If Functions Are Aspirin, Then How Do You Create The Headache?

This Week’s Skill

Determining if a relationship is a function or not.

A relationship that maps one set to another can be confusing. Questions like, “What single element does 2 map to in the output set below?” are impossible to answer because 2 maps to more than one element.

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By contrast, a function is a relationship with certainty. Take any element of the input and ask yourself, “Where does this function say that element maps?” You aren’t confused about any of them. Every input element maps to exactly one element in the output.

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Pearson and McGraw-Hill’s Algebra 1 textbooks simply provide a definition of a function. Pearson’s definition refers to a previous worked example. McGraw-Hill has students apply the definition to a worked example immediately afterwards. Khan Academy dives straight into an abstract explanation of the concept. In none of these cases is the need for functions apparent. Students are given functions without ever feeling the pain of not having them.

What a Theory of Need Recommends

If we’d like students to experience the need for the certainty functions offer us, it’s helpful to put students in a place to experience the uncertainty of non-functional relationships first. Here is what I’m talking about.

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Put the letters A, B, C, and D on your back wall, spaced evenly apart.

Ask every student to stand up. Then give them a series of instructions.

Transportation

If you walked to school today, stand under A.
If you rode your bike to school today, stand under B.
If you drove or rode in a vehicle today, stand under C.
If you got to school any other way, stand under D.

Duration

If it took you fewer than 10 minutes to get to school today, stand under B.
If it took you 10 or more minutes to get to school today, stand under D.

Class

If you’re in seventh grade, stand under A.
If you’re in eighth grade, stand under B.
If you’re in ninth grade, stand under C.
If you’re in any other grade, stand under D.

These instructions are all clear and easy to follow. Students are certain where they should go. Then give two other sets of instructions.

Clothes

If you’re wearing blue, stand under A.
If you’re wearing red, stand under B.
If you’re wearing black, stand under C.
If you’re wearing white, stand under D.

Birthday

If you were born in January, stand under A.
If you were born in February, stand under B.
If you were born in March, stand under C.
If you were born in April, stand under D.

Perhaps you see how these last two examples generate a lack of certainty. Students were lulled by the first examples and may now feel a headache.

“I’m wearing white and red. Where do I go?”

“I was born in August. There’s no place for me to stand.”

Now we gather back together and apply formal language to the concepts we’ve just felt. “Mathematicians call these three relationships ‘functions.’ Here’s why. Why do you think these relationships aren’t functions?” Invite students to interrogate the concept of a function in different contexts. Try to keep the focus on certainty – can you predict the output for any input with certainty? – rather than on the vertical line test or other rules that expire.

Next Week’s Skill

Graphing linear inequalities. It’s extraordinarily easy to turn questions like “Graph y < -2x + 5" into the following series of steps:

  1. Graph the line.
  2. If the inequality includes the boundary, make the line solid. Otherwise, make the line dashed.
  3. Test a point on either side of the line. Use (0,0) if possible.
  4. If that point is a solution to the inequality, shade that side of the line.
  5. If that point isn’t a solution to the inequality, then shade the other side of the line.

Students can become quite capable at executing that algorithm without understanding its necessity or how it figures into algebra’s larger themes.

What can you do with this?

BTW

Kate Nowak encouraged me to look at other textbooks beyond McGraw-Hill and Pearson’s. She recommended CME, which, it turns out, does some great work highlighting this need for functions. It asks students to play a “guess my rule” game, one which has a great deal of certainty. Each input corresponds to exactly one output. Then the CME authors offer a vignette where a partner reports multiple outputs for the same input, making the game impossible to play. Strong work, CME buds.

About 

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.

39 Comments

  1. Why isn’t the birthday example a function?

    It can be a function where the domain is the first four months of the year.

    Or it can be a function where the domain is the months of the year, but is undefined for values of May — December. We’d all agree that y = 10/x is a function, yet it too is undefined when x = 0.

  2. Is the game really impossible to play when there are multiple outputs? Seems like you could still figure out if the relation is a circle or whatnot.

  3. IMO, the hands-down best “headache” for graphing inequalities is the type of situation that would lead to “linear programming” (to use the technical term for a potentially simple but powerful concept).

  4. Chester Draws

    July 9, 2015 - 1:41 pm -

    Roy,

    In New Zealand we don’t even teach graphing inequalities until we do linear programming.

