Our first approach in preparing a new lesson is often to ask, “Where does this skill apply in the world of work or in the world outside the classroom?” There may well be a great answer for some skills, but this strategy generalizes very poorly to lots of mathematics. So instead, I try first to ask myself, “Why did we invent this skill? How does this skill resolve the limits of older skills? If this skill is aspirin, then what is the headache and how do I create it?”
Two examples from my recent past.
Combining Like Terms
Why did we invent the skill of combining like terms in an expression? Why not leave the terms uncombined? Maybe the terms are fine! Why disturb the terms?
One reason to combine like terms is that it’s easier to perform operations on the terms when they’re combined. So let’s put students in a place to experience that use:
Evaluate for x = -5:
3x + 5 + 2x2 – 7 + 8x – 5x2 – 11x + 4 – 5x + 3x2 + 4 + 3x – 6 + 2x + x2
Put it on an opener. The expression simplifies to x2, giving students an enormous incentive to learn to combine like terms before evaluating.
[I’m grateful to Annie Forest for bringing the example to mind. She also adds a context, if that’s what you’re into.]
Parentheses
When students first learn to graph points, the parentheses are the first convention they throw out the window. And it’s hard to blame them. If I told you to graph the point 2, 5, would you need the parentheses to know the point I’m talking about? No.
So why did mathematicians invent parentheses? What purpose do they serve, assuming that purpose isn’t “tormenting middle school students thousands of years in the future.”
It turns out that, while it’s very easy to graph a single point with or without parentheses, graphing lots of points becomes very difficult without the parentheses. So let’s put students in a place to experience that need:
Graph the coordinates:
-2, 3, 5, -2, 8, 1, -4, 0, -10, 4, -7, -3, -2, 7, 2, -5, -3
You can’t even easily tell if there are an even number of numbers!
[My thanks to various workshop participants for helping me understand this.]
Closer
The need for combining like terms is Harel’s need for computation and the need for parentheses is Harel’s need for communication. I can’t recommend his paper enough in which he outlines five needs for all of mathematics.
My point isn’t that we should avoid real-world or job-world applications of mathematics. My point is that for some mathematics that is actually impossible. But that doesn’t mean the mathematics was invented arbitrarily or for no reason or for malicious reasons. There was a need.
Math sometimes feels purposeless to students, a bunch of rules invented by people who wanted to make children miserable thousands of years in the future. We can put students in a place to experience those purposes instead.
Previously
We explored these ideas in a summer series.
19 Comments
Kyle Pearce
May 12, 2016 - 8:21 am -Sometimes I can get caught up over-complicating things when trying to make my math lessons meaningful. This post models beautifully how simplistic creating that need for mathematics can be. Thanks for sharing.
Andrew Stadel
May 12, 2016 - 8:23 am -“We can put students in a place to experience those purposes instead.”
Well said. Creating purpose is so crucial. And not just in math class. However, I’m glad you’re stressing the importance of need and purpose!
Thankfully, students are very good at reminding us of this when they ask, “Why do we need to do this?” or something along those lines. It’s hard at first, to not take this personal, especially when we might not know the “historical” answer to some mathematical notation/rule. I appreciate how you’re encouraging us math teachers to seek out that need/purpose in order to better serve our students.
I’m wondering, what classroom conversations might arise if we sincerely asked students to think of logical reasons we need to put parentheses around ordered pairs? For me, this might be the best route when I’m stumped at creating the “headache” for students or crafting a scenario that disturbs a student’s “intellectual need”. Chances are good, students might think of something more creative and logical than me.
David Wees
May 12, 2016 - 8:47 am -I bristled at the use of the word “skill” as to me that implies that this idea has no mathematical lineage, that it is disconnected from other ideas.
My first question was, “How is this mathematical idea connected to things kids already know?” and my second was, “How is this mathematical idea connected to things kids will learn?”
In this particular case, combining like terms of polynomials relies on understanding that addition is closed in its domain, or alternatively, you can only use addition when the units being added are the same (units here being a description of the domain of the quantities being added). This first question is basically an extension of a question like, “Bill has four apples and three oranges. Sally has two apples and five oranges. How many apples and how many oranges do they have?” If you say, “They have 14 pieces of fruit”, you’ve changed the domain in question.
Maybe this means that a way of framing the intellectual need here is, “How do we make this operation on these terms operate consistently with what we know already about addition?”
The coordinate example here makes more sense, but even here mathematical consistency (wherein the symbols and language used to describe mathematical ideas are included in the set of things we call ‘mathematics’) seems relevant. In this case, what we call points are ordered pairs and are related to unordered sets like {0, 2}. Ordered pairs are a kind of set, so we want to denote them similarly to sets, but they have order, so we want the notation to be different. Finally, in both cases we want the notation to imply grouping (which is what your example relies on). Now this doesn’t mean that relating points to sets is going to convince children of the intellectual need, but I’d like to consider it as a factor in determining how I teach this idea.
Andrew Stadel
May 12, 2016 - 8:54 am -@David
“How is this mathematical idea connected to things kids already know?” and my second was, “How is this mathematical idea connected to things kids will learn?”
These are such strong questions! Ones I wish I was more aware of earlier in my teaching career!
Dan Meyer
May 12, 2016 - 9:08 am -David Wees:
Are you importing that definition of “skill” from anywhere in particular? It seems parochial to me somehow.
I don’t disagree. I’m trying to put students in a position to experience that consistency. (Rather than hear me testify about it, for example.) That consistency affords computational efficiency, which resulted in this particular example. Is there another way to put students in a position to experience the advantages of consistency?
Sarah Miller
May 12, 2016 - 12:31 pm -As I was watching Curious George with my three-year-old the other day, it occurred to me that in many of the episodes (and trust me, I’ve seen A. Lot. of episodes!), George “invents” something to fix a ‘headache,’ and has to modify his invention or idea as problems come up. But of course, the thing he invents has already been invented and really, they are just trying to teach why that thing is the way it is. But I thought it was an interesting connection to the aspirin/headache approach.
Kevin Hall
May 12, 2016 - 12:40 pm -Another way to generate need for combining like terms: if you do Jo Boaler-type explorations of visual patterns, you often get students creating different expressions for the same pattern. Only by combining like terms can they verify that their apparently different expressions are really equivalent. For example, consider the famous pool border tiles problem, in which students write expressions for the number of tiles in the border around a square with dimensions S by S. If you click the link, you’ll see that Strategy 1 leads to the expression 4S+4, while Strategy 2 gives 2(S+2)+2S. They’re equivalent, but it’s not obvious that they are unless you either know how to combine like terms or resolve the dispute by breaking out some algebra tiles or something. I’ve found it a good way to generate need for combining like terms.
Fred Thomas
May 12, 2016 - 12:54 pm -Perhaps there are important math skills for which it is impossible to find an application outside math classes, but combining like terms is not one of them. I am especially troubled by the introduction of a rather tedious and uninteresting task (evaluating a poorly written expression for a single value of x) and using that as the motivator. Except for those students already motivated to learn math, a task which looks like nothing except a math assignment or a math test question provides very little real motivation. In addition, if this expression only needs to be evaluated for x = -5, it might actually be faster to plug-and-chug term-by-term rather than going to the trouble (and the risk of error) that goes with combining like terms.
Our approach at Math Machines (illustrated by a one-minute video at https://www.youtube.com/watch?v=w7ZQYZ5RArw) is to provide a context which illustrates an actual application and which provides students with immediate, authentic feedback. Programming a computer or spreadsheet to do many repetitions of the calculation for different values of x IS an application where combining terms makes an authentic, job-related difference, and we let students test both the correctness of their answer and the decrease in required CPU time. There are other reasons for combining like terms, including the fact that it is much easier to understand and use the nature of the expression when it is expressed in a more easily understood form. Most students would be hard pressed to describe the behavior of the original expression in words, for example, while the second is much easier to understand.
If I understand the Common Core correctly, its authors specifically avoided using words like “combining like terms” and “simplifying polynomials,” opting instead to emphasize the more general Mathematical Practice 7, “Look for and make use of structure.” Combining like terms can and should be justified to students in that more general way, not just as a technique for making tedious math a little less tedious.
Dan Meyer
May 12, 2016 - 1:05 pm -@Sarah, examples, please! YouTube links!
@Kevin, nice. Dragging your anecdote back to my framework for a second, you’re using the need for communication (of results) to motivating a simplified form whereas my initial attempt was the need for computation.
@Fred, I wish I understood your application. For students like me who don’t have an intuitive understanding of color theory, the application may be less motivating and more confusing. In general, that’s my struggle with real world applications. What’s real is relative.
Fred Thomas
May 12, 2016 - 2:00 pm -Dan, I understand and agree with your comment that reality (and familiarity) are relative. One of our biggest concerns with using color was the fact that an average class has at least 1 or 2 students (usually male) who are color blind. That’s part of the reason we always display graphs as well. The only part of color theory that comes into this at all is the principle most students learned with crayons that red + green appears yellow. The typical confusion is in the understanding of mathematical functions–that 2 expressions which appear very different can actually be equivalent for all values of x. Nothing motivates every student, but we have found that a large number enjoy producing dynamic color displays–even people like me with very little artistic sense.
