Transcript Document

Number Sense and Mental Math
A Small but Mighty Instructional Task
SIM London Region
February 2015
Number Sense
• What does it mean to have Number Sense?
• How do you develop Number Sense?
Person #1
Person #2
Number
Sense
Person #4
Person #3
Number Sense and Numeration.
Page 8
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Number sense refers to a general understanding of number and operations as well
as the ability to apply this understanding in flexible ways to make mathematical
judgements and to develop useful strategies for solving problems. In this strand,
students develop their understanding of number by learning about different ways
of representing numbers and about the relationships among numbers. They learn
how to count in various ways, developing a sense of magnitude. They also develop
a solid understanding of the four basic operations and learn to compute fluently,
using a variety of tools and strategies.
A well-developed understanding of number includes a grasp of more-and-less
relationships, part-whole relationships, the role of special numbers such as five
and ten, connections between numbers and real quantities and measures in the
environment, and much more.
Experience suggests that students do not grasp all of these relationships
automatically. A broad range of activities and investigations, along with guidance
by the teacher, will help students construct an understanding of number that
allows them to make sense of mathematics and to know how and when to apply
relevant concepts, strategies, and operations as they solve problems.
Best Evidence Synthesis on Effective
Pedagogy in Mathematics
Effective mathematical pedagogy is a coherent
system rather than a set of discrete,
interchangeable strategies. This pedagogical
system encompasses:
• A non-threatening classroom environment
• Instructional tasks
• Tools and representations
• Classroom discourse
Effective Pedagogy in Mathematics/Pangarau by Glenda Anthony
& Margaret Walshaw, New Zealand (2007)
Effective Teaching and Learning
The learning of mathematics has been defined to
include the development of five interrelated
proficiencies that, together, constitute
mathematical proficiency (NRC 2001):
• Conceptual understanding
• Procedural fluency
• Strategic competence
• Adaptive reasoning
• Productive disposition
Number Sense – Mental Math – Number Talks
Is there a connection?
Four Components of the
Pedagogical System
Mathematical Proficiencies
• Non-threatening classroom
environment
• Instructional task
• Tools and representations
• Classroom discourse
1.
2.
3.
4.
5.
Conceptual understanding
Mathematical fluency
Strategic competence
Adaptive reasoning
Productive disposition
Mathematical
Processes
•
•
•
•
problem solving
reasoning and proving
reflecting
selecting tools and
computational
strategies
• connecting
• representing
• communicating
What are Number Talks?
- 5 to 15 minute classroom
conversations
- built around purposefully
crafted computation problems
The problems are designed to
elicit specific strategies that focus
on number relationships.
The problems are solved mentally.
Your Turn
How would you mentally solve this problem - no
paper or pencil
26 + 60
25 + 66
126 + 60
125 + 66
Key Components of Number Talks
- Classroom environment &
community
- Classroom discussions
- Teacher's role
- Role of mental math
- Purposeful computation problems
Classroom discussions
Students clarify their own thinking,
consider and test other strategies to see if
they are logical, investigate and apply
mathematical relationship, build a
repertoire of efficient strategies, make
decisions about choosing efficient
strategies for specific problems
Teacher's role
A shift in our role from “sage on the
stage” to “guide on the side”. Much
different from how we may have been
taught math. Teacher is a learner
alongside the students. Gain insights
into alternative strategies.
Role of mental math
- focus on number relationships and use these
relationships to develop efficient, flexible
strategies with accuracy. Solving problems
mentally puts the emphasis on using what they
know and understand about numbers as well as
how they are related.
-to help strengthen students’ understanding of
place value. Initially problems are written
horizontally to encourage students to think of
numbers as whole quantities instead of columns
of numbers
Number Talks p.13
Purposeful computation problems
This is an essential part of number
talks. The teacher’s goals and
purposes for the number talk should
determine the numbers and
operations that are chosen. This
involves thoughtful and careful
planning and selection of examples
of number talks.
Let’s Try some Multiplication
35 X 20
35 X 24
32 X 15
For multiplication the ability to break numbers apart in
flexible ways is even more important than in addition or
subtraction.
Students need a repertoire of models for multi-digit
multiplication. Introduce different representations (one at
a time) as ways to explore multiplication until you are
comfortable that the class has a collection of useful
ideas.
At the same time do not force students who reason
very well without drawings to use models when they
are not needed.
Van de Walle p.114
An array model is as important to multiplication and
division as the number line model is to addition and
subtraction.
The visual representation of rows and columns helps
students as they develop their proportional reasoning.
The array identifies the parts (factors) and the whole
(total are of product) and can be used to demonstrate
and prove student strategies.
NT p.233
http://www.eworkshop.on.ca/edu/resources/guides/NSN_vol_4_Division.pdf
Page 20 http://www.eworkshop.on.ca/edu/resources/guides/Guide_Math_K_6_Volume_5.pdf
From Jo Boaler
• Number sense is inhibited by over-emphasis on
the memorization of math facts in classrooms
and homes
• Math facts are a very small part of mathematics
but unfortunately students who don’t memorize
math facts well often come to believe that they
can never be successful with math and turn away
from the subject
• Math anxiety and the blocking of working
memory are associated with time tests to
measure number facts
Fluency Without Fear: Research Evidence on the Best Ways to Learn Math
Facts by Jo Boaler (youcubed)
Why do we do NUMBER TALKS?
Students need to:
-have the ability to reason about
quantitative information
-possess a sense of number
-check for the reasonableness of
solutions
Fluency with numbers is
knowing how a number can
be composed and
decomposed and using that
information to be flexible and
efficient with solving
problems.
