In many math curricula, functions are treated as something complicated and mysterious and they don’t get introduced until some time in middle school. But in fact, the basic idea is very simple and beautiful. A few days ago I introduced the concept of a function to Katie, and again the credit for the idea goes to Moebius Noodles, this time the book. First, I tried to explain to her what a function is in words. I drew a square on a piece of paper with an ‘in’ door and an ‘out’ door. I told her that you have to decide what type of objects can go in and what comes out. I’m not sure that she fully understood and I realized that if I was to keep her attention, then examples were needed, and fast.
Since I wanted Katie to right away think that functions are really cool, my first example was: girls can go in and princesses come out. Katie was in disbelief. ‘That doesn’t exist,’ she said. I told her that functions can be real or make believe. So naturally I had to give her a ‘real life’ example next. To illustrate the importance of specifying a domain, I gave her two examples: a dishwasher and a washing machine. In both functions dirty objects go in and clean ones come out, but you wouldn’t put dishes into a washing machine or clothes into a dishwasher. She seemed to be convinced and amused.
For the next 15 minutes (which happened to coincide with bath time 🙂 ), we took turns coming up with functions. A lot of Katie’s functions involved animals. Sometimes she tried to list individually which animals can go in and which other animals they get turned into (eg, giraffe goes in, pig comes out). My three favorites from her were: 1) boys go in and boys come out (the identity function!), 2) goats go in and sheep come out, sheep go in and goats come out, and 3) one function goes in and another one comes out (the last two she didn’t come up with on that first day but at different times during the next few days).
I did not want to explicitly introduce functions of numbers right away, but to start getting her used to the idea, on some of my turns I used functions like: one animal goes in and two come out, two animals go in and three come out, etc. She seemed to particularly like this one.
Even in this short period of time, functions have already been a great source of both learning and entertainment for Katie. There were several occasions when she would be doing something and then suddenly she’d stop and say, “mommy, can I tell you about a function I came up with?” I highly recommend introducing this concept to your kids early and I would love to hear about other peoples experiences with this rich topic!
Musings of a Mathematical Mom 
“3) one function goes in and another one comes out” – ooh, a functional! How cool and playful is that! She got the point that ANYTHING can be an input, and applied it to functions in a self-referential manner. Incredible!
http://en.wikipedia.org/wiki/Functional_(mathematics)
Giraffe-pig example is typical of kid thinking. We can distinguish functions that the other person can reasonably PREDICT from a few items, from functions that are totally unpredictable. It may be fun to make very whimsical, unpredictable (random) functions once or twice, but they are sort of boring for guessing games. It’s a good math discussion to have.
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Wow, composition of functions! My students had so many troubles with this concept, but Katie discovered them by herself. And the most important fact is that she continues to think about new concepts like she did with patterns before. You have to keep introducing new stuff to her, to feed her brain.
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The “toy breaker function”: perfectly good toys go in and they come out broken.
The “toy medic” function is its antidote.
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I like your idea of introducing inverses, and this seems like a great example for it. You can also have real life examples of this; little kids are the toy breakers and parents the toy medics. However, not all toys can be fixed by these medics (which might be a good thing since otherwise there would be no consequences for breaking the toys and kids would have no incentives to stop doing it 🙂 ).
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Really great idea to introduce inverse functions. As for Unrepairable toys, well, not all functions have their inverses 🙂
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This is a great post – some great ideas here for discussion with really young kids. Another way to play around this topic is the “black box” game – basically, you tell them what goes in and what comes out and they have to guess the function. It would obviously work well with numbers but you could even do some kind of simple pattern – e.g., lion cub goes in, lion comes out, duckling goes in, duck comes out – and they would have to guess that the function is baby animal goes in, big animal comes out. This may be for slightly older kids (although Katie might be ready) – I’ve been meaning to try it with my five year old group, so once I do, I’ll let you know how it goes.
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I have thought about this variation but have not tried it with Katie yet. Maybe I’ll wait to hear how your experiment goes first 🙂 ).
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Considering the fact that Katie knows and likes patterns, why not to try to discuss a function as a way to turn one pattern to another (or one set of patteerns to another). I think she is ready for it.
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Can you give me an example that you think would be a good one?
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The simplest one that comes to mind is: we have a pattern of four different colors – red, blue, green, yellow. The function turns red to green, blue to yellow, green to red, yellow to blue.
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One more idea. Using the fact that she knows odd-even numbers (actually, this is unnecessary), discuss a girl that wears sneakers with lights. Lets say, she has left red-light sneaker and right blue-light one. If we count steps, each number will produce either red or blue light. BTW, that is how you can introduce periodic functions 🙂
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One of my favorite games when I was about your daughter’s age (or maybe a little older) was rays of light:
http://boardgamegeek.com/boardgame/4534/rays-of-light
It’s kinda like battleship. Each player has a 4×4 grid where they put a red, blue, yellow and black chip. Then you take turns asking what color occurs when you shine a light in different ways (function taking numbers–the entry–to the color) and trying to reproduce what the opponent’s board is.
Advanced play has mirrors too.
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I’m going to be the odd-person out on this. I love your ideas and admire the way your think about things, and I do understand how you are getting your daughter to think about function as trans-formative operations, which they are, Now here’s my BUT… aren’t functions in math all about describing relationships? 1 cup of rice can make 4 servings. 1 gallon of gas can go 9 miles (in a hummer) When the time is right will she be able to switch her thinking from the magic function machine into the ratio machine? It seems to me that concepts that children learn at a young age can really settle in deep, and be hard to shift. Just something to consider. Like I said, i am a fan of yours and enjoy reading your ideas. I hope it’s okay to post a dissenting opinion.
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Paula, dissenting opinions are most welcome – they spur discussions :-).
Now I claim that what you’re calling a relation and what I’m calling a transformation are one and the same. If 1 cup of rice can make 4 servings then 3 cups can make 12 servings – the transformation “multiply by 4” takes one to the other. What I was/am trying to show my daughter and other kids I teach is that functions aren’t just about numbers. It is an abstract concept that can apply to numbers, but also to many other things around us. However, if you check out the post https://aofradkin.wordpress.com/2014/06/09/the-function-of-2048/ which happened almost a year after this one, I think that it shows that she will have no trouble applying the concept to numbers.
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Transformations and correspondences (relationships) are two (of several!) basic meanings of functions. Because transformation is an action and corresponding is an object, transformation is easier to grasp for newbies, young kids, poets, etc. So it’s better to start with that meaning and then “reify” (objectify) it into the correspondence meaning.
For example, if you build a function machine that transforms every orange into two halves (to share among you and me), it’s a transformation for a kid, at first. But when you later make “before and after” table with 1, 2, 3… oranges on little plates on the left, and 2, 4, 6… halves on little plates on the right, it will be a correspondence (relationship) representation.
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Thanks Maria for the clarifying insight! It is true that a function is defined by its set of input/output pairs and in fact almost all mathematical functions out there do not have a nice and concise “transformation” rule. However, kids do not encounter these functions for a while (if ever!), and when they do, hopefully their overall understanding of math is sufficiently sophisticated to transition to the new way of looking at it. This happens a lot in math. Also, I feel that the more ways one has of visualizing or thinking about a concept, the richer one’s understanding of it.
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