Structure
Writing the final exam for my course for pre-service teachers gave me a chance to think about the structure of the course. Like in most math classes, the structure of the course is very closely tied to the structure of the textbook.
The book develops the first semester's material in a very mathematical, algebraic sense. It starts by defining a set. (Or, more accurately, asserting the existence of sets and noting, "in abstract mathematical logic and set theory, a set is an undefined term.") Then it moves up to the commutative monoid of whole numbers under addition, defining addition in terms of cardinality of unions of disjoint sets. Really, it does. Next, it extends the properties to make the whole numbers into a commutative ringoid. From there you can guess the progression: completions bring us the ring of integers and the fields of rational and real numbers (although, technically, the book stays within the algebraic closure of the rationals).
Along the way we encounter functions, one-to-one correspondences, arithmetic in other bases, divisibility, the Euclidean algorithm, and primes. There is an optional digression into modular arithmetic. (I did not take that path, as my students struggle enough with arithmetic in domains.) The book is structured like a groups-first abstract algebra class for students who have never seen arithmetic before. Beneath the overarching algebraic themes, the book will go on for pages explaining the mundane mechanics of how to do long division or add fractions.
The theory escapes my students entirely. Since this is a 200-level class (but one with no pre-reqs), there is a clear message that the course is supposed to have a certain amount of theory. And it would be irresponsible to teach a class focused entirely on how to perform the algorithms of arithmetic. But the theory the way it's presented in the book is not helping my students to understand the structure or the mechanics of our number system.
A better course for the students I have would have less abstract themes. The idea of grouping and regrouping comes up in bases, fractions, and divisibility. The idea of parts and wholes keeps coming up. So often we look at equivalent representations of a quantity, be it in simplifying fractions or in applying the properties of addition and multiplication. A book with a clear emphasis on these ideas would support a much richer structure for my course.
The book develops the first semester's material in a very mathematical, algebraic sense. It starts by defining a set. (Or, more accurately, asserting the existence of sets and noting, "in abstract mathematical logic and set theory, a set is an undefined term.") Then it moves up to the commutative monoid of whole numbers under addition, defining addition in terms of cardinality of unions of disjoint sets. Really, it does. Next, it extends the properties to make the whole numbers into a commutative ringoid. From there you can guess the progression: completions bring us the ring of integers and the fields of rational and real numbers (although, technically, the book stays within the algebraic closure of the rationals).
Along the way we encounter functions, one-to-one correspondences, arithmetic in other bases, divisibility, the Euclidean algorithm, and primes. There is an optional digression into modular arithmetic. (I did not take that path, as my students struggle enough with arithmetic in domains.) The book is structured like a groups-first abstract algebra class for students who have never seen arithmetic before. Beneath the overarching algebraic themes, the book will go on for pages explaining the mundane mechanics of how to do long division or add fractions.
The theory escapes my students entirely. Since this is a 200-level class (but one with no pre-reqs), there is a clear message that the course is supposed to have a certain amount of theory. And it would be irresponsible to teach a class focused entirely on how to perform the algorithms of arithmetic. But the theory the way it's presented in the book is not helping my students to understand the structure or the mechanics of our number system.
A better course for the students I have would have less abstract themes. The idea of grouping and regrouping comes up in bases, fractions, and divisibility. The idea of parts and wholes keeps coming up. So often we look at equivalent representations of a quantity, be it in simplifying fractions or in applying the properties of addition and multiplication. A book with a clear emphasis on these ideas would support a much richer structure for my course.
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