic structure*. I am unclear on just what the power of the idea is or how algebraic structure differs from any other structure. For example, what is then the difference between logical or linguistic structure and algebraic structure? How is that structure helpful for insight and problem solving? That’s almost like saying the structure of our bodies is a big idea. For that matter, isn’t the structure the binary and reversible relations? If not, what’s the difference?

*Function* is, on the face of it, the most obvious big idea, given that it dominates instruction, as you mention. And it certainly is one of the first times that students are expected to grasp the idea that a function models a relationship involving variability. But the more I think about it, I am less sure of it. Consider that the drawn-out concept is “showing that something is a function of something else.” But once you get that f(x) is just shorthand for that idea, then what? It seems more “foundational” than “big” to just use the word function. Is there another way to phrase what the power of the idea is?

Ah: lurking in the background is a familiar question for us. Does the idea remain big once you are no longer a novice? “Big” according to whom?

I am harkening back to our conversation about rigor. Ultimately, we run into the same problem: is bigness a relative term? A big idea to an expert is likely to be inscrutable to most novices, and a big idea to a novice may be trivial to an expert.

So, I wish to up the ante. To me a big idea is big for both: I am looking for those ideas that are big – powerful and fecund – for both novice and expert.

I always return to this simple example from soccer: *Create dangerous space on offense; collapse dangerous space on defense* is a big idea at every level of the game, from kid to pro. And it is transferrable to all space-conquest sports like lacrosse, hockey, and basketball. Truly big.

Do your 4 ideas have that same power, if perhaps implicitly? I remain unsure on some of them. And part of my lack of certainty is that I remain unsure about algebra as a whole. Is it, as a whole, a big idea? Or is it just a useful set of tools? Part of my criticism of algebra courses as currently designed is that they provide no intellectual priorities or direction – the courses are actually framed, taught, and assessed as merely a set of tools (like teaching soccer only via drills and never playing games). So, until and unless we see algebra courses framed more like courses in Philosophy, C++, Economics, or The Short Story (which arguably have more of an overall intellectual purpose than algebra courses) I am unclear on what the best big ideas for algebra are.

Jumping back to arithmetic makes my clack of clarity clearer, perhaps. Many texts and standards identify *place value* as a big idea. I don’t see that as a big idea at all. Place value is a derivative idea from the iterative structure of a number system. (If anything, place value confuses kids rather than helping them!) I’d be more likely, therefore, to offer the associative, commutative, and distributive properties as big ideas in arithmetic since they help you understand and use the structure to solve problems.

But I am not a mathematician. Thoughts from the experts?

PS: This back and forth has generated a lively and friendly exchange with a number of people online. But the best response came here. Read it and then read my reply – particularly if you like the soccer angle as much as the math angle!

Max Ray (@maxmathforum)

said:I tried to take a stab at some of the smaller big ideas, and the idea that big ideas CAN and SHOULD apply to lots of fields of math, and it’s applying those ideas within a particular set of constraints to a particular set of objects is the math: http://blog.chrislusto.com/?p=753#comment-113

But also math is not exactly like soccer because as Chris says, math is about “Creat[ing] structure when you’re building; look[ing] for structure when you’re exploring”. So not only are you using general, overarching big ideas about representing to play with matrices but you’re also creating new big ideas by juxtaposing and comparing and resolving ambiguity and creating links among seemingly different domains like F = ma or E = mc^2 or algebraic relationships can be represented geometrically by arranging two number lines perpendicularly.

It’s sort of like if you came up with a way of applying some fundamentally chess-ish move to soccer and winning. But I have no idea what that would be, since I play neither chess nor soccer. Help?

grantwiggins

said:Funny, Chris’ idea applies to soccer: you are always looking to create a structure to get the ball past the other team in a fluid environment; and you look for structure (and space in the structure) to find possibilities. It isn’t therefore clear how such an idea would help a math student looking for priorities and lighthouses.

Let’s go to geometry. It is a big idea that everything depends upon the assumptions made; and that the assumptions Euclid posed permit us to do what we want: prove the Pythagorean theorem, similar triangles, and other proofs that reveal a consistency throughout all of space, culminating in the proof of the 5 regular solids. In other words, Euclid had a mission, and the big ideas lead the way to proving it. Same with Newton. Yet, most of where modern math goes is too technical and abstract for kids (Dedekind, Hilbert, Russell, Cantor, Godel). We now reject any ontological value or larger purpose to the math as we free it from needing to fit our experience. But why make everyone study it? My daughter is taking pre-cal in college and the course is no different from the pointless course I taught 40 years ago. They even do logs. The textbook may be the worst I have ever seen: not one iota of larger context or purpose to learning functions and formulae.

Algebra and pre-cal don’t ‘go’ anywhere interesting intellectually as taught, so they don’t have such big ideas – that’s really my point. It’s like requiring 2 years of C++ – or chess!!. I would certainly not require all American students to pass either to graduate from HS. Surely a course in stats would be a better requirement for most people.

Max Ray (@maxmathforum)

said:Maybe you and Chris have hit on the core big idea at the heart of most human production: when exploring, create conflict; when building, resolve it. Whether you’re exploring your opponent’s defense to find dangerous space and building your defense to collapse dangerous spaces in soccer or seeking and creating ambiguous objects and then creating definitions and theorems and relations that resolve those ambiguities in math, that may be one way to describe human creativity. It’s very useful in soccer to define that conflict as “dangerous space” because it’s useful to novices and beguiling to experts. William Byers who wrote “How Mathematicians Think” would argue that mathematicians are looking for ambiguity and resolving ambiguity through bringing previous unrelated fields, concepts, etc together with new definitions or theorems.

There’s a second point then which is who can be engaged in seeking out and resolving mathematical ambiguity, and whether that’s a useful big idea for novices, and if so, what that means for how math is taught. Is seeking out and resolving ambiguity related to the mathematical literacy students need to be savvy shoppers, informed consumers of media, successful in their vocations and avocations, etc? Is seeking out and resolving mathematical ambiguity central to the learning of math?

I think it is. I think it’s part of learning arithmetic and understanding fractions and place value and order of operations. I think resolving the ambiguity that “slope” is both steepness of a line and rate of change of a comparing quantity is key to being graph-literate and at the heart of understanding one of the most powerful mathematical ambiguity-creators-and-resolvers, the Cartesian plane. And that the tools to resolve that ambiguity are the same ones mathematicians apply to orders of infinity and other such bizarre topics.

Some of the lightposts that let young mathematicians know that their relationships, hypotheses, and proposed definitions are on the right track are (and now I’m cribbing from and also revising my post on Lusto’s blog):

* That they are using different representations and making connections and finding things that are the same in different representations

* That they are focused on planning for and predicting the future, either “will this always work?” or “what can this tell me about what will happen?” or “can I deal with this new thing happening?” or “what new questions do I have?” or “how will this help me solve more problems in the future?”

* That they are being as precise as possible… no “well, almost, but it could be…” is wiggling into their work and if it is, that they are excited about resolving it!

* That they aren’t “breaking math” i.e. that as they extend their knowledge of mathematical objects into new domains that their creations are consistent with things they already know and rely on, like if you define multiplication of points on the plane that it not change the result for multiplication of points on the number line, or if you come up with a new way to define “even” that it still keeps 0, 2, 4, 6, 8… etc. even and 1, 3, 5, 7, 9… etc. as odd, stuff like that.

I’m pretty confident that this is work done by young mathematicians and seasoned ones. That seasoned ones take it to somewhat bizarre-seeming and only-interesting-to-mathematicians extremes: http://mathwithbaddrawings.com/2013/08/13/the-kaufman-decimals/ or http://math.uchicago.edu/~mann/Lakatos.pdf but even the youngest ones are doing the same kinds of reasoning, probing, using multiple representations, seeking and resolving ambiguities, attending to precision, and trying to make predictions/descriptions/models that are powerful for the future: http://ummedia04.rs.itd.umich.edu/~dams/umgeneral/seannumbers-ofala-xy_subtitled_59110_QuickTimeLarge.mov or https://sites.google.com/site/constancekamii/videos. I see the same thinking that young students are excited to use to construct their understanding of number and arithmetic happening among PhD mathematicians playing with these Kaufman decimals and I want that kind of argument and clarification to be a hallmark of developing numeracy, exploring data and statistics, and doing math in schools.

