by Mark Saul
Maybe it is obvious, but it is something I’ve come to appreciate only after years of experience: mathematics is logic driven, and teaching and learning mathematics is centered on teaching and learning logic. I find this to be true philosophically, but also practically, in my teaching. And even in my own learning.
Philosophically, this point of view has deep roots. Plato’s Academy. Russell and Whitehead. Frege, Tarski. And that’s all I want to say about this area, which is outside my expertise. I leave it to those who think more deeply about the philosophy of logic to forge connections between my experience and their work. I think it is probably enough here to think about the ‘logic’ as concerning just the simplest propositional calculus: implication, negation, and perhaps quantifiers.
Because what I want to say is that in my teaching, the closer I look at difficulties that students have the more likely it is that the difficulty is with these basic aspects of logic. And (conversely!) if students leave my classroom having understood these logical connectives more robustly, I consider that I have succeeded.
Okay. “Deep learning”. (In education, this phrase as a more general and less technical meaning than its use in computer science.) For me, this has a particular and specific mathematical meaning. It refers to learning based on logic, on the connections among statements. Which, I think of as coextensive with mathematics itself.
If we take this point of view, the whole landscape of mathematics is laid out before us, as from the top of a mountain. Too, this view resolves many disputes about the relative importance of skills vs. concepts, etc.
That is: if we are connecting statements, we are doing mathematics. If we are not connecting statements, we are not doing mathematics. We are doing something else. And the statements do not have to be about number or length or functions. Those are the objects on which logic acts in a mathematics classroom.
Of course, logic acts on other objects in other classrooms. We make arguments and build logical structures in studying chemistry, in reading literature, in learning a new language. But the mathematics classroom is the place where we focus directly on these activities, where logic is most quickly and most accurately developed.
As I have noted, this view of course has philosophical roots stretching back to antiquity. More recently, it is the view of Bertrand Russell: “Pure mathematics is the class of all propositions of the form ‘p implies q’…” (see https://todayinsci.com/R/Russell_Bertrand/RussellBertrand-Mathematics-Quotations.htm) . But it also has very practical applications to pedagogy. If a student is struggling, it is logic he is probably struggling with. If you untie the logical knot, lay out the train of thought—particularly of implications—that leads one to the actions taken to solve a problem, then the student will understand and be able to work the problem.
Of course, by ‘lay out the train of thought’ I do not mean ‘give a lucid explanation’. I mean get the student to construct the chain of implications in his or her head. For some (usually graduate) students this may mean giving a very clear lecture. For other populations, it means ‘guide on the side’. I am not claiming that this meaning of ‘deep learning’ implies a particular pedagogy. But it sets a standard for the success of any pedagogy.
I can be even more specific. Russell’s definition of mathematics points to the center of learning of logic: the notion of implication. If my students, after graduating from high school, really understand what it means for one statement to imply another, have been trained to look for such implications, and can judge whether the implication is valid or not—if they can do all that, I don’t care if they know the formula for sin (x+y) or how to measure an inscribed angle. Or even how to perform long division. As Underwood Dudley has provocatively shown us (https://doi.org/10.1080/07468342.1997.11973890), the claim to practicality of mathematics (in the sense of specific mathematical results) is often exaggerated.
But the importance of mathematics, seen as the study of implications, cannot be exaggerated. It is a characteristic of our species. It is what has led us to dominate our environment. It has also led to some incredibly inhuman events. I leave to more serious philosophers to decide whether the phenomenon of human reason is ‘good’ or ‘bad’—or neither. The point is, it is profoundly human.
In making this statement, I disagree with the view that we must ‘humanize’ or ‘re-humanize’ mathematics. Mathematics is, almost by definition, human. It is its uses, and its teaching, whose humanity we must examine.
To be even more precise, and even technical: the definition of implication rests on the distinction between a statement and its converse. So I can go still further in my wild claims to know if I’ve succeeded. If a student, five years after graduation, can distinguish a statement from its converse, in even the most bewildering of logical environments, then I have succeeded with him or her. Don’t think this is so easy: I have caught important mathematicians, or they have caught themselves, confounding a statement with its converse. And of course I have caught myself.
I am not asserting that if you know about the converse then you know mathematics. I am asserting that if you don’t know about the converse, then you do not know mathematics. Or, less aggressively: if you mistake a statement for its converse you are making an error in mathematics.
So, for example in geometry we often teach about the classification of quadrilaterals: trapezoid, parallelogram, rectangle, etc. Students will often say things like: “If we know a parallelogram has equal diagonals, then we know it is a rectangle… or it could be a square.” Venn diagrams, illustrating set inclusion, can certainly help untangle the confusion. But there is also a deeper lesson to draw from this error, one that transfers to, and taps into, other experiences. This deeper lesson emerges in phrasing the statement in canonical ‘if..then’ form: if a parallelogram is a square, then it is a rectangle. But if it is a rectangle, it may or may not be a square.
I find this an important guiding principle in pedagogy at all levels. Even when we teach young children with hands-on tactile experiences, what we are teaching them is about objects which will, sooner or later, be objects subject to reason. For me, this resolves the endless debates about mechanical skills, about fluency or automaticity. And it resolves it in two ways. First, the object of fluency is to be able to reason more easily—more fluently, if you like. When do you concentrate on fluency (‘drill and kill’)? When lack of fluency gets in the way of reasoning. And when do you reach for the calculator? When doing without it will derail your train of thought.
The second way a focus on logic resolves issues about fluency is more directly pedagogical: fluency is best acquired by making logical connections among statements. For example, if a child knows that 8+8 equals 16, she doesn’t have to memorize that 7+9 also equals 16 or that 8+9 equals 17, or that 80 + 80 = 160 or that…
This is the meaning I take from Liping Ma’s “knowledge packages” (1999). I have written elsewhere about how I think her very useful work can be given more meaning (https://www.ams.org/journals/notices/201405/rnoti-p504.pdf). That article was one step towards the view I express here, which I’ve come to only aftera decades of experience. To some readers it may be perfectly obvious, and to others perfectly ridiculous. I would be interested in hearing both reactions.
Liping Ma, Knowing and Teaching Elementary Mathematics, Lawrence Erlbaum Assoc. Inc., Mahwah, New Jersey; London, 1999.