ncdpi curriculum and instruction mathematics “teaching for understanding”
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NCDPICurriculum and Instruction
Mathematics
“Teaching for Understanding”
“Teaching for Understanding”
Phil DaroMath SCASS
February 12, 2013
Problem: Mile wide –inch deep curriculum
Cause: Too little time per concept
Cure: More time per topic
“LESS TOPICS”
Why do students have to do math problems?
a. To get answers because Homeland Security needs them, pronto.
b. I had to, why shouldn’t they?
c. So they will listen in class.
d. To learn mathematics.
What is learning?
• Integrating new knowledge with prior knowledge; explicit work with prior knowledge
• Prior knowledge varies across students in a class (like fingerprints); this variety is key to the solution, it is not the problem.
• Thinking in a way you haven’t thought before and understanding what and how others are thinking.
To Learn Mathematics • Answers are part of the process, they are not the
product.
• The product is the student’s mathematical knowledge and know-how.
• The ‘correctness’ of answers is also part of the process. Yes, an important part.
“Answer Getting vs. Learning Mathematics”
United States:
• “How can I teach my kids to get the answer to this problem?”
Japan:
• “How can I use this problem to teach the mathematics of this unit?”
“The Butterfly Method”
Discussion
• How might these ideas challenge teachers in your district or school?
• How can we move from “answer getting” to “learning mathematics”?
• What evidence do you have that teachers might not know the difference?
Blogstop.com
“Faster Isn’t Smarter”by
Cathy Seeley
“Hard Arithmetic is not Deep Mathematics”
p. 83
“Hard Arithmetic is not Deep Mathematics”
• What issues or challenges does this message raise for you?
• In what ways do you agree or disagree?
• What barriers might keep students from reaching these standards, and how can you tackle these barriers?
Instructional Task I
• What rectangles can be made with a perimeter of 30 units? Which rectangle gives you the greatest area? How do you know?
• What do you notice about the relationship between area and perimeter?
Instructions
• Discuss the following at your table
– What thinking and learning occurred as you completed the task?
– What mathematical practices were used?
– What are the instructional implications?
Compared to….
5
10
What is the area of this rectangle?
What is the perimeter of this rectangle?
“Who’s doing the talking, and who’s doing the math?”
Cathy Seeley, former president, NCTM
The Mathematical Practices develop character: the pluck and persistence needed to learn difficult content. We need a classroom culture that focuses on learning…a try, try again culture. We need a culture of patience while the children learn, not impatience for the right answer. Patience, not haste and hurry, is the character of mathematics and of learning.
How can we move from “answer getting” to “learning
mathematics”?
“Modeling in Mathematics”
by
CCSSO and Math SCASS(Council of Chief State School Officers)
(The State Collaborative of Assessment and Student Standards)
What is modeling? A word with different meanings
1. “Modeling a Task”
- An instructional strategy where the teacher shows step by step actions of how to set up and solve the task
Mathematical Task:
2 + ___ = 8
Use step by step actions to “model” how to solve this
task
What is modeling? A word with different meanings
2. “Model with Manipulatives”
- Start with the math then use manipulatives to demonstrate and understand how to solve the problem.
mathToothpicks as
a model
What is modeling? A word with different meanings
3. “Model with Mathematics”
- Start with the task and choose an appropriate mathematical model to solve the task
Four birds sat on a wire, 2 flew away. How many birds
remain on the wire?
Choose a grade appropriate mathematical model to solve the task:e.g. writing the number
sentence 4 – 2 = 2
What is modeling? A word with different meanings
4. “A Model with Mathematics”
What is modeling? A word with different meanings
1. “Modeling a Task”
2. “Modeling with Manipulatives”
3. “Model with Mathematics”
4. “A Model with Mathematics”
What is modeling? A word with different meanings
1. “Modeling a Task”
2. “Modeling with Manipulatives”
3. “Model with Mathematics”
4. “A Model with Mathematics”
What makes something a modeling task?
• Are there criteria for “modeling tasks”?
• What are the skills involved?
Task 932: (Unpublished)
How well posed is well enough posed?
• Should a student still have questions after they read the task?
• Should students have to find their own information outside of what is given in the problem?
• Should assumptions be stated, or reasoned differently by each individual student?
Five Problems to Ponder
• Painting A Barn
• The Ice Cream Van
• Birthday Cakes
• Graduation
• Sugary Soft Drinks
The Barn
Task 85: Ice Cream Van N-Q.A.1
Would all the birthday cakes eaten by all the people in Arizona in one year fit inside the University of Phoenix football stadium?
Cody Patterson Original
Birthday Cakes
Graduation
Sugary Soft Drinks
How many packets of sugar are in a 20 ounce bottle of soda?
http://threeacts.mrmeyer.com/sugarpackets/
Collecting and Selecting Information
1. All and only
relevant information
is given
Images: http://www.dbarn.net/, http://blog.pinkcakebox.com/25th-birthday-cake-2007-09-15.htm, www.NYC.gov, Dan Meyer, http://balfour.rbe.sk.ca/node?page=3, http://www.realmagick.com/ice-cream-vans/
5. Determine what
information is needed and
find the information
yourself
4. Given information,
but you decide what
is useful
2. Brainstorm what you need and then are
given it
3. Told what you need, you go
and find it
Matching Activity
• Match each task with a “Collecting and Selecting Information” description.
