2. “When parents and teachers alike believe
that hard work pays off and when
mathematics is taught and learned by
using all the strands of proficiency
mathematics performance improves for
all students.”
Helping Children Learn Mathematics
(2002) Kilpatrick and Swafford

3. At The Crossroads—
Meeting the Challenges of a Changing World —
ISM 2012

4. The world we know is
changing
 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.
 Business and industry want
employees with these skills! OECD
ISM 2012

5. Our Challenge
 The top 10 in-demand jobs in 2010 did not
exist in 2004.
 Estimated that today's learner will have 10-14
jobs by the age of 38.
 The amount of new technical information is
doubling every 2 years For students starting
a 4 year technical degree this means that half
of what they learn in their first year of study
will be outdated by their third year of study.
ISM 2012

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.” APEC
ISM 2012

7. Scenario
My child doesn’t know his number
facts. Why aren’t we teaching the
basics and teach REAL math.
ISM 2012

8. Myths and Facts
Why Has Mathematics
Instruction Changed?
ISM 2012

9. Myth Fact
 Some people  Every adult and
just can’t do every child can
math. do mathematics.
ISM 2012

10. Myth Fact
 Literacy is  Literacy is far
reading and broader, includin
writing. g an
understanding of
graphs, tables, s
tatistics, probabil
ity...
ISM 2012

11. Myth Fact
 Students learn  Students learn
math best math best when
through they are active
teacher- participants in
directed the learning
lessons. process.
ISM 2012

12. Myth Fact
 The use of  When used
calculators appropriately, cal
will have a culators improve
negative computation
impact on skills.
students’
computation
skills.
ISM 2012

13. Myth Fact
 Manipulatives  Manipulatives
are for young help connect
abstract thought
children.
to something
tangible ---
for learners of
ALL ages.
ISM 2012

14. Myth Fact
 The school  Rigor does not
equate to
program is not
difficulty.
rigorous.
ISM 2012

15. What is mathematics?
Mathematics is a way of thinking about,
understanding, explaining, and
expressing phenomena.
Mathematics is about inquiry and
insight. Computation is (usually) a
means to an end.SM 2012
I

16. ISM 2012

17. How fluent are you in
mathematics?
Add 346 and 484
62
- 38
1½ x 2¼ 2 ÷⅜
43 X 37 196 ÷ 18
ISM 2012

18. Reading Fluency
Efficiency
FLUENCY
Prosody Accuracy
Fluency is the ability to read with sufficient ease and accuracy that
one can focus attention on the meaning and message of text.”
Adams, 2002
ISM 2012

19. MATH FLUENCY (Russell, 1999)
Efficiency: Accuracy:
Student does not get A working knowledge
bogged down into too many steps of number facts, combinations,
or lose track of logic or strategy. and other important number
(WORKING MEMORY) relationships.
(AUTOMATIC RETRIEVAL)
FLUENCY
Flexibility:
Knowledge of more than
one approach to problem solve.
Allows student to choose appropriate
strategy and to double check work.
(EXECUTIVE FUNCTIONING)
ISM 2012

20. What is your number sense?
Add 346 and 484
62
- 38
1½ x 2¼ 2 ÷⅜
43 X 37 196 ÷ 18
ISM 2012

21. What is Number Sense?
A “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”
(Howden, 1989). ISM 2012

22. Learning
 The knowledge base for
classroom assessment
is closely tied to
contemporary theories
of learning -----about
how knowledge is
organized in the mind
and about how
participation in
communities of practice
shapes understanding.
How People Learn
Bransford, Brown, & Cocking,
1999, NRC
ISM 2012

23. Learning Progressions
describe skills,
understandings, and
knowledge in the
sequence in which
they typically develop:
a picture of what it
means to ‗improve‘ in
an area of learning.‖
(Masters & Forster, 1996).
ISM 2012

24. Climbing the Number Sense
Ladder
ISM 2012

25. Relevant Research
 “When students compute with
strategies… they choose because they
are meaningful …they are able to
remember and apply their knowledge.”
 Baroody, Arthur J. “The Development of Basic
Counting, Number, and Arithmetic Knowledge among Children
Classified as Mentally Handicapped.” International Review of
Research in Mental Retardation, edited by LaraineMasters
Glidden. pp.51-103. New York: Academic Press, 1999.
ISM 2012

26. How Students Learn
 “Can engage in
instructional activities
but teaching has not
occurred until student
learning has occurred“
 “…covering the
material and explaining
it well is NOT the same
as the student learning
it.”
NRC
ISM 2012

