My takeaways 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:
 Traintrack 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 gradelevel 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: \((a1)(a+1)=a^21\)
 The product of numbers that are four units apart is four
less than the square of the number in the middle: \((a2)(a+2)=a^24\)
 The product of numbers that are six units apart is nine
less than the square of the number in the middle: \((a3)(a+3)=a^29\)
 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: \((ab)(a+b)=a^2b^2\)
 You can then use this idea for quick mental math
computation:
 Forward: \(79^2=80*78 + 1=6241\)
 Backward: \(125^2123^2=(125+123)(125123)=248*2=496\)
What is Mathematical Structure and What Does Attention to Structure Afford Problem Solvers in the 712 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 closedform 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 yintercept. 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
inputoutput 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
yintercept 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
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