Conceptual Questioning

8/2/2019 Conceptual Questioning

1/16

Conceptual Questioningfor the Mathematics Classroom

1

INTRODUCTION

Look at the equations in the boxes below, asking students to find a value for . Mathematically these equations areidentical, but to many of my students these appeared as distinct problems, each needing a unique solution.

3 = 7 3 + = 7 7 = + (3)

I found students were accustomed to memorizing rote procedures without a true understanding of how or why a step

was being done. In the first example above, students arrived at a solution by, taking the middle number (3) and adding

it to the last number (7). So when presented with an example missing a middle number like the one in the second

box they struggled to approach the problem.

Teaching mathematics is much like building a three-legged stool. The first leg is conceptual understanding. In the

example above this would have been an exploration of the property of equality and it affect on expressions. The second

leg is an application of the concept (i.e. skills or procedures) that let the students see a most-efficient route to a

solution. In the example above this might have been a procedure similar to SADMEP, telling students which operationsshould come first to get at an answer quickly. The third leg is a problem solving approach which teaches the students

what questions to ask and what to look for when solving any problem. In the example above this might have been

explaining that by isolating a variable, the students could find its value. The importance of a three-legged stool is that no

matter how strong any one leg is, without all three present, the stool will not stand. Thus in a transformationa

mathematics classroom, each of these three avenues of instruction must be progressed simultaneously.

In almost all the research I have read, it has been emphasized for every educator to cultivate the art of asking questions

as a means to better teach and assess student understanding. Textbook exercises are useful, but no text can provide a

universal series of questions that provoke every student to think through a new concept. Questions that lead students to

understanding must start at their current level and provoke them to think forward to the mathematics at hand. Thus the

ideas that drive my classroom instruction are

Concept Over Procedure

A solid understanding of a mathematical concept is far more powerful that the rote memorization of

various procedures, because conceptual understanding allows students to apply prior knowledge to new

situations.

From a practical stand-point it is impossible to teach every type of question a student might see on an

exam. Echoing the mantra, it is better to solve the same problem three different ways than to solve

three different problems the same way, students are better served with a deep understanding of core

concepts which will allow them to access a large array of unique problem scenarios.

The Answer Is Not Enough

Particularly in math, students are seeking a single, right answer to a problem. However simple

numerical answers tell teachers very little. In fact in my classroom, the answer is the least important

part. I care more about the explanation, which illuminates a students thinking process.

As an example, say a student correctly reduced =

. What the numerical answer does not tell us is that

a student may have done so by cancelling the sixes , instead of understanding the equality of thevalues through division. Asking for an explanation in detail would have quickly highlighted this error.


2/16


2

QUESTIONING STRATEGIES

In practice I attempt to push students to a deep conceptual understanding by asking questions that get beyond the

surface conventions and procedural steps. Below I have included a list of the questioning strategies I use during various

parts of my lessons to push student thinking further.

Say Why (or How)Say Why is a questioning strategy that can be employed during any part of the lesson to push students for a more

complete explanation, ultimately verbalizing their thinking process. Say Why involves asking a student (or students) why

a particular answer is correct or why a step in being performed.

The key to effectively implementing Say Why is continuing to dig for the answer you are looking for by playing ignorant

or the devils advocate and have the student teach you. If a student has difficulty verbalizing his/her thoughts, it can

help to facilitate the zone of proximal development (ZPD) if you ask a stronger student to first verbalize the thought

process and then have the weaker student rephrase the explanation.

A sample dialogue might look like this

Teacher John, what do you get when you evaluate (3) ?John I get 27.Teacher How did you arrive at that answer?

John I multiplied 3 by 3 by 3.Teacher Why did you do that, John?

John Because the exponent told me to multiply the base three times.

Teacher Great, and why is the answer not positive 27, I thought when multiplying negatives theanswer was positive?

John A negative multiplied by a negative is positive, but then a positive times a negative

switches the answer back to a negative.


3/16


3

Always, Sometimes or NeverThe Always, Sometimes or Never (ASN) strategy encourages students to synthesize information and think about a range

of possibilities in a given context. ASN particularly helps students to see mathematical trends, as it forces them to think

through a number of different constructs. ASN is ideally used toward the end of a lesson (during the independent

practice and closing) after students have had time to master the foundational material.

If students are struggling to begin to formulate an answer for an ASN question, it may be best to prompt their thinking

by providing a concrete example that is contradictory to the trend.

A sample dialogue might look like this

Teacher Is it always, sometimes or never true that when you square a number is gets bigger?

Sandra I do not understand.

Teacher Lets start by picking a number Sandra.

Sandra Okay, I pick 4. And 4 is 16, so yes it is bigger.Teacher What about 7?Sandra Well, 7 is going to be 49. Again bigger.Teacher What if I chose 0.5.Sandra If I do (0.5) then I get 0.25 ah wait, that is smaller.Teacher Excellent Sandra, so when will the result be bigger and when will it be smaller?

Sandra Squaring a number makes it bigger only if it is a whole number. Squaring fractions will

make them smaller.


4/16


4

Same or DifferentThe Same or Different questioning strategy lets students think through multiple representations of the same value or

information. This strategy helps students see that even though a value or problem may look dissimilar to your example,

it can be approached the same way. Same or Different is also a powerful method to help students to see equality. This

strategy can be used during any part of the lesson to break students out of the mold of rote thinking.