    That way the need for them arises naturally.

    Re functions:

    You can tease students by playing “Guess my number” with non-functions — I would normally offer a small lolly as prize for winning, but it’s not obligatory.

    It goes: “my number squared is 16”. A student says 4 and I say, no, it’s -4.

    Next one is “my number squared is 25”. They guess -5, and I say, no, +5.

    They complain that I only allow one answer, so we change the rules to allow two. Next question is “sine of my number is 0.5”.

    If they are up to complex numbers, “my number to the power 3 is -1” also works.

    They can quickly work out which offered options give certainty of answer (“my number logged in 2”) and which do not.

  5. Why use the word certainty here? For example, in the birthdays example there is nothing uncertain for the person born in August. They just stay put. Lots of one to many relation are equally certain. If I talk about my sisters I am very certain who I am talking about. Only if I talk about one of my sisters chosen randomly am I introducing uncertainty and that is another topic.
    The term you seem to be referring to is specificity not certainty.

    The example of the clothing colors that leaves students not knowing what to do doesn’t seem like a good example of a well defined relation.

    Why not try to explain the real challenge – why are you going spend so much time on just the subset of relations called functions? Why is excluding just one to many relations so worthwhile? You still have many to one relations, what was so special about excluding just one type of relation? Why is it useful to make the somewhat arbitrary definition of arcsine?

  6. A further example. The quadratic formula is a very useful formula it takes 3 numbers and spits out one or two. Given three input numbers here is no uncertainty about what it will split out. It’s not a function but hopefully you agree it is useful and interesting.

    Why are you going to focus on a subset of formulae that exclude this one?

    Surely there is a better motivation than some teacher wants to play a guessing game that arbitrarily excludes interesting cases such as, for example, the quadratic formula or circles.

  7. Just saw a twitter link to this (x^x^x… power tower) that implicitly has a nice example of making an implicit definition, but then having to check whether it is a function (well defined).

    @Scott: picking up your example of a circle, let’s say the unit circle. If your output is allowed to be 2 values, then you still have defined a function, but now it is not from [-1,1] -> [-1,1] but [-1,1] -> subsets of [-1,1] (or even subsets of [-1, 1] with 1 or two elements)

    @Stan: similarly, you can redefine a one-to-many (or many-to-many) relation between sets X (domain) and Y (pseudo-range) as being a function instead from X to P(Y) (the power set of Y, the set of all subsets of Y). This even allows you to extend a relation that only links a proper subset of X with elements of Y, since you can send the rest to the empty set.

    In each of these cases, we gain a lot by being precise about what relation/function we are considering.

    Also, the sister/sisters example is very vivid for me. For years, a co-worker would tell me stories about “his brother.” I formed a very strong conception that these were all about a single person who got up to a wide variety of crazy stunts and was stunned to find out that he actually had 4 brothers and the stories were collected experiences across all 4.

    Finally, the quadratic formula is a great example because *it isn’t a function* so we can’t use it on its own to define a value. There are plenty of school examples where this catches students, but here is my favorite:

    consider the continued fraction with all coefficients 1, so
    x = [1,1,1,1,1,…]

    Assume x exists as a real number, what is its value? Well,
    x = 1 + 1/x, so it is the root of a quadratic polynomial. We can use the quadratic formula to find the roots of this polynomial.
    which one is x?

  8. Josh Roberts

    July 9, 2015 - 8:43 pm -

    When I get into functions and domain and range we talk about inputs and outputs. My favorite analogy for a function is a Pepsi machine (insert . We talk about the buttons on the machine and how the first button is Pepsi and so is the second. Then the third is Diet Pepsi and the fourth is Mountain Dew or whatever. The buttons are the input and the sodas are the outputs.

    The idea of one input giving only one output is clear and students also learn quickly that two buttons (inputs) can produce the same soda (output) and that it is still a properly working soda machine (function).

    We then talk about what it would mean if you pushed the Mountain Dew button and sometimes got a Mountain Dew, sometimes got a Cactus Cooler, and sometimes got a Dr. Pepper.

    This then becomes a reference point for whenever we discuss if a relation is a function or not. “Remember the Pepsi machine.”

