This web page supports a two-part session in which we sample some of the technology-based lessons being developed as part of the Forging Mathematical Connections Project, supported by the NSF program for Innovation for Undergraduate STEM Education (IUSE award 1712280).
We'll deal with several problems and try to find some solution for them. (We might call these meta-problems; they are not the kind you're used to seeing in math textbooks.)
Math-education researchers suggest that students’ understanding of abstract concepts is enhanced by connecting those concepts to students’ concrete experiences, and that explicitly linking and fading from concrete experiences to the abstract (usually symbolic) form can help students make important mathematical connections and strengthen their understanding of the target concept. We want to leverage technology to create concrete experiences of variables and functions.
Accordingly, the lessons we’re using in this extended session are designed to gradually link students’ concrete experiences to their understanding of binomial multiplication, of geometric transformations, and of the rate of change of functions.
Background on Concreteness Fading
Fyfe, E. R., McNeil, N. M., Son, J. Y., & Goldstone, R. L. (2014). Concreteness fading in mathematics and science instruction: A systematic review. Educational Psychology Review, 26(1), 9-25.
“The benefits of concreteness fading may be explained by the fact that it starts with a well-understood concrete format, and explicitly links and fades it to the abstract symbols. Concrete materials are advantageous initially because they allow the math concept to be grounded in easily understood, real-world scenarios.”
Fyfe, E. R., McNeil, N. M., & Borjas, S. (2015). Benefits of “concreteness fading” for children's mathematics understanding. Learning and Instruction, 35, 104-120.
Technology and Concreteness Fading
Interactive technology (particularly “constructionist” technology such as Sketchpad, Web Sketchpad, Desmos, etc.) can be used to create a middle ground between students’ physical, sensorimotor experience and the abstractions of mathematical concepts.
The model, and the tools themselves, can eliminate non-essential limitations of physical tools while emphasizing those mathematical aspects of the model and/or tools that are important to targeted abstract understandings.
Thus far, Web Sketchpad (WSP) is unique in its support for custom tools. Though it lacks a “standard” set of tools, it enables the activity author to create tools that are closely tailored to the needs of particular activities.
2 Identify Functions
This lesson addresses the fundamental mathematical concept of function, in a way that expands students’ horizon as to what constitutes a function while emphasizing the essential mathematical elements that characterize functions.
The main objective of this lesson is that students create their own definition of function.
For pedagogical reasons, we avoid the set-theoretic definition, which is excessively abstract at this stage, and which intentionally omits any sense of motion or continuity, even though motion and continuity characterize most of the functions that students will study.
(Historically, the set-theoretic definition was developed to address certain “pathological functions” such as the function `f(x)`which has the value 1 when `x` is rational and the value 0 when `x` is irrational. There is no reason for any but the most advanced of high school students to learn about such functions and to consider the necessity of modifying the function definition in a way that takes them into account.)
Time: 5 minutes in pairs to manipulate and observe; 5 minutes to discuss.
3 The Rotate Family
In this lesson students construct and investigate the behavior of the rotate function. We'll explore this lesson in its natural habitat. Here we list the part of the lesson to look at, and the time to spend on each.
This activity is designed to raise questions that students will answer during the course of the lesson. To channel Dan Meyer, we might say “If Rotate Function behavior is the aspirin, this Warm-up activity creates the head-ache.”
Time: 5 minutes in pairs to experiment; 5 minutes to discuss your observations.
In this activity you’ll use WSP tools to create and observe some simple rotate functions.
Time: 10 minutes to work in pairs, experimenting and creating a design; 5 minutes to discuss your observations.
Discussion question: What is the intended pedagogical role of each tool?
In this activity, you’ll trace rotate functions to create the missing arms of the star on page 1.
Time: 10 minutes to work in pairs, experimenting and creating; 5 minutes to discuss your methods and results.
How do the different activities in this lesson support the following function concepts?
Time: 10 minutes
5 From Flatland to NumberLand
This lesson is located at https://geometricfunctions.org/fc/unit4/flatland-to-numberland/
The lesson combines three important investigations:
Time: 5-minute demo of pages 3–6 of Reduce the Dimension.
Time: 10 minutes in pairs exploring and discussing Lineland Mess-up.
Time: 15 minutes in pairs constructing and discussing Number the Domain.
Time: 15 minutes in pairs constructing and discussing Compose in Numberland.
6 Create a Dynagraph
This lesson is located at https://geometricfunctions.org/curriculum/cartesian-connection/create-a-dynagraph/
In this activity you will create your own dynagraph. This is based on the previous construction (composing dilation and translation on a number line), but you'll separate the two functions to put each on its own horizontal axis, thus separating the independent and dependent variables and making them easier to observe and analyze.
Time: 15 minutes in pairs constructing and experimenting with your Dynagraph.
Time: 5 minutes discussing dynagraph behavior.
Time: 5 minutes individually playing the dynagraph game.
7 Connect to Cartesian
This lesson is located at https://geometricfunctions.org/curriculum/cartesian-connection/connect-to-cartesian/
In this activity you will create the Cartesian graph of a linear function using only geometric transformations.
Time: 15 minutes in pairs constructing and experimenting with your construction.
Time: 5 minutes for the concluding discussion.
The Fine Print
Record every major revision, in reverse chronological order