1. Raising the ‘Cognitive Demand’ & Effective Questioning
Introductions
Purpose of the day
Norms for discussion
February Bus Custer Professional Development
Grade 7 & 9 Mathematics

2. 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?
Two-three minutes of small group conversations;
Then have an open discussion; Q&A: facilitators circulate the room;

3. The Cognitive Demand Spectrum
Memorization
Tasks that
require
memorized
procedures
in routine
ways
Procedures Without Connections
to understanding, meaning or concepts
Procedures With Connections
to understanding meaning & concepts
Doing Math
Tasks that require
engagement with
concepts, and stimulate
students to make
connections to meaning,
representation, and other
mathematical ideas
Stress the idea “Connections” refers to students actually making connections to other concepts, meanings and representations; other mathematical ideas – drawing connections between concepts studied
Stress that it is ideal to have a balance between all of these task types: The key is to diversify instruction so that students are always being challenged cognitively, and so that you are utilizing a combination of procedures with, without, and truly engagaing mathematical problems.
Memorization Tasks: reproducing previously learned facts, rules, formulas, definitions; exact reproduction of previously seem material; no connection to the concepts/meaning that underlie the facts, rules, definitions etc.
Procedures Without Connections: algorithmic; procedure use is evident based on prior instruction or placement of task; limited cognitive demand for successful completion; no connection to the concepts that underlie the procedure;
Procedures With Connections: Focus on the use of procedures to further develop a deeper understanding of concepts/meaning; usually represented in multiple ways; procedures have close connections to underlying conceptual ideas, as opposed to narrow algorithms;
Doing Math: complex, non-algorithmic thinking; requires exploration into the nature of the concepts/relationships; requires students to access relevant knowledge and experiences and make appropriate use of them; requires considerable cognitive effort and may involve some anxiety due to the unpredictable nature of solution process required;
Supporting Teachers in Their Classrooms to Increase Student Achievement
David Foster, National Supervisors of Mathematics; Atlanta 2007

4. Problems Solved by Explicitly Using Processes
Making Connections
Problems Solved by Explicitly Using Processes
Particular attention should be brought to the comparison of the US (Canada – although we consistently score higher on assessments) and Japan – results from a TIMSS Video Study; (Trends in International Math and Science Study)
The data shows percentage of tasks/problems that are solved by either showing results, procedures only, stating concepts –fact recall, or making connections;
Idea is that we point out the substantially high percentage of procedural tasks on US assessments;
The idea is to point out other countries who use high percentages of tasks that require students to draw upon, and make connections through the learning process;
Supporting Teachers in their
Classrooms to Increase
Student Achievement.
David Foster, Director
Silicon Valley Math Initiative
CMC-N Admin. Conference
Asilomar, Dec. 2006

5. 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?

6. Martha’s Bedroom vs. Rabbit Pens
ALIKE
• Both require Area and
Perimeter calculations
• Both require students to “explain your thinking”
• Both are word problems, set
in a “real world” context
Different
• Rabbit Pens requires a systematic approach
• Rabbit Pens leads to
generalization and
Justification
• The “thinking” in Rabbit Pens is complex - requires more than applying a memorized
formula

7. ‘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.’
If a teacher says that he/she does “problem-solving’ with their students, what exactly might this mean?
Debrief the various activities that groups feel qualify as problem-solving activities.
In order for students to experience ‘problem-solving, the activities must facilitate learning opportunities where students have the opportunity to struggle for a solution using a method of their choice.

8. Memorization
What are the decimal and percent equivalents for the fractions:
and
Openly discuss the relevance of this problem and its categorization as Memorization; - Fact recall – no connections to the meaning of fractions; other representations, the deeper understanding behind student knowledge;

9. Procedures ‘Without’ Connections
Convert the fraction to a decimal and a percent.
Procedures ‘With’ Connections
Openly discuss the difference between these two tasks with regards to the connections that the students must make – representations etc.
Using a 10x10 grid, identify the decimal and percent equivalents of 3/5

10. 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.
“Explain how to determine,” is a critical element of checking for student understanding – not necessarily all about getting to a solution but the process of explaining how one would/could solve a problem should be valued as equally if not more important to the learning process;

