Quote Dubinsky (1989): "A person's mathematical knowledge is her or his tendency to respond to certain kinds of perceived problem situations by constructing, reconstructing and organizing mental processes and objects to use in dealing with the situations."

"Applying this point of view to mathematics (or any other subject) consists of determining the nature of the specific processes and objects that are constructed and how they are organized when one studies mathematics"

Ways of thinking about functions:

- prefunction - does not understand any real ways of using function concepts
- action - repeatable mental or physical manipulation (e.g., plug in numbers and calculate); static; one step at a time
- process - think of function as a single dynamic transformation

This article proceeds with the view that in mathematics, there is a duality in mathematical constructs being a process or an object. That is, conceive of things operationally (process) or structurally (object). Historical examples include the expansion of number systems: positive to negative (operational: subtraction as adding a negative to structural: negative numbers as objects), and real to complex (i=sqrt(-1) as an operational convenience to an actual object)

An included reference suggests finding another article: Kieran, C.: 1992, 'The learning and teaching of school algebra', in D. A. Grouws (ed.), The Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, pp. 390-419. I'll have to see if I can find this one, as it is cited for the sentence, "[reification] was also used to introduce some order into the quickly growing bulk of findings about algebraic thinking."

Interesting phrase: "the ability to grasp the structural aspect is not easy to achieve" and "those crucial junctions in the development of mathematics where a transition from one level to another takes place are the most problematic."

Another interesting way to think about how mathematics is organized: (1) Logical, or the way it fits together; (2) Historical, or the way in which it was developed; and (3) Cognitive, or the processes in which people learn.

Modes of Algebra

1.1) Algebra as Generalized Arithmetic: The Operational Phase

-- solve for the unknown, but not using symbols (grade school algebraic thinking)

-- rhetoric algebra

-- principally reversing processes

1.2) Algebra as Generalized Arithmetic: The Structural Phase

1.2.1) algebra of a fixed value (unknown)

-- Notational convenience, but treat variable as a fixed value

:::: becomes a mental challenge to think of formula as both a process and result

:::: example given: 2+3 represents process, 5 represents result. But x+3 represents both, no separate "result"

:::: compare to the challenges of new number types required to think about division, subtraction, and extracting square roots

** Nice comment: "Once we manage to overcome this difficulty, it is quickly forgotten. ... Our eyes are easily blinded by habit and by our own ontological beliefs. Nevertheless, much evidence for the difficulty of reification may also be found in today's classroom, provided those who listen to the students are open-minded enough to grasp the ontological gap between themselves and the less experienced learners."

1.2.2) Functional algebra (of a variable)

:::: View formula as object

:::: Parameters represented as symbols not numbers.

2) Abstract Algebra

Give examples of interview questions. Students at early stages of thinking think about formulas as recipes for computations (process) but do not perceive them as valid objects. "The equality sign is interpreted as a 'do something signal' (Behr et al 1976; Kieran 1981)"

Here's something I see all the time in calculus classes: "It [the = symbol] serves here as a 'run' command. When treated in this way, the equality symbol looses [sic] the basic characteristics of an equivalence predicate: it stops being symmetrical or transitive. Indeed, young children seem to have no qualms about solving word problems with the help of a chain of non-transitive equalities. For instance, when asked 'How many marbles do you have after you win 4 marbles 3 times and 2 marbles 5 times?', the child would often write: 3*4=12+5*2=12+10=22."

Equations of the form 2x-3 = 11 can be interpreted as a formula whose result is 11 (which can be solved by inverse operations); equations of the form 2x-3=5x-9 appear to be two different formulas, and inverse operations do not make sense.