Video / Practice Discussion
PLC Guide 1
Pre-Session
1. View video “Reason Abstractly and Quantitatively “
The traditional mathematics textbook has emphasized abstract, symbolic manipulations. For example,
students are shown how to solve 2-step equations and then are given a set of equations to solve. At the end
of the problem set there are always one or two real-life application problems. However, the students usually
did not understand or do these problems. I believe that this approach is exactly the opposite of how we
ought to be teaching mathematics. Instead, we ought to place the math in a real context and help students
see the connection between the math and the real-life meaning. I suggest this mathematical practice
standard is written backwards. The standard would be better written as, “Reason Quantitatively and
Abstractly”. The natural order of constructing meaning is to go from quantitative understanding to
mathematical abstractions.
In this video the students are considering a problem called “Snow Fences”. It is one of the problems I selected
to help my students understand 2-Step equations and how we write, solve, and use them to understand reallife problems. The students have previously completed a unit on linear patterns and a unit on functions.
The class is a Math 8 Team class. Half of the students are Special Education students and the other half are
regular students. Mrs. McPhie and I team teach the class. One of the students is doing the videotaping. I
apologize for how jerky the video is. It wasn’t so bad watching the video on the little camera screen, but I
have to admit that I got a headache watching the video on a big screen.
2. Read the Mathematical Practice overview
3. Review the task below.
Snow Fences
The Wyoming State Department of Transportation wants to build snow fences along a windy stretch of
interstate. The snow fences allow blowing snow to pile up against the fences and thus prevent the snow from
drifting onto the highway. Snow fences are already in place for a 24 mile stretch of the road and the
transportation department believes it can build more snow fences at a rate of 2 miles per hour.
1.
2.
3.
4.
5.
6.
How many total miles of snow fences will there be after the transportation department crew works for 8
hours? Show how you get your answer and label your work.
How many total miles of snow fences will there be after the crew works s for 10 hours? Show how you get
your answer and label your work.
In your work for numbers 1 and 2, what numbers are constant and what do these numbers represent in the
problem context?
In your work for numbers 1 and 2, what numbers will you need to represent with variables and what do
these numbers represent in the problem context?
Define the variables and write a function rule that gives total miles of snow fences as a function of the
number of hours worked.
After how many hours will the snow fences be a total of 52 miles? Explain what you did to get your answer.
Introductory Activity – Video discussion
Discussion 1. Why do you believe that traditional textbooks have emphasized abstract, symbolic
Questions
manipulations? For example, why do they emphasize problem A over problem B.
A. Find the slope of the line
using rise over run.
B. Find the rate of increase of total cost
as a function of the number of shirts.
Total Cost
Number of Shirts Ordered
A. Solve, 2x + 24 = 52.
Show all steps.
(Hint: always undo the addition
first.)
B. Why would you subtract off
the existing snow fencing before
determining how many hours it takes
to build the new snow fencing?
A. Find the slope of the line between
(2,8) and (-5,-2)
B. The hurricane is 180 miles off shore
at 8:00 PM. The hurricane is 36 miles
off-shore at 2:00 AM. How fast is
the hurricane traveling toward shore?
2. How does emphasizing abstract, symbolic manipulations add to students’ perception
that mathematics has no meaning in their lives?
3. How does emphasizing abstract, symbolic manipulations add to students’ perception
that they can’t do mathematics?
4. What other examples of algebraic rules, formulas, or procedures where we
could place the math in a real-life context first; let the students gain a quantitative
understanding of the situation; and then move to the more efficient, abstract
manipulations.
5. How do we find the resources we need to embed mathematical rules, formulas, or
procedures in real-life contexts?
Discussion – What does this mean in my classroom?
Discussion 1. Do you believe that students would understand mathematics better if they were
Questions
allowed more time to consider numbers in a context?
2. How can you find opportunities in every mathematics class to help students reason
Quantitatively and abstractly?
Application – Create a lesson plan implementing the practice
Choose an algebraic rule, formula, or procedure that you typically teach. Plan a lesson
that will allow you to place the rule, formula, or procedure in context and allow students
to work quantitatively and make connections between the numbers in context and the
abstract manipulations.
Be very clear about what you want the lesson to accomplish.
Consider working with a colleague. Brainstorm ideas together. Implement the lesson
plan. Debrief how the lesson went together.
Reflections – Write in Journal
1. What went well in the video lesson?
2. In the video, what evidence did you see of the students’ understanding of abstract reasoning?
3. What would you do differently?
4. Did this video change the way you look at teaching students mathematics? In what way(s)?
Other observations and feelings: