Do The Multiplication Tables Help or Hinder?

They can do both. Children come to primary school with a fairly developed, if  somewhat limited, sense of number. Thanks to their brain’s capacity to seek out patterns, they can already subitize, and they also have learned a pocketful of simple counting strategies through trial and error. Too often,  arithmetic instruction in the primary grades purposefully avoids recognizing these intuitive abilities and resorts immediately to practicing arithmetic facts.

If the children’s introduction to arithmetic rests primarily on the rote memorization of the addition and multiplication tables and other arithmetic facts (e.g., step-by-step procedures for subtraction), then their intuitive understandings of number relationships are undermined and overwhelmed. In effect, they learn to shift from intuitive processing to performing automatic numerical operations without caring much about their meaning.

On the other hand, if instruction in beginning arithmetic takes advantage of the children’s number sense, subitizing, and counting strategies by making connections to new mathematical operations, then the tables become tools leading to a deeper understanding of mathematics, rather than an end unto themselves.

Some students may have already practiced the multiplication tables at home. My suggestion would be to assess how well each student can already multiply single-digit numbers. Then introduce activities using dots or pictures on cards that help students practice successive addition (the underlying concept of multiplication). The idea here is to use the students’ innate sense of patterning to build a multiplication network without memorizing the tables themselves. Of course, this may not work for every student, and for some, memorizing the tables may be the only successful option.

Can We Teach Number Sense?

Those who view number sense as an intrinsic ability will argue that the elementary components are genetically programmed, have a long evolutionary history, and develop spontaneously without explicit instruction as a young human interacts with the environment. However, most of these researchers do not view number sense as a fixed or immutable entity. Rather, they suggest that the neurocognitive systems supporting these elementary numerical abilities provide just the foundational structure needed for acquiring the expanded abilities cited by mathematics educators. And they recognize that both formal and informal instruction can enhance number sense development prior to entering school.

Berch (2005) notes that the abilities and skills associated with the expanded view of number sense cannot be isolated into special textbook chapters or instructional units, and that their development does not result from a set of activities designed specifically for this purpose. He agrees with those mathematics educators who contend that number sense constitutes a way of thinking that should permeate all aspects of mathematics teaching and learning. It may be more beneficial to view number sense as a by-product of other learning than as a specific goal of direct instruction.

Gersten and Chard (1999) suggest that the innate qualities of number sense may be similar to phonemic awareness in reading development, especially for early experiences in arithmetic. Just as phonemic awareness is a prerequisite to learning phonics and becoming a successful reader, developing number sense is a prerequisite for succeeding in mathematics. They further propose that number sense is the missing component in the learning of early arithmetic facts, and explain the reason that rote drill and practice do not lead to significant improvement in mathematics ability.

Because Gersten and Chard (1999) believe that number sense is so critical to success in learning mathematics, they have identified five stepping-stones that allow teachers to assess a child’s understanding of number sense. Their five levels are

• Level 1. Children have not yet developed number sense beyond their innate notions of numerosity. They have no sense of relative quantity and may not know the difference between “less than” and “more than” or “fewer” and “greater.”
• Level 2. Children are starting to acquire number sense. They can understand terms like “lots of,” “six,” and “nine,” and are beginning to understand the concepts of “less than” and “more than.” They also understand lesser or greater amounts but do not yet have basic computation skills.
• Level 3. Children fully understand “less than” and “more than.” They have a concept of computation and may use their fingers or objects to apply the “count up from one” strategy to solve problems. Errors occur when the child is calculating numbers higher than five, because this requires using the fingers of both hands.
• Level 4. Children are now relying on the “count up” or “counting on” process instead of the “counting all” process they used at the previous level. They understand the conceptual reality of numbers in that they do not have to count to five to know that five exists. Assuming they can count accurately, children at this level are able to solve any digit problem.
• Level 5. Children demonstrate retrieval strategies for solving problems. They have already automated addition facts and are acquiring basic subtraction facts.

Next week we will take a closer look at teaching number sense at all grade levels.

References:

Berch, D. B. (2005, July/August). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38, 333–339.

Gersten, R., Chard, D., Jayanthi, M., & Baker, S. (2006). Experimental and quasi-experimental research on instructional approaches for teaching mathematics to students with learning disabilities: A research synthesis. Signal Hill, CA: Center on Instruction/RG Research Group.

 

Learning to Count

Although infants are born with the same rudimentary number sense observed in rats and chimpanzees, they possess two arithmetic capabilities that quickly separate them from other animals. One is the ability to count. The other is to use and manipulate symbols that represent numeric quantities.