Sarah: I put a screen-capture jpg at http://www.mathmachines.net/temp/ColorFun.jpg
Kevin: That precisely what we want. “Simplifying expressions” and such is really all about selecting a suitable form for communicating. That includes communicating to other math students/teachers (which is what many seem to mean by “standard form”), but we think it should also include communicating mathematically with computers. It also includes getting things into a summary form for one’s own mental processes.
l hodge
May 12, 2016 - 5:04 pm -It seems to me that the first question is more of an extension of 10 + 200 + 30 + 500 than oranges and apples. Oranges & Apples would be more like 3x + 2y + 5x. Just as we wouldn’t say 10 + 300 is 400, we wouldn’t say x + 3x^2 is 4x^2.
Mia L.
May 12, 2016 - 5:55 pm -This is a real interesting idea. As a college student persuing a career as a mathematics teacher, I have been struggling to find ways to look at some concepts in the real world so my future students will find it important. This is an interesting viewpoint. Thanks for sharing!
Lane Walker
May 13, 2016 - 5:13 am -Anne’s Youtube example is priceless in my experience. The emotional hook provides a faster refresh of the “need” than the symbolic alone. For the value of an emotional connection, research is on Anne’s side. A 60-second portion of the video Anne provided is likely to save time in recall later.
Emotion enhanced retention of cognitive skill learning
by Steidl, Stephan; Razik, Fathima; Anderson, Adam K
Emotion (Washington, D.C.) (1528-3542), 02/2011, Volume 11, Issue 1, pp. 12 – 19
Brain mechanisms of emotion and emotional learning
by LeDoux, J E
Current opinion in neurobiology (0959-4388), 04/1992, Volume 2, Issue 2, p. 191
danburf
May 13, 2016 - 7:56 am -I blogged on this about a month or so ago also! I hooked students a little more by telling them I would race them to see who could finish the fasted, I gave ’em a 30 second head start just to even out the playing field.
https://danburf.wordpress.com/2016/04/11/adding-polynomials/
Miles Calabresi
May 13, 2016 - 8:14 am -I remember on an old post that a previous comment suggested avoiding the expression “creating need” so we don’t look like we’re rationalizing the existence of an otherwise useless set of skills or facts (and indeed, we’re not). You correctly pointed out that the need already exists, but you continue using the term “create” — why not say something like “highlighting” the need? We’re not so much creating the need for them as pointing it out in a plausible context. It already exists; they might just not know about it yet because they haven’t faced a problem where the need is apparent.
The headache needs a cure, but students might not know the headache exists until they try a worthwhile activity (for instance, running track, if I may stretch the metaphor) that brings it into focus. Is that not more of the business we want to be in? Guiding the student through worthwhile activities, letting them discover already existing headaches and then presenting the aspirins that have been developed so that they can see the usefulness of these cures themselves? Sorry to bring up an old topic, but “creating headaches” sounds a little more dishonest than what you’re actually doing.
Dan Meyer
May 13, 2016 - 1:38 pm -Miles Calabresi:
I suppose my meaning is “creating the need in the student’s head.” The need may already exist in the world, or the development of math, but not in the student’s head. I’ll have to think harder about my choice of words. Thanks for the comment.
Bill S
May 13, 2016 - 6:04 pm -Or could it be that
-2, 3, 5, -2, 8, 1, -4, 0, -10, 4, -7, -3, -2, 7, 2, -5, -3
is meant to be
(-2, 3, 5, -2, 8, 1, -4, 0, -10, 4, -7, -3, -2, 7, 2, -5, -3),
a single point in 17-dimensional space?
education realist
May 14, 2016 - 7:42 am -While I often use activities to reveal the need for a particular process, I had an idea last month on showing how two dimensions expand to three, and why we need it. It was an ad hoc lesson that turned into a pretty epic discussion for one of my algebra 2 classes.
https://educationrealist.wordpress.com/2016/05/06/great-moments-in-teaching-the-third-dimension-part-i/
So epic that I’m working on part 2. It’s long, but an easy read.
Kristen Peterson
May 19, 2016 - 1:18 pm -“Why did we invent this skill?” is such a powerful question to begin lesson planning with. Whether the answer leads to a practical ‘real-life’ skill or a necessary mathematical skill, offering our students a purpose for why we are learning promotes more engagement and better retention. I love the idea of presenting the headache to the students first, before revealing the aspirin. I also applied this idea to like terms, but took it a little further with polynomials. I started teaching polynomials as representations of numbers. I started by expanding a three digit number into scientific notation (965 = 9 x 10^2 + 6 x 10^1 + 5 x 10^0) and then replacing the 10 with a variable. I told my students that polynomials to numbers in different bases, we just don’t know what the base is — that really baffled them! Their connection between mathematical representations and numbers has been growing stronger since I implemented this strategy. Now my lessons on polynomials are able to constantly be connected back to what we would do with numbers. Like when we add two numbers we add the place values, so when we add two polynomials we can only combine terms with the same exponent since that represents the place value. Thanks for the inspiration!