Gelman's Counting Principle

Gelman’s Counting Principle

How do we know whether children can count ?

If knowing how to count just means reciting

the numbers (i.e., ‘‘one, two, three...”) up to “five” or

“ten,” perhaps pointing to one object with each numeral,

then many two-year-olds count very well.

Of course, counting only tells you number of things if you do it correctly,

 

following the three ‘how-to-count’ principles identified by Gelman and

 

Gallistel (1978). These are :

 (1) The one-to-one principle, which says that ‘‘in

enumerating a set, one and only one [numeral] must

be assigned to each item in the set.”

 

(2) The stable-order principle, which says that ‘‘[Numerals] used in

counting must be used in the same order in any one

count as in any other count.” and

 

(3) The cardinal principle, which says that ‘‘the [numeral] applied to the

 

final item in the set represents the number of items in the set.”

 

As Gelman and Gallistel pointed out, so long as the

child’s counting obeys these three principles, the numeral

list (‘‘one,” ‘‘two,” ‘‘three,”... etc.) represents the cardinalities

1, 2, 3,... etc.

However, observations have shown that three-year-old

children often violate the one-to-one principle by skipping

or double-counting items, or by using the same numeral

twice in a count.

Children also violate the stable-order principle, by producing different

 

numeral lists at different times.

The cardinal principle can also be viewed as something more profound

 

– a principle stating that a numeral’s cardinal meaning is determined by

 

its ordinal position

in the list. This means, for example, that the fifth numeral in

any count list – spoken or written, in any language – must

mean five. And the third numeral must mean three..

If so, then knowing the cardinal principle means having

some implicit knowledge of the successor function – some

understanding that the cardinality for each numeral is generated by

 

adding one to the cardinality for the previous numeral.

Thus Gelman feels that preschoolers learn to count as a result of the

 

perfecting of counting procedures.

 

Do you agree ?

 

Marina Ho-BSc08

Fast Math Tricks - How to multiply 2 digit numbers up to 100 - the fast ...

The multiplication method my teachers never taught me. I wish I had learnt this way earlier. I was taught the top down method.

Pick’s Theorem

This formula was created by Georg Alexander Pick in 1899.

 

Using the Geoboard to Assist :

 

·        Draw a polygon on the Geoboard

·        Determine the number of boundary points and the number of interior points and calculate its area

·        Repeat with different polygons

·        Draw polygons without any interior points and find the area

·        Draw polygons with one interior point and find the area

·        Draw polygons with two interior points and find the area

·        Prepare a table with number of boundary points, number of interior points, and area. Discover that the relationship between B the number of boundary points, i the number of internal points, and the area of the geoboard polygon is given by Pick's formula A = I + ½ B – 1

 

The diagram below explains this further.

 

I understand this better now! Wow :)

Marina Ho
BSc08

 

 

By The Numbers - Subitizing

Watch this video on what subitizing means.

Elementary Math - Day 3 Blog

What is Subitizing ?

 

Perceptual subitizing

Perceptual subitizing is closest to the original definition of subitizing: recognizing a number without counting. For example, children might "see 3" without using any learned mathematical knowledge.

 

Conceptual subitizing

But how is it that people see an eight-dot domino and 'just know" the total number? This is Conceptual subitizing. They are able to see the relationship between numbers when dealing with different quantities. These people are capable of viewing number and number patterns as units of units (Steffe and Cobb 1988).

Spatial patterns, such as those on dominoes, are one kind.

 

Subitizing and counting

Young children may use perceptual subitizing to make units for counting and to build their initial ideas of cardinality.

 

 

 

 

 

 

By : Marina Ho - BSc08

Seed Institute

 

 

Elementary Math - BSc08

What are Nominal Numbers ?

Numbers used to name something eg. Blk 543, a bus number 963 or your IC No. 7612345B.

How we have often mistaken the bus numbers to be nine hundred and sixty three, instead of "9" "6" "3".





What are Ordinal Numbers ?

Ordinal numbers are used very often to label the position of children lined up in a row, eg. 1st, 2nd, 3rd, 4th, 5th etc.

In the picture below children can also learn to count from the 2nd person (Sam) and also backwards in position, to determine how many children are before Sam. Thus ordinal numbers can be used children a different way of understanding number counting.

Marina  Ho - BSc08
Seed Institute

Elementary Math - Daily Blogs

Learning About Numbers for Children

Children have to learn about counting in 2 different ways; Rote counting and Rational Counting.
When children learn rote counting they are counting in sequence without knowledge of quantity. However, when children learn rational counting, they learn how to count wth knowledge and understanding. They count with cardinality in mind.

The above video show rational counting, where children have to learn 1 to 1 correspondence in counting different objects.
This shows how children need to be shown concrete objects or have a pictorial representation to have a realtionship about what a number quantity means.

Understanding Mathematics

 

 Parents who endeavour to know if a school math program is

 

well planned should examine whether 6 areas are present.

 

These are;  high standards and strong support for students,

 

curriculum that is focused and coherent, teaching that

 

challenge and support students. Next would be learning with

 

understanding from prior knowledge, assessment for

 

students to find new information, and lastly technology like calculators and computers are

 

used to enhance learning.

The Math classroom should also have 6 components :

-          an environment that gives students an equal opportunity to learn

-          teaching that has a balance of both conceptual understanding and procedural fluency

-          ensures active student participation in problem solving, reasoning, communications, connections and representation

-          use of technology to improve understanding

-          emphasis on reasoning of math

Parents also need to know what it means to understand math in today’s classroom. Relational understanding means knowing what to do and why, while the opposite is instrumental understanding (just doing without understanding). Thus students who have proficiency in math would have mastery in 5 areas :

-          conceptual understanding : knowledge about the relationship between ideas in a topic.

-          Procedural fluency : able to use rules and procedures in carrying out math processes.

-          Strategic competence : able to use different methods to eventually solve a problem.

-          Adaptive reasoning : able to reflect, evaluate and adapt or use different methods.

-          Productive disposition : when solving a problem you should rely on a how to and applying your knowledge rather than a recall method of how you can solve it.

I trust these points in mind would make parents see Math in a different light.. Math is not to be approached through rote drilling and countless worksheets but with crystal clear thought and understanding.

By : Marina Ho Mee Lin
SEED - BSC08