4. Mathematical thinking . . .
A gateway to higher mathematics?
OR
A wall blocking path for
students?
Parent Session SAS 2013
5. The Future
75 % of jobs will be in STEM
Not just STEM careers,
it is STEM in every job
Technology as a “global knowledge economy” is the
future, and it requires different skills.
future
Business and industry want employees with these skills!
5
6. 21st Century Learning
“We are responsible for preparing students to address
problems we cannot foresee with knowledge that has not
yet been developed using technology not yet invented.”
Ralph Wolf
Parent Session EARJ 2013
7. Jobs of the Future
The TOP 10
jobs in 2015
are not yet
invented.
Parent Session EARJ 2013
8. 21st Century Skills Framework
Thinking and Learning Skills
•
•
•
•
Critical Thinking & Problem Solving Skills
Creativity & Innovation Skills
Communication & Information Skills
Collaboration Skills
Parent Session EARJ 2013
10. What Does It Mean
to Understand Mathematics?
Knowing ≠ Understanding
Understanding is the measure of quality and
quantity of connections between new ideas and
existing ideas
ASB MCI2
Problem Solving
11. “Understanding is the key to
remembering what is learned and
being able to use it flexibly.”
- Hiebert, in Lester & Charles,
Teaching Mathematics through
Problem Solving, 2004.
12. A Thought
“People who do not understand mathematics today are
like those who could not read or write in the industrial
age.”
Robert Moses
ASB MCI2
Problem Solving
13. The Bridge To Understanding
Representation
“SEEING” Stage
Concrete
“DOING” Stage
Abstract
“SYMBOLIC” Stage
Parent Session EARJ 2013
16. Conceptual vs. Procedural Knowledge
Conceptual (connected networks)
Knowledge and understanding of
logical relationships and
representations with an ability to talk,
write and give examples of these
relationships.
Procedural (sequence of actions)
Knowledge of rules and procedures
used in carrying out routine
mathematical tasks and the symbols
used to represent mathematics.
-- David Allen
The question of which kind of knowledge is most important is the wrong question to ask. Both kinds of
knowledge are required for mathematical expertise...
Instead, we should focus on designing teaching environments that
help students build internal representations of procedures that
become part of larger conceptual networks.
James Heibert and Tom Carpenter, Learning and Teaching with Understanding, 1992
18. Priorities in Mathematics
Grade
Priorities in Support of Rich Instruction and
Expectations of Fluency and Conceptual Understanding
K–2
Addition and subtraction, measurement using
whole number quantities
3–5
Multiplication and division of whole numbers
and fractions
6
7
8
Ratios and proportional reasoning; early
expressions and equations
Ratios and proportional reasoning; arithmetic
of rational numbers
Linear algebra
5/29/12
19. Key Fluencies
Grade
Required Fluency
K
Add/subtract within 5
1
Add/subtract within 10
Add/subtract within 20
2
3
Add/subtract within 100 (pencil and
paper)
Multiply/divide within 100
Add/subtract within 1000
4
Add/subtract within 1,000,000
5
Multi-digit multiplication
6
Multi-digit division
Multi-digit decimal operations
7
Solve px + q = r, p(x + q) = r
8
Solve simple 2×2 systems by inspection
19
20. Number Sense …
Howden (1989) described it as “good intuition about
numbers and their relationships. It develops gradually
as a result of exploring numbers, visualizing them in a
variety of contexts, and relating them in ways that are
not limited by traditional algorithms.”
.
21. Prior Understandings
2.G.2. Partition a rectangle into rows and columns of
same-size squares and count to find the total number of
them.
www.JennyRay.net
21
22. Distributive Property
& Area Models
3 x 7 =__
3
5
+
2
15
+
6
3x7=
3 x (5 + 2) =
(3 x 5) + (3 x 2)= 15 + 6 = 21
www.JennyRay.net
22
34. Where’s the Math?
Models help students explore concepts and build
understanding
Models provide a context for students to solve
problems and explain reasoning
Models provide opportunities for students to generalize
conceptual understanding
37. Multiplication – number line model
Aoife earns €12 per hour. What would she earn in 2, 3, 4,
3/4 hours?
Notice “of “ becoming multiplication
3/4 x12 = 12 x ¾ =9
38. Multiplication – Area Model
Cara had 2/5 of her birthday cake left from her party. She ate ¾ of the leftover cake. How much
of the original cake did she eat?