It is critical that students learn to
represent numbers in as many ways
as they can. Each of these
representations broadens the
meaning of the number for the
student. Numerals are only one form
of representation.
Marian Small, Making Math Meaningful p. 139
Conservation of number and one-to-one
correspondence are essential foundations in
building number sense for young children.
Conservation of number is the
understanding that the quantity of a
given number of objects remains the
same regardless of how it is spatially
arranged.
Number Talks by Sherry Parrish p.37
Even when the arrangement has
been changed, a child who has
conservation of number will still
recognize that he still has five
marbles without having to recount.
Number Talks by Sherry Parrish p.37
How many dots do you see?
How do you see them?
Grade 1 FI Classroom June 2013
These were the student guesses.
How many dots do you see?
How do you see them?
Examples of other Dot
Images to make 7
SUBITIZING - to immediately recognize a
collection of objects as a single unit.
e.g. looking at 5 pips on a die and immediately
recognizing it as 5
Using five- and ten-frames, or rekenreks
provide opportunities for children to build
recognition of numbers and their parts.
Number Talks by Sherry Parrish p.39
Numbers are most meaningful
to a student when they are
related to anchor or benchmark
values that are well understood.
Marian Small, Making Math Meaningful p. 139
FIVE FRAMES
TEN FRAMES
Counting All
Counting On
http://www.ronblond.com/MathGlossary/Division01/Rek
enrek/REKENREK/
How many beads do you see?
How do you see them?
How many beads do you see?
How do you see them?
How many beads do you see?
How do you see them?
CHECKING BACK - REFLECTION
Number Sense and Mental Math
How does it connect with:
• the curriculum expectations?
• the pedagogical principles?
• the mathematical proficiencies?
• the task features?
• the mathematical processes?
• developing a positive mindset?
• Number Sense and Numeration page 8?
What does the Curriculum Say?
Full Day Learning
Grade 1
Grade 2
Grade 3
-demonstrate an
understanding of
numbers, using
concrete materials to
explore and
investigate counting,
quantity, and number
relationships
– solve problems
involving the addition
and subtraction of
single-digit whole
numbers, using a
variety of mental
strategies (e.g., one
more than, one less
than, counting on,
counting back,
doubles)
-solve problems
involving the addition
and subtraction of
whole numbers to
18, using a variety of
mental strategies
(e.g.,“To add 6 + 8, I
could double 6 and
get 12 and then add
2 more to get 14.”)
– solve problems
involving the addition
and subtraction of
two-digit numbers,
using a variety of
mental strategies
(e.g., to add 37 + 26,
add the tens, add the
ones, then combine
the tens and ones,
like this: 30 + 20 =
50, 7 + 6 = 13, 50 +
13 = 63)
-multiply to 7 x 7 and
divide to 49 ÷ 7,
using a variety of
mental strategies
(e.g., doubles,
doubles plus another
set, skip counting)
Number Sense – Mental Math – Number Talks
Is there a connection?
Four Components of the
Pedagogical System
Mathematical Proficiencies
• Non-threatening classroom
environment
• Instructional task
• Tools and representations
• Classroom discourse
1.
2.
3.
4.
5.
Conceptual understanding
Mathematical fluency
Strategic competence
Adaptive reasoning
Productive disposition
Mathematical
Processes
•
•
•
•
problem solving
reasoning and proving
reflecting
selecting tools and
computational
strategies
• connecting
• representing
• communicating
What Teachers are Saying about
Number Talks
• I think number talks has for me been the best
idea I have come across since teaching. This has
been a whole new way of learning for the
students and myself! It has opened up so many
doors for the students to gain confidence, access
more strategies, and make Math more
understandable. This ripple effect has made me
look at all the Math curriculum and begin to
teach it in a variety ways (not just the standard
way I was taught). I see students more engaged
and ready to take on more tasks.
Derek Bouma
What Teachers are Saying about
Number Talks
• "Since implementing number talks, I have seen increase
engagement and confidence in my students. All students
have an entry point, regardless of their ability level, and I
see participation from students who have a negative
attitude towards math and low confidence. Our staff has
begun using number talks on a school wide basis and it is
so invigorating to hear the other teachers share their
excitement and the wonderful things they are also seeing
with their students. We are looking forward to seeing what
happens after our students have been exposed to number
talks for several years in a row and how that will change
their attitudes and skill levels." - Jennifer Pedersen
What Students are Saying about
Number Talks
•
"Number talks are fun and they help me learn easier strategies to make the
questions easier." - Aiden
•
"Number talks are fun but sometimes a little challenging but I am learning math
easier because of it." - Jaxon
•
"Number talks help me learn new strategies that help me with other math." - Kate
•
"It helps with my facts, it makes multiplying and dividing easier. It also improves
what I think about math." - Melia
•
"Number talks are amazing. I think I am getting better." - Jadyn
•
"I like number talks because it helps me build my mental math skills and it is
helping with my math." - Tyler
What Students are Saying about
Number Talks
• I think number talks are good for us because it helps us
with our learning, it helps us become better at math. I have
noticed myself grow and become better at math than I was
before. I like that the questions get harder each time and
build on each other.
• Number talks helps me grow mathematically. I always
found that I do better with a visual and a voice tutorial and
number talks incorporates both.
• I think that number talks are great! They help learning and
they don’t make people feel bad by not getting the answer
right or quick enough. I found myself using new strategies
for getting the answer since we have done number talks.
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