  9. Re: Next week’s skill…

    Ask students to find two numbers whose sum is less than or equal to ten (or, alternatively, points that satisfy your 2x + y < 5 above). The headache is caused by asking them to list 'em all. The aspirin is another way to communicate all of these points — the graph determined by the five steps listed above. Rather than present the steps, have students plot their points as a class. The fact that there’s a line and some shading involved should jump out, as in this post from John Scammell.

  10. This isn’t a “headache” by itself, but for inequalities we try to stress the “truthiness” of something, just like equations:

    The graph of 2x + 3y = 12 is the set of points that make the statement true

    The graph of y > 2x – 5 is the set of points that make the statement true

    One problem I like is having each kid pick a point, then running it through a “test” like y > x^2. They plot their point green or red depending on whether or not it passes the test — and a rough shape of the graph emerges. The entire concept that you can shade a full side of things once you know a point matches them isn’t obvious but becomes viable after these types of problems.

    Even your version (test a point, shade a side) is better than what some texts do about shading “up” or “down” based on greater than or less than, rules which quickly vanish when pieces of equations move around or non-linear inequalities come about. One of my largest annoyances about linear inequalities is rules or instructions that work -only- for linear inequalities and nothing else, which strike me as completely useless.

  11. Joshua,
    You miss my point. Sure lots of practical problems use the quadratic formula but only one solution is valid. But this suggests that when you have these problems knowing about this relation is what you want.

    That seems like a good motivation to learn about mathematical relations rather than functions. The same applies to all the trick questions that are examples of relations. These suggest a motivation for studying relations rather than functions.

    The value in studying the mathematics of functions is more than just being able to recognize when a problem has only one valid solution even though the quadratic formula is a handy tool for working on the problem.

    I am trying to make two points – the motivation for studying functions is more than just learning a taxonomy and as in the quadratic formula certainty verses uncertainty is not a very clear way of distinguishing functions and more general relations.

  12. My simplistic theory of learning is that WHAT we teach should respect the consensus of a larger community of experts (e.g., the Common Core) and that HOW we teach needs to incorporate explicit linkages to both the context of mathematics as a discipline and the multiple contexts in which non-mathematicians might use a specific skill or concept outside math class. I remember memorizing definitions of functions, domain and range as a high school student and believing at the time that the concepts had no value whatsoever either in math or in the broader world. I did not learn to appreciate the importance of the ideas (including the necessity of a single output) until Bob Chaney and I began using engineering control systems as a teaching tool for both math and science.
    Automated control systems are an extremely important component of engineering, technical and medical careers. They can be as simple as the inequality which opens a door under one set of circumstance and closes it under another set. They can also be as complex as the automated decisions which guide a driverless car or as critical as the equipment which delivers the right amount of oxygen to a premature infant. In every case, the system design and operation demand a single output at any one time.
    One of our early successes with our SAM vehicle (a calculator-controlled version of Papert’s Logo Turtle) came in asking trig students to find the two math functions which would guide SAM back home after it had moved randomly along 2 sides of a triangle. It is fairly easy when the first two sides were at right angles, although most students initially want to plug is the formula for the triangle’s interior angle while SAM needs to turn through the exterior angle. It gets harder when the angle (as well as the sides) were random, and the possibility arose of SAM being in any quadrant. Students experience the headache of realizing that their calculators were converting arctangents to a single output in a way that did not always bring SAM home. Amazingly enough, they are sufficiently motivated to keep working on the problem until they find a better way of giving SAM the single instruction it needs. The immediate physical feedback of watching SAM go astray increases the headache and is critical to maintaining motivation until they do get it right.

  13. If we have to go all modern math and use the term “relation” then it should be noted that a relation is between two sets and a “mapping” is from one set to another. then mappings can be classified as either “one to one” or “one to many” or “many to one”. then a function is a mapping which is not “one to many”.
    This describes the more real world view of a function as a machine converting an input into an output, where a given input always yields the same output, today, tomorrow, and forever. It is a consequence of this that the graph of a function does not loop back on itself. Asking questions such as “Which of these graphs represents a function?” is not productive of any real understanding.