2/5 cake
Divide into
quarters
¾ of 2/5
3 2 6 3
× =
=
4 5 20 10
Multiplication making smaller
http://www.learner.org/courses/learningmath/number/session9/part_a/try.html
Area of 3x2 out of
area of 4x5
39. A muffin recipe requires 2/3 of
a cup of milk. Each recipe
makes 12 muffins. How many
muffins can be made using 6
cups of milk?
Adapted from Multiplicative Thinking. Workshop 1. Properties of
Multiplication and Division. http.nzmaths.com, 2010.
40. The Additive Thinker
A muffin recipe requires 2/3 of a cup of milk.
Each recipe makes 12 muffins. How many
muffins can be made using 6 cups of milk?
42. A muffin recipe requires 2/3 of a cup of milk.
Each recipe makes 12 muffins. How many
muffins can be made using 6 cups of milk?
43. Content + Practices
“The Standards for Mathematical Practice describe
varieties of expertise that mathematics educators at all
levels should seek to develop in their students. These
practices rest on important ‘processes and
proficiencies’ with longstanding importance in
mathematics education.”
(CCSS, 2010)
Parent Session EARJ 2013
45. 1
Make sense of
problems
and
persevere in
solving them
2
Reason
abstractly and
quantitatively
5
4
Model with
mathematics
Standards for
Mathematical
Practice
6
Look for and
make use of
structure
Construct
3
viable
arguments and
critique the
reasoning of
others
7
Attend to
precision
Use
appropriate tools
strategically.
8
Look for
and express
regularity in
repeated
reasoning
46. Grouping the Standards of Mathematical Practice
1. Make sense
of problems
and persevere
in solving
them.
6. Attend to
precision.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and
critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
Overarching habits of
mind of a productive
mathematical thinker.
7. Look for and make use of structure.
8. Look for and express regularity in
repeated reasoning.
William McCallum University of Arizona- April 1, 2011
Reasoning and
explaining
Modeling and
using tools.
Seeing structure
and generalizing.
47. 1. Make sense of problems and persevere in solving
Do students:
• EXPLAIN?
• ANALYZE?
• Make CONJECTURES?
• PLAN a solution pathway?
• MULTIPLE representations?
• Use DIFFERENT METHODS to check?
• Check that it all makes sense?
• Understand other approaches?
• See connections among different approaches?
48. 2. Reason abstractly and quantitatively
Do students:
• Make sense of quantities & their relationships?
• Decontextualize?
• Contextualize?
• Create a coherent representation?
• Consider units involved?
• Deal with the meaning of the quantities?
49. 3. Construct viable arguments and critique the reasoning
of others.
Do students:
Understand & use stated assumptions, definitions, and previous
results?
Analyze situations, recognize & use counterexamples?
Justify conclusions, communicate to others & respond to
arguments?
Compare the effectiveness of 2 plausible arguments?
Distinguish correct logical reasoning from flawed & articulate the
flaw?
Look at an argument, decide if it makes sense,& ask useful
questions to clarify or improve it?
Make conjectures& build a logical progression?
50. 4. Model with mathematics
Do students:
• Apply the mathematics they know everyday?
• Analyze relationships mathematically to draw conclusions?
• Initially use what they know to simplify the problem?
• Identify important qualities in a practical situation?
• Interpret results In the context of the situation?
• Reflect on whether the results make sense?
51. 5. Use appropriate tools strategically.
Do students:
• Consider available tools?
• Know the tools appropriate for their grade or course?
• Make sound decisions about when tools are helpful?
• Identify & use relevant external math Sources?
• Use technology tools to explore & deepen understanding
of concepts?
52. 6. Attend to precision.
Do students:
• Communicate precisely with others?
• Use clear definitions?
• Use the equal sign consistently & appropriately?
• Calculate accurately & efficiently?
53. 7. Look for and make use of structure.
Do students:
• Look closely to determine a pattern or structure?
• Use properties?
• Decompose & recombine numbers & expressions?
• Have the facility to shift perspectives?
54. 8. Look for and express regularity in repeated reasoning.
Do students:
• Notice if calculations are repeated?
a%
of
b
=b
• Look for general methods & shortcuts?
• Maintain process while attending to details?
• Evaluate the reasonableness of intermediate results?
