Lesley University Day 1

My take-aways from Day 1 of Lesley University's Summer Math Institute: Mathematical Representation: Looking at and Making Use of Structure  

Successfully Structuring Math Lessons, Courses, and Programs

Jim Matthews

- In the 180 days you teach, how many lessons are traditional vs. one (or more) of these formats?

• Guided discovery
• Reasoning and proof
• Data collection and analysis
• Interesting application
• Interdisciplinary connection
• (don’t immediately know what to do) Problem Solving
•  Counterintuitive Phenomenon

- Two problems, that among other things, fall under the counterintuitive phenomenon:

• Train-track problem for Pythagorean Theorem. Question: How far off the ground will the tracks be where they meet?
• Rip a sheet of newspaper in half and place the pieces on top of each other. Repeat for a total of 52 rips. How tall will the stack be?
• Key: have students make a prediction before solving. Ex: Would a telephone poll fit underneath the traintracks? An algebra textbook? A piece of paper?

- What do mathematicians do? They do verbs (in red) much more than nouns.

How do you teach structural thinking to students?

Amy Lucenta and Grace Kelemanik

- Structural thinking sits in the noticing not in the answer. This is why we are always going back to the noticing stage.

• In sharing stage: “We noticed _____, so we ______”
• In reflection stage: “Noticing ____ helped count/calculate quickly because _____.” And “Knowing _____ comes in handy when quick counting/calculating because _____.”

- Contemplate then Calculate specifically designed to push structural thinking, whereas number talks are about multiple strategies and highlight something about a particular operation. The first three examples here all have one structural shortcut that makes them easier to solve. The last example has many structural shortcuts that make calculate easier.

- The most productive Contemplate then Calculate tasks have multiple shortcuts that leverage structure

- To choose tasks, first think about the structural thinking that is helpful in current grade-level content, then think about where that type of structural thinking can be used in earlier content. Those are the tasks that you want to start with. (My next steps are to do this thinking with eighth grade content)

Algebraic Formulas Make Sense!

Natalya Vinogradova

- algebra is generalized arithmetic

- algebra is a beautiful and efficient expression of an idea, but you have to develop the idea firstà visual representation described in words.

To illustrate these ideas: factoring the difference between two squares:

- The product of numbers that are two units apart is one less than the square of the number in the middle: \((a-1)(a+1)=a^2-1\)

- The product of numbers that are four units apart is four less than the square of the number in the middle: \((a-2)(a+2)=a^2-4\)

- The product of numbers that are six units apart is nine less than the square of the number in the middle: \((a-3)(a+3)=a^2-9\)

- The product of numbers that are 2b units apart is \(b^2\) units less than the square of the number in the middle. And so we can generalize: \((a-b)(a+b)=a^2-b^2\)

- You can then use this idea for quick mental math computation:

• Forward: \(79^2=80*78 + 1=6241\)
• Backward: \(125^2-123^2=(125+123)(125-123)=248*2=496\)

What is Mathematical Structure and What Does Attention to Structure Afford Problem Solvers in the 7-12 Classroom?

Roser Giné

- From Mason, Stephens, and Watson:

• Mathematical Structure: “the identification of general properties which are instantiated in particular situations as relationships” 
• Structural Thinking: “a disposition to use, explicate, and connect properties in one’s math thinking”

So an example of structure and structural thinking with quadratics:

1. First, we can identify some of the properties of quadratics by completing the following table for the general equation \(p(x)=ax^2+bx+c\)

2. Next, write a closed-form equation for the quadratic function given by the following table of values:

Based on the properties of quadratics I saw in the first table, I can tell that a=7 based on the second differences and that c=-2 based on the y-intercept. My inclination to find b then is to just choose a random point and solve for b.

However, Roser suggested a different way to determine b.

Once we know that a=7, we have the equation \(f(x)=7x^2+bx+c\), so therefore \(f(x)-7x^2=bx+c\). In other words, the difference between our unknown quadratic function f(x) and the function \(g(x)=7x^2\) is the linear function h(x)=bx+c. So let’s actually look at input-output pairs of that function:

x	\(h(x)=f(x)-7x^2\)
0	-2
1	7
2	-12
3	-17

We can see that the slope in this table is -5 and the y-intercept is -2. Therefore, b=-5 and c=-2.

Finally, we can see this linear relationship of the difference between the functions f(x) and g(x) in the graph below. We know that the two quadratics will be translations of each other because they have the same quadratic term, which determines how skinny/wide the “u” is.

Structural thinking that happened here:

- When we take away the leading terms we are left with a linear relationship

- comparing the two graphs

- properties used:

• that a determines how open the parabola is
• linear functions have constant 2^nd differences