  14. I think this is great. Those advocating starting with mathematical, abstracted examples and non-examples of functions (quadratic formula, equations of circles, etc), I think maybe don’t grok your typical 8th grader’s facility with quickly evaluating expressions in his head. (Some of them are great at it, sure, but most are iffy and some are still not good at it — and don’t we want every kid in the room to get what a function is, if we’re going to the trouble of teaching it?) For a brand new concept like “what’s a function?” it’s much better to hook it to something they can easily relate to, like Dan does here, rather than making evaluating the quadratic formula a prerequisite for understanding the new thing. PLUS, “what’s a function?” often comes chronologically before kids see equations of anything other than lines.

    If someone’s going to take this lesson idea and build it out, I’d be interested in seeing what people come up with for what comes after this. How do you build the bridge from the understanding developed in this lesson to being able to approach all the types of function/not-a-function questions you want kids to be able to handle.

  15. After having students play the “guess my rule” game, I always ask the class if anyone ever asked the same number twice. Obviously they wouldn’t because they would get the same answer. This is one way to lead into the definition of a function.

    I’ve also done the vending machine analogy, which I think is great, but with one issue I’ve not been able to resolve. Many of my students think it’s not a function when the button says one thing but gives another. As long as it’s consistently giving the wrong drink, it’s a function. This is confusing.

  16. Back in the early ’80s, I found a book on precalculus that was comic-strip based (can’t recall the name off-hand, but it wasn’t one of Larry Gonick’s “cartoon guides”. I recall clearly that the author presented functions as guarantees (a little guarantee popped out every time the characters “ran the function machine.”

    That really hammered home the notion of functions as predictable. I offer students example like getting paid an hourly wage of $20, working 40 hours in week 1 and grossing $800, working 40 hours in week 2 and grossing $800, and then working 40 hours in week 3 and grossing only $600. When they go to the payroll department (or directly to the boss) to complain, they’re told, “Oh, well, we had a bad week, so we can only afford to pay you $15 and hour this time.” Naturally, this would be an outrageous violation of a guarantee.

    Then I give the example of a table of temperatures collected every hour on the hour from 1 AM to midnight on a certain day and ask whether, given the data, they could confidently state what the temperature was for each of the 24 hours that measurements were taken. Most quickly realize that they could do this since any given hourly measurement corresponds to exactly one number. Finally, I ask if they could reverse this process: given a temperature, could they (likely) be able to predict with certainty which of the 24 hours corresponded to that number.

    Of course, the above is quite simplistic and doesn’t capture anything close to the full picture of functions or why mathematicians (and those in applied fields) find them to be important and worthy of study. But it does tend to get students to see the headache.

    By the way: 1) when people say “Pearson book” or “McGraw-Hill book,” I’m at a loss. I have a pile of current Pearson intermediate algebra books on my floor that I got last fall when thinking about changing from the publisher I was using. I also looked at other texts for that course from that and other publishers. As the business has been reduced to perhaps three major publishers, they all have multiple texts, multiple (current) editions, from multiple authors. So saying “Pearson book” and the course isn’t describing a function;

    2) the particular text from Pearson I’m using now for intermediate algebra (Bittinger, Beachy, & Johnson, 12th edition) does something on the topic we’re discussing that I don’t recall seeing elsewhere: it introduces and defines functions first and only then defines and explores relations.

    Frankly, I find that quite bizarre. Seems to me that relations are the most general case, with functions being one natural restriction on them. Of course, there are special and important kinds of relations (equivalence relations being foremost in my mind) and subclasses of functions. But I can’t see why one would introduce functions before relations. So I don’t. I’ve never been shy about changing the order of topics to suit my thinking and purposes, though I’ve met many teachers, particularly in K-6, who find that to be terribly daring. Just wonder if anyone has found other authors/books teaching functions before introducting relations.

  17. Eric I:

    Why isn’t the birthday example a function?

    It can be a function where the domain is the first four months of the year.

    Or it can be a function where the domain is the months of the year, but is undefined for values of May — December. We’d all agree that y = 10/x is a function, yet it too is undefined when x = 0.

    The domain in this example is the students themselves. Some members of the domain lack a mapping to the range.

    Roy Wright:

    IMO, the hands-down best “headache” for graphing inequalities is the type of situation that would lead to “linear programming” (to use the technical term for a potentially simple but powerful concept).

    There is a categorical difference between “headaches” (as I’m using the term) and “real world applications.” Those categories may overlap but I’m trying to be precise about the concepts.

