Photo of Renee LaRue available here.

Co-author Nicole Engelke Infante

Classic problem: Fence along a barn. Minimize amount of fencing given a fixed amount of fencing.

Literature: 4 versions of this problem. More difficult when students have to set up problem from words.

Carlson and Bloom (2005) Problem-Solving Framework

Tall and Vinner (1981) Concept image

7 students (pilot with 3 students), just before final exam involving optimization.

Recordings of students thinking aloud.

Questions about rectangles – what happens to perimeter if area changes?

5 students solved without intervention, 3 perfectly, 2 forgot about barn. Other 2 needed much help.

Responses showed gaps in reasoning.

Six key maths concepts that played a role:

1. Use 2x + y or 2y + x? 2l + 2w = 2y + 2x (matching variables to what they think they must mean).
2. Function notation. Haphazard use of equal signs. Student said you can’t write 2x + y as f(x) because y = f(x).

Opening photo of co-author Owe’s colourful toenails – a different colour for each of the 10 Delta conferences!

Photo available here

Paper available here

Traditionally, engineers demand from mathematics fluent use of techniques.

With technology, is this as important as conceptual insight?

Procedural (mechanical) knowledge in mathematics, e.g. ‘For this function, what is the equation of the tangent line at this point?’

Conceptual knowledge link relationships between verbal, visual, symbolic representations. E.g. Match graph of derivative to a written description.

SA – Sweden project.

Quantitative analysis: Junior students in 3^rd semester; senior students in 7^th semester.

Qualitative analysis: interviews with Swedish and SA engineers.

Swedish engineer

• Procedural maths needed as a basis for mathematics (concerns that he understood procedural knowledge as basic maths background for applications, equivalent to learning the language of maths).
• Conceptual understanding = engineering judgement, broader than maths.
• New engineers should be able to be independent, deal with a whole problem, be self-confident.

Emilie Naccarato presenting, co-author Gulden Karakok

https://sites.google.com/site/emiliernaccarato/_/rsrc/1410927046099/home/Me.jpg?height=320&width=213

Goal: talk about themes occurring in flipped classrooms in tertiary maths in USA

Flipped class:

• Some course content on technology accessed outside class
• More time in class for related, meaningful activities
• Same contact, face-to-face hours
• (notice the loose way a flipped class can be arranged – many ways to arrange)

Existing studies

Descriptions of implementation and student perceptions.

Looking at student performance in flipped vs traditional courses. Big differences in studies – no consensus on what works best (but implementations vary a lot).

Need to link expectation and goals of a flipped class to the implementation – can’t simply compare all versions of a flipped class.

How do you assess differences when learning goals and implementations are so different?

You need alignment. Schoenfeld (2000) emphasized “identifying important topics and specifying what it means to have a conceptual understanding of them. With this kind of information … [they] could then decide which aspects of understanding were most important, which they wanted to assess, and how.” (p.…