There were two particularly useful comments in response to this problem:

The moment of inertia for rotating a I-beam about its long axis has no practical relevance in structural engineering. This is a fake-world problem, of no interest either mathematically or to engineers.

Even if this task *did* have practical interest for structural engineers, its presentation here will move the needle on student engagement only a fraction of a degree. The issue here isn’t *the usefulness of the application to professionals* but *the tedious, pre-determined work students do*.

When I saw the two boards, I wanted to go get a board and try standing on it. How much weight could we put on the board in each position before it broke? That would be an engaging problem.

I don’t know. That *might* be an engaging problem.

There are 100 different directions that question can go in terms of *the work students do* in class and only a handful of them will actual leave kids mathematically powerful and capable.

Watch me ruin the problem:

The maximum load a board can hold before it snaps is given by the formula:

[formula involving cross-sectional area and mass]

Dan weighs 90 kilograms and the dimensions of the board are 2 inches by 4 inches by 70 inches. Will the board hold his weight?

I have no confidence this task will result in the sense of accomplishment and connection the editors of the NYT seem to think it will.

There are other ways to present this kind of task, though. Which is my point. **The â€œreal worldâ€-ness or â€œjob worldâ€-ness of the task is one of its least important features.**

## 16 Comments

## Kenneth Tilton

December 11, 2013 - 9:58 am -I think one would have to change the problem to be one that could be tested in the classroom. And now challenge the student to predict a result, or achieve a desired result by first calculating the outcome. Perhaps to keep things manageable the teacher controls the test apparatus while students come up with predictions, whch the teacher then executes.

I am thinking of all the fun we had in science messing with pendulums, inclined planes, bouncing balls… the neat thing is the eureka moment when a calculation works out and the ball goes thru the hoop or whatever.

Jes thinkin out loud.

## mr K

December 11, 2013 - 10:16 am -I remember learning this as an engineer, and the interesting part was doing the calculus to derive the formula – there was a power there in being able to determine the load capabilities of any shape, not just I beams. That led an understanding of why I beams are shaped the way they are, and why you don’t make them out of concrete, and why driveshafts are hollow instead of solid.

There are a ton of potential Aha Moments in the STEM business. It’s a shame that problems presented like this stomp them into misery.

## Dan Meyer

December 11, 2013 - 10:23 am -Kenneth TiltonThis is the line of thinking that separates a lousy, boring instance of this problem from a productive, engaging one.

## Jason Dyer

December 11, 2013 - 3:07 pm -I’m more interested in: how do we fix this problem?

Here’s a starting go: why is it you (sometimes) can put your hand in a fan without losing any fingers?

In what situations could you lose fingers?

What if the fan was in the shape of an I-beam?

Something like that?

## gasstationwithoutpumps

December 11, 2013 - 8:04 pm -I agree that it is possible to make even real-world problems dull and boring. I designed a circuits course for bioengineering majors last year, because the one EE taught was incredibly dull and boring, largely due to fake-world problems like the one you described (though from EE not from structural engineering).

It is possible to add the trappings of a real-world problem to a fake one, but that doesn’t make it a real problem. Getting the trappings wrong makes it more obviously not a real-world problem than getting the trappings right. (For a “real-world” version of what you started form, one could do a boring problem about the stiffness or strength of an I-beam, which would be the sort of thing a structural engineer might have to calculate. It would still be boring as hell, unless you really were trying to figure out how to fix a sagging bridge.)

For my circuits class, I did not design the class around the math or the “concepts” I wanted them to learn. Instead I selected the 10 design projects they would do, and figured out what I would need to teach them so that they could actually do those projects. I had to go through a lot more than 10 projects myself to figure out which ones were too easy, which ones too hard, which ones too expensive (none are too cheap for me!), and which ones would require too many concepts outside the scope of what I wanted to teach.

The key was ensure that everything the students were taught in the course was something that they needed right then, in order to do the design task that they were facing. I covered only about 60% of the math that the standard circuits course covers (it is normally taught like an applied math course, with almost no attempt to do any engineering), but my students at the end of the course were able to design simple amplifiers for sensors relevant to bioengineers (like an EKG), and *knew* that they could do it, because they had done it. Students coming out of the standard circuits course can pass a tougher math test (for a few weeks anyway), but need to take more courses before they can design anything.

I’m drifting way off topic here—the point I was trying to make is that real-world topics can be a powerful incentive for learning, if they empower the students to do things that they care about. But the empowerment must be immediate, not “someday you’ll want to know this”.

## Jane Taylor

December 12, 2013 - 12:44 pm -We just finished a unit in which we collected data and fit different functions (linear, power, exponential, etc) to the data to find the best model. It isn’t really the breaking point that is the engaging part, I guess, but maybe the process of collecting data in order to create a function that described the relationship between weight applied and displacement of the board for various widths or positions of the board. The stated problem gives them the function. Could they develop their own function? In this case, maybe not. The relationship is fairly complex. But that is what I was really thinking.

## Brian Miller

December 12, 2013 - 2:45 pm -The problem you cited doesn’t seem real world to me because it does not give any mention of why someone in the real-world would do such a calculation. The teacher giving this problem would have to add that piece of information I suppose.

Here is a real-world scenario that is “real-world” in the sense that my fiance had to do this calculation by hand the other week during her job as a project manager at a green building firm:

She had one site that was 3/4 an acre and a price from that site for $54,000 for some work. Then she had another site that was 2.5 acres and needed to know how much that same work would be for the larger site.