    Chester Draws:

    In New Zealand we don’t even teach graphing inequalities until we do linear programming.

    Not even single variable inequalities?

    Joshua:

    Also, the sister/sisters example is very vivid for me. For years, a co-worker would tell me stories about “his brother.” I formed a very strong conception that these were all about a single person who got up to a wide variety of crazy stunts and was stunned to find out that he actually had 4 brothers and the stories were collected experiences across all 4.

    Awesome.

    Tim Hartman:

    I’ve also done the vending machine analogy, which I think is great, but with one issue I’ve not been able to resolve. Many of my students think it’s not a function when the button says one thing but gives another. As long as it’s consistently giving the wrong drink, it’s a function. This is confusing.

    Helpful clarification. There is a categorical difference also between “headaches” and “metaphors.” I didn’t think hard enough about this particular metaphor, though.

    And thanks to Chris and Bowen for saving me the trouble of writing next week’s post. Teachers sometimes note in response to the lesson idea both of them propose that it isn’t real world, or that a student might ask, “When will I ever use this?” I’m curious about their responses.

  18. Chester Draws

    July 10, 2015 - 3:51 pm -

    In New Zealand we don’t even teach graphing inequalities until we do linear programming.

    Not even single variable inequalities?

    We teach writing and solving single variable inequalities, starting in our Year 10 (US 9th Grade). That includes testing for meaning too in Year 11, such as writing inequations from word problems.

    However we don’t test that skill graphically at all — though I imagine that many teachers use a number line to explain it, and some may test it internally.

    The concept of graphing inequalities on a number line has few real world applications at all. This year, in fact, we have had instructions that the Algebra testing in Year 11 is going to move even further away from process skills like that and towards problem solving.

  19. Dan you’re wrong about the birthday thing, Eric is correct.
    Simply because elements of the domain are not described by the mapping does not disqualify the existing mapping from being a function.

    Consider y=sqrt(x) or y=log(x) as algebraic examples. Yes the real number domain is severely restricted for these relations. Does this make them somehow disqualified from being functions? Of course not.

    Be careful in what you’re saying here, most especially with your examples. This is exactly what leads to confusion for many students.

  20. The domain of those functions is implicitly restricted. The domain of the birthday example is implicitly all the students in the class.

  21. I’m with Dan on this issue. The general definition of functions most high school algebra texts use requires that every member of the domain be paired with exactly one member of the range. Members of the range that get “hit” for the image set, and we can also refer to the domain as the preimage. But that sort of terminology doesn’t show up much in high school books and neither do injective, surjective, or bijective, as opposed to one-to-one, onto, and one-to-one correspondence. Of course, if Bourbaki were king of the universe. . .

    There are authors who speak of the natural domain of a function. For example on p. 7 of the first edition of MATHEMATICS for the DIGITAL AGE and PROGRAMMING in PYTHON, Maria and Gary Litvin write: “If we define a function by a formula and do not specify the domain explicitly, it is assumed that the domain is the set of all numbers for which the formula makes sense.

    This is called the ‘natural domain’ of a function defined by a formula.”

    In category theory, to describe a relation that leaves one or more members of the domain unassigned to a member of the range is to by definition fail to define/describe a category, as is to assign a member of the domain to more than one member of the range. I think it’s pretty clear that this parallels exactly the typical definition for function.

    I think the issue isn’t that “Dan is wrong” and the “Eric is correct” as mike suggests, but rather a too-inflexible approach to what matters here. And in terms of teaching students about functions and relations, we very quickly get at the issue of restricted domains. The textbook I’m currently using for intermediate algebra does this almost immediately after introducing the main definitions by looking at examples of rational functions (in Chapter 2) even though those functions and other non-polynomial functions aren’t introduced formally until Chapter 5.

    One quibble, Dan: I think you meant “implicitly” rather than “explicitly” in both sentences above. [Good catch. Fixed. –dm]

  22. This is an amusing introduction. I like it. But, is it aspirin? Some students don’t know what letter to stand under in the last two examples. They are then given formal vocabulary for the situations. Isn’t this analogous to describing symptoms to the doctor and being told that what you are experiencing is called a headache?

