What were you thinking?

Asking questions that reveal understanding

Dr. Frank WilsonSue Steele

Chandler-Gilbert Community College

The Problem

Students often learn to memorize and mimic instead of think and apply

If assessment questions focus only on procedural mastery, we may erroneously conclude a student with a weak conceptual understanding has achieved conceptual mastery

Case Study: College Algebra

Students are given and are asked to describe the effect of each of the parameters on the graph of f.

Students who correctly identify the effect of the parameters have varying degrees of success in implementing the transformation on the graph of f

Student A

Student A (cont.)

Looking for• Vertical reflection• Vertical compression

by factor of 2/3• Shift left 4 units• Shift up 2 units

Student B

Student B (cont.)

Looking for• Vertical reflection• Vertical compression

by factor of 2/3• Shift left 4 units• Shift up 2 units

Student C

Student C (cont.)

Looking for• Vertical reflection• Vertical compression

by factor of 2/3• Shift left 4 units• Shift up 2 units

Student D

Student D (cont.)

Looking for• Vertical reflection• Vertical compression

by factor of 2/3• Shift left 4 units• Shift up 2 units

Observations To effect a vertical reflection on a graph, a

student must have a conceptual understanding that points are reflected across the horizontal axis.

To effect a vertical compression on a graph, a student must understand that output values are being reduced (scaled) by the given factor

To effect a horizontal/vertical shift, a student must understand direction, magnitude, and recognize that a shift does not alter the shape of the graph only its position on the axes

Case Study: Arithmetic

A volunteer group is putting together care packages to hand out at a local food kitchen. Each volunteer is given 30 molasses cookies, 48 chocolate chip cookies, and 36 sugar cookies to divide into boxes such that each box contains the same number of each kind of cookie.

What is the greatest number of boxes the volunteer be able to fill?

Student A

Student B

Student C

Student D

Student E

Student F

Observations At the ‘developing’ level, a student is able

to discern that if finding a GCF in the form of a basic factor tree, the solution is easy to determine.

At the ‘competent’ level, a student is able to visualize the meaning of the GCF with diagrams or a factor tree that factors to prime factors.

At the ‘effective’ level, a student uses Venn Diagrams to demonstrate their thinking of the GCF

Observations At the ‘mastering’ level, the student

determines that the greatest number of boxes is based on the minimum number of cookies that can be placed into individual boxes.

Based on the student that thinks outside of the box and focuses on the real-world situation, the student asks, “Why would you package more than one of each cookie in the care package?”

Writing conceptual questionsActivity

Question prompts

Write two questions that, if answered correctly, will demonstrate that a student has mastered the concept of the slope of a line

Write two questions that, if answered correctly, will demonstrate a conceptual understanding of the product of two fractions

Sample Questions – Slope The equation may be used to convert

temperature from C degrees Celsius into F degrees Fahrenheit. If air temperature increases by 10 degrees Celsius, by how many degrees Fahrenheit will the temperature increase?

To answer this question, the student must understand that each 5 degree increase in Celsius corresponds with a 9 degree increase in Fahrenheit. So a 10 degree increase in Celsius corresponds with an 18 degree increase in Fahrenheit.

Sample Questions – Slope

The graph of the equation y = 4 is shown in the figure. What is the slope of the line? Explain. To answer this

question, the student must understand that slope is a comparison of the change in the output to the change in the input. There is 0 change in the output for any change in the input so slope is 0.

Sample Question – Product of Fractions A pizza is half pepperoni

and half mushroom and onion. Mike, who doesn’t like onions, ate two thirds of the pepperoni side of the pizza. What fraction of the entire pizza did Mike eat?To answer this question, the student must determine 2/3 of 1/2

Mike ate 2/6 (1/3) of the entire pizza.

Sample Question – Product of Fractions Forever 21 offered a Buy 2 Get 1 Free

sale on selected items. A customer notices that the price of dresses on the clearance rack have already been reduced by 25% (1/4). To take advantage of the Buy 2 Get 1 Free sale she buys three equally priced dresses from the clearance rack. On average, what fraction of the original price (before all discounts) does she end up paying for each dress?To answer this question, the student must determine 2/3 of 3/4.

Sample Question – Product of Fractions

We start with the original price and remove 1/4 of the price.

We buy three dresses: the first two at 3/4 price and the last for free.

We pay 6/12 (1/2) of the original price per dress.

Dress 1Dress 2

Dress 3

Best Practices from Literature

In effective teaching, teachers use a variety of question types to assess and gather evidence of student thinking, including questions that gather information, probe understanding, make the mathematics visible and ask students to reflect on and justify their reasoning.

Principles to Action, NCTM, p. 41

Principles to Actions, NCTM, pg 41

Questions?