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Middle School Performance Tasks and Student Thinking for Mathematics. CFN 609 Professional Development | March 29, 2012 RONALD SCHWARZ Math Specialist, America’s Choice,| Pearson School Achievement Services. Got change?.

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middle school performance tasks and student thinking for mathematics

Middle SchoolPerformance Tasks andStudent Thinking for Mathematics

CFN 609Professional Development | March 29, 2012

RONALD SCHWARZMath Specialist, America’s Choice,| Pearson School Achievement Services

got change
Got change?

Try to figure out a way to make change for a dollar that uses exactly 50 coins. Is there more than one way?

performance tasks and student thinking
Performance Tasks and Student Thinking


  • Reflection
  • Progressions
  • Properties
  • Misconceptions
  • Tasks
  • Looking at Student Work
practice and experience
Practice and Experience

Have you tried out any of the strategies, tasks or ideas from our previous sessions, and if so, what were the results?

reflection and review
Reflection and Review

What are one or two ways that US math instruction differs from that in higher-performing countries?

reflection and review1
Reflection and Review

Describe and list some of the Standards for Math Practice.

standards for mathematical practice
Standards for Mathematical Practice

1 Make sense of problems and persevere in solving them.

2 Reason abstractly and quantitatively.

3 Construct viable arguments and critique the reasoning of others.

4 Model with mathematics.

5 Use appropriate tools strategically.

6 Attend to precision.

7 Look for and make use of structure.

8 Look for and express regularity in repeated reasoning.

expertise and character
Expertise and Character

Development of expertise from novice to apprentice to expert

The Content of their mathematical Character

reflection and review2
Reflection and Review

How are levels of cognitive demand used in looking at math tasks?

levels of cognitive demand
Levels of Cognitive Demand


  • Memorization
  • Procedures without connections


  • Procedures with connections
  • Doing mathematics
reflection and review3
Reflection and Review

What do we mean by formative assessment and what are some strategies involved in it?

some habits of mind
Some Habits of Mind
  • Visualization, including drawing a diagram
  • Explanation, using their own words
  • Reflection and metacognition
  • Consideration of strategies

A train one mile long travels at a rate of one mile per minute through a tunnel that is one mile long. How long will it take the train to pass completely through the tunnel?

some more habits of mind
Some More Habits of Mind
  • Listening to each other
  • Recognizing and extending patterns
  • Ability to generalize
  • Using logic
  • Mental math and shortcuts
two jars
Two Jars

You have a 3-liter jar and a 5-liter jar. Neither of them have any markings and you do not have any extra jars. You can easily measure out exactly 3, 5 and 8 liters. Is it possible to measure out exactly 1 liter? If so, how? What about 4 liters? 6 liters? 7 liters?


A tree doubled in height each year until it reached its maximum height in 20 years. How many years did it take this tree to reach half its maximum height?

some strategies for approaching a task
Some Strategies for Approaching a Task
  • Make an organized list
  • Work backward
  • Look for a pattern
  • Make a diagram
  • Make a table
  • Use trial-and-error
  • Consider a related but simpler problem
express this sum as a simple fraction in lowest terms
Express this sum as a simple fraction in lowest terms:

1_+ 1_+ 1_+ 1_+ 1_+ 1_

1×2 2×3 3×4 4×5 5×6 6×7

and some more strategies
And Some More Strategies
  • Consider extreme cases
  • Adopt a different point of view
  • Estimate
  • Look for hidden assumptions
  • Carry out a simulation
tennis tournament
Tennis Tournament

A tennis tournament has 50 contestants, with these rules: no tie games and the loser of each game is eliminated, the winner goes on to play in the next round. How many games are needed to determine a champion?

the most important ideas in the ccss mathematics that need attention
The most important ideas in the CCSS mathematics that need attention

Properties of operations: their role in arithmetic and algebra

Mental math and (algebra vs. algorithms)

Units and unitizing

Operations and the problems they solve




Practice Standards

fractions progression
Fractions Progression

Understanding the arithmetic of fractions draws upon four prior progressions that inform the CCSS:

  • Equal partitioning
  • Unitizing
  • Number line and
  • Operations
unitizing links fractions to whole number arithmetic
Unitizing links fractions to whole number arithmetic
  • Students’ expertise in whole number arithmetic is the most reliable expertise they have in mathematics
  • It make sense to students
  • If we can connect difficult topics like fractions and algebraic expressions to whole number arithmetic, these difficult topics can have a solid foundation for students
nine properties are the most important preparation for algebra
Nine properties are the most important preparation for algebra

Just nine: foundation for arithmetic

Exact same properties work for whole numbers, fractions, negative numbers, rational numbers, letters, expressions.

Same properties in 3rd grade and in calculus

Not just learning them, but learning to use them

using the properties
Using the properties
  • To express yourself mathematically (formulate mathematical expressions that mean what you want them to mean)
  • To change the form of an expression so it is easier to make sense of it
  • To solve problems
  • To justify and prove
properties are like rules but also like rights
Properties are like rules, but also like rights
  • You are allowed to use them whenever you want, it’s never wrong.
  • You are allowed to use them in any order
  • Use them with a mathematical purpose
linking multiplication and addition the ninth property
Linking multiplication and addition: the ninth property

Distributive property of multiplication over addition

a × (b+c) = (a×b) + (a×c)

a(b+c) = ab + ac

find the properties in the multiplication table
Find the properties in the multiplication table

There are many patterns in the multiplication table, most of them are consequences of the properties of operations:

Find patterns and explain how they come from the properties

Find the distributive property patterns

what is an explanation
What is an explanation?

