Video / Practice Discussion

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 real- life 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. 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. 2. 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. 3. In your work for numbers 1 and 2, what numbers are constant and what do these numbers represent in the problem context? 4. 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? 5. Define the variables and write a function rule that gives total miles of snow fences as a function of the number of hours worked. 6. 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 B. Find the rate of increase of total cost using rise over run. as a function of the number of shirts.

y 180 y

100Total Cost

x 0 2 4 6 8 10 12 14 16 18 20 22 Number of Shirts Ordered

A. Solve, 2x + 24 = 52. B. Why would you subtract off Show all steps. the existing snow fencing before (Hint: always undo the addition determining how many hours it takes first.) to build the new snow fencing?

A. Find the slope of the line between B. The hurricane is 180 miles off shore (2,8) and (-5,-2) 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: