by Mark Saul<\/p>\n
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.\u00a0 I find this to be true philosophically, but also practically, in my teaching.\u00a0 And even in my own learning.<\/p>\n
Philosophically, this point of view has deep roots.\u00a0 Plato’s Academy.\u00a0 Russell and Whitehead.\u00a0 Frege, Tarski.\u00a0 And that’s all I want to say about this area, which is outside my expertise.\u00a0 I leave it to those who think more deeply about the philosophy of logic to forge connections between my experience and their work.\u00a0 I think it is probably enough here to think about the ‘logic’ as concerning just the simplest propositional calculus: implication, negation, and perhaps quantifiers.<\/p>\n
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.\u00a0 And (conversely!) if students leave my classroom having understood these logical connectives more robustly, I consider that I have succeeded.<\/p>\n
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Okay. \u00a0\u201cDeep learning\u201d.\u00a0 \u00a0(In education, this phrase as a more general and less technical meaning than its use in computer science.) \u00a0For me, this has a particular and specific mathematical meaning.\u00a0 It refers to learning based on logic, on the connections among statements.\u00a0 Which, I think of as coextensive with mathematics itself.<\/p>\n
If we take this point of view, the whole landscape of mathematics is laid out before us, as from the top of a mountain.\u00a0 Too, this view resolves many disputes about the relative importance of skills vs. concepts, etc.<\/p>\n
That is: if we are connecting statements, we are doing mathematics.\u00a0 If we are not connecting statements, we are not doing mathematics.\u00a0 We are doing something else.\u00a0 And the statements do not have to be about number or length or functions.\u00a0 Those are the objects on which logic acts in a mathematics classroom.<\/p>\n
Of course, logic acts on other objects in other classrooms.\u00a0 We make arguments and build logical structures in studying chemistry, in reading literature, in learning a new language.\u00a0 But the mathematics classroom is the place where we focus directly on these activities, \u00a0where logic is most quickly and most accurately developed.<\/p>\n
As I have noted, this view of course has philosophical roots stretching back to antiquity.\u00a0 More recently, it is the view of Bertrand Russell: “Pure mathematics is the class of all propositions of the form ‘p implies q’\u2026” (see https:\/\/todayinsci.com\/R\/Russell_Bertrand\/RussellBertrand-Mathematics-Quotations.htm<\/a>) .\u00a0 But it also has very practical applications to pedagogy.\u00a0 If a student is struggling, it is logic he is probably struggling with.\u00a0 If you untie the logical knot, lay out the train of thought\u2014particularly of implications\u2014that leads one to the actions taken to solve a problem, then the student will understand and be able to work the problem.<\/p>\n Of course, by ‘lay out the train of thought’ I do not mean ‘give a lucid explanation’.\u00a0 I mean get the student to construct the chain of implications in his or her head.\u00a0 For some (usually graduate) students this may mean giving a very clear lecture.\u00a0 For other populations, it means ‘guide on the side’.\u00a0 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.<\/p>\n I can be even more specific.\u00a0 Russell’s definition of mathematics points to the center of learning of logic: the notion of implication.\u00a0 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\u2014if 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.\u00a0 Or even how to perform long division.\u00a0 As Underwood Dudley has provocatively shown us (https:\/\/doi.org\/10.1080\/07468342.1997.11973890<\/a>), the claim to practicality of mathematics (in the sense of specific mathematical results) is often exaggerated.<\/p>\n But the importance of mathematics, seen as the study of implications, cannot be exaggerated.\u00a0 It is a characteristic of our species.\u00a0 It is what has led us to dominate our environment.\u00a0 It has also led to some incredibly inhuman events.\u00a0 I leave to more serious philosophers to decide whether the phenomenon of human reason is ‘good’ or ‘bad’\u2014or neither.\u00a0 The point is, it is profoundly human.<\/p>\n In making this statement, I disagree with the view that we must ‘humanize’ or ‘re-humanize’ mathematics.\u00a0 Mathematics is, almost by definition, human.\u00a0 It is its uses, and its teaching, whose humanity we must examine.<\/p>\n To be even more precise, and even technical: the definition of implication rests on the distinction between a statement and its converse.\u00a0 So I can go still further in my wild claims to know if I’ve succeeded.\u00a0 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.\u00a0 Don’t think this is so easy: I have caught important mathematicians, or they have caught themselves, confounding a statement with its converse.\u00a0 And of course I have caught myself.<\/p>\n I am not asserting that if you know about the converse then you know mathematics.\u00a0 I am asserting that if you don’t know about the converse, then you do not know mathematics.\u00a0 Or, less aggressively: if you mistake a statement for its converse you are making an error in mathematics.<\/p>\n So, for example in geometry we often teach about the classification of quadrilaterals: trapezoid, parallelogram, rectangle, etc.\u00a0 Students will often say things like: “If we know a parallelogram has equal diagonals, then we know it is a rectangle\u2026 or it could be a square.”\u00a0 Venn diagrams, illustrating set inclusion, can certainly help untangle the confusion.\u00a0 But there is also a deeper lesson to draw from this error, one that transfers to, and taps into, other experiences.\u00a0 This deeper lesson emerges in phrasing the statement in canonical ‘if..then’ form: if a parallelogram is a square, then it is a rectangle.\u00a0 But if it is a rectangle, it may or may not be a square.<\/p>\n I find this an important guiding principle in pedagogy at all levels.\u00a0 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.\u00a0 For me, this resolves the endless debates about mechanical skills, about fluency or automaticity.\u00a0 And it resolves it in two ways. First, the object of fluency is to be able to reason more easily\u2014more fluently, if you like.\u00a0 When do you concentrate on fluency (‘drill and kill’)?\u00a0 When lack of fluency gets in the way of reasoning.\u00a0 And when do you reach for the calculator?\u00a0 When doing without it will derail your train of thought.<\/p>\n 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.\u00a0 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\u2026<\/p>\n This is the meaning I take from Liping Ma’s “knowledge packages” (1999).\u00a0 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<\/a>). \u00a0That article was one step towards the view I express here, which I\u2019ve come to only aftera decades of experience.\u00a0 To some readers it may be perfectly obvious, and to others perfectly ridiculous.\u00a0 I would be interested in hearing both reactions.<\/p>\n REFERENCE<\/p>\n Liping Ma, Knowing and Teaching Elementary Mathematics, Lawrence Erlbaum Assoc. Inc., Mahwah, New Jersey; London, 1999.<\/p>\n <\/p>\n