11. NCTM - Principals and Standards for School Mathematics
“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

12. 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?
Work on the following problem individually at first, then move into small groups to discuss strategy and solutions.
Encourage participants to solve the problem using as many different strategies as possible.
Once enough time has been given, and participants have identified a variety of solution methods, select a sampling of participant work to display on overhead, chart paper, Board etc.
Discussion:
Re: solution strategies/methods
Emphasize: Good problems can be solved in many different ways and they lead to building connections and understanding of mathematical concepts.
Discussion: What, in your mind, constitutes a ‘good’ math problem?
- debrief and validate
Team Leaders should administer this problem with their class in order to practice modeling effective questioning and teaching through problem solving; We can reflect on this experience at our next Leader session; Facilitators can also draw upon their experiences with their classroom use of this problem during their day of facilitation;

13. BREAK

14. Good Math Problems Require estimation, calculation and reasoning
Open-ended; many possible solution strategies
Inviting to students who are apprehensive about mathematics
Students want to find solutions
There are no readily available procedures to find these solutions
Students must ‘risk-take’ and make an attempt to solve the problem

15. 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.”
We solve problems everyday, on a regular basis; at work, at home etc.
In the mathematical context, problem solving usually refers to those application problems at the end of a chapter or lesson that requires students to apply a procedure to a real-world application question.
Solving problems based on real-life situations can strengthen students’ understanding of concepts and help them make sense of abstract ideas…Helps Build Connections!

16. 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?’
Have the participants debrief the idea of problem-solving and building connections while discussing the provided two questions.
Ask for volunteer suggestions after time is given for group discussion; create a list on over-head, chart-paper, board etc.
Stress: the idea of being able to integrate one’s previous experience, current knowledge, and intuition into the problem-solving process is key.
Some ideas to add to the list if not brought up:
1) students must understand how the problem is stated/presented
2) students must be familiar with multiple ways of solving the problem
3) students should have past experience with problem-solving or even similar problems
4) students can identify various intermediate steps needs before a final solution can be determined.

17. 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.
By presenting ‘rich,’ situational problems, we ask students to combine concepts and skills and to develop an understanding of the overall relationship between these concepts and skills.
Classroom atmosphere is critical; students must feel comfortable to explore strategies and to share strategies with others.
Students must learn to value multiple perspectives.
Time allotment to the problem-solving process is delicate – different students will require different amounts of time – differentiation
Must refrain from rescuing the students!
Create a culture of problem-solving within the classroom by providing many opportunities for students to practice problem solving.

18. Teaching & Learning ‘Through’ Problem-Solving
What is the mass of a fully loaded Boeing 747?
Have the participants answer the following problem in small groups and then discuss with a larger group table.
Discuss solutions and have volunteers justify or explain their reasoning behind their problem-solving process and solutions.
The Key:
Ask participants what mathematical concepts and procedures are involved with this problem? What could be the Big Idea(s) of the content behind this problem?
Ask participants, if they were to give this problem to their students, how would THEY connect their students’ thinking to the mathematical content that was intended to be taught through this problem?
Another key idea (s): What is good about this problem?
How would you solve it?
Emphasize the process of problem solving as opposed to only valuing the product;
Students can share their rationale, learn strategies from other students; all/most students have an entry point into the problem solving process;

19. What constitutes a ‘Rich’ math problem?
“Rich” Math Problems
What constitutes a ‘Rich’ math problem?
A Definition
As an entire group, brainstorm ideas that arise from this question; initiate conversation around rich, situational problems. – chart paper – have participants come up with their ideas/thoughts regarding ‘rich’ math problems;
After discussion, refer to handout and site the reference as a definition of a rich problem that seemed to be universal throughout our research. All encompassing definition.
Discuss the definition and get feedback from the participants.
“Solving a rich, situational problem involves mathematical reasoning as well as the development and implementation of strategies that require students to use their prior knowledge. In so doing, they must perform a series of operations in order to decode, model, verify, explain and validate information. This is a dynamic process that involves anticipating results, retracing certain steps in the problem-solving procedure and exercising critical judgment.
These situations require the students to choose and combine a number of previously acquired concepts and processes. The instructions or constraints do not divide the work into sub-questions or subtasks, nor do they suggest the processes or concepts to be used.
A context is provided for the problem in order to interest and motivate the students and help them draw upon their prior knowledge or experience related to the situation.”
Stress that this/these definitions are just samples of the research that exists regarding rich problems;