%o
fa
55. Shift in Mathematics #1
Deeper Learning Fewer Concepts
•How Parents Can Help Students at Home
Students must …
Parents can …
Spend more time on fewer concepts
Know what the priority work is for the
grade level
Represent math in multiple ways
Ask, “Can you show me that in another
way?”
Apply strategies, not just get answers
Focus on how the child is tackling the
problem over what the answer is
56. Shift in Mathematics #2
Focus on Strong Number Sense and Problem Solving
•How Parents Can Help Students at Home
Students must …
Parents can …
Be able to apply strategies and use
core math facts quickly
Ask the child’s teacher what core math
facts should be practiced at home Ask
students which strategies they are
using
Compose and decompose numbers
Help children break apart and put
together numbers to make problem
solving easier
57. Shift in Mathematics #3
Focus on Communication of Thinking and Language Rich Classrooms
•How Parents Can Help Students at Home
Students must …
Parents can …
Understand why the math works—
explain and justify
Ask questions to find out whether the
child really knows why the answer is
correct
Talk about why the math works—
explain and justify
Ask children to explain how they solved
the problem and why they chose the
strategies they used
Prove that they know why and how the
math works—explain and justify
Ask children to show how they know
they have the correct solution Talk
about alternative strategies
Use academic vocabulary to explain
their reasoning and critique that of
others
Expect children to use the language of
math
Talk about math
58. Shift in Mathematics #4
Perseverance and Grappling with Mathematics
•How Parents Can Help Students at Home
Students must …
Parents can …
See mistakes as learning opportunities
Help their children use their mistakes
as windows into their thinking
Understand that there is usually more
than one way to solve a problem
Celebrate and value alternative
responses Ask, “Is there another way
to solve this?”
Spend more time solving a single
problem in a deep way
Expect fewer problems but more
writing and explaining in homework
60. •
•
•
•
•
Eighth Grade Test questions---1895 Arithmetic [Time, 1.25
hours]
1. Name and define the Fundamental Rules of Arithmetic.
2. A wagon box is 2 ft. deep, 10 feet long, and 3 ft. wide. How
many bushels of wheat will it hold?
3. If a load of wheat weighs 3942 lbs., what is it worth at
50cts/bushel, deducting 1050 lbs. for tare?
4. District No. 33 has a valuation of $35,000. What is the
necessary levy to carry on a school seven months at $50 per
month, and have $104 for incidentals?
5. Find the cost of 6720 lbs. coal at $6.00 per ton.
Parent Session EARJ 2013
61. Eighth Grade Test
•
•
•
•
•
6. Find the interest of $512.60 for 8 months and 18 days at 7
percent.
7. What is the cost of 40 boards 12 inches wide and 16 ft. long at
$20 per metre?
8. Find bank discount on $300 for 90 days (no grace) at 10
percent.
9. What is the cost of a square farm at $15 per acre, the distance
of which is 640 rods?
10. Write a Bank Check, a Promissory Note, and a Receipt
Parent Session EARJ 2013
66. A thought…
•
We can best close the achievement gap by eliminating
the opportunity gap. If we,
as mathematics teachers K-12, each make it our
personal goal for every student to have the opportunity
to learn mathematics in ways that promote the habits
of mind espoused
•
in the standards for mathematical practice,
we will be successful in helping all students to
be successful in learning and doing mathematics.
Parent Session EARJ 2013
The first alternative to the standard algorithm is multiplying using partial products. This strategy is about using decomposition and place value, two critical aspects of all mathematics.
What challenges could you see for a student using this method? (Allow time for discussion)
Here is the same partial products strategy used within an area model. Students draw a rectangle and divide given the number of digits in each number. For example 62 is divided into 60 (six tens) and 2. Once the student has each number decomposed, he then multiplies each column by each row. For example, 10 times 2 is equal to 20, while 10 times 60 is equal to 600. The answer is represented by the area of a rectangle with sides 62 and 18.
Let’s try one. Here is the problem 54 x 37. Given the number of digits in 54 and 37, what will be the dimension's of my rectangle? ( 2 by 2)
What is the next move? Decompose each number and note on the rectangle.
Click
Next we multiply each column and row.
Click to show product of each.
Finally we add all of the products together to find the answer to our initial problem.
Click
Handout #2: Susie’s SMPs
Follow along, underline or highlight words.
The paragraphs are dense, I had to separate them and make an outline so that I could understand them
(show my outline)
