Connecting the Dots Backwards: Not Knowing What We Don’t Know, Part 2
Note from the author: This article is based on a previous article published online by Math Horizons magazine. That version can be accessed here: https://www.maa.org/sites/default/files/pdf/Mathhorizons/supplement/MH-CoreyWeb.pdf
If you recall from part 1, I had a son that wanted to engineer a fantastic body suit, but he didn’t want to go to college to be an engineer because he would have to learn a lot of information that he wouldn’t need to create the suit. Suppose he took my advice, based on my arguments in part 1, that he should go and get an engineering degree. I guarantee you that he would be coming back to me constantly saying that I was wrong. Why? Because he could simply take ideas he is learning in his engineering classes and shove them in my face and say “Dad, when am I ever going to use this?” And, I would have to sit there without a satisfactory answer.
Don’t get me wrong, I would still stick by my advice that he should get a degree. The problem is something else. I teach mathematics and more than a few times have students asked this question in class. Students wonder about this question a lot. I began typing in this question into a search engine bar and it showed ten popular completions for “when am I ever going to use . . .” and every single one of them dealt with school mathematics. But billions of people use mathematics everyday, arithmetic to multivariable calculus. This has become a paradox that I have spent time thinking about – math is certainly useful, yet why is it so hard to explain or show students how it can be useful to them?
A Thought Experiment
Take a moment and think about the last time you used multiplication outside of a school environment. If you can’t remember the last time, try to recall at least some time recently where you used multiplication. It doesn’t have to be a case where you actually wrote down a multiplication problem or worked it out on paper. I am sure that you use the idea of multiplication with its closely related mathematical cousins of finding areas, counting things in groups or arrays, scaling things up or down, or working with proportions. It has just become a part of our natural thinking that we don’t consciously think “OK, this is a multiplication problem, how was I taught to do this in school?”
This may be the first piece of understanding our paradox. Math is useful, and in actual use, the large majority of the use is done mentally and much of that subconsciously. When you look at a graph, a number, a formula, a chart, a quantitative situation, or anything of the sort, you draw on a myriad of mathematical connections made during hours upon hours of math classes and math homework (and other situations) to immediately make meaning (or begin making meaning) of what you are experiencing. Rarely could you actually go back and figure out when you actually learned the skills, even one skill, you are using in the moment of making sense of anything, not just mathematics. For example, consider: Where did you learn to think the thought(s) you had after reading the last paragraph? It is nearly impossible to go back and find the moments you learned specific skills, especially those that create the schema which we use to make sense of the world. The result is often a loss of credit to much deserving teachers who have helped students learn, simply because the source of the knowledge which they use becomes nearly intractable.
Similarly, much of the skill that is used by an expert engineer is something that would be hard to write down, and if you do write it down then it is hard to learn/teach. Educators have struggled mightily to help students become good problem solvers, that is, help students be able to solve problems (in whatever field) without being told how to solve them beforehand, especially where some creativity is involved. We can’t just explain to a student how to be creative, or how to be a good problem solver and then suddenly they have the skills and cognitive tools to be expert in those areas.
Taking the Thought Experiment Further
Let’s get back to our thought experiment. When I first did this experiment I thought of two recent situations, calculating the area of a raspberry patch and figuring out if I had picked up enough cans of cream of mushroom soup at the grocery store. Yesterday I asked a couple other people to do this same experiment and they reported using multiplication to figure out if a box of diapers would last through the month and to estimate if their favorite baseball player would end the season with 100 RBI’s. Now consider the following possible conversation:
Student: When will I ever use multiplication?
Teacher: I used it just the other day to calculate the area of my raspberry patch.
Student: Yeah, like I am ever going to be doing that in my life time.
Teacher: Well I had a friend who just told me yesterday that he used multiplication to figure out if his favorite baseball player might get one hundred RBI’s by the end of the season?