    The approach Bowen described for inequalities has worked well for me. Solutions & non-solutions can be plotted using different colors on Desmos. Ask if there are solutions in every quadrant? On the y-axis? Interesting solutions? Etc. If it doesn’t come up, at some point ask about making a fence between the solutions & non-solutions.

    Likely a couple of points will be incorrectly coded as solutions (or non-solutions). Someone will eventually notice that they don’t fit the pattern, at which point you can fix them.

  23. l hodge:

    This is an amusing introduction. I like it. But, is it aspirin? Some students don’t know what letter to stand under in the last two examples.

    This activity angles at Harel’s “need for communication.” The last two experiences feel categorically different from the previous ones. But why? They’re just another set of instructions. That disequilibrium is resolved by drawing boundaries around the two experiences and calling them what they are.

  24. I wonder if students given the headache of not knowing where to stand would make the aspirin themselves. Like- I’m going to stand under the color of my shirt or the dominant color that I’m wearing. Could we teachers use this urge to naturally make things make sense to develop why functions are useful?

  25. Katie Waddle

    July 11, 2015 - 5:01 pm -

    I’d also take a look at what CPM does with function, my go to is the coke machine example. You wouldn’t want a coke machine with a button that sometimes gives you coke and sometimes gives you doctor pepper. When you press the button, you want to know what’s going to come out. That’s a function.

  26. I am missing the fantastic motivation (headache) of the rule guessing games in motivating the distinction of functions from other relations.

    It would be just as easy to play this game with a relation. I could draw on my piece of paper some polygon and you call out an x or y coordinate and I give you the corresponding coordinate. I could give you all of them or just one requiring you to try the same value several times to know if there is more than one. You have to guess what polygon I drew.

    There would still be a rule to guess even with a non-function relation answer.

    The game with functions is easier as you only have to try each x value once. But why is this particular way to make the game easier important? Dealing with only one to one relations would be even easier. But for some reason, not given here so far, exploring relations which are functions is very important.

    What do you say to a student who says why should I care about the distinction between functions and other relations – because a coke machine?

  27. CPM talks about a soda machine. There are usually 2 buttons for coke bc it’s popular. Either button gets you the same result. This machine “works”. (Function)

    Then they press orange soda and get orange soda. Another kid comes up and presses same button and gets sprite. Is this machine “functioning” correctly?

    I think this is a simplistic lesson that a majority of kids would understand. It’s from core connections algebra I. I previously showed example and non example and showed vertical line test. I think the soda machine analogy allows for students to make sense of it in a real life scenario.

  28. Hmm. What if there’s a soda machine with an unmarked button. Joe puts money into the machine, hits that button, gets a Coke, walks off. June puts money in the machine 20 minutes later, pushes that button, gets a Sprite, walks off. Is the machine “functioning” correctly?

  29. @Michael Paul Goldenberg I would say that it’s not functioning correctly. If you press an unmarked button, you should get only one type of drink if it’s functioning correctly. In math terms, if it gives me a coke and sprite different times, than the same input is matched to more than one unique output, therefore not a function.

  30. I’m afraid I have to vehemently disagree with your reasoning about the birthday example Dan. Using your own logic, the game “implicitly” restricts itself to only those students whose birthdays appear on the board. This doesn’t have anything to do with whether the game is a function or not. Something that would be of use to understand for students. Perhaps that’s what you were driving at in the first place, but from your response, I’m not so sure.

  31. Also from Maria & Gary Litvin’s MATHEMATICS FOR THE DIGITAL AGE and PROGRAMMING IN PYTHON (2008):

    “Example 3

    Suppose P is the set of all the students in the classroom, and the function birthday(p) takes a person p as its input and return’s p’s birthday as the output. It is convenient to view this as a mapping from the set P (all students in the classroom) INTO the set D (all 366 possible birthdays, including February 29 for those special people who were born on that day in a leap year). The range of the function birthday – the days on which the actual birthdays of the students in the classroom happen to fall – is a subset of D.” (p. 4)

    I would personally suggest that here the “bigger set” D is the range and that the range described above is the image of birthday. But that’s just quibbling over terminology. What matters is the idea that the target set is a subset of a (possibly) “larger” containing set. We map the domain either into or onto a set which might be identical to the “larger” containing set, and when they are in fact identical we call the mapping “onto” or “surjective,” in Bourbaki Talk.

