elementary school performance tasks and student thinking for mathematics
DESCRIPTION
Elementary School Performance Tasks and Student Thinking for Mathematics. CFN 609 Professional Development | April 2, 2012 RONALD SCHWARZ Math Specialist, America’s Choice,| Pearson School Achievement Services. Got change?. - PowerPoint PPT PresentationTRANSCRIPT
Slide 1
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
CFN 609Professional Development | April 2, 2012
RONALD SCHWARZMath Specialist, America’s Choice,| Pearson School Achievement Services
Elementary SchoolPerformance Tasks andStudent Thinking for Mathematics
Slide 2
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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?
2
Slide 3
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Performance Tasks and Student Thinking
AGENDA• Reflection• Progressions• Properties• Misconceptions• Tasks• Looking at Student Work
3
Slide 4
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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?
Slide 5
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Reflection and Review
What are one or two ways that US math instruction differs from that in higher-performing countries?
Slide 6
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Reflection and Review
Describe and list some of the Standards for Math Practice.
Slide 7
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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.
7
Slide 8
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Expertise and Character
Development of expertise from novice to apprentice to expert
The Content of their mathematical Character
Slide 9
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Reflection and Review
How are levels of cognitive demand used in looking at math tasks?
Slide 10
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Levels of Cognitive DemandLower-level• Memorization• Procedures without
connectionsHigher-level• Procedures with connections• Doing mathematics
10
Slide 11
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Reflection and Review
What do we mean by formative assessment and what are some strategies involved in it?
Slide 12
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Some Habits of Mind
• Visualization, including drawing a diagram
• Explanation, using their own words
• Reflection and metacognition• Consideration of strategies
Slide 13
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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?
13
Slide 14
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Some More Habits of Mind
• Listening to each other• Recognizing and extending
patterns• Ability to generalize• Using logic• Mental math and shortcuts
Slide 15
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Getting Comfortable with
Non-Routine Problems
15
Slide 16
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Pleasure in Problem-Solving
16
Slide 17
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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
17
Slide 18
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Grid Lock
18
Slide 19
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
And Some More Strategies
• Consider extreme cases• Adopt a different point of view• Estimate• Look for hidden assumptions• Carry out a simulation
19
Slide 21
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Bananas
If three bananas are worth two oranges, how many oranges are 24 bananas worth?
21
Slide 23
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Nine properties are the most important preparation for algebra
Just nine: foundation for arithmeticExact 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
Slide 24
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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
Slide 27
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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
Slide 28
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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.
Slide 29
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Mental Math
72-29= ?In your headComposing and decomposingPartial productsPlace value in base 10Factor x2 +4x + 4 in your head
Slide 30
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
NYSED Grade 4 Sample Tasks
Slide 31
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Misconceptions
How they arise and how to deal with them
Slide 32
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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
Slide 33
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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
Slide 34
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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
Slide 35
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Place Value in Grade 2
Slide 36
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Dividing Fractions
36
12
13
÷
Slide 37
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Dividing Fractions
“Ours is not to question why,just invert and multiply.”
37
12
13
÷
Slide 38
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Dividing Fractions
WHY?38
12
13
÷
Slide 39
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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
Slide 40
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Second grade
When you add or subtract, line the numbers up on the right, like this:
23+9
Not like this 23+9
Slide 41
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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
Slide 42
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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
Slide 43
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Multiplying makes a number bigger.
Slide 44
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Percents can’t be greater than 100%.
Slide 45
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Graphs are pictures.
Slide 46
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
To multiply by ten, put a zero at the end of the number.
Slide 47
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Numbers with more digits are bigger.
Slide 48
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
x means the number that you have to find.
Slide 49
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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
Slide 50
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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
Slide 51
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
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
Slide 52
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Math Tasks
52
Slide 53
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Sources ofNon-Routine
Problems
53
Slide 54
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Slide 55
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Slide 56
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Slide 57
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Slide 58
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Slide 59
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Sources ofNon-Routine
Problems
59
Slide 60
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Math Olympiad for Elementary and Middle School
60
Slide 61
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Looking at Student Work
61
Slide 62
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Student Work
62
Slide 63
Cop
yrig
ht ©
20
10
Pearso
n E
du
catio
n, In
c.or its a
ffilia
te(s). A
ll righ
ts rese
rved
.
Pearson Professional Development
pearsonpd.com
RONALD SCHWARZ, [email protected]