[Fake World] It’s “Real” If They Can Argue About It

Hans Freudenthal changed the conversation from “real world” to “realistic world“:

The fantasy world of fairy tales and even the formal world of mathematics can provide suitable contexts for a problem, as long as they are real in the student’s mind.

This complicates our task. It’s easy to create real world tasks that aren’t real in the student’s mind. It’s harder to create realistic tasks.

Here’s one way to test if the context is “real in the student’s mind”:

Can they construct an argument about it?

From Jennifer Branch’s presentation handout at CMC-South [pdf], I’ve pulled a series of questions she calls “Eliminate It!”


None of these are “real” in the sense that most of us mean the word. But each of these groups is “real” to different students. Triangles are real. Pentagons are real. Diameters are real. We know they’re real because those students can construct an argument about which one doesn’t belong. That ability to argue proves their realness.

(Of course, the value of the task is that different arguments can be made for each member of the group.)

On the other hand, consider:


These elements are definitely “real.” They’re metals. But are they realistic? Are they real in your mind? Can you construct an argument about their substance?

If not, how is it in our best interests to promote a definition of “real” that admits “magnesium” but denies “pentagons”?

2013 Nov 26. Similarly, it’s “real” if they can sort it meaningfully.

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. The elements are realistic and should promote an argument in a Chemistry class. Therefore the problem is that each subject area is trying to teach its core vocab and organizing principles and then (hopefully) asking students to reason about these terms. Both “secant” and “Mg” are unfamiliar to the uninitiated and create a barrier for students to begin the actual task we care about, which is reasoning. In English class there is a lower barrier to entry for this reasoning, because while discussing specific characters/books might require a student to learn arbitrary details (by reading the book), most students can reason about human behavior in general. So while I agree that all math, science, history, characters of the English canon, etc. are potentially discussable because they can be made real to anyone who understands them, the ‘real’ issue is how to get students to understand these ideas enough to make them real in their mind so they can discuss them, and why we are asking kids to only discuss these ideas.

  2. Do students consider the third group (x^2 + 4 etc) to be real, even if they could argue about it?

    Maybe the word “real” is holding us down, and we need some variation of “mentally coherent in a way that resembles tangible objects”?

  3. As a chemist, I think the chemistry exercise might be a good one. You could look up physical and chemical properties. Boring? Perhaps, but no more or less meaningful than the mathematics question to the non-expert in each field.

  4. Glad you’re having this discussion. I remember back when I was not doing high school math homework my teacher tried to convince me I would need math — “What if you want to build cabinets for your kitchen? Or compute the height of a house?” she said.

    How sad and small that seemed, compared to the year of my 14 year-old life she wanted to take away with nightly exercises. I promptly failed the class.

    Why we teach students math and why students might want to learn math are two separate questions, but neither is answered by our weird fixation on cabinet building as a potential future. Broadly, the social reason why certain types of math exist in high school tends to be that by the time a student finds out what types of math will be relevant to what they want to do in life (e.g. engineer or reporter? etc) it’s really too late to backfill more advanced mathematics. So kids learn calculus in high school (as opposed to, say, discourse linguistics) so that options can be kept open until they are old enough to decide.

    The real question is given this is what you have to do, how do you make it a meaningful experience for the students *now*; real-world is one way into that, but there are many others.

  5. I always hate my comments on your blog immediately after I post them. So I want to add this — even when I taught a statistical literacy class (which is extremely “real-world”) the real test was about habits of mind. In other words, the real test was not whether we were solving real-world problems in class, but whether certain habits of mind were being developed that get triggered in the real world.

    This is a really poorly formed thought, but I do wonder if “developing good and pleasurable habits of mind” might be a better frame than “real-world”.

  6. Mike — what’s particularly weird about the cabinet building is that you are unlikely to use much math should you actually build cabinets. I’ll never forget watching a student senior project in which he had read up on carpentry and learned how to build a shed. How disappointing to see that even in that frame full of triangles, professionals used neither trig nor the pythogorean theorem to get things square.

  7. IanR:

    As a chemist, I think the chemistry exercise might be a good one. You could look up physical and chemical properties. Boring? Perhaps, but no more or less meaningful than the mathematics question to the non-expert in each field.

    Exactly my point. Only I’m trying to keep the lingo consistent here so instead of “meaningful” I’m going to swap in “realistic” or “real in your mind.”

    If I knew my periodic table any better (or at all) I’d probably be able to make an argument about those four elements. Because they’d be real in my mind.

    Mike Caulfield:

    The real question is given this is what you have to do, how do you make it a meaningful experience for the students *now*; real-world is one way into that, but there are many others.

    Right. Charts like this will do very little for a kid having a meaningless time in math now. (I mean, what kind of value proposition is that? “Do this stuff you hate for 16 years and we’ll reward you later with a job.” Yikes.)

  8. Let me ask this. Below is a question from one of our old national math tests for 9th graders (the test is given every year to 9th graders over several days). Which, if any, of the “realistic” categories being discussed do you feel this fits under? Is it perplexing? How could it be made better if not? Note that there were two easier leading questions about the volume of a cylinder as a type of “scaffolding” or easy entry to the ideas. I have skipped those here.

    Problem (roughly translated):
    You have a flat piece of metal 24 cm x 6 cm and would like to cut a 1 dl measuring cup from this. You will cut the sides and the bottom (there is a picture showing the basic ideas of a circle next to a rectangle and an image of an aluminum measuring cup).

    Is it possible to do this? Support your conclusion with calculations and reasoning.

  9. @Dan: On the real-life chart

    Probably this got commented on when you first posted it, but the chart mainly seems to reinforce students don’t need Algebra II (unless they want to be math teachers!)

  10. After reading this entry, I imagined my students grasping mathematics as the girls do in this episode of Big Bang Theory.


    They inquire into why the boys are so into Comic-con and comic books. So they sit down and peruse some. At first they see no use for the books, until one makes a statement, and someone contradicts it. From there they are pulled in, and a humorous discussion ensues.

    I like seeing my students productively create arguments especially when it comes to “realistic” concepts comparing unit rates, percentages and discounts, the use of tax or interest.

    The challenge, as stated above is to find things that draw the kids in (even if they don’t realize it at first), and to question.

  11. I’ve been battling this same question. How do I get the kids to care or see relevance? I loved math from a very early age and, sometimes, I have to stop myself from asking why everyone doesn’t.

    I stumbled upon what I think is a great idea for my remedial math class. In a nutshell, I need my walls painted and wanted to tie it into class. I realized that, I think, I can teach most, if not all, of the linear portion of the algebra curriculum through this, all while making it “real”.

    Anyway, I’m going to try to document it at http://benjaminclaymorris.blogspot.com/.

    As soon as I started blogging in 2012, the rug got pulled out from under me. I want to be able to chronicle this, however. I think there is a TON of potential here. I’d love for some of y’all to give some constructive criticism.

  12. It seems as though you are suggesting that a criteria to be considered “realistic” is that the problem is accessible to the students. After all, a couple people with some background in chemistry commented on the value of the Eliminate it chemistry task you created. The reason that the task was “realistic” to them is because they could activate some prior knowledge.

    Having said this, it seems to me that the criteria for being realistic falls in line with cognitively demanding tasks (something that I am a big fan of). I wonder if a lot of what we would deem as worthwhile mathematical tasks is encompassed by the criteria of cognitively demanding tasks. In my experience, such tasks are at least engaging from a problematic standpoint and they allow students to argue, discuss, and create a number of different solutions.