Same or Different can take a visual or algebraic approach. Below are two examples.

Visual Approach

Which of these represent the same value?

Algebraic Approach

Which of these does not equal ?A. B. C. ()D.


5/16


5

Find the ErrorThe Find the Error questioning strategy asks students to find an error in sample work or to explain why an answer may

be incorrect. Find the Error presses students to think through their own solution process as well as become familiar with

how others may have used a different strategy to solve the same problem. Find the Error is most powerful at

highlighting common errors that students may make. Find the Error is best used during the guided or independentpractice and the closing of the lesson, after students are comfortable with their own solution approach.

Depending on the readiness of students, one can increase the rigor of Find the Error problems by removing the explicit

steps containing the error, pushing students to imagine various outcomes. Below are two examples, both showcasing

the Find the Error strategy. The first asks students to spot an error in provided work. The second steps up the rigor by

asking students to make logic-leaps to predict what error was made.

Explicit Approach

Look at Janes work below.

Where did she make a mistake?

( 4)( + 2)

First: x x = x2

Outer: x 2 = 2x

Inner: -4 x = 4x

Last: -4 2 = -8

Janes Solution x2

+ 6x 8

Non-Explicit Approach

Look at the problem below.

What error did Jane make?

( 4)( + 2)

Janes Solution x2 + 6x 8


6/16


6

What the WhatThe What the What questioning strategy asks students to chart a path from a question to a given solution, in essence

filling in the missing work. This deemphasizes the numeric answer for the students, instead placing importance on

showing and explaining the solution process. The What the What strategy can be used during any part of a lesson from

the opening as an engagement strategy (How did this happen?) which activates prior knowledge to the closing wherestudents must synthesize knowledge to explain a solution to a never-before-seen problem.

For students who may struggle to get started, What the What can also be used in conjunction with a sample explanation

which is either incomplete, poorly worded, or terse to give students a jumping off point.

With Sample Explanation

A cylindrical jug filled with water is leaking at a

constant rate. After 20 minutes the jug will be as

full. Omar calculated that it would be 25 more

minutes until the jug was only full. How did hearrive at this solution?

Omar:

I first found common denominators

to find how much was leaking per

minute

Without Sample Explanation

A cylindrical jug filled with water is leaking at a

constant rate. After 20 minutes the jug will be as

full. Omar calculated that it would be 25 more

minutes until the jug was only full. How did hearrive at this solution?


7/16


7

Below is a student sample highlighting the Say Why strategy.

The student is responding to the following prompt.

Clair is having carpet installed in two rooms of her house. The dimensions of the rooms

are shown below. Which room has the larger carpet? Explain how you know.

+ 2

+ 5 3

Student Sample 1 (Say Why)


8/16


8

Below is a student sample highlighting the Say Why strategy.


Clair is having carpet installed in two rooms of her house. The dimensions of the rooms

are shown below. Which room has the larger carpet? Explain how you know.

+ 2

+ 5 3

Student Sample 2 (Say Why)


9/16


9

Below is a student sample highlighting the Always, Sometimes or Never (ASN) strategy.


Look at the two exponential functions below. Will the -intercept of function A always,sometimes or never be larger than function B?

A B

() = 3 1 () = 5 1

Student Sample 3 (Always, Sometimes or Never)


10/16


10

Below is a student sample highlighting the Always, Sometimes or Never (ASN) strategy.


Look at the two exponential functions below. Will the -intercept of function A always,sometimes or never be larger than function B?

A B

() = 3 1 () = 5 1

Student Sample 4 (Always, Sometimes or Never)


11/16


11

Below is a student sample highlighting the Same or Different strategy.


Look at the two swimming pools below. The dimensions are shown. Do these swimming

pools have the same area? How do you know?

3524

2534

Student Sample 5 (Same or Different)


12/16


13/16


13

Below is a student sample highlighting the Find the Error strategy.


Larry wants to build a fence around his square garden. He measures one side of his

garden and finds it is 7 meters long. Larry mistakenly buys 49 meters of fencing and hasa lot left over. What was Larrys mistake?

Student Sample 7 (Find the Error)


14/16


14

Below is a student sample highlighting the Find the Error strategy.


Larry wants to build a fence around his square garden. He measures one side of his

garden and finds it is 7 meters long. Larry mistakenly buys 49 meters of fencing and hasa lot left over. What was Larrys mistake?

Student Sample 8 (Find the Error)


15/16


15

Below is a student sample highlighting the What the What strategy.


Dave has been fighting with his brother and wants to divide their bedroom in half,

diagonally. Dave and his brothers room is shown below. Dave estimates he will need tobuild a pillow barricade 52 feet long. How did Dave get this?

5 ft.

5 ft.

Student Sample 9 (What the What)


16/16


16

Below is a student sample highlighting the What the What strategy.


Dave has been fighting with his brother and wants to divide their bedroom in half,

diagonally. Dave and his brothers room is shown below. Dave estimates he will need tobuild a pillow barricade 52 feet long. How did Dave get this?

5 ft.

5 ft.

Student Sample 10 (What the What)

Conceptual Questioning

Documents

Transcript of Conceptual Questioning