I’m not sure if I would chunk school maths into Algebra I, II, Geometry, Pre-Calc, Calculus but I might be tempted to do something similar in the sense that each high-school math course could be reasoning about different mathematical objects in that heady moment that high school represents, where students are taking up the gauntlet of rigor, learning about professional ways of knowing, and trying them on for size. I would very likely start with statistics and data analysis and predicting and modeling the world around us (which would cover lots of the important stuff Algebra I and II and PreCalc and would be all about finding trends and modeling and interpolating and extrapolating and exploring confidence and sampling and experiment design and hypothesis testing). But I would also want some “algebra” that is about getting to know the numbers and operations as an algebraic structure and exploring the power of identities and inverses and what happens when you treat weird things like matrices and functions and points on the plane just like you treated numbers in elementary school. The project or goal of that course might be something like developing an algebra of matrices or complex numbers that preserves what we already know and like about the algebra of real numbers. And I would definitely want kids to explore the ambiguities at the heart of analysis, to meet and wonder at infinity and infinitessimals and zoom way in on functions and zoom way out on them again and try to classify them by their zoomed in and zoomed out behaviors. One obvious goal of that class could be to understand infinity and infinitessimals enough to find the underlying connection between accumulating infinitely many tiny pieces of area under a curve and the slope of the line tangent to a curve at a single (infinitely small) point on that curve. And your description of the goal of Euclid’s work gives a great trajectory for a Geometry course (and then there’s non-Euclidean Geometry — fun!). It’s not that I think kids will have professional uses for the things they learn after statistics, data analysis, and basic numeracy but I do believe that if they learn to be sniffers out and coherent resolvers of mathematical ambiguity through precise definitions and clear statements and powerful representations, they’ll be ready for an appreciation-level exploration of the mathematical objects most beloved by mathematicians. Just like their appreciation-level study of literature, science, history, sport, etc. And maybe not every kid will get there, but I hope that more will, and a more diverse group of students will see themselves as mathematical thinkers, and that more Americans will have positive feelings about math’s beauty and utility.

Patrick Honner

said:I’m greatly enjoying this conversation–it continues to give me lots to think about!

In addition to the many fun exchanges on Twitter we’ve had about this, there’s a lot of worthy material in the comments here: I love how people are sharing their idea of big ideas. In particular, I love “What do you do when you don’t know what to do?” and “Rules have reasons’.

Without committing myself to an “official” response here (I’ve still got lots of

big ideasto chew on), I would like to reiterate the point Chris Lusto made with great eloquence and wit on his blog: in soccer,create dangerous spaceis indeed a big idea. But it’s a really, really big idea. It’s so big that, as you point out, it applies to other sports like hockey, lacrosse, basketball.In fact, it’s even bigger than that:

create dangerous spaceis a big idea in games, too, like chess, checkers, go, and others. This is in fact a huge idea!As an instructor, I think it’s important to know and communicate huge ideas like this to students, whether they’re learning algebra or soccer. But I think we need more narrowly defined “big” ideas to build a course around.

xiousgeonz

said:Oh, yea, not sure this counts as “algebra” either, but my folks who know how to “cross multiply” but don’t grok what proportions mean do all kinds of weird things…

xiousgeonz

said:Seems to me that clarifying that you wanted a “thesis statement” as opposed to a “topic” might have led to a better result — oops, this isn’t English 101 The big ideas that my guys struggle with if they’re solidly wedded to procedures are things liek:

The equals sign isn’t there to point at the answer; it separates different ways of describing the same value. Frequent errors are when students do the same thing to “both sides” … of… a reasonable amount of space.

Those variable things… have two kinds of roles. The idea that “x + x = 2x” is true for all numbers often gets totally lost in the “you’re supposed to get X by itself” goal.

Negative means opposite… not sure that counts as “algebra” or not, but subtracting integers is hangin’ up my pre-algebra folks this week, per usual…

Those function things — that f(x) doesn’t mean f times x even though it really looks exactly like that and usually nobody’s bothered to explain that detail to ’em, so they file it under “another example of the math teachers not making sense, but what else is new?”… but that functions are neat patterns that can explain complicated things and let us predict things.

Matthew G Jones (@mattguitarj)

said:Functions are undoubtedly central to mathematics. But is it function itself that is a big idea in middle/high school introductory algebra? Or perhaps the question is, What about the concept of function is a big idea in algebra? I submit that the big idea is that functions provide useful ways to model behavior. In algebra, the functions usually model quantitative behavior. Even more specifically, the big idea is probably that linear functions model situations in which there is a constant rate of change. If a student who completes algebra is able to recognize many disparate situations as exhibiting a constant rate of change, and is then able to determine a linear function, to use that function to make predictions about the situation, and to interpret the results, that student has grasped a big idea from algebra.

grantwiggins

said:I like the idea of saying that the big idea is: functions permit a modeling of phenomena that involve variable quantities.

gasstationwithoutpumps

said:“Algebraic structure” includes precisely things like “the associative, commutative, and distributive properties” along with the definitions of addition and multiplication. These are group theory, field theory, and other branches of abstract algebra for math majors. They are big ideas, and they are not limited to just arithmetic. In linear algebra we see that matrix multiplication is associative, but not commutative, for example.

grantwiggins

said:But we need a better phrase, then, to capture exactly what the ‘idea’ is. What is the powerful idea, in a nutshell of those elements?

gasstationwithoutpumps

said:For whom is the phrase trying to capture the idea? Like with Feynman’s example, where “energy” is immediately evocative of a powerful and subtle idea for a physics teacher, but not for a first grader, for a mathematician “algebraic structure” is immediately evocative of a powerful and subtle idea. Note that Feynman did NOT continue his talk by defining “energy”, but by arguing that first graders should be taught something else more concrete (as first graders are taught more concrete math concepts, like counting physical objects and addition of small natural numbers).

I found Honner’s explanation much clearer and more accurate than your attempts to rephrase them. For example, “binary relations provide useful equivalences, based on complete reciprocity” reduces binary relations to just equivalences, which are only one type of binary relation, and adds the notion of “reciprocity” which Google defines as “the practice of exchanging things with others for mutual benefit, esp. privileges granted by one country or organization to another”—a mostly irrelevant concept here.

grantwiggins

said:No, I meant reciprocity in the sense Piaget used reversible operations. That’s why I edited it later.

Fair enough criticism. I am trying to help the teachers, though, not the experts. A phrase like ‘algebraic structure’ means nothing to a typical 9th grade math teacher in a typical school. How else might you phrase it, then, to help them help students with useful and illuminating ideas?

Todd Pytel

said:The following are some of the unit-level ideas we use in our freshman algebra course at Senn High School in Chicago. I often struggle with finding the right balance between really big, transdisciplinary ideas and slightly lower-level ones that more directly relate to day-to-day instruction. So I’m not sure whether these fit your desired level of “bigness”.

* Rules have reasons. (From our unit on power laws and polynomials, teaching students how to create mathematical arguments to justify rules, and then use those rules in future arguments.)

* Generalization is the process of describing and verifying patterns. (From our unit on algebraic expressions and properties.)

* Context matters – what something means depends on what’s around it. (From our unit on ratio, proportion, and percent, showing students how these ideas are essentially about comparisons and never a single value in isolation.)

I’d love to hear your comments on these.

Todd Pytel, A’99

grantwiggins

said:love the email address – sophrune.org. Only Johnnies know…

I like ‘rules have reasons’ though I think it needs to be more pointed for the math. The second one does;t work for me – it;s the definition of generalization, really. What is the power of such generalities? That’s what the understanding should be. Context matters – for sure. That works.

l hodge

said:I like “function” – identifying and using a functional relationship. This needn’t involve the tedious process of defining a function, f(x) notation, or even an equation. For example you could plot a few points for year versus college tuition. Sketch a curve that “fits” the points as best you can. Use other points on your sketch to make reasonable estimates or predictions about costs in other years. Of course we needn’t limit ourselves to modelling with functions either. .