• Place them in order as they should appear on a continuum based on;
– What information is needed?
•
Collecting and Selecting Information
1. All and only
relevant information
is given
Images: http://www.dbarn.net/, http://blog.pinkcakebox.com/25th-birthday-cake-2007-09-15.htm, www.NYC.gov, Dan Meyer, http://balfour.rbe.sk.ca/node?page=3, http://www.realmagick.com/ice-cream-vans/
5. Determine what
information is needed and
find the information
yourself
4. Given information,
but you decide what
is useful
2. Brainstorm what you need and then are
given it
3. Told what you need, you go
and find it
Collecting and Selecting Information
What information is needed?
1
543 2
Find the information needed.
1
53
4
2
“All Around the School”
A class was studying metric and customary measurement, comparing quantities of one unit of measure to quantities in the other. (2003)
Question: If all the students in the school hold hands, will they create a chain long enough to circle the school?
Compared To……
Our school is 485 meters around. There are 535 students in the school, and the average arm span of a child is 2 meters. Can we circle the school if we hold hands and make a human chain?
Lunch
Blogstop.com
“Faster Isn’t Smarter”by
Cathy Seeley
“Constructive Struggling”p. 88
What is learning?
• Integrating new knowledge with prior knowledge; explicit work with prior knowledge
• Prior knowledge varies across students in a class (like fingerprints); this variety is key to the solution, it is not the problem.
• Thinking in a way you haven’t thought before and understanding what and how others are thinking.
16 3 =
• What concept is addressed in this situation?
• What strategies could be used to develop conceptual understanding?
Show 15 3 =
1. As a multiplication problem
2. Equal groups of things
3. An array (rows and columns of dots)
4. Area model
5. In the multiplication table
6. Make up a word problem
Show 15 3 = 1. As a multiplication problem (3 x [ ] = 15 )2. Equal groups of things: 3 groups of how many
make 15?3. An array (3 rows, ? columns of 3 make 15?)4. Area model: a rectangle has one side = 3 and
an area of 15, what is the length of the other side?
5. In the multiplication table: find 15 in the 3 row6. Make up a word problem
Show 16 3 = 1. As a multiplication problem
2. Equal groups of things
3. An array (rows and columns of dots)
4. Area model
5. In the multiplication table
6. Make up a word problem
“Who’s doing the talking, and who’s doing the math?”
Cathy Seeley, former president, NCTM
The Mathematical Practices develop character: the pluck and persistence needed to learn difficult content. We need a classroom culture that focuses on learning…a try, try again culture. We need a culture of patience while the children learn, not impatience for the right answer. Patience, not haste and hurry, is the character of mathematics and of learning.
Blogstop.com
“Faster Isn’t Smarter”by
Cathy Seeley
“Faster Isn’t Smarter”p. 93
PersonalizationThe Tension:
personal (unique) vs. standard (same)
Standards are a Peculiar Genre
1. We write as though students have learned approximately 100% of what is in preceding standards. This is never even approximately true anywhere in the world.
2. Variety among students in what they bring to each day’s lesson is the condition of teaching, not a breakdown in the system. We need to teach accordingly.
3. Tools for teachers…instructional and assessment…should help them manage the variety.
Why Standards?
• Social Justice
• Good curriculum for all students
• Start with the variety of thinking and knowledge students bring
• On-grade learning in the cluster of standards
• Extra time and attention outside of class time.
Minimum degree of varying prior knowledge in the average classroom
Student AStudent BStudent CStudent D Student E
Lesson START Level
CCSS Target Level
Degree of prior knowledge in the average classroom
Student AStudent BStudent CStudent D Student E
Planned time
Needed time
Lesson START Level
CCSS Target Level
Student AStudent BStudent CStudent D Student E
I - WE - YOU
Lesson START Level
CCSS Target Level
CCSS Target
Student AStudent BStudent CStudent D Student E
I - WE - YOU
Lesson START Level
CCSS Target
Answer-Getting
You - We – I Instruction based on prior
knowledge
Student AStudent BStudent CStudent D Student E
Lesson START Level
Formative Assessment
Day 2Target
Four Levels of Learning
I. Highest Standard: Understand well enough to explain to others
II. Good enough Standard: Understand enough to learn the next related concepts
III. Low Standard: Can get the answers
IV. No Standard: Noise
Four levels of learningThe truth is triage, but all can prosper
I. Understand well enough to explain to othersAs many as possible, at least 1/3
II. Understanding enough to learn the next related concepts
Most of the rest
III. Can get the answers without understanding Sometimes we have to settle for low, but don’t aim low
IV. NoiseAimless
Teach at the speed of learning• Not faster
• More time per concept
• More time per problem
• More time per student talking
• Fewer problems per lesson
Blogstop.com
“Faster Isn’t Smarter”by
Cathy Seeley
“Crystal’s Calculator”p. 159
Illustrative Mathematics
Example Problemsillustrativemathematics.org
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