Inverse problems in education

Forward problems are problems with a well-defined answer: throwing a fair die, what’s the probability of getting a 4? Inverse problems look at data and create what necessarily has uncertainty: looking at this data, was it generated with a fair die?

Most problems given in mathematics classes (outside of statistics) are forward problems, with well-defined answers. Yet, most real-life problems are inverse problems. We don’t know the actual equations of the world, and even if we did, our measurement of reality would have uncertainty.

Pure mathematics is important, but I maintain complete allergy to error is unhealthy and gives a distorted view of mathematics. Consider, for instance —

[Vimeo link]

This is my favorite of Dan Meyer’s videos.

If you go through the calculations correctly for working out how long it takes Dan to get up the stairs, the answer comes short by about a second and a half.

Yet, this is still a perfectly valid problem. Where did the extra time come from? This is a useful discussion and matches the sorts of discussions scientists and engineers have often.

(Note the long step, which would naturally extend the time slightly.)

There’s also inverse problems where nobody could truly know the answer (but we can get a pretty good idea anyway with mathematics). I’ve mentioned previously my favorite problem from teaching statistics:

Based solely on the number of wrecks, is there anything mystical going on in the Bermuda Triangle?

By its very nature, “is anything mystical going on?” is a unfalsifiable claim, hence the problem is necessarily an inverse one. The students used a shipwreck database to decide if the number of wrecks in the area is abnormally high. (They found it was safer inside the Bermuda Triangle than outside it.)

The teacher can also manufacture an inverse problem where the teacher knows the answer but the students are not given enough to make a truly definitive answer.

For example, here are two excerpts from 19th century American humorists:

Under favorable circumstances the Roller-Towel House would no doubt be thoroughly refitted and refurnished throughout. The little writing-table in each room would have its legs reglued, new wicks would be inserted in the kerosene lamps, the stairs would be dazzled over with soft soap, and the teeth in the comb down in the wash-room would be reset and filled. Numerous changes would be made in the corps de ballet also. The large-handed chambermaid, with the cow-catcher teeth and the red Brazil-nut of hair on the back of her head, would be sent down in the dining-room to recite that little rhetorical burst so often rendered by the elocutionist of the dining-room—the smart Aleckutionist, in the language of the poet, beginning: “Bfsteakprkstk’ncoldts,” with a falling inflection that sticks its head into the bosom of the earth and gives its tail a tremolo movement in the air.
On receipt of $5 from each one of the traveling men of the union new hinges would be put into the slippery-elm towels; the pink soap would be revarnished; the different kinds of meat on the table will have tags on them, stating in plain words what kinds of meat they are so that guests will not be forced to take the word of servant or to rely on their own judgement; fresh vinegar with a sour taste to it, and without microbes, will be put in the cruets; the old and useless cockroaches will be discharged; and the latest and most approved adjuncts of hotel life will be adopted.

On the fourth night temptation came, and I was not strong enough to resist. When I had gazed at the disk awhile I pretended to be sleepy, and began to nod. Straightway came the professor and made passes over my head and down my body and legs and arms, finishing each pass with a snap of his fingers in the air, to discharge the surplus electricity; then he began to “draw” me with the disk, holding it in his fingers and telling me I could not take my eyes off it, try as I might; so I rose slowly, bent and gazing, and followed that disk all over the place, just as I had seen the others do. Then I was put through the other paces. Upon suggestion I fled from snakes; passed buckets at a fire; became excited over hot steamboat-races; made love to imaginary girls and kissed them; fished from the platform and landed mud-cats that outweighed me—and so on, all the customary marvels. But not in the customary way. I was cautious at first, and watchful, being afraid the professor would discover that I was an impostor and drive me from the platform in disgrace; but as soon as I realized that I was not in danger, I set myself the task of terminating Hicks’s usefulness as a subject, and of usurping his place.
It was a sufficiently easy task. Hicks was born honest; I, without that incumbrance—so some people said. Hicks saw what he saw, and reported accordingly; I saw more than was visible, and added to it such details as could help. Hicks had no imagination, I had a double supply. He was born calm, I was born excited. No vision could start a rapture in him, and he was constipated as to language, anyway; but if I saw a vision I emptied the dictionary onto it and lost the remnant of my mind into the bargain.

Which one is Mark Twain? I gave another known Mark Twain excerpt to the students and had them do statistical analysis to justify their answer as A or B.

It’s a messy and “impure” problem and even can be partly reckoned with via English class skills. Statistics deals with such worries all the time, yet many American students never see such a problem until possibly their senior year and often not until college.

Even ignoring statistics and just considering modeling problems like the first one, mathematics teachers seem deeply uncomfortable with the possibility of error. Mathematics is only infallible when contained within its own world.

(Incidentally, there’s a letter by a “Donald Ross” that some people think is by Mark Twain: Things a Scotsman Wants to Know. It qualifies for the category of “inverse problems nobody will know the answer to unless someone builds a time machine”.)