Would you say that it is a big idea to think of two points, or a few points, as being part of a larger collection of points, and, to think of that collection of points (the function) as an entity in itself.

grantwiggins

said:I like the last sentence – that seems to be to get at the power of function.

jim cibulka

said:Kirkwood High School’s big ideas in freshman physics:

1. Balanced forces cause constant velocity.

2. Unbalanced forces cause acceleration.

3. Energy is the ability to cause change.

4. In a closed system, the total amount of energy stays constant.

5. Waves transfer energy?

Big for novice and expert alike? What do you all think?

Thanks for what you do, Grant!

jimc

grantwiggins

said:Well, my physics is a little rusty, but the first 4 seem pretty solid to me. Not sure about the waves one because it seems a bit ‘small’ or qualified or even closer to ‘knowledge’. Maybe something about energy transfer more broadly? The Energy one reminds me of Feynnman’s famous complaint about science textbooks that say “energy” makes it move:

There is a first grade science book which, in the first lesson of the first grade, begins in an unfortunate manner to teach science, because it starts off an the wrong idea of what science is. There is a picture of a dog–a windable toy dog–and a hand comes to the winder, and then the dog is able to move. Under the last picture, it says “What makes it move?” Later on, there is a picture of a real dog and the question, “What makes it move?” Then there is a picture of a motorbike and the question, “What makes it move?” and so on.

I thought at first they were getting ready to tell what science was going to be about–physics, biology, chemistry–but that wasn’t it. The answer was in the teacher’s edition of the book: the answer I was trying to learn is that “energy makes it move.”

Now, energy is a very subtle concept. It is very, very difficult to get right. What I meant is that it is not easy to understand energy well enough to use it right, so that you can deduce something correctly using the energy idea–it is beyond the first grade. It would be equally well to say that “God makes it move,” or “spirit makes it move,” or “movability makes it move.” (In fact, one could equally well say “energy makes it stop.”)

Look at it this way: that’s only the definition of energy; it should be reversed. We might say when something can move that it has energy in it, but not what makes it move is energy. This is a very subtle difference. It’s the same with this inertia proposition.

Perhaps I can make the difference a little clearer this way: If you ask a child what makes the toy dog move, you should think about what an ordinary human being would answer. The answer is that you wound up the spring; it tries to unwind and pushes the gear around.

What a good way to begin a science course! Take apart the toy; see how it works. See the cleverness of the gears; see the ratchets. Learn something about the toy, the way the toy is put together, the ingenuity of people devising the ratchets and other things. That’s good. The question is fine. The answer is a little unfortunate, because what they were trying to do is teach a definition of what is energy. But nothing whatever is learned.

Suppose a student would say, “I don’t think energy makes it move.” Where does the discussion go from there?

I finally figured out a way to test whether you have taught an idea or you have only taught a definition.

Test it this way: you say, “Without using the new word which you have just learned, try to rephrase what you have just learned in your own language.” Without using the word “energy,” tell me what you know now about the dog’s motion.” You cannot. So you learned nothing about science. That may be all right. You may not want to learn something about science right away. You have to learn definitions. But for the very first lesson, is that not possibly destructive?

I think for lesson number one, to learn a mystic formula for answering questions is very bad. The book has some others: “gravity makes it fall;” “the soles of your shoes wear out because of friction.” Shoe leather wears out because it rubs against the sidewalk and the little notches and bumps on the sidewalk grab pieces and pull them off. To simply say it is because of friction, is sad, because it’s not science.

from What is Science? You can find it here: http://www.fotuva.org/feynman/what_is_science.html

JDodds@cnyric.org

said:Hi Grant,

In the mathematics department that I worked in at Dennis-Yarmouth High

School in Massachusetts, we developed some essential questions that we

used across our classes. They were:

Why is the process often more important than the answer?

Why ask why?

What do you do when you don’t know what to do?

Is there anything in life that cannot be represented mathematically?

If you combine those questions with the big idea that “Mathematics allows

us to model our world”, I think you begin to capture the essence of

mathematics including Algebra. And of course, the first 3 questions are

not limited to mathematics.

My 2 cents.

Jeremy Dodds

District Data Coordinator

CNY Regional Information Center

6075 E. Molloy Road

PO Box 4866

Syracuse, New York 13221-4866

jdodds@cnyric.org

CNYRIC Phone: (315) 433-8314 Fax (315) 433-2221