    As for how this reflects on Dan’s example, I’ll suggest that the notion of a “natural domain” as the Litvins and others use it reflects an implicit restriction. But that’s a matter of chance to some extent. Unless there are at least 366 students, we know that the domain may be too small to map to all 366 members of D. That’s fine, and no one should/would complain.

    But Dan purposely places an unnatural restriction, arbitrarily listing a range (or image) set that is too small relative to any reasonable domain set. This forces the students to restrict the domain for reasons that don’t really make sense. And so what is described doesn’t have the feel of a function, even if we can torment our definitions to “make it so.” I think that’s what Dan was trying to come up with, and for my part I think it worked. You need to have a way to assign exactly one arrow from each member of the domain to a member of the range. Otherwise, whether you have some unassigned domain members or some domain members assigned more than one arrow (i.e, more than one corresponding member of the range), you don’t have a function.

  32. But you do actually have a function, given the domain is “implicitly” defined as Dan attempts to say.

    The problem lies in this. ” This forces the students to restrict the domain for reasons that don’t really make sense.” That’s quite right – it doesn’t make any sense. Confusing for students as they’ll undoubtedly run up against such functions that do make sense(logarithms, etc.) in different contexts later in their studies, as any year 11/12 student knows who has been bored to death by the universality of Families of Functions, of which logs, etc. are certainly members.

    (This by the way, is an excellent reasons to be very cautious around this issue, since it would seemingly directly fall under Dan’s original argument regarding “Rules That Expire”: http://www.nctm.org/Publications/teaching-children-mathematics/2014/Vol21/Issue1/tcm2014-08-18a_pdf )

    Far more powerful then would have been the birthday example with ALL months listed, to illustrate that(given n>12) a range may be much smaller than a domain and yet provide a function, resulting in domain pile-ups at somewhat regular range intervals(….. one might even chance to suggest a periodic distribution – hello trig).

    That is, unless the entire intent was to spark an investigation into mathematical functions that have domain restrictions that DO make sense for very good reasons, but again, I’m fairly certain that was not the intent here.

  33. Michelle raises an interesting point about students providing aspirin. For example, make more signs to stand under: “AB”, “AC”, etc. Pretty much any mapping can be considered a function if you are creative in thinking about the domain & range.

    Harel… Need for communication… What would Clay Davis say to that? In the introduction, the situation is concrete and the disequilibrium is fleeting – easily addressed and articulated by the students with everyday language. We are wondering into mathematical la-la land if we insinuate that the students do not have sufficient language to communicate the underlying ideas in the introduction.

  34. Ava Erickson

    July 13, 2015 - 5:39 pm -

    Thanks for this. I was also thinking that you should check out CPM (College Preparatory Mathematics) at cpm.org. They often do a great job at motivating the need for new mathematics but I’ve only looked at one or two of their text books.

  35. I have an abstract thing and an algebra thing related to linear inequalities.

    First I have them pick a location: their house, the school, the movie theater, local stadium, whatever. We describe the location in terms of nearby major intersections, like you would if you were giving someone directions. If you can reference a location from two parts of an intersection (north, south, east, west), you’ve narrowed down the region significantly and give your friend an idea of where they’re going. Adding streets increases the accuracy of your region.

    Then I give them a task, pick a bunch of locations and describe how you’d find them using at least two references. Screenshot the map, doodle on top of it, and share it. Lacking any sort of screens, print them some maps you make with various landmarks on them and ask for the same thing.

    http://infinitesums.com/commentary/2013/11/8/more-inequality-explorations

    Does it create a headache? Maybe a little bit. But my kids speak to each other in intersections all. the. time. when it comes to finding things in the area.

    Later, get more specific with your references. Maybe the region is defined by an absolute value function and a constant. Maybe a couple quadratics. Or if it’s Algebra 1, a couple of lines. Have them graph a dozen or so points and give them some systems to draw (or you know, desmos). Give them some conditions (less than line A, greater than line B) and see what they think, then change the condition, and then get picky and ask how you’d prove that without a picture.

  36. This post helped me make a connection of my own — functions are similar to mutually exclusive events. The ambiguity of a result (where to stand) is cleared up when each result option has no overlap (is mutually exclusive with) the other result options for the given inputs (students). I’m not sure how I would use that idea to further student understanding yet, but there may be something there.