27. Learning Mathematics
 For all students to
become
mathematically
proficient, major
changes must be
made in instruction,
assessments,
teacher education,
and the broader
educational system.
◦ Adding It Up (NRC)
ISM 2012

28. ISM 2012

29. Building Mathematical Concepts
Concrete Pictorial Abstract
Manipulatives Representation Symbols
IIII 4+4=8
2x4=8
IIII
ISM 2012

30. The Bridge To Understanding
Representation/Pictorial
―SEEING‖ Stage
Concrete Abstract
―DOING‖ Stage ―SYMBOLIC‖ Stage
ISM 2012

31.  According to Piaget (1949, 1958), children
learn primarily by manipulating objects until the
age of 12.
 If children are not taught math with hands-on
methods, between years 1 and 12, their ability
to acquire math knowledge is disturbed at the
point when hands-on explorations were
abandoned in favor of abstractions. This
clearly sets them up for mathematical
disabilities in the next developmental period of
formal propositional operations.
Center for Teaching/Learning Mathematics, 1986
ISM 2012

32. Mathematical thinking . . .
A gateway to higher mathematics?
OR
A wall blocking path for
students?
ISM 2012

33. Knowing Understanding
“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.
ISM 2012

34. ISM 2012

35. Students Can Do Basics, ...
347 + 453 90% 864 – 38 73%
… But Students Cannot Solve Problems
Ms. Yost’s class has read 174 books, and Mr.
Smith’s class has read 90 books.
33% How many more books do they need to read to
reach the goal of reading 575 books?
Source: NAEP 2009
ISM 2012

36. Explain your thinking
• A furniture maker was putting stools
and chairs together. The stools were
the three- legged kind and the chairs
had four legs. If the seats were the
same for the stools and chairs, how
many of each could he make if he had
21 seats and 72 legs?

37. Explain your thinking
• 2/3 of my students play the piano.
• 1/2 of the students who play the
piano also play the violin.
• 1/4 of those students also play the
flute.
• What fraction of my students play all
three instruments?
ISM 2012

38. Math Literacy =
Just as knowing the
definitions of words
does not make a
person
literate, knowing the
rules and
algorithms to solve
mathematics
problems does not
make a person
mathematically
literate.
ISM 2012

39. Conceptual
Understanding
Procedural Problem
Fluency Solving
ISM 2012

40. ISM 2012

41. Connecting
Procedural Knowledge
with Conceptual Understanding
• What misconception(s) is
the student holding?
Measurement
• What key concept is the
Place Value student missing?
• How might this
misconception have
Operation Sense developed?
and Algebra
Decimal • What types of activities
Fractions and conversations help
Common students construct correct
Fractions ISM 2012
concepts?

42. ISM 2012

44. PROGRESSION

45. PROGRESSION

46. A Learning Progression Model:
Learning Progression: A learning
progression is a sequenced set of
subskills and enabling knowledge
that, it is believed, students must
master en route to mastering a more
remote curricular aim. (Popham 2008)
ISM 2012

47. Scenario
Parent: How can I help my child with
math? It is so different from when I
went to school.
ISM 2012

48. Yes, You Can
Help Your Child
You really can!

49. Changes
• Textbooks and learning
materials
• Types of questions posed
• Types of responses needed
• Assessment and grading
• Uses of class time

50. Yes, I can …
Talk about math in positive ways
I’m sure you will I could never do
understand if the math either.
you… Don’t worry about
Let’s figure it out the math.
together.

51. Yes, you can …
Support your child through homework
challenges by listening and asking
questions

52. Yes, I can …
• Allow my child to
struggle through the
process of problem
solving.
• Discuss mistakes as
learning opportunities.

53. Questions to Ask …
• What do you need to find out?
• Tell me what you know?
• Show me what you started?
• What can you try first?
• Can you make a drawing or picture?
• Will a list or table help?

54. Prompts …
• Ask your child to show you how he/she is
finding the answer.
• Share your method.
• Explain to each other why your different
methods are successful.
Providing shortcuts for getting the answer
might hinder deeper student understanding.

55. Manipulatives and
Technologies
Their Role in Teaching and
Learning Mathematics
ISM 2012

56. Research …
“Manipulatives support the conceptual
development of important mathematical
ideas for tactile and visual learners.”
ISM 2012

57. Research …
• Students learn in the presence of
technology.
• Technology should influence the
mathematics content taught and how it is
taught.
ISM 2012

58. ISM 2012