20. “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
“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
Just more research on Rich Problems

21. 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?
Have participants attempt this problem through a “Think aloud” with a partner; Explain how one would go about solving this problem – discuss strategy and solution process;
Then discuss afterwards:
Would you consider this a ‘rich’ problem? Explain.
2) Does this problem fit into your current curriculum? If so, where?
3) What would be the big ideas or intended math content that you would want to get out of this problem?
4) What concepts would have been covered before this problem was given to students?
5) Where would you be going with your class after this problem is debriefed?
6) How would you debrief this lesson with your students; How would you build connections to the intended math concepts and procedures during this debriefing?
Reinforce that not every concept can be introduced “through” problem solving; Stress that there are multiple uses and intentions of rich problems:
- reinforcement problems; extension problems; assessment problems – beginning, middle, end of unit etc.

22. LUNCH

23. Effective Questioning
Raising the Cognitive Demand
Traditionally, it is in our nature to want to help our students find success; sometimes the most effective technique in which we can do this is the most time consuming, challenging, and risky – effective questioning!!

24. 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?
As a whole group, discuss this question and get feedback; initiate conversation around this question.
We use questioning as a tool to help students reflect on their learning, assess student understanding of a problem or concept, and to help students make progress on a problem without rescuing or giving them the answers.
Students use questioning for:
clarification or to further understanding
to question themselves in order to reflect on their understanding
to help pursue new ideas and to broaden their understanding
Initiate conversation regarding the second question; ask for participant volunteers and sharing.

25. 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

26. 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.
Have the participants, in pairs, think of a teaching moment or question where the questions that were asked were done so merely to get the answers on the table – said aloud.
Have them write the actual question down on paper and think of some of the questions that their practice would naturally bring out. Have them write these questions down too.
Discuss the nature of questions that they would have asked for their example. Have each pair create questions that require a higher order of thinking – raise the cognitive demand of their lesson.
Possible examples to initiate this task:
Grade 7 – ½ + 2/3 Questions: Add these two fractions.
What is the common denominator?
What do I do next? What’s the answer? Are we good?
Grade 9 – (x+2)(x-3) Questions: Multiply these binomials.
What do I do first? Then what? Do I leave all the terms at the end? What must we do with them?
What’s the final answer? Are we all good?

27. 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?
Model this mini-lesson exemplifying Direct Instruction; help the participants see themselves in the instruction and delivery of content. Closed questions, at the overhead; leading questions with one and only one answer for each.
Keep the pace quick; as each question is asked, look for one person to provide an answer and then move on.
Discuss with the entire group their observations of the mini-lesson; Could they see themselves in any part of the instruction?
Debrief the last question with the group, having participants share their thoughts on how they could make this mini-lesson more engaging, more of a learning opportunity etc.

28. 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

29. 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?
Re-teach the mini-lesson using effective questions that serve to engage students, clarify student thoughts, and refocus students who volunteer answers or suggestions.
The Goal – model effective questioning technique and intended purpose for each question that is asked;
Get to the deeper understanding and the big ideas of the problem;
Stress ‘equality statements’ and maintaining the truthfulness of the statement; stress variables, solving for unknowns, coefficients etc.

30. Examples: Engaging Refocusing Clarifying
“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?”

31. Looking at Assessment Questions
Complete the griney grollers activity
(see handout)
Please after question #7.
Discuss questions 1-7 as a group
STOP
What types of questions were found in #1- 7?
What was their purpose? (if any)
How meaningful were they in assessing student understanding and/or learning?
Now, continue with #8 thinking about how we can ask questions that increase the cognitive demand on our students.
Using this example, what types of questions could we ask that would promote higher levels of student thinking and understanding?