Student: What the heck is an RBI? I don’t even like baseball.
Teacher: Well you can use it when you are shopping. You could use it to figure out if the package of diapers you are considering buying will last you until your next pay check?
Student: Diapers? Are you serious? If having kids means I will be using math then I am not going to have kids.
Teacher: (under their breath) I knew I should have been a doctor.
OK, I am really just kidding about that last line, but you can see how frustrating it might be trying to convince students by such a technique, and remember, this is about a topic that is used extensively with actual recent uses by real people.
Application problems may convince students that math is NOT useful rather than the alternative. Any application problem that a teacher picks will be outside the interest and field of almost all students, thus showing one more piece of evidence that they will not use that particular topic anytime in their lifetime. I call this the paradox of application. This becomes the second insight into the larger paradox we are trying to understand: applications so often involve such specific contexts that they miss the reality of almost all students.
The Eye of the Mind
Our current knowledge might impact us the most through what we are able to “see.” What we are able to “see” has far reaching effects on our experience in life.
I have a colleague who had a friend that was a mathematics professor. This professor had a tropical fish aquarium. The aquarium must be kept at a certain temperature for the fish to survive, so special light bulbs are used to heat the water. His light went out and he needed to travel to a nearby town to get a new light. During the fifteen minute drive he set up a differential equation based on the size of his aquarium to calculate the correct wattage of the bulb he needed to maintain proper water temperature. He had the differential equation solved by the time he got to the store so he was able to get the right bulb. I could not do that and you could probably not do that. But we didn’t know enough mathematics to even recognize that finding the wattage in this way was even a possibility. Most of us would not have even “seen” the mathematics problem.
I know a doctor that has a strong background in mathematics. As a practicing physician he said that he uses the ideas of limit, derivative, and integral every day in his work (although he rarely writes out a problem to solve, he uses the ideas to reason). He uses them time and time again as new situations arise where he uses these ideas to make sense of what is going on to help him make decisions about diagnoses and treatments. These core concepts of calculus are invaluable to his work. This is not unique to this doctor. I met another doctor that expressed a similar sentiment.
Many other doctors who are not fluent in these ideas do not use these valuable tools. They cannot use them for they are not theirs to use. Yet, according to my doctor friend many of them think that calculus was a waste of time. When he tells them that calculus would help them to overcome some of the struggles he sees they are having, they don’t believe him because they don’t “see” any calculus problems in their work.
When I see one of my sons put on their shirt inside out and backwards I think of the structure of the symmetry group generated by the actions on his shirt. When I am playing on the trampoline with my kids and we have balls on the trampoline I think how the paths of the balls could be modeled by hyperbolic geometry and how the model is also used in applying the special theory of relativity to the paths of light across long distances.
It doesn’t go the other way! People don’t stand on the trampoline and ask themselves what connections this has to hyperbolic geometry. They don’t see a ball roll across a trampoline and ask themselves if it has anything to do with the way that light travels through the solar system. They don’t usually think about studying abstract algebra concepts when they see an inside out shirt. They can’t “see” these connections so they are non-existent to them. Yet these are the same people that say that advanced math has no connection to real life.
By getting an engineering degree my son would be expanding his ability to see the world and his work in different ways; ways that would be incredibly helpful in designing his body suit. But just because I can’t tell him any of the specifics on how this might happen, should he stop studying?
Conclusion
What we learn, what we really learn and understand, impacts our lives in ways that we don’t fully realize. All of our previous knowledge, previous experience, and previous thoughts, impact what we currently think. Consider this statement which agrees with an important point made by modern cognitive science:
“Every thought that one permits through his [or her] mind leaves its trace. Thoughts are things. Our lives are governed a great deal by our thoughts.”
What is learning but the gradual molding of our mind, heart, and hands by these thoughts and experiences which leave their impression to make us something better and more capable than we were before?
Editor’s note: This is part two of a two part series. Click here for part 1.
