Developing Higher Level ThinkingAfter School PDNovember 2010

Cleaning UpObject of the race

◦Move down and around the chair and back to the start line

◦You must stay on the paper◦If you walk on the floor, you have to

start over

Warm-Up

What is π ?

Two Approaches to Teaching

Instrumental RelationalMemorizing and

routinely applying procedures and formulas

Does not focus on “why” you solve problems the way you do

Explaining, reasoning, relying on multiple representations

Helping students develop their own understanding of content

Comparing and Contrasting Instrumental vs. Relational Learning

Benefits for instrumental

understanding are:

◦usually easier to

understand

◦ the rewards are more

immediate and

apparent

◦ students receive

quicker answers due to

the fact that less

knowledge is involved

Advantages of relational

understanding include:

◦adaptable to new tasks,

◦easier to remember

(although harder to

learn, there is less re-

learning involved),

◦relational understanding

is a more natural path

to promote independent

learning

What does the research say?“Teach for meaning initially, or risk

never getting students beyond a superficial understanding that leaves them unprepared to apply their learning.”

Leinwand, Steve, and Steve Fleishchman. "Teach Mathematics Right the First Time." Educational Leadership. 62.1 (2004):

What are some easy ways to do this?Open Ended Questions

◦Can be a great way to differentiate instruction

◦Can also be a good way to assess your students

Writing Strategies◦Can help students develop meaning

and make connections◦Can also be a good way to assess

your students

Open Ended QuestionsDescribe the picture below by

using a mathematical equation

To which fact family does the fact 3x4 =12 belong to?

X X X X

X X X X

X X X X

Open Ended Question StrategiesTurn Around A QuestionAsk for similarities and

differencesReplace a number with a blankAsking for a number sentenceChange an existing question

From : Small, Marian. Good Questions: Great Ways to Differentiate Mathematics Instruction. New York: Teachers College Press, 2009. Print.

1) Turn Around A Question

Instead of asking what is the median of this set of data:

3, 5, 6, 8, 10, 11, 11

Ask, “I have a data set with 7 numbers and median of 8. What is the data set?”

2) Ask for Similarities and Differences

How are lines in standard form and lines in slope intercept form similar? How are they different?

3) Replace a number with a blankTraditional Percent question:

◦72 is 75% of what number?

Replace it with:◦Fill in values for the blanks to make

this statement true◦72 is _____ % of _____

Replacing a Number with a BlankIf s=4 and t=5, these statements

are true.3(s+t)=27 2s+3t=23 2t-

2s=2

Choose values for p and q. Write three true statements using those variables. See if a partner can figure out what your values are.

4) Asking for a number sentenceStudents are asked to create a

number sentence that includes certain words or numbers ◦Example: Create a question

involving multiplication or division of decimals where the digit 9, 4, and 2 appear somewhere

◦Example: Create a question involving the height of a triangle using the numbers 3 and 6

5) Change an existing question

Old:◦A cookie has a diameter of 1.75

inches. Express the diameter as a fraction in simplest form.

New: ◦A cookie has a diameter between 1

and 2 inches. Express the diameter as a fraction in two different ways.

Open Ended Question StrategiesTurn Around A QuestionAsk for similarities and

differencesReplace a number with a blankAsking for a number sentenceChange an existing questionTake 5 minutes and write your own open ended question that you could use in the next 2 weeks. Be ready to share with the group.

More GuidelinesMake the question meaningful—

focused on curricular goals◦Have just the right amount of ambiguity◦Students should have a general idea what

you’re looking for◦Open enough so all students can engage

with the question and have an answerFollow up discussion

◦Call on those with simpler answers first◦Value all answers◦Build connections between answers

Wrapping Things UpHow do you think open-ended

questions can fit with your curriculum?

How can you structure it so that students can share their responses?