Recognizing the number of objects in a small collection is part of innate number sense. It requires no counting because the numerosity is identified in an instant. Researchers call this process subitizing (from the Latin for “sudden”). But when the number in a collection exceeds the limits of subitizing, counting becomes necessary.

Counting

No one knows when and how humans first developed the idea of counting beyond the innate sequence of “one, two, and many.” Perhaps they began the way young children do today: using their fingers. (This system is so reliable that many adults also do arithmetic with their fingers.) Our base- 10 number system suggests that counting began as finger enumeration. The Latin word digit is used to mean both numeral and finger. Even evidence from brain scans lends further support to this
number-to-finger connection.

When a person is performing basic arithmetic, the greatest brain activity is in the left parietal lobe and in the region of the motor cortex that controls the fingers (Dehaene, Molko, Cohen, & Wilson, 2004). Part of the parietal lobe and the section of the motor cortex that controls finger movement is highly activated when a person is doing arithmetic.

This raises an interesting question. Is it just a coincidence that the region of the brain we use for counting includes the same part that controls our fingers? Or is it possible that counting began with  our fingers, and the brain later learned to do counting without manipulating them? Some researchers speculate that if our human ancestors’ first experience with numbers was using their fingers, then the region of the brain that controls the fingers would be the area where more abstract mental arithmetic would be located in their descendants (Devlin, 2000).

Assuming fingers were our first counting tools, we obviously ran into a problem when counting collections of more than 10 objects. Some cultures resorted to using other body parts to increase the total. Even today, the natives of the Torres Straits Islands in New Guinea denote numbers up to 33 by pointing to different parts of their body, including fingers, arms, shoulders, chest, legs, and toes. Naming the body part evokes the corresponding number. Thus, the word six is literally “wrist,” and nine is “left breast.” They use sticks for numbers larger than 33 (Ifrah, 1985). But this process is hopeless for numbers beyond 30 or so. Eventually, some cultures used a physical tally system, such as making notches on a bone or stick. Notched bones have been discovered that date back about 40,000 years. According to the fossil record, this is about the same time that humans started to use symbolic representations in rock carvings and cave paintings (Devlin, 2000).

Finger counting and physical tallies show that these cultures understood the concept of numerosity, but that does not imply they understood the abstract concept of number. Archeologists, such as Denise Schmandt-Besserat (1985), suspect that the introduction of abstract counting numbers, as opposed to markings, appeared around 8,000 BC and were used by the highly advanced Sumerian society that flourished in the Fertile Crescent of what is now Iraq and Syria. They used tokens of different shapes to represent a specific quantity of a trade item, such as a jar of oil or loaf of bread. They used symbolic markings on clay tablets to keep running totals of items in commerce. It was not really a separate number system, but it was the first use of a symbol system that set the stage for the functional, abstract numbers we use today.

Our present numbering system was developed over two thousand years by the Hindus, and attained its present form in about the sixth century. In the seventh century, it was introduced to Europe by Persian mathematicians and thus became known as the “Arabic system.” This ingenious invention now enjoys worldwide acceptance for several reasons.

• Each number has its own word, and the number words can be read aloud. Saying a number, such as 1776 (one-thousand seven-hundred and seventy-six), clearly reveals the numeric structure of units, tens, hundreds, and thousands.
• The numerical system is not just symbols but also a language, thereby allowing humans to use their innate language fluency to handle numbers.
•  It is concise and easily learned.
•  We can use it to represent numbers of unlimited magnitude and apply them to measurements and collections of all types.
• It reduces computation with numbers to the routine manipulation of symbols on a page.

In fairness, I should mention that the original idea of denoting numbers by stringing together a small collection of basic symbols to form number words came from the Babylonians around 2000 BC. But the system was cumbersome to use because it was built on the base 60, and thus did not gain wide acceptance. Nonetheless, we still use it in our measurements of time (60 seconds make one
minute, etc.) and geography (60 seconds make one degree of latitude and longitude).

References:

Dehaene, S., Molko, N., Cohen, L., & Wilson, A. J. (2004, April). Arithmetic and the brain. Current Opinion in Neurobiology, 14, 218–224.

Devlin, K. (2000). The math gene: How mathematical thinking evolved and why numbers are like gossip. New York: Basic Books.

Ifrah, G. (1985). From one to zero: A universal history of numbers. New York: Penguin Books.