Adding Critical Thinking to the Curriculum at the Elementary Level
Elementary School Performance Tasks and Student Thinking for Mathematics
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Transcript of Elementary School Performance Tasks and Student Thinking for Mathematics
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CFN 609Professional Development | April 2, 2012
RONALD SCHWARZMath Specialist, America’s Choice,| Pearson School Achievement Services
Elementary SchoolPerformance Tasks andStudent Thinking for Mathematics
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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?
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Performance Tasks and Student Thinking
AGENDA• Reflection• Progressions• Properties• Misconceptions• Tasks• Looking at Student Work
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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?
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Reflection and Review
What are one or two ways that US math instruction differs from that in higher-performing countries?
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Reflection and Review
Describe and list some of the Standards for Math Practice.
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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.
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Expertise and Character
Development of expertise from novice to apprentice to expert
The Content of their mathematical Character
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Reflection and Review
How are levels of cognitive demand used in looking at math tasks?
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Levels of Cognitive DemandLower-level• Memorization• Procedures without
connectionsHigher-level• Procedures with connections• Doing mathematics
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Reflection and Review
What do we mean by formative assessment and what are some strategies involved in it?
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Some Habits of Mind
• Visualization, including drawing a diagram
• Explanation, using their own words
• Reflection and metacognition• Consideration of strategies
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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?
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Some More Habits of Mind
• Listening to each other• Recognizing and extending
patterns• Ability to generalize• Using logic• Mental math and shortcuts
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Getting Comfortable with
Non-Routine Problems
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Pleasure in Problem-Solving
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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
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Grid Lock
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And Some More Strategies
• Consider extreme cases• Adopt a different point of view• Estimate• Look for hidden assumptions• Carry out a simulation
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Bananas
If three bananas are worth two oranges, how many oranges are 24 bananas worth?
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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
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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
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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
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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.
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Mental Math
72-29= ?In your headComposing and decomposingPartial productsPlace value in base 10Factor x2 +4x + 4 in your head
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NYSED Grade 4 Sample Tasks
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Misconceptions
How they arise and how to deal with them
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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
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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
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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
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Place Value in Grade 2
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Dividing Fractions
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÷
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Dividing Fractions
“Ours is not to question why,just invert and multiply.”
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12
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÷
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Dividing Fractions
WHY?38
12
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÷
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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
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Second grade
When you add or subtract, line the numbers up on the right, like this:
23+9
Not like this 23+9
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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
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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
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Multiplying makes a number bigger.
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Percents can’t be greater than 100%.
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Graphs are pictures.
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To multiply by ten, put a zero at the end of the number.
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Numbers with more digits are bigger.
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x means the number that you have to find.
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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
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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
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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
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Math Tasks
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Sources ofNon-Routine
Problems
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Sources ofNon-Routine
Problems
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Math Olympiad for Elementary and Middle School
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Looking at Student Work
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Student Work
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Pearson Professional Development
pearsonpd.com
RONALD SCHWARZ, [email protected]