Problems Worth Talking About, Posing Purposeful Questions
Problems Worth Talking About, Posing Purposeful Questions Linda Venenciano, Seanyelle Yagi, Fay Zenigami Curriculum Research & Development Group University of Hawai‘i at Mānoa National Council of Teachers of Mathematics Conference April 6, 2017
1
Warm-up Driving from town A to town D, you pass first through town B and then through town C. From A to B is 10 miles farther than from B to C; from B to C is 10 miles farther than from C to D. If A is 390 miles from D, how many miles is A from B?
Rachlin, S., Matsumoto, A., Wada, L., & Dougherty, B. (2001). Problem Set 1-3, Problem 1. In Algebra I: A Process Approach (p. 13). Honolulu: University of Hawaii, Curriculum Research & Development Group
A Closer Look at the Problem How can this problem be used to initiate a classroom environment for discourse? What mathematics topics could be connected to this problem? What questions could be used to prompt class discourse?
Teacher’s Notes for the Problem The intent of this problem is to promote creating and using diagrams as a strategy for problem solving. Have students critique the reasoning driving each strategy and construct arguments for how the strategies may be connected. Since this is an introductory problem, emphasis on algebraic equations is not necessary.
Discussion Prompts
How did you get started? What solution strategies worked? Have you answered the question? How do you know if your answers are reasonable?
An Example from the Classroom
Scan for the transcript
Transcript of 8th grade student solution presentations during mathematics class. Recorded on September 2000
Problem: Driving from town A to town D, you pass first through town B and then through town C. From A to B is 10 miles farther than from B to C; from B to C is 10 miles farther than from C to D. If A is 390 miles from D, how many miles is A from B? 1 2 3 4 5 6
Presentation 1 Student 1:
That's our diagram. So we knew that from A to D was 390 miles so we divided that by three. So each thing would be 130 miles. B was 10 more miles than B to C. And from B to C was 10 more miles than C to D. For that, we just transferred 10 miles to the top.
7 8 9 10 11
divided by three – I divided 390 by 3 and that equaled 130 miles. Then A to B that's X+10 equals 130 plus 10 miles which equals 140 miles.
16 17 18
Presentation 2
19 20
Student 3:
Um, well, I got that too.
21
Teacher:
Who just said that? Rachel?
22
Student 3:
I did mine the same way as she did.
23
Teacher:
So you did a equation. You used X?
24
Student 3:
Ya
25
Teacher:
OK. Dom?
26
Presentation 3
27 28
Student 4:
I (inaudible) did one that was just a little more different. Do you want me to go up and show it?
29
Teacher:
Sure!
Student 4:
I drew a map like, A, B, C, and D. And A to D they told you that was 390 miles. And they told you, you know, B– A to B was um, B to C plus ten miles. And B to C was like the distance from C to D plus another 10 miles. And so then I just put X here, 'cause I didn't know, like, what that was.
Student 2:
From B to C I put X and then A to B, since it's 10 more miles than B to C, I put X+10 and C to D is 10 less miles so I put X-10.
30 31 32 33 34 35
Student 2:
Then I put X+10, plus X, plus X-10 which equals 390. That's from A to D. Then I put 3X+0 which equaled 390. 'Cause there's three Xs. And so I
36
12 13 14 15
Curriculum Research & Development Group * University of Hawai‘i, Mānoa Annual Meeting of the National Council of Teachers of Mathematics, April 6, 2017 Problems Worth Talking About, Posing Purposeful Questions
Transcript of 8th grade student solution presentations during mathematics class. Recorded on September 2000 37 38 39 40 41 42 43
Student 4:
And then I added them all together and I just put 1X 'cause that doesn't make any difference. I got – with the plus tens – and the plus 10 plus 10– got 3x+30=90 (sic) and to find out what the X was, I subtracted 30 from that and that and I got 3X=360. And then I divided 3 from each one and I got X=120. So I know that C to D is 120 miles.
Student 4:
And since B to C is the distance of C to D plus ten miles, I knew that that was 130 miles. And then A to B is the distance of B to C plus another 10 miles, so I knew that the distance from A to B is 140 miles.
44 45 46 47 48
49
Curriculum Research & Development Group * University of Hawai‘i, Mānoa Annual Meeting of the National Council of Teachers of Mathematics, April 6, 2017 Problems Worth Talking About, Posing Purposeful Questions
Tasks that Connect to Other Topics Lead Students to: Know mathematics as a progression of interconnected topics Review or practice previously learned concepts and skills, and preview upcoming ones Be resourceful and creative when solving a problem
6
Mathematical Tasks and Discourse Mathematical tasks and discourse were found to be “robust features” (p. 420) of the classroom that need to be considered in looking at relationships between teaching and learning. As instructional features, mathematical tasks and discourse, influence learning by affecting the kind of cognitive processes in which students engage. (Hiebert & Wearne, 1993)
7
Not all Tasks are Created Equal What are your objectives for engaging students in mathematical discourse? What qualities of a problem/task lend themselves to class discourse? How does the design of the task help address the learning goal(s) for the lesson? Do you want students to develop understanding about a concept? Memorize a procedure? Something else?
8
Not All Questions Support Class Discourse Set A
Set B
What was your answer?
How did you begin to think about this problem?
How did you get that?
How did you decide _______ ?
What were your steps?/Did you show your work?
How do our methods compare?
Do you have any questions?
What questions do you have?
Did you check your answer?
Could there be other possible (correct) answers? Why? Does this problem remind you of a problem we’ve seen before?
Teacher’s Notes for the Problem Students consider external factors as they relate to a real world experience. Have students consider how speed changes as a function of the time, and other factors such as, assuming constant rate regardless of direction. Use these ideas to help students model mathematics in different ways.
Discussion Prompts
What assumptions did you make before solving this problem? What did you need to calculate? What needed to be measured? How did you determine Barry’s speed?
Tasks that are Challenging yet Accessible Lead Students to: Engage in the mathematics via multiple entry points Employ different strategies as they work on the problem Draw on what what they know as they work through the task Look for and use a solution pathway that is not previously known or apparent Explore and understand the nature of mathematical relationships, processes, and concepts
12
Tasks that Promote Communication Lead Students to: Explain beyond procedural steps and checking answers Pose questions, make conjectures, justify, and extend the problem situation Problem solve productively and make the processes visible to all Participate actively Listen critically
14
A Good Problem Solving Task … Connects to other topics Is challenging yet accessible Promotes communication
15
Mahalo Linda Venenciano [email protected]
16
1
Warm-up Driving from town A to town D, you pass first through town B and then through town C. From A to B is 10 miles farther than from B to C; from B to C is 10 miles farther than from C to D. If A is 390 miles from D, how many miles is A from B?
Rachlin, S., Matsumoto, A., Wada, L., & Dougherty, B. (2001). Problem Set 1-3, Problem 1. In Algebra I: A Process Approach (p. 13). Honolulu: University of Hawaii, Curriculum Research & Development Group
A Closer Look at the Problem How can this problem be used to initiate a classroom environment for discourse? What mathematics topics could be connected to this problem? What questions could be used to prompt class discourse?
Teacher’s Notes for the Problem The intent of this problem is to promote creating and using diagrams as a strategy for problem solving. Have students critique the reasoning driving each strategy and construct arguments for how the strategies may be connected. Since this is an introductory problem, emphasis on algebraic equations is not necessary.
Discussion Prompts
How did you get started? What solution strategies worked? Have you answered the question? How do you know if your answers are reasonable?
An Example from the Classroom
Scan for the transcript
Transcript of 8th grade student solution presentations during mathematics class. Recorded on September 2000
Problem: Driving from town A to town D, you pass first through town B and then through town C. From A to B is 10 miles farther than from B to C; from B to C is 10 miles farther than from C to D. If A is 390 miles from D, how many miles is A from B? 1 2 3 4 5 6
Presentation 1 Student 1:
That's our diagram. So we knew that from A to D was 390 miles so we divided that by three. So each thing would be 130 miles. B was 10 more miles than B to C. And from B to C was 10 more miles than C to D. For that, we just transferred 10 miles to the top.
7 8 9 10 11
divided by three – I divided 390 by 3 and that equaled 130 miles. Then A to B that's X+10 equals 130 plus 10 miles which equals 140 miles.
16 17 18
Presentation 2
19 20
Student 3:
Um, well, I got that too.
21
Teacher:
Who just said that? Rachel?
22
Student 3:
I did mine the same way as she did.
23
Teacher:
So you did a equation. You used X?
24
Student 3:
Ya
25
Teacher:
OK. Dom?
26
Presentation 3
27 28
Student 4:
I (inaudible) did one that was just a little more different. Do you want me to go up and show it?
29
Teacher:
Sure!
Student 4:
I drew a map like, A, B, C, and D. And A to D they told you that was 390 miles. And they told you, you know, B– A to B was um, B to C plus ten miles. And B to C was like the distance from C to D plus another 10 miles. And so then I just put X here, 'cause I didn't know, like, what that was.
Student 2:
From B to C I put X and then A to B, since it's 10 more miles than B to C, I put X+10 and C to D is 10 less miles so I put X-10.
30 31 32 33 34 35
Student 2:
Then I put X+10, plus X, plus X-10 which equals 390. That's from A to D. Then I put 3X+0 which equaled 390. 'Cause there's three Xs. And so I
36
12 13 14 15
Curriculum Research & Development Group * University of Hawai‘i, Mānoa Annual Meeting of the National Council of Teachers of Mathematics, April 6, 2017 Problems Worth Talking About, Posing Purposeful Questions
Transcript of 8th grade student solution presentations during mathematics class. Recorded on September 2000 37 38 39 40 41 42 43
Student 4:
And then I added them all together and I just put 1X 'cause that doesn't make any difference. I got – with the plus tens – and the plus 10 plus 10– got 3x+30=90 (sic) and to find out what the X was, I subtracted 30 from that and that and I got 3X=360. And then I divided 3 from each one and I got X=120. So I know that C to D is 120 miles.
Student 4:
And since B to C is the distance of C to D plus ten miles, I knew that that was 130 miles. And then A to B is the distance of B to C plus another 10 miles, so I knew that the distance from A to B is 140 miles.
44 45 46 47 48
49
Curriculum Research & Development Group * University of Hawai‘i, Mānoa Annual Meeting of the National Council of Teachers of Mathematics, April 6, 2017 Problems Worth Talking About, Posing Purposeful Questions
Tasks that Connect to Other Topics Lead Students to: Know mathematics as a progression of interconnected topics Review or practice previously learned concepts and skills, and preview upcoming ones Be resourceful and creative when solving a problem
6
Mathematical Tasks and Discourse Mathematical tasks and discourse were found to be “robust features” (p. 420) of the classroom that need to be considered in looking at relationships between teaching and learning. As instructional features, mathematical tasks and discourse, influence learning by affecting the kind of cognitive processes in which students engage. (Hiebert & Wearne, 1993)
7
Not all Tasks are Created Equal What are your objectives for engaging students in mathematical discourse? What qualities of a problem/task lend themselves to class discourse? How does the design of the task help address the learning goal(s) for the lesson? Do you want students to develop understanding about a concept? Memorize a procedure? Something else?
8
Not All Questions Support Class Discourse Set A
Set B
What was your answer?
How did you begin to think about this problem?
How did you get that?
How did you decide _______ ?
What were your steps?/Did you show your work?
How do our methods compare?
Do you have any questions?
What questions do you have?
Did you check your answer?
Could there be other possible (correct) answers? Why? Does this problem remind you of a problem we’ve seen before?
Teacher’s Notes for the Problem Students consider external factors as they relate to a real world experience. Have students consider how speed changes as a function of the time, and other factors such as, assuming constant rate regardless of direction. Use these ideas to help students model mathematics in different ways.
Discussion Prompts
What assumptions did you make before solving this problem? What did you need to calculate? What needed to be measured? How did you determine Barry’s speed?
Tasks that are Challenging yet Accessible Lead Students to: Engage in the mathematics via multiple entry points Employ different strategies as they work on the problem Draw on what what they know as they work through the task Look for and use a solution pathway that is not previously known or apparent Explore and understand the nature of mathematical relationships, processes, and concepts
12
Tasks that Promote Communication Lead Students to: Explain beyond procedural steps and checking answers Pose questions, make conjectures, justify, and extend the problem situation Problem solve productively and make the processes visible to all Participate actively Listen critically
14
A Good Problem Solving Task … Connects to other topics Is challenging yet accessible Promotes communication
15
Mahalo Linda Venenciano [email protected]
16
