1. Reconceptualizing Mathematical Objects as Mediating Discursive Metaphors Aaron Weinberg Ithaca College

2. Example

5. Leslie: See, I used to think you had to flip it over the y equals x axis. What is that for? Do you know what I'm talking about? Tyler: Oh, that is inverse. L: That's inverse... So if you flip it over it will be like... T: Oh, see… then it goes out that way, see? Yeah, it's just the opposite of this graph L: Like that? T: Yeah, no... just solve for y equals... so negative square root of x, and you solve for y for this? L: What do you mean solve for y? T: We could just plug in points to see. Negative square root of x? L: Is that possible? T: Take the square root first, and then put the negative.

6. Example How are students thinking about functions? How are students thinking about representations? How are students participating? What is the relationship between – Representations – Conceptions of functions – Dialogue Are the students successful? How can we describe their activity?

7. Functions in APOS Prefunction: Does not assign much meaning Action: One step at a time Process: Dynamic repeatable transformation – Think about the function as a whole – Combine several processes – Reverse individual processes Object: – Functions as inputs – Discuss general characteristics – Work with non-computable functions

8. Evaluating Encapsulation Prefunction: The student does not have very much of a function concept at all Action: Emphasized the act of substituting numbers for variables and calculating to get a number, but did not refer to any overall process of beginning with a value (numerical or otherwise) and doing something that resulted in a value Process: The input, transformation, and output were present, integrated and fairly general Breidenbach, Dubinsky, Hawks, & Nichols, 1992

9. Evaluating Encapsulation ConceptionExamples PrefunctionA mathematical equation with variables. A mathematical statement that describes something. ActionA function is an equation in which a variable is manipulated so that an answer is calculated using numbers in place of that variable. A function is an expression that will evaluate something when either variables or numbers are plugged into the function. ProcessA function is some sort of input being processed, a way to give some sort of output. A function is an algorithm that maps an input into a designated output.

10. Drawbacks and Concerns Is it a complete description? Strict hierarchical ordering – Discrete categories – Linear development Prioritize object conception Acquiring cognitive structures – Diminishes social, historical, and cultural lenses – Contexts and toole – Novice-expert approach – Culturally-embedded self-description

11. Why APOS is Important Help make sense of student work Make pedagogical decisions Helps focus on concept development “Necessary” for some concepts Goal: Reconceptualize developmental levels and mathematical objects

12. Analyzing Discourse Mediational Toolkit Actions, Processes, Objects as Metaphors Language use Objectified Discourse Semantics Intramental Activity “Aboutness” Facet (Representation)

13. Analyzing Discourse Mediational Toolkit Actions, Processes, Objects as Metaphors Language use Objectified Discourse Semantics Intramental Activity “Aboutness” Facet (Representation)

14. Mediation of Thought Examples – “Seeing” stick – Driving around New Jersey – Graphs of functions Symbols and language are tools We use tools to engage in social action – Culturally and historically situated Discursive technology – Tools and action influence our thinking

15. Mediational Toolkit Speaker’s use of linguistic metaphor and associated actions Tool choice based on – Different functions – Context – Authority – Addressee Situated in speech genre and context

16. Evaluating Encapsulation ConceptionExamples PrefunctionA mathematical equation with variables. A mathematical statement that describes something. ActionA function is an equation in which a variable is manipulated so that an answer is calculated using numbers in place of that variable. A function is an expression that will evaluate something when either variables or numbers are plugged into the function. ProcessA function is some sort of input being processed, a way to give some sort of output. A function is an algorithm that maps an input into a designated output.

17. Mediating Metaphor Action Describes the function using a metaphor of a series of operations or events, or as a computation without describing a systematic relationship between an input and output Performs computations or explicitly evaluates the function without also describing a systematic variation Manipulates the function by manipulating a statement that describes explicit calculation (such as a symbolic representation) Process Describes the function using a metaphor of a non-explicit calculation, systematic relationship, or machine that connects an input (or input values) and output (or output values) Refers to the results of a computation performed on an input value without explicitly performing the computation Describes a systematic dependency of an output value on an input value without performing a computation. The systematicity need not be explicitly stated but can be expressed by the student referring to a rule or implied systematic relationship Manipulates input values as a set (e.g. describing a change that affects all input values) to produce a change in output values Combines the function with another function using an arithmetic operation, applying the operation pointwise (e.g. adding two functions pointwise) Object Describes the function using a metaphor of a concrete or physical object Describes general attributes of the function Manipulates or operates on the function as a whole without explicitly manipulating its input and output values or explicit computational process Uses the function as an input to an operator or another function without performing explicit computation Combines the function with another function using an arithmetic operation, applying the operation simultaneously to all values of the function

18. Mediating Metaphor Action Performs computations or explicitly evaluates the function without also describing a systematic variation Process Describes a systematic dependency of an output value on an input value without performing a computation. The systematicity need not be explicitly stated but can be expressed by the student referring to a rule or implied systematic relationship Object Manipulates or operates on the function as a whole without explicitly manipulating its input and output values or explicit computational process

19. Mediating Metaphor Action Plugging in values to evaluate a function Process “If a function is increasing is its inverse also increasing? “It’s increasing. Yeah, cause, think about it. The domains and the ranges switches, so if the domain is increasing, which it is, the range is increasing and if they switch spots for…” Object Manipulating the graph as a whole

20. Discreteness and Granularity Categories not discrete or ordered –Adds flexibility without creating new categories –Reduces predictive power Unit of analysis –Utterance –Task –Student

21. Quick Results Relationship between metaphor and representation Interviews with 15 students –6 pairs –3 individual Number of instances No statistical significance

22. Metaphor and Facet FacetActionProcessObject Colloquial127612 Graphical322345 Notational2438 Numerical72280 Symbolic592219 Verbal21736 Percent of Coded Instances by Metaphor

23. Process-Object Frameworks Help us make sense of student work Help guide development of instructional materials Metaphors allow flexibility –Non-linear –In-between metaphors –Social and cultural perspective Focus on participation vs. acquisition