Transcript number sense - RoddyAtUWS

Mark Roddy, PhD
Seattle University
Number and Number Sense
A lecture for Primary Mathematics and
Numeracy 2 (101584)
11 September 2014
University of Western Sydney
Session Objectives
Students will develop their
grasp of:
• the concepts of number
and number sense as
these terms are used in
mathematics education;
• the benefits of number
sense;
• Effective pedagogy;
• Relevant resources.
The Plan
• Introduction
• Pig
• What is number and what
is number sense?
• How do I teach it?
• Where are the resources?
• What are the benefits of
number sense?
PIG
and Number Sense
…a nice opportunity for students to generate data
through their own activity and to do meaningful,
self-directed maths with these data – to make
sense of their experience and to understand it
more fully through mathematics.
¿Number? ¿Number Sense?
OED - number: “an arithmetical value, expressed
by a word, symbol or figure, representing a
particular quantity and used in counting and
making calculations.”
77 out of 217 pages devoted
to “Number and Algebra”
Number:
the kitchen sink of mathematics
Statistics
Probability
ELSE!
Algebra
Measurement
Geometry
Number Sense
“…Number sense is also about knowing that 6 + 7 and
7 + 6 both produce a sum of 13; that 25 × 7 is less than
200; that the quotient for 1/2 ÷ 1/4 is larger than 1/2;
that the stadium could not seat 400,000 people; that
tripling the square footage wouldn’t make for the most
economical home to heat; etc.”
As students estimate, talk about numbers, compute,
use mental math, and judge the reasonableness of
their results, they become more flexible in working
with numbers. A sense of number emerges that is built
on [solid] foundations, which yields responses such as,
‘I knew 3/4 was more than 3/5 because the pieces
were bigger in fourths.’ This is what all math teachers
want.”
Number Sense—Right Now! Francis (Skip) Fennell (2008).
Number sense begins at home….
“How old are you?”
“You can invite three friends.”
“How tall are you?”
“How much more is there?”
“There’s a half moon out tonight.”
Number sense is developed at school.
“Students develop
efficient strategies for numerical calculation,
recognize patterns, describe relationships, …”
• Whole Numbers; patterns and patterning
place value
• Addition and
composing
decomposing
numbers
Whatanddo
teachers
do in
Subtraction;
order to enable students
• Multiplication and commutative,
associative,
distributivesense?
laws
to develop
number
Division
facts and fluency
• Fractions and Decimals;
• Patterns (and Algebra).
NSW Syllabus (2012)
MA2-6NA uses mental and informal written
strategies for multiplication and division
“If I know … then I know …”
• e.g. you draw 6 and 4 => 6x4=24 so, ….
• 4x6=24 and,
Teachers orchestrate experiences
that allow students to construct
• 4+4+4+4+4+4=24 and,
understanding.
• 6+6+6+6=24 and,
• (4+4+4)+(4+4+4)=(12)+(12)=24 and,
• (6+6)+(6+6)=(12)+(12)=24 and (6+6)=(4+4+4)! and,
• (2x6)+(2x6)=24 and,
• 2x2x6=24 and ….!
MAe-6NA – groups, shares and counts
collections of objects …
4x
+
The Join Machine (for addition)
The 4x Join Machine (for multiplication)
Concrete -> Symbolic -> Abstract
MA1-6NA - uses a range of mental strategies
& concrete materials for mult. and division.
3x4=?
Concrete -> Symbolic -> Abstract
MA2-6NA - uses mental and informal written
strategies for multiplication and division …
12 x 13 = ?
Concrete -> Symbolic -> Abstract
MA3-6NA - selects and applies appropriate
strategies for multiplication and division …
26 x 48 = ?
partial products
algorithm
standard
algorithm
Concrete -> Symbolic -> Abstract
http://thinkmath.edc.org/resource/multiplication-and-division
MA2-7NA - represents, models and compares
commonly used fractions and decimals
1
2

