The Penny Determination Algorithm

Some of my students fabricated their data, that much was obvious. The question was, which ones? I passed this graph out to my students and asked them to come up with a definition for cheating.

Most of them circled the same four extraordinary outliers but very few students could come close to verbalizing why they chose those outliers. It was just … obvious. The formula for statistical error was so intuitive they couldn’t verbalize it.

Something I enjoy about computational thinking (the focus of my position with Google) is that it asks you to explicitly verbalize processes that may only exist in your intuition, processes that are too obvious for words.

Case in point: which of these is a US penny?

100% of my students would correctly select I. But if I asked them, “Why? How do you know?” and requested a “Penny Determination Algorithm,” the sort of thing you could give to an alien race to identify our one-cent currency, it would drive them crazy.

“I don’t know. It’s just … obvious.”

My particular students may not be ready for that conversation but it would be a good one. We’d prioritize algorithms that used cheap inputs. After all, we could scan every coin and apply some kind of edge-finding filter in Photoshop, but that would be too expensive and time-consuming for the machine that counts your change at the supermarket. Once students completed their algorithms, we’d trade them around the room and try to throw exceptions at each other.

These conversations are very difficult to have with students whose teachers for the last eight years have a) defined the inputs for their students (“The area of a triangle depends on its base and its height.”) and b) given them the algorithms (“The area is half the product of the base and height.”).

Those students just want you to give them a sack of thirty objects so they can use the algorithm you gave them to answer the question, “which of these is a penny?” They don’t want to answer the question, “how do you know?” They wouldn’t know where to start.

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


  1. I see two sides to this issue. On the one hand, if you don’t teach the algorithm, then they may get more inspired and work to actually learn what’s going on rather than parrot off something that a calculator could do.

    On the other hand, if they can’t even do that algorithm when you spell it out for them, how can you expect them to learn the background? Plus, many of the students who struggle with following the algorithms will be the ones who end up with jobs where that is what they are asked to do, “If you have this situation, then do step 1, then step 2, etc.” So, while we hope to get kids who are motivated enough or try to help them see the big picture, the “lowest common denominator” may be driving what you’re seeing.

  2. Aren’t you basically asking your students to solve an image based CAPTCHA?

    (I can’t find a working link at the moment, but the CAPTCHA’s that use, for instance, a grid of cat and dog pictures and ask you to click on the ones that are dogs)

  3. This reminds me of instructional strategies for teaching vocabulary, which I always found really challenging, that require students to come up with their own strong definitions for certain terms or sets. For example, given a set of scatterplots (some of which are positively correlated, some of which are negatively correlated, and some of which are not clearly correlated in any way), students sort the plots into groups and then come up with rules or definitions for how they made their distinctions. The teacher provides additional examples that test students’ definitions so that they have to clearly delineate the boundaries between groups. For more challenging groupings or for students who need additional support, tell them how many categories they should have (given relations and functions, tell them they should create two groups and explain how they did it).

    I’ve found that this works well for helping students develop an understanding of how we sort and classify things (and why)– they’re later able to say “oh! this is like that one example that we did that we couldn’t decide which category it belonged it but finally said it was a _.”

  4. grace: The teacher provides additional examples that test students’ definitions so that they have to clearly delineate the boundaries between groups.

    Thanks for this. I really like your division of responsibility between students and teachers. The students come up with the definition. The teacher tests it. Or, in the words of Oliver Wendell Holmes: “The young man knows the rules, but the old man knows the exceptions.”

  5. Posts like this are why I can’t stop reading this. If I ever meet you, Dan, then the first round is definitely on me.

    Here I finally see some kind of connection between your work and my own. I’m mostly a language teacher, but this year I have a Sociology class with 18-19 year-olds (kind of like winning a lottery). Here I get to ask your question: “How do you know that?”. Asking this question is about the most important thing we have to do. If the constraints of schools mean that we can’t spend lots of time asking it, then there must be something seriously wrong with schools. I mean, we should all be grace, shouldn’t we?

    Interestingly, you have touched on a problem in semantics – in a sense the nature of meaning. When we use language, it seems that we are not actually using mental definitions, conscious or unconscious (exception made for technical and academic terms). For instance- it is fiendishly difficult to come up with a good, simple, characteristic-based definition of ‘dog’ that allows us to easily eliminate all non-dogs, but easily include a three-legged dog, but we do this in life without trouble. So, if we don’t walk around with definitions in our heads, what do we have in there that allows us to identify dogs and pennies so easily? There is an ongoing debate about whether or not we are actually using mental algorithms when we do things like identifying a penny. This is another point where math is relevant, as Roger Penrose based his famous arguement for the mind’s non-algorithmic character on Godel’s theorem. (We can recognise arithmetic as true when this is difficult with algorithms.)

  6. Have you read Blink by Malcom Gladwell? One of the interesting aspects of the book is the idea that we are not very good at understanding how our intuitions work. “Looking under the hood” of our intuitions can be exceedingly difficult, but two of the great things about math are that it can help us 1) find order in the world around us, and 2) describe the order that we can see.

  7. “These conversations are very difficult to have with students whose teachers for the last eight years have a) defined the inputs for their students and b) given them the algorithms.”

    I feel you on this Dan. I am a science teacher, teaching integrated science (a hodge podge of physics and chemistry), and much of what you say here resonates with me.

    When I have spoken to some of my incredibly intelligent and respected colleagues about this exact issue, I am met with responses of “The students just don’t want to;” “They hate to think;” and not much else. No, “let’s forge a new way through where our students aren’t allowed to let us do all the hard work for them.”

    As soon as students see a formula, they feel like they can take a big sigh of relief, turn off their brains for a moment, and plug and chug. So much so that they forget that the formulas are a type of shorthand, which represents the relationships between the concepts that we are trying to explore.

    I think that on my next go around on this gravity unit, I would start out differently, discuss how gravity is affected by changes in the mass of the objects in question and the distance between them solely in words, and perhaps derive some of the formulas with the students as a way to represent those concepts using a few symbols.

    Much of what I have read of your lesson plans thus far I think endeavor to ask students provocative questions (questions that spark their curiosity and imagination) that they are intrinsically motivated to solve because they simply want to know the answer and can use math as a tool to answer. i.e. Wouldn’t this all be easier if we had some way of thinking about these concepts in an organized fashion?

    So cheers to you, Dan. I think that you are really on to something. I have been grappling with the obvious notion that traditional teaching falls short, but have struggled with finding how to go about teaching in a truly constructivist manner.

  8. I’m pretty sure that my “actual” human brain is using a series of cheap tricks, before it gets down to the nitty gritty of actually looking for features.

    If you take a look at this image;

    I can feel my brain trying to coerce it into some form of 3×3 grid before giving up and looking for more detail in the colour and texture of the objects.

    In addition if you agree with Richard Feynmans speculation on how we think;

    its likely that there are a whole bunch of learned strategies for “cheap input” algorithms.