In fact, what she ended really needing to know was how much 1 acre was worth, so she could scale it to all her other jobs. To scale it she had initially multiplied 54,000 by 1.25 and that didn’t work and she wanted to know why it didn’t work. I think that could be a good question for a student.

Real-world guarantees nothing, but teachers being able to express the reason someone needs the “real-world” math is a factor worth looking at.

## Dan Meyer

December 12, 2013 - 6:54 pm -Jane Taylor:You’ve identified an enormous part of the problem with the task as written: the model is pre-determined and we don’t know

howit was determined. It contributes to a student’s sense that math is a neverending black box.## Chris Painter

December 13, 2013 - 12:30 pm -It seems like the common thread throughout most comments is that students need to have a chance to interact with the situation and the mathematics involved. It does no good to throw predetermined mathematics at them and ask them to work with the relationship in a simplistic way.

This does remind me however, of a previous post as it seems connected (and also connected to a post on Michael Perhsan’s blog). I wonder if one of the significant flaws of fake world math problems is that they are, at their heart, easy. All we are asking students to do is plug in values and calculate… not very challenging just time consuming. Side note: I do acknowledge that making sense of the diagram (reading the problem if you will) is challenging but that doesn’t mean the mathematics and problem solving are challenging also.

## Kenneth Tilton

December 13, 2013 - 1:54 pm -This thread reminded me of the day I sat down with a spreadsheet and started building a model of my small business.

Let’s see, a direct mailing of 20k is this much postage, the list of addresses costs that much, the printing and mailing, then the orders come in: the increased wages to my part-time fulfillment helper, the revenue, my estimated income taxes, the step-function in which I have to re-order inventory 2,000 boxes at a time… I was impressed by the size of the sheet when I got done.

Those are pretty simple formulas but a bunch of them and expressing them all would be a good exercise — self-corrected by the spreadsheet software running the calculations for them.

The algebra is not so hard, but the real-world thing is clear and students are generating the formulas as long as the description is word-problemish and does not give away the formula.

jes thinkin out loud.

## Jason Dyer

December 13, 2013 - 3:00 pm -@Kenneth: My first year teaching I did some teaching algebra via spreadsheets.

One of the students, realizing that once the formulas are put in the calculations happen automatically when numbers are changed, said: “Now I get it! Algebra is like cheating for math!”

## Roberto Catanuto

December 14, 2013 - 12:57 pm -I generally agree that “real-world” doesn’t imply “meaningful” all the time, not most of the time either.

I now wonder (and ask for inspirations):

– is it possible to give an estimate of how much a topic, or a single problem is “meaningful” for a commonly-not-engaged-in-math-teenager-student?

– if so, how?

– is it possible to set about a way to connect Math to students personal interests? We speak about Math all the time, but rarely about students.

We teach students much more that we teach Math.

Thank you for your comments.

## Kenneth Tilton

December 14, 2013 - 1:43 pm -@jason Your student had a way with words. Should go far in blogging!

@robert I have been thinking about this and I think we can easily calculate how meaningful most math is to students’ everyday lives: zero.

Even worse, I have another measure for you: if we somehow get them 60% excited about the relevance of math, how much better will they do at simplifying x^2+x-2 over x-1? Kersplat goes the enthusiasm.

When we answer the relevance question we are setting ourselves (and the students) up for failure.

I like Arthur Benjamin’s “math is cool” message as one answer to “why math?”: http://youtu.be/SjSHVDfXHQ4

But the kids won’t fall for that either. :) The biggest win will be when we get better at teaching pure math, by which I mean students get better at learning it. When all students experience the fun of math that many of us experienced, they will not even ask why they have to learn it.

You are right to bring the students back into the discussion. The real problem is that many of them are not learning a very learnable subject: pure algebra. We should be working on that.

## Roberto Catanuto

February 23, 2014 - 8:00 am -The point of “real-worldliness” is something I strongly agree upon.

It is not always a guarantee for the effectiveness of Math’s learning. Thanks again for raising this debate.

I just want to add that this issue is known way back to the 19th century. For example, from the revolutionary approach to Geometry teaching of a French mathematician (Alexis Claude Clairaut, 1871).

From one of his greatest works – Elements of Geometry – he pointed out that teaching Geometry to young students beginning with highly abstract concepts is too arid a way to invite and inspire them into the beauty of Math.

After that, if you simply infuse your lessons with possible practical uses of those abstract concepts in the ”real-world”, you just get students bored about the Theory and confused about the Practice.

His work has been widely cited by other great and innovative Math educators as E. Castelnuovo, who’s been in turn cited and appreciated many times by C. Gattegno.

Robert.

## David Srebnick

August 27, 2014 - 2:15 am -Most of the time when I see the words “real world problem” in print it means “problems with concrete, familiar objects,” or perhaps, “problems with easy-to-understand, concrete models, lots of information given, and a direct path to solution.” The kind of problems that only occur in textbooks.

I wholeheartedly agree that solving these types of problems requires little if any mathematical understanding beyond the mechanics of solving the equation, and is certainly not the type of problem that one would have to solve in a real-life context.

And I agree that regardless of the amount of information that is given to a student, the problem should at least be something that an engineer or mathematician would be interested in.

My question is, does a well defined, concrete, “unreal-world” problem have any value? Do you think that it is helpful for some students to work through a few problems like this in order to help them understand how the math relates to a physical model?