1
3
1
4
1
5
1 2 4
5
  
 0.5  0.50
2 4 8 10




1
6

1
8
1
10

Etcetera … Volunteers?
Gilbert, A. (2002). Teaching the Three R's: Through Movement Experiences
1
100
1
a) Locate the running time on a DVD.
a) Convert this into hours and minutes.
c) You arrive at your friend’s house on
Saturday, 2PM, to watch the movie.
What time should your mum pick you
up?
d) Pick some of your favourite DVDs. How many could you watch
in 12 hours? Explain your thinking.
e) Pick 6 different DVDs and create a timeline based on the year
each one was made.
5 cm per Second
Princess Bride
1985
The Castle Off the Map
1990
1995
2000
2010
2015
Sit down and think about this question:
2
What is your absolute favorite activity to play at the park?
Circle one of the activities:
1) Slides 2) Swings 3) Monkey bars 4) Climbing nets 5) Tunnels
IIII
II
III
I
Now survey the students in your maths trail group about which of the five above activities
is their favorite and record it using a tally.
Now, in the space below, graph your findings. (Be sure to label your graph.)
5
• Which is the most popular?
4
• Which is the least popular?
• How many people did you
survey?
3
2
• How many people liked the
same activity as you?
1
0
Slides
Swings
Monkey Bars
Climbing Nets
Tunnels
2
What fraction of the people liked the slides best?
Write down and draw as many ways as you can think
of, to represent the amount of people who like the
slide over the total amount of people you surveyed.
4 2

10 5
3
1) Estimate the length of
the Flying Fox, then use the
trundle wheel to check
your guess.
2) Working in pairs, one
person will run and another
will keep time, race to the
end of the Flying Fox.
Document your times.
3) Now, working in pairs again, race on the Flying Fox. Compare
the times.
4) Was there much difference from running to using the Flying
Fox? Why would one be faster than the other?
5) Can two people on the Flying Fox go faster than one? Why or
why not?
4
If the T stand
(for the swings) is
rolled 20 times,
what position will
it be in?
T
T
T
T
T
T
T
T
T
T
MA2-9MG
Measures, records, compares and estimates lengths,
distances and perimeters in metres, cms and mms, ….
MA2-18SP
Selects appropriate methods to collect data, & constructs,
compares, interprets and evaluates data displays, including
tables, picture graphs and column graphs.
MA2-2WM (Problem Solving)
Selects and uses appropriate mental or written strategies,
or technology, to solve problems.
Let’s take a look, shall we?
Concrete -> Tabular -> Graphical -> Abstract
(After AIMS “Model of Learning” http://www.aimsedu.org/resources/modelL.html )
http://growbeast.wikispaces.com
How does a Grow Beast grow?
Self-contained SPED classroom grades 1-3
http://roddyatuws.wikispaces.com
Students are …
•
•
•
•
•
•
•
Making choices;
Making predictions;
Measuring;
Recording data;
Constructing charts;
Drawing conclusions;
And enjoying the process!
In short, they are using maths in making sense of
their experiences with a fun toy.
Sample Resources
• NLVM (National Library of Virtual Manipulatives)
• NCTM > Illuminations > Interactives
• Arcademics • 101Questions
10 cm in 10 minutes => 1 cm/min
… really??
2 highs and 2 lows/day
=> 6 hours between
Each high and low ….
15 ft high and -1 ft low => 16 ft change
16 ft is ~5m = 500 cm
So 500 cm/360 min …
and ….
and 6 hours is 6 x 60 = 360 minutes
So ….
For too many of our students, particularly those
in the upper grades, maths is just a subject to be
endured, or worse, avoided. With number
sense, mathematics can be a stimulus to pay
attention to the world all around us. It can
enable us to pose questions and to look for
answers that make sense. If we can help our
students see mathematics as a way of making
sense of the world, we have done them a
service that can last a lifetime.