ED 5480

October 3, 2011

Natural and Un-Natural Math Processes

Logic and Skills Invoked while Learning to Count

& Calculate

ED 5480

Present day understanding of math has developed since time immemorial. Considering this what are some possible math-based applications that ancestral humans (i.e. cavemen) possessed? Counting Inventory and Distribution Connection through artistic patterns Measuring time

Pregnancy Crop planting/harvest Camp or go home

Hunting (spear throwing etc.) Ceremonial burial plots – avoid overlapping

Count the number of objects in each of the next few slides

1

2

3

4

5

6

7

8

9

1 2 3

4 5 6

7 98

3 6 9

9 24 6

6 2 7

Which of the nine previous slides did you actually count the individual objects and which could you accurately gage without actually counting?

The process that gives one the ability to comprehend the number of a collection of items without actually counting them is called “Subitizing”.

With most people this process is reliable for up to four objects.

Investigations have shown that when people are challenged to count four objects or less the only part of the brain that is active is the visual cortex (the part of the brain responsible for processing visual information).When challenged to count more than four objects numerous brain networks were recruited by the counter to process the information

There are two types of subitizing, Perceptual & Conceptual.

Perceptual – involves recognizing numbers without using

other mathematical processes, such as counting.

Conceptual – allows one to know the number of objects by

recognizing a familiar pattern.1 4

The regions of the brain involved in processing numerical data are the same regions involved in language processing.

How does this information help you as a teacher?

Working Memory & CountingRecite the number list below out loud – Those who speak Mi’kmaq or Maliseet please use your traditional language

Take ~ 20 sec to memorize the list

Now reproduce the list on paper

How did you do?

7 5 9 11 8 3 7 2

Maliseet 1-10

1- PESKW 2 - NIS 3 - NIHI 4 - NEW

5 - NAN 6 - KAMACIN 7 - OLUWIKONOK 8 - UKAMOLCIN 9 - ESKWONATEK 10 - KWOTINSK

Chinese would likely be able to completely reproduce the sequence. Any idea why?

Hint: Chinese numbers are very brief to recite

What may be some other obstacles caused by the English language to the development of a number sense?

Skill sets that develop as a result of involvement with learning activities in mathematics (from Birch, 2005)

Ability to recognize something has changed in a small collection when, without knowledge, an object has been removed or added to the collection

Elementary abilities or intuitions about arithmetic A mental number line on which analog representations

of numerical quantities can be manipulated An innate capacity to approximate numerosities (concept

of quantities) Ability to make numerical magnitude comparisons Ability to decompose numbers naturally Ability to develop useful strategies for solving complex

problems Ability to use the relationships among arithmetic

operations to understand the base-10 number system Ability to use numbers and quantitative methods to

communicate, process, and interpret information Awareness of levels of accuracy and sensitivity for the

reasonableness of calculations Desire to make sense of numerical situations by looking

for links between new information and previously acquired knowledge

Skill sets that develop as a result of involvement with learning activities in mathematics (from Birch, 2005)

{continued}

Knowledge of the effects of operations on numbers Fluency and flexibility with number and understanding of

number meanings Recognition of gross numerical errors Understanding of numbers as tools to measure things in

the real world Inventing procedures for conducting numerical

operations Thinking or talking in a sensible way about general

properties of a numerical problem or expression, without doing any precise computation

Did you notice that the previous slides said nothing about memorizing things, such as math rules or times tables?

Why are multiplication tables difficult to learn?

1. Associative memory2. Pattern recognition3. Language

In early development of understanding multiplication students flourish with an intuitive understanding of repeated addition. As students progress to larger and larger numbers this becomes cumbersome and rote learning of multiplication facts are often encouraged, thus abandoning the intuitive path of learning. As a result, calculations now require the acquirement and storage of a large database of numerical knowledge.

This is what normally happens in the learning process, but it is not natural. There are ancestral survival benefits from counting and adding, there are no benefits by developing a multiplicative intuition.

1. Associative memory – The human brain excels at connecting facts through association. For example,

Where were you the day the planes crashed into the twin towers?

I bet most everyone had a reasonable answer!

With multiplication tables this innate desire to associate information is often detrimental.

7 X 8 = ?

Was your first thought 54 or 56?

How did you know the correct answer?

2. Pattern Recognition– Patterns can often interfere with each other. Consider the following:

• Carl Dennis lives on Allen Brian avenue• Carl Gary lives on Brian Allen avenue• Gary Edwards lives on Carl Edwards avenue

Where does Carl Gary live again?

This is simply an introduction to the multiplication tables.

Carl = 3 Dennis = 4Allen = 1

Brian = 2 Gary = 7Edward = 5

“lives on” represents equals (=)

• Carl Dennis lives on Allen Brian avenue• Carl Gary lives on Brian Allen avenue• Gary Edwards lives on Carl Edwards avenue

3

3

3

4

77

11

22

5 5

==

=

X

XX

3 X 4 = 123 X 7 = 217 X 5 = 35

3. Language– Memorizing through reciting is a powerful technique to retain what you are attempting to memorize. Teachers often have students recite their times tables (or poems) to activate this “verbal memory”

Tell me a poem you learned as a child.

The regions of the brain used for exact arithmetic calculations are the same regions used for verbal memory, most likely because of the verbal representation of numbers.

So what hazard is their to this approach?

When attempting to do estimations in math the human brain uses different regions than what is used in verbal memory. Strengthening the verbalization-to-math-facts line does nothing to enhance the comprehension of the math facts.

Is memorizing the multiplication table ever a benefit?

YES!If the tables become tools to a deeper understanding of mathematics itself and not an end unto themselves.

If a child’s understanding of arithmetic rests primarily on rote memorization then their intuitive understandings of number relationships are undermined and overwhelmed.

Today we have been talking largely about avoiding short-cuts for short-term goals (rote memorization) and focusing on long-term goals for deeper comprehension and appreciation of mathematics.

Journal Topic Suggestion – Reflect on a time in your life when a person of authority (i.e. school teacher, coach, parent, grand-parent) insisted that you do something the “right” way instead of the fastest way.