Raising the ‘Cognitive Demand’ & Effective

Questioning

February Bus Custer Professional Development

Grade 7 & 9 Mathematics

The Cognitive Demand• Let’s define ‘Cognitive Demand’ as it relates to classroom

instruction.

• Are our classroom lessons grounded on cognitively high demanding activities? If so, what does that look like?

• What types of things would students be doing in a cognitively demanding learning environment?

The Cognitive Demand Spectrum

Memorization

Tasks thatrequirememorizedproceduresin routineways

Procedures Without Connections

to understanding, meaning or concepts

Procedures With Connections

to understanding meaning & concepts

Doing Math

Tasks that requireengagement withconcepts, and stimulatestudents to makeconnections to meaning,representation, and othermathematical ideasSupporting Teachers in Their Classrooms to Increase Student Achievement

David Foster, National Supervisors of Mathematics; Atlanta 2007

Making ConnectionsProblems Solved by Explicitly Using Processes

Supporting Teachers in theirClassrooms to IncreaseStudent Achievement.David Foster, DirectorSilicon Valley Math InitiativeCMC-N Admin. ConferenceAsilomar, Dec. 2006

Consider These Two Area Problems

Martha’s Carpeting Task

Martha is re-carpeting her bedroom, which is 6 meters long and 4 meters wide. How many square meters of carpet will she need to purchase, and how much baseboard will she need to run around the edge of the carpet? Explain your thinking.

The Fencing Task

Ms. Brown’s class will raise rabbits for their science fair. They have 24 meters of fencing with which to build a rectangular pen. If Ms. Brown’s students want their rabbits to have as much room as possible, how long should each of the sides of the pen be? Explain your thinking.

For each problem, what kind of thinking is required?

Martha’s Bedroom vs. Rabbit Pens

ALIKE

• Both require Area andPerimeter calculations

• Both require students to “explain your thinking”

• Both are word problems, setin a “real world” context

Different

• Rabbit Pens requires a systematic approach

• Rabbit Pens leads togeneralization andJustification

• The “thinking” in Rabbit Pens is complex - requires more than applying a memorizedformula

‘Problems’ vs. Exercises

• In groups, discuss what you believe it means to ‘problem-solve’ with your students.

• Consider the range of activities that may be considered ‘problem-solving.’

Memorization

What are the decimal and percent equivalents for the fractions:

2

1and

4

1

Procedures ‘Without’ Connections

• Convert the fraction to a decimal and a percent. 5

3

Procedures ‘With’ Connections

• Using a 10x10 grid, identify the decimal and percent equivalents of 3/5

Doing Mathematics

Shade 6 small squares in a 4 x 10 rectangle. Using the rectangle, explain how to determine each of the following:

a) the percent of area that is shaded.

b) the decimal part of area shaded.

c) the fractional part of area shaded.

“Problem solving is an integral part of all mathematics learning. In everyday life and in the workplace, being able to solve problems can lead to great advantages. However, solving problems is not only a goal of learning mathematics but also a major means of doing so. Problem solving should not be an isolated part of the curriculum but should involve all Content Standards…”

NCTM - Principals and Standards for School Mathematics

http://standards.nctm.org/document/chapter3/prob.htm

The Locker Problem• A school has 1,000 lockers and 1,000 students. The students decide to

have fun one day, so they take turns opening and closing the lockers, according to the following plan.

• The first student opens every locker.

• The second student closes every second locker.

• The third student opens every third closed locker.

• The fourth student closes every fourth open locker.

• The students continue in this manner until all 1,000 students have

had their turn.

• When all the students are finished, how many lockers remain open?

BREAK

Good Math Problems

• Require estimation, calculation and reasoning

• Open-ended; many possible solution strategies

• Inviting to students who are apprehensive about mathematics

The Idea Behind ‘Problem-Solving’

• Focus is on: Reasoning, Engagement, & the Struggle

• Students are required to think of the situation that is presented & seek answers that make sense.

• In the classroom, ‘problem-solving’ connects the real-world to mathematics.

• Promotes a deeper understanding of math and the real world.

• Video – Annenberg Series – “Problem-Solving.”

Discussion & Debrief

• What factors influence our ability to solve math problems?

• What can we do as teachers, to help students make connections, and to increase their success with ‘problem-solving?’

Teaching ‘Through’ Problem-Solving• Students learn math concepts and procedures ‘through’

solving problems.

• Process:

(1) Identify intended content you want to teach

(2) Select a problem that requires consideration of content

(3) After it is solved, connect student thinking to math

content you are teaching.

• Must refrain from rescuing the students while in the ‘problem-solving’ process.

