Foundational Reasoning Abilities that Promote Coherence in Students' Function Understanding

Oehrtman, Michael; Carlson, Marilyn P.; Thompson, Patrick W

doi:10.5948/upo9780883859759.004

Cited by 133 publications

(90 citation statements)

References 22 publications

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“…This way of thinking could support reasoning about a function defined parametrically, such as (x, y) = (sin10t,cos 20t),0 ≤ t ≤ 1. Thompson used this example to suggest a way of thinking about curves in space, such as (x, y, z) = (sin10t,cos 20t,t),0 ≤ t ≤ 1, by imagining that t is actually an axis, "coming straight at your eyes" (Oehrtman, Carlson, & Thompson, 2008). Thompson proposed that this way of thinking about a curve in space could help the student visualize the graph of the function defined by z = f (x,y), by thinking about y or x as a parameter.…”

Section: Extension To Graphs In Three Dimensionsmentioning

confidence: 99%

Students’ images of two-variable functions and their graphs

Weber

Thompson

2014

Educ Stud Math

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This paper presents a conceptual analysis for students' images of graphs and their extension to graphs of two-variable functions. We use the conceptual analysis, based on quantitative and covariational reasoning, to construct a hypothetical learning trajectory for how students might generalize their understanding of graphs of one variable functions to graphs of two variable functions. To evaluate the viability of this learning trajectory, we use data from two teaching experiments based on tasks intended to support development of the schemes in the HLT. We focus on the schemes that two students developed in these teaching experiments and discuss their relationship to the original HLT. We close by considering the role of covariational reasoning in generalization, consider other ways in which students might come to conceptualize graphs of two-variable functions, and discuss implications for instruction.

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Section: Extension To Graphs In Three Dimensionsmentioning

confidence: 99%

Students’ images of two-variable functions and their graphs

Weber

Thompson

2014

Educ Stud Math

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“…In recognizing IGFs one has to have a picture of the function as an entity. In the literature this perspective of the function, seeing the function as a whole, is also addressed as the object or global perspective (Confrey & Smith, 1995;Even, 1998;Gray & Tall, 1994;Oehrtman et al, 2008;Sfard, 1991). There is also another perspective of the function, namely, to see a function as an input-output machine.…”

Section: Global and Local Perspectivesmentioning

confidence: 99%

“…Students have difficulties in seeing a function both as an input-output machine and as an object (Ayalon, Watson, & Lerman, 2015;Gray & Tall, 1994;Oehrtman, Carlson, & Thompson, 2008;Sfard, 1991). Graphs appeal to a gestalt-producing ability, and in this way can help to consolidate the functional relationship into a graphical entity (Kieran, 2006;Moschkovich et al, 1993).…”

Section: Introductionmentioning

confidence: 99%

Graphing formulas: Unraveling experts’ recognition processes

Kop

Janssen

Drijvers

et al. 2017

The Journal of Mathematical Behavior

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a b s t r a c tAn instantly graphable formula (IGF) is a formula that a person can instantly visualize using a graph. These IGFs are personal and serve as building blocks for graphing formulas by hand. The questions addressed in this paper are what experts' repertoires of IGFs are and what experts attend to while recognizing these formulas. Three tasks were designed and administered to five experts. The data analysis, which was based on Barsalou and Schwarz and Hershkowitz, showed that experts' repertoires of IGFs could be described using function families that reflect the basic functions in secondary school curricula and revealed that experts' recognition could be described in terms of prototype, attribute, and part-whole reasoning. We give suggestions for teaching graphing formulas to students.

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“…Many see function as a key mathematical topic at the secondary school level (Brenner et al, 1997;Llinares, 2000). Others suggest that it represents the foundation for the whole subject of mathematics (Hitt, 1998;Oehrtman, Carlson, & Thompson, 2008). We chose linear function as a fair comparable topic because the learning goals associated with the topic were similar between England and Shanghai as discussed later in the section 'Linear Function in Curricula' at Results part.…”

Section: Introductionmentioning

confidence: 99%