11 Responses

  1. Maybe it is my background in physics, but working out the stairs problem IS correctly identifying the pieces involved and then solving those pieces. When you say “direct” problems I think you are referring to what are called “template” problems. The stairs problem would be a compound problem that involves more than one piece yet the pieces are each template problems and are solved using prior art.

    I generally categorize problems as either simple (template) or compound (stairs) and as requiring either prior art or original art. Drill and exercises are generally simple / prior art while contextual problems are generally compound / prior art. Really good (hard) problems can be either simple or compound, but the critical factor is that they require original art. The AMC exam for example involves a lot of original art. Classic honor’s texts generally have a large number of word problems and these will cover these categories. Lower level texts unfortunately stay in the shallow end of the pool and stick with exercises (simple and prior art).

    The issue I have found with “Mark Twain” problems is that teachers provide so much assistance that it is like holding someone’s fingers and guiding them through a piano piece by Schumann. Obviously, such an exercise, while maybe engaging, instills little if any instrumental acumen. Imo, it takes a lot of preparation by the student, prior to the Mark Twain problem, to gain acumen from the Mark Twain problem.

    Bob Hansen

    • There’s lots of axes to categorize problems, I’m sure. We need (or probably there already is) a taxonomy somehow. The fact there’s not enough information in the stair problem to get an exact number is what makes it an inverse problem.

      I’ve noticed a lot of word problems also stay in the shallow end of the pool by using a very standardized setup which becomes predictable. (Another reason I like the stair problem is even though the setup is quite logical it doesn’t work mathematically like most of the boat/river problems I’ve come across.)

      The Mark Twain problem required very little intervention from me (all of the students — not very experienced ones mathematically, I should note — figured it out correctly; not all used the same metrics).

  2. The main area where I’ve seen this done with students at early ages is with games, particularly NIM. This falls into the later category though of situations where the teacher knows the answer and the students are trying to figure it out. But there are other games that are not as simple that we can come up with that exploit mathematical thinking and uncertainty.

    An area that I’d like to see more of at early ages is Operations Research, particularly Linear Programming. Some of those problems, like the diet problem or the farmer’s problem are well studied and a lot more is known, but there are so many other problems and the tools used in this type of work require mainly just something like Gaussian Elimination and the understanding of inequalities.

    • I thought most games of NIM had a set method of solution? The Mark Twain example is one where even a “perfect” mathematician (with the given information) would not be able to absolutely say for 100% which text is the correct one. That’s what makes it an inverse problem.

      • True, but when I gave this talk last month students were wondering about playing NIM-variants, many of which I had never studied and could only encourage them to play the games, analyze the results and see if you see any patterns.

        But relating this back to the real world problems. I like the NIM example because, although the variants may be more difficult than the NIM problems, the NIM game itself gives a starting position to begin thinking about the problem. Likewise in many real world problems, the problem isn’t just dropped from the sky. Generally there’s a story behind where the problem comes up and related literature which can prove very helpful in solving these new problems.

        So in your Twain example, although I may never be able to know with certainty, I’d probably start by doing some Text Analytics on both documents, as well as Twain’s work. Maybe TD-IDF, word frequency and the likes.

        Generally my main goal(s) in doing this type of stuff is to introduce students to non-standard classroom mathematics, ala the type of math that I fell in love with in college, as well as to show students how to solve “real world” problems with mathematics. The phrase “real world’ is debatable, but once they learn the process we can replace “NIM games” with “topic modeling” or “association rules”. I just find that games are generally able to hold the attention of students for a longer period of time than say a problem dealing with the fairness of airline scheduling.

  3. I think you have a fair point here, though I’ve always loved the pure, perfect, self-contained aspect of maths where you KNOW when you have the right answer, because that is not the type of maths that the vast majority of people will use in their lives. They will use the messier application of it on real data. It reminds me of Dan Meyer’s talk about how useful a discussion it is to talk about sources of error in their working.

    I’m also a little disappointed because I’ve been really excited for weeks when I invented a lesson on analysing writing styles (here: ), but it turns out you’ve got there before me! Mine is slightly different in that it kind of works backwards from your Mark Twain lesson, so I guess I’ll take that at least.

    Love the Bermuda triangle idea btw.

  4. Have you written up the shipwreck task somewhere? (Sent it into IMP?) It seems there are a lot of possibilities depending on the format of the shipwreck data (latitude and longitude, I’d assume) which seems locked off to the public. Provocative.

    • We desperately need more stats problem for the IMP, but has legal restrictions on data use in a way that doesn’t encourage me to send a lesson.

      My one map-based Illustrative Mathematics lesson (The Longest Walk) is incidentally not an inverse problem, but works like one in practice. For the purposes of the target audience they would need more mathematical knowledge and be able to program a computer to obtain the correct answer.

  5. Reblogged this on evangelizing the [digital] natives and commented:

  6. […] Forward problems are problems with a well-defined answer: throwing a fair die, what's the probability of getting a 4? Inverse problems look at data and create what necessarily has uncertainty: look…  […]

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