Discourse and Mathematical Argumentation in the Classroom

Wednesday, June 6, 2012

Computing seems to be an instructional priority despite the fact that the field of mathematics goes far beyond procedures. At its core, mathematics involves exploring, arguing, and justifying.We need to find a way to go beyond procedures in order to teach for conceptual understanding.

8 Standards for Mathematical

PracticeMake sense of problems and persevere in solving them.Reason abstractly and quantitatively.Construct viable arguments and critique the reasoning of others.Model with mathematics.Use appropriate tools strategically.Attend to precision.Look for and make use of structure.Look for and express regularity in repeated reasoning.

Discourse that Promotes Conceptual

Understanding by Elham Kazemi

3 things that are clearer to you regarding discourse, mathematical argumentation, and sociomathematical norms

2 connections to your own classroom1 question or point you would like to discuss further

Kazemi’s Main Points

Discuss in pairs…Kazemi wrote that “press for learning was measured by the degree to which teachers

emphasized students’ effort

focused on learning and understanding

supported students’ autonomy

emphasized reasoning more than producing correct answers.”

What does mathematical argumentation look like in a classroom?How can we incorporate mathematical argumentation into our lessons?

My Research Study

Mathematical Context: Arithmetic PropertiesGoals:

Identify and describe the characteristics of students’ mathematical arguments related to the arithmetic propertiesQuantify growth when instruction promoted mathematical argumentation

Two Fourth-Grade ClassroomsOne had more opportunities for argumentation and an emphasis on explaining and exploring.One followed the traditional textbook and worksheets.

Depth of Knowledge

Traditional Textbook Tasks

Modified-Lesson TasksIf you take any number and multiply it by zero, you will always get a number larger than 100. Do you agree or disagree? Explain.

5 + 1 = _____ + 5

“What should go in the blank so that the following number sentence is true? Who can explain why their answer is correct?”

Teacher Moves and Sociomathematical

NormsLanguage Frames(Ross, Fisher, &

Frey, 2009)I believe _____ (statement) because _____ (justification).

Based on _____, I think _____.

I noticed _____, when _____.

I disagree with ____ because ____.

Sentence Starters (Loper and Baker, 2009)

What do you think?

What is your claim?

Why do you think that?

What is your evidence?

My claim is…

What teacher moves do you notice?

My Teacher MovesI had 23 teacher moves but only six of them

occurred before mathematical arguments:repeats or restatesinvites ideas or observationsinvites students to agree or judgeprompts for justification, explanation, or convincingprompts for elaboration or clarificationgives a false claim

Facts about my Teacher Moves

The most commonly used teacher move (repeating or restating) did not occur before many arguments.The teacher move that most frequently occurred before a student argument was a prompt for justification or explanation (about 65%) even though it was not a common teacher move overall (occurring only about 7 % of the time).

Mathematical Errors

What are some ideas about teacher moves for a mathematical error introduced by a student?

What have you done in the past?

What are your ideas about introducing false claims?

What might have been a good response to the student’s false

claim?

What are the students’ reactions to my false claim?

What could I have done next to follow up?

Did including mathematical

argumentation make a difference in student

learning?Pre- and Post-Assessment DataIndividual interview assessments with

students in both classrooms focused on:CorrectnessComputational vs. Number Pattern StrategiesRecognition of a Property

CorrectnessPercentage of students whose score improved

Percentage of students whose score stayed the

same

Percentage of students whose score decreased

Focus Classroom (N=18)

67% 33% 0%

Traditional Classroom (N=13) 31% 38% 31%

Computation vs. Number Pattern Strategies

Percentage of students who shifted toward NPS

Percentage of students who shifted

toward C

Percentage of students who did not

shift

Focus Classroom (N=18)

83% 6% 11%

Traditional Classroom (N=13)

31% 23% 46%

Pre- and Post- Assessment Results Summary

Some students in both classrooms showed improvement on the assessment. Students in the focus classroom applied their understanding of properties more frequently than in the contextual classroom.Students in the focus classroom applied more number pattern strategies to the assessment tasks.

Discussion Points

What are the benefits and disadvantages of including mathematical argumentation? What could be done to alleviate the disadvantages?How can you include mathematical argumentation in your knowledge package topic?

Depth of Knowledge and Standards for

Mathematical PracticeIn order to teach at higher levels of the DOK framework, we need to be including mathematical argumentation and justification.

Mathematical argumentation is one of the 8 Standards for Mathematical Practice.

What are the characteristics of tasks aligned with standards for mathematical practice, high DOK levels, and mathematical argumentation?

http://vimeo.com/30924981

HomeworkRead What exactly do “fewer, clearer, and higher standards” really look like in the classroom? Using a cognitive rigor matrix to analyze curriculum, plan lessons, and implement assessment.

Karin K. Hess, Dennis Carlock, Ben Jones, John R. Walkup

Reflection