32. 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”

33. 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?
Discuss both questions with the group together; give them time to share in small groups and then debrief as a larger group together.
Ask the group to dive into their textbooks and consider layout and formatting such as the DTM, Communicating Key Ideas, CYU Chapter problems, Extend and Assessment questions etc.
Questions to prompt discussion and thought: if your students do not demonstrate an understanding of the intended concepts after the DTM, could we integrate a rich problem to reinforce the concept discovery assuming students have gained some problem solving strategies from the DTM?
Could there be other uses for a rich problem throughout the textbook sections? Let’s think extension, summative assessment/formative assessment, differentiation (curriculum compacting) etc.

34. Textbook Activity: A Focus on Understanding
Grade 7 – DTM: Sections 1.3 & 1.4
Grade 9 – DTM: Sections 3.1 & 3.4
Discussion
Provide the group time to complete their respective DTM sections.
During the discussion, address what the group believes the purpose of the DTM is?
What would be the big ideas that you would want your students to get from this activity?
What key questions could you ask in order to get a better understanding of whether or not your students grasped the Big Ideas?
What could you do if you realize that your students didn’t grasp the intended purpose of the DTM?
Attempt to bring forth the “Assessment For Learning” piece.
Draw out the value of Rich Problems and their use in this situation.
Reinforce the fact that Rich problems have several key intensions: 1) reinforcement of concepts / strategies
2) solidify deeper understanding of concepts
3) an extension of learning

35. DTM & Rich Problems Grade 7: Grade 9:
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?
‘Model-Teach’ the provision and administration of a Rich problem for both grade levels that corresponds to the particular sections of the DTM. (3-part lesson)
Have participants complete the problem independently first, and then discuss in small groups. Circulate around the room asking effective questions – engaging, refocusing, clarifying
Debrief:
Discuss the strategies that participants used; ask for volunteers to share work;
Introduce intended terminologies and procedures/concepts
- Ask group to imagine that they assigned this problem to their students. What could be explored after this problem: ex: grade 7 – Patterning, Relationships, Divisibility, # Sense
grade 9 – Equality, inequalities, intersections, programming?
Refer the group to the CYU questions and have then explore questions: grade 7 – 11, 13, 19, 24 (pg )
grade 9 – 8, 13, 15, 16 (pg )
Suggest that the sequence of the CYU questions was strategically selected; The Goal: KEEP THE COGNITIVE DEMAND HIGH throughout the lesson or series of lessons
It is understood that not every student will be capable of doing all of these assigned questions… Discuss the importance of the Differentiation piece and how our textbooks support differentiation strategies;
Stress how the textbook is set up to support all of the different dynamics in your classroom -the textbook itself differentiates your instruction – not all students must do every question, the same questions etc.; lead into the differentiation piece of instruction;

36. 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?
Ask participants if Rich problems can be found in our textbooks? Can we make problems that we like, Rich? How? In this explanation, consider the criteria for what makes a problem rich. If the CYU questions do not have a Rich problem, change one or two to make it so; Have the participants examine the textbook to see whether it supports this type of instruction and learning; Ask participants to find cognitively demanding problems, determine where they fit within the outcomes, and to find other questions that could support the intended content, but at various levels on complexity;
Could we use the Cognitive Demand Spectrum and find questions that we could use for a specific section that would fall under each category along the spectrum. Could we adjust a procedural task to make it more of a ‘problem?’
Determine what has to be done in order to prepare students for learning through problem solving; How do we help students to not resort to “I don’t get it!” when presented with challenging problems or struggle in mathematics?
How is the text book structured in support of raising the cognitive demand in your classroom? Dynamics are certainly quite different classroom to classroom; How does the text help teachers keep the demand high on all students, no matter what their ability levels are? Tell the participants to be prepared to share their lesson (s)
Overall Day Debrief:
Other sources for Rich Problems? Classroom implementation (the how-to piece); The use of effective questioning while teaching ‘through’ problem solving; How will these activities impact your daily classroom teaching? If at all!!!
How could these techniques impact student learning? Student success?
The END!