Teaching & Learning ‘Through’ Problem-Solving

• What is the mass of a fully loaded Boeing 747?

“Rich” Math Problems

• What constitutes a ‘Rich’ math problem?

• A Definition

“Rich problems have multiple entry points, force students to think outside the box, may have more than one solution, and open the way to new territory for further exploration.”

NSDL – Middle School Portal

http://msteacher.org/epubs/math/math11/problems.aspx

“Good problems give students the chance to solidify and extend their knowledge and to stimulate new learning. Most mathematical concepts can be introduced through problems based on familiar experiences coming from students' lives or from mathematical contexts.

NCTM – Principals and Standards for School Mathematics

http://standards.nctm.org/document/chapter3/prob.htm

Teaching ‘Through’ Problem Solving

• Bill’s Snow Plow can plow the snow off the school’s parking lot in 4 hours. Jane’s plowing company can plow the same parking lot in just 3 hours. How long would it take Bill and Jane to plow the school’s parking lot together?

LUNCH

Effective Questioning

Raising the Cognitive Demand

What Role does Questioning play in our

daily lessons?

What can teachers do to encourage students to use inquiry to reflect on their learning and deepen their understanding of math concepts?

Effective Questioning:• Helps control the flow of information

• Keeps students focused on important mathematical ideas

• Helps students make sense of mathematics

• Moves discussions from discrete, unrelated responses to in-depth dialogue

• Supports and encourages student thinking

* Ultimately, effective questioning helps to raise the ‘cognitive demand’ placed on our students during our daily lessons

Discussion: What is the Purpose?

• One can categorize questions in many ways, but an important part of questioning is:

The Teacher’s Purpose for asking the questions

• That is, what is the teacher trying to accomplish?

• In partners, think about a specific problem where questions are designed merely to get answers ‘on the table.’

• Jot down the example and some questions that may be asked.

• Now create questions requiring higher level thinking for your example…raise the cognitive demand of your questions.

Examining a Lesson

• Model the mini-lesson:

• Discuss your observations

• How could this mini-lesson be modified in order to increase it’s effectiveness? Student Engagement?

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Classifying Questions based on Purpose

• Engaging: invite students into a discussion; keep them engaged in conversation; invite

them to share their work, or get answers on the table

• Refocusing: help students get back on track or move away from a dead-end strategy

• Clarifying: help students explain their thinking or help you understand their thinking

Re-examining A Lesson:

• Discuss your Observations

• Is there a ‘grey-area’ with respect to the intended purpose of the questions asked?

• Reflect on your own practice, would an observer in your class know the intended purpose for each of the questions that you ask?

Examples:Engaging

• “What strategies might we use to solve this problem?”

• “If you wanted to graph this function, how would you label the axis?”

Refocusing

• Similar Figures – “What does it mean for two figures to be similar?”

• Ratio – “What quantities are you comparing?”

Clarifying

• “How did you figure out your answer?”

• “Why did you start with that number?”

Looking at Assessment Questions

• Complete the griney grollers activity

• (see handout)

• Please after question #7.

• Discuss questions 1-7 as a group

STOP

Review Handouts: (Your Questioning Toolbox)

• “Supporting Teachers in Asking Questions and Choosing Tasks”

• “Developing Mathematical Thinking with Effective Questions”

• “Questions that Probe Understanding”

• “Motivating Every Student Through Effective Questioning”

• “Summary of Questioning Techniques”

Looking Deeper into our Resources

• How is our textbook set up to support effective questioning & learning and teaching ‘through’ problem-solving?

• How could we implement teaching ‘through’ problem-solving as a strategy to supplement the layout of our textbook?

Textbook Activity: A Focus on Understanding

• Grade 7 – DTM: Sections 1.3 & 1.4

• Grade 9 – DTM: Sections 3.1 & 3.4

• Discussion

DTM & Rich Problems• Grade 7:

What is the least number that is divisible by 7 and has a remainder of 1 when divided by each of the following numbers 2, 3, 4, 5, & 6 (separately)?

• Grade 9:

Mr. Rose invested $3500 in two business enterprises. In one enterprise, he made a profit of 6%; in the other, he suffered a loss of 4%. His net profit for the year was $160. How much did he invest in each enterprise?

Textbook Activity

• Explore a unit or section of the textbook that you are currently teaching or will be teaching soon.

• Explore the questions that are provided through the lens of ‘cognitively demanding’ tasks/problems.

• Can we adjust procedural tasks to make them cognitively more demanding?

• Does the textbook provide questions with varying levels of complexity? Can all students be challenged at their own level?