Why you think it’s true and why you think it makes sense.

Saying “distributive property isn’t enough, you have to show how the distributive property applies to the problem.

mental math
Mental Math

72-29= ?

In your head

Composing and decomposing

Partial products

Place value in base 10

Factor x2 +4x + 4 in your head


If three bananas are worth two oranges, how many oranges are 24 bananas worth?


How they arise and how to deal with them

misconceptions about misconceptions
Misconceptions about misconceptions
  • They weren’t listening when they were told
  • They have been getting these kinds of problems wrong from day one
  • They forgot
  • Their previous teachers didn’t know the math
more misconceptions about the cause of misconceptions
More misconceptions about the cause of misconceptions
  • In the old days, students didn’t make these mistakes
  • They were taught procedures
  • They weren’t taught the right procedures
  • Not enough practice
  • Teachers’ misconceptions perpetuated to another generation (where did the teachers get the misconceptions? How far back does this go?)
  • Mile-wide inch-deep curriculum causes haste and waste
  • Some concepts are hard to learn
whatever the cause
Whatever the Cause
  • When students reach your class they are not blank slates
  • They are full of knowledge
  • Their knowledge will be flawed and faulty, half baked and immature; but to them it is knowledge
  • This prior knowledge is an asset and an interference to new learning
dividing fractions1
Dividing Fractions

“Ours is not to question why,

just invert and multiply.”






hose a takes 30 minutes to fill a tub with water1
Hose A takes 30 minutes to fill a tub with water.

What fraction of an hour is that? How many tubs could hose A fill in one hour?

Suppose another hose could fill a tub in twenty minutes. What fraction of an hour is that? How many tubs could that hose fill in one hour?
A third hose takes forty minutes to fill a tub. What fraction of an hour is that? How many tubs could the third hose fill in one hour?
What is the connection between the amount of time it takes for a hose to fill a tub, and the number of tubs it can fill in one hour?

The relationship between the time it takes for a hose to fill a tub, and the number of tubs that can be filled in one hour is called a reciprocal relationship. Each is also called the multiplicative inverse of the other.


Compare the relationship between a number and its multiplicative inverse with the relationship between a number and its additive inverse. What is similar?

a 30 minutes b 45 minutes

A: 30 Minutes B: 45 Minutes

Hose A takes 30 minutes to fill a tub with water. Hose B can do the same in 45 minutes. If you use both hoses, how long will it take to fill a tub?

stubborn misconceptions
Stubborn Misconceptions
  • Misconceptions are often prior knowledge applied where it does not work
  • To the student, it is not a misconception, it is a concept they learned correctly…
  • They don’t know why they are getting the wrong answer
second grade
Second grade

When you add or subtract, line the numbers up on the right, like this:



Not like this



third grade
Third Grade

3.24 + 2.1 = ?

If you “Line the numbers up on the right “ like you spent all last year learning, you get this:

3.2 4

+ 2.1

You get the wrong answer doing what you learned last year. You don’t know why.

Teach: line up decimal point.

Continue developing place value concepts


Frequently, a ‘misconception’ is not wrong thinking but is a concept in embryo or a local generalization that the pupil has made. It may in fact be a natural stage of development.

Malcolm Swan

teach from misconceptions
Teach from misconceptions
  • Most common misconceptions consist of applying a correctly-learned procedure to an inappropriate situation.
  • Lessons are designed to surface and deal with the most common misconceptions
  • Create ‘cognitive conflict’ to help students revise misconceptions
    • Misconceptions interfere with initial teaching and that’s why repeated initial teaching doesn’t work
key features of the a well designed intervention
Key features of the a well-designed intervention

Lean and clean lessons that are simple and focused on the math to be learned

Rituals and routines that maximize student interaction with the mathematics

Emphasis on students, student work, and student discourse

Teaches and motivates how to be a good math student

Assessment that is ongoing and instrumental in promoting student learning

malcolm swan example
Malcolm Swan example
  • Goldilocks problems that lead to concepts through work on misconceptions (faulty prior knowledge)
  • Discussion craftily scaffolded
  • Instructional assessment on all cycles, especially within lesson
  • Tasks easy as possible to engage as activities that also hook straightaway to questions that lead to concept
  • “encouraged uncertainties” at the door of insights
odd one out1
Odd One Out
  • a) 20, 14, 8, 2…
  • b) 3, 7, 11, 15…
  • c) 4, 8, 16, 32…
social and meta cognitive skills have to be taught by design
Social and meta-cognitive skills have to be taught by design
  • Beliefs about one’s own mathematical intelligence
    • “good at math” vs. learning makes me smarter
  • Meta-cognitive engagement modeled and prompted
    • Does this make sense?
    • What did I do wrong?
  • Social skills: learning how to help and be helped with math work => basic skill for algebra: do homework together, study for test together
diagnostic teaching1
Diagnostic Teaching

Goal is to surface and make students aware of their misconceptions

Begin with a problem or activity that surfaces the various ways students may think about the math.

Engage in reflective discussion (challenging for teachers but research shows that it develops long-term learning)

Reference: Bell, A. Principles for the Design of Teaching Educational Studies in Mathematics. 24: 5-34, 1993

pearson professional development
Pearson Professional Development


RONALD SCHWARZ, facilitator