Forging Connections
Guest Presentation: Mathematical Problem Solving for Technology

1 Introduction

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).

Problem Solving

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.)

  • Pedgogical Problem: Many students, even college math majors, have a poor understanding of variables and function concepts.
  • Concrete/Abstract Problem: Our efforts to connect students’ concrete experiences to abstract mathematical concepts are often ineffective and misguided.
  • Curricular Problem: US curricula divide math up into overly rigid categories. For the sake of our students, and for the sake of mathematical elegance and beauty, we need to connect these categories.
  • Variables Problem: Variables hardly ever move! They certainly don't move in equations like `y = x^2 -2x -3`, and they don't move in Cartesian graphs. (In fact, they don’t even appear explicitly!)
  • Transformation Problem: Students don't connect geometric transformations with functions.

Concreteness Fading

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.


The complete lesson and downloadable worksheet are here.

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?

Tool Use

  • Tap a tool to activate it. (Double-tap to make it stay active so you can use it repeatedly.)
  • You must match each glowing object, or tap the green checkmark, to complete each use of the tool.
  • To match a glowing tool object, drag it to your desired location, tap at your desired location, or press near your desired location and finish by dragging.
  • While glowing objects remain, you can still adjust the way you have matched previous objects.
  • When you match the last glowing object, even by mistake, the tool is finished. If you make a mistake, press the Undo arrow.

Rotation Stars

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?

  1. Definition of a function
  2. Concept image of a function
  3. Domain and range
  4. Rate of change of the dependent variable with respect to the independent variable
  5. Fixed point of a function
  6. Composition of functions

Time: 10 minutes

4 Homework

  1. Download the worksheet from the Identify Functions lesson and complete the table for page 1. Be sure to include both direction and speed when you describe the rate of change of each function.
  2. Try the Rotate Games activity from the Rotate Family lesson. How well can you do at each of the four games? Make a screen capture of a good result from each game. (Hint: You must press Reset before changing the level. Don't go to advanced levels too quickly.)
  3. Open the Unit 1 lesson on Function Dances, and view the Nicholas Brothers video and the two student videos. Describe a lesson you have taught, or that you might teach, in which it's valuable to get students up out of their seats and moving around.
  4. Write several paragraphs about the relationship between technology use and concreteness fading. Specifically, discuss these three questions:
    1. How were the activities you sampled today effective in connecting your sensorimotor experiences with relevant mathematical concepts and understanding?
    2. How would you modify one or more of these activities to make such connections more effectively
    3. Describe a different lesson, either one you have taught or one you might teach, in which you can make use of technology-leveraged concreteness fading to help students utilize sensorimotor experiences to develop an important mathematical understanding.

5 From Flatland to NumberLand

This lesson is located at

The lesson combines three important investigations:

  1. Restrict the domain of each geometric transformation family (reflect, rotate, translate, glide reflect, dilate) to a number line, and determine which of these families can be arranged to have their range on the same nummber line.
  2. For each family, consider how the numbers corresponding to the independent and dependent variables are related.
  3. Compose functions from the two families that work best with the number line, and observe how the composed function's numeric behavior relates to the rules used for the two functions.

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

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

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


These activities require web access using a browser that supports HTML5 and JavaScript. (That means almost any current browser.) No purchase is required, and there’s no advertising anywhere.

Release Information

[an error occurred while processing this directive]

Update History:

Record every major revision, in reverse chronological order

14 Feb 2018: Simplified this template.
24 Jan 2018: Created this lesson template.