My good friend Erik Postma (@postmath) sent me this tweet the other day.
— Erik Postma (@postmath) October 17, 2016
This is a ‘classic’ problem about making sense with mathematics and highlights a problem with math education, with the general arguments that there is too much emphasis on rote calculation and working through canned problems and not enough emphasis on sense-making. If you’ve watched the video and / or read the article by the eloquent Junaid Mubeen (@fjmubeen) you’ll understand that this is all types of silliness. To that, I yield. Yes, there are problems with math education and videos like these bring to the surface the gap between reasoning and mechanics. I want to make some other arguments, however.
Every Subject …
Every subject suffers from the problem of “sense-making”. We struggle with this in History class — teach dates and events for memorization or explore why things played out the way the did? A lot of students’ experience with history has been about solely and soullessly memorizing dates.
We struggle with this in Writing class — the Four Paragraph Essay offers a logical template for organizing an essay, but it’s a guideline not a rule. Unfortunately, many students are taught to write in this formulaic way with “why” having been lost.
We struggle with this when spelling common words — its vs it’s; your vs you’re; ’tis vs t’is. We can either commit to memory the spelling and the use case or we can understand why the apostrophe is where it is.
We struggle with this in the three main sciences: Biology, Chemistry, and Physics. Biology class turns into endless memorization. Stoichiometry (a staple of high school Chemistry) is a horrible mess for students. And Physics? Ha! Ask any physics teacher / professor about the insane answers students give — negative lengths, objects larger than the universe, cars travelling faster than light, etc.
We struggle with this in personal finance. I’ve taught personal finance units and let me tell you, there is plenty of not-sense-making. You are offered a job. You can choose your salary: $25/hr or an annual salary of $60,000, which do you take assuming that a normal work week is 40 hours? Do you know how many students have responded with $25/hr because “that’s a lot per hour and you can get overtime”? This does not make economic sense!
This idea of a lack of sense-making is not particular to math. If I had to guess, Physics classes probably have a higher incidence of a lack of sense-making than Math classes. We can argue that in the cases of the sciences or personal finance the stumbling block is on the nonsensical math performed. But is it really a math problem or is it a reading comprehension problem? How do we want to dice and categorize this?
Broadly, we can say that it’s a problem with general education. Pinning this on math education is a type of populism. We pick on math because that’s the subject that students have to take no matter what from an early age. Hence, we have more crazy stories we can tell that can easily resonate with the masses.
So my first point here is that if there is a problem, it’s a general problem with education and not necessarily an idiosyncrasy with math education.
Now, on to the second point.
The Trust and Social Conditioning Angle
Students, generally, trust the teacher in the role of educator. As students grow up and go through the school system, formal or not, there tends to be a bias towards solving problems. That is, problems have solutions. The idea of “no solution” doesn’t really show up as a regular theme in math courses until the introduction to system of equations. Within the sciences, there is a cultural bias in the apotheosis of inventors — they did something that nobody thought was possible! And outside of the sciences, there is enough freedom of expression that the idea of not being able to do something is not an artifact of the discipline but more an oppression by the cultural rule makers. And yet, we laud the defiance of Picasso, as a for instance.
So in this light, when asked the question about the shepherd’s age, there is an instinctive bias towards trust — trust in a solution that is not “no solution”.
There’s also a meta-social conditioning on problem solving that I would argue leads to students wanting a solution. We like to ask kids brain teasers and trick questions. Continuing with my armchair psychology, maybe we do this to help develop “lateral thinking skills”. Maybe we do this to help train kids about what implicit assumptions they are making. Brain teasers and trick questions sometimes come require the person answering to deduce the hidden information in the question or to invoke some extraneous information that solves the problem — Deus ex machina. So what reflex do we expect when faced with an unsolvable problem or one with not enough or irrelevant information? I would argue that it’s within reason for a student to try to inject extra information to come up with a solution. And that extra information can come in the form of a nonsense equation / formula.
That some students do not fall for this and simply say “there’s not enough information” is evidence that critical thinking is being taught and learned. Maybe this is happening more regularly at home or maybe kids develop critical thinking skills at different points in their lives, maybe both.
At the heart of all this is trust. Students, generally, trust their teacher. Students trust that they are being given reasonable tasks, where “reasonable” comes in the form of solvable. And a question like this one about the shepherd’s age is exploiting an implicit trust factor. Maybe they need to be disavowed of their trust.
But when and how?
I’m not sure what the consequence would be of arming a child with “no solution” as a viable, regular answer to questions. Maybe there is no harm. Or maybe, to excuse themselves from a problem, they invoke “there is no solution” and wash their hands of it. Maybe that’s me being crazy.
Still, there is another problem — proving that no solution exists is not trivial. And in mathematics, one of the first places we expose (all) students to “no solution” is in the light introduction to linear algebra by way of systems of equations. The linear algebra context is purely a mathematical one without any other type of context (ie a word problem about the shepherd’s age).
The inability to find a solution doesn’t mean that a solution doesn’t exist. This is what makes it difficult to ask unsolvable problems. How do I know I can stop searching for a solution? Why can’t I deduce the shepherd’s age from the information given?
As I said, I grant that math education can be done better — I mean, this is mathmisery.com. But screwball answers to the shepherd’s age problem is less damning of math education as it is a general surfacing of a gap in reasoning skills, which seems to have been deemed as a responsibility of math education solely, wholly. I also maintain that we should consider the social aspects surrounding a child’s education. Many students grow up in a trusting environment with their teachers as well as an environment of “anything is possible” / “you can be anything”. A “no solution” problem is antithetical to the can-do messages that students receive regularly and routinely.
Students should learn the possibility of no-possible-solution problems, but in a way that doesn’t betray their trust in educators nor in a way that negatively impacts their optimism on tackling the impossible.