Tuesday, October 5, 2010

Number Properties

Numbers are abstract and difficult to define. 
Numbers are difficult to define, so use properties of numbers (axioms) to do arithmetic. Counting, which is a use of numbers, in an ineffective method to do arithmetic, yet counting and counting strategies seem to dominate U.S math programs in lower grades. While counting has its place, especially at the PreK and kindergarten levels, by 1st grade, we should focus instruction on the properties of numbers and the rapid acquisition of background knowledge (factual and procedural knowledge) in long-term memory. If you get the idea that the American approach has shortcomings, then you would be right. We do not teach arithmetic [or algebra] well. 
First-grade students should learn the properties of numbers, starting with equality and other properties of whole numbers [e.g., 0 is a whole number, zero property of addition (2 + 0 = 2), etc.], which, in turn, enable students to do arithmetic and, later on, algebra. Understanding implies “doing.”  Moreover, they should memorize the number facts and work extensively with operational procedures and axioms to build background knowledge needed for problem-solving. 
Understanding number ideas means knowing how numbers work (i.e., using number properties to make true mathematical statements: 5 + 0 = 5, 2 + 3 = 3 + 2, 12 = 10 + 2, 4 + 6 = 8 + 2, etc.  Doing arithmetic implies that students think in terms of making true statements. Furthermore, most number properties in basic arithmetic are simple ideas and easy to learn, such as the three equality properties of numbers: 5 = 5 (reflexive); if 10 = 3 + 7, then 3 + 7 = 10 (symmetric); and if 2 + 3 = 5 and 4 + 1 = 5, then 2 + 3 = 1 + 4 (transitive). (Incidentally, first grade students can verify number properties using cubes and equal-arm balances.) The same number properties apply to algebra. In first grade, I used words students could understand and told students to “Think Like A Balance.An equation is like a balance. 
To learn more about number properties, see Endnotes 1 & 2. 
In arithmetic, algebra, and calculus, problem-solving is a function of prior knowledge and experience. To do arithmetic and other math well, at the least, requires the automaticity of facts and procedures; i.e., essential background [prior] knowledge in long-term memory. In short, the fundamentals of arithmetic and algebra should be taught for mastery but often aren’t. Unfortunately, the pace of math instruction in the United States is very slow, so it should be no surprise that our kids lag behind their peers from other nations. Our math programs lack rigor. Lastly, learning arithmetic and algebra well (mastery) takes repetition and lots of practice, yet practice has been out of favor in American classrooms for decades. 
Lastly, arithmetic is sequential (one idea builds on old ideas, etc.) and “the brain,” says John Medina (Brain Rules), “is a “sequential processor, unable to pay attention to two things at the same time.” In short, our brains thrive on a step-by-step hierarchy. (See “left/right brain bunk” below under Tangential Thoughts) Thus, arithmetic should be taught hierarchically to optimize the building of background knowledge, starting with number facts, number properties, and procedures. The focus in 1st grade should be on equivalent relationships and the number properties that govern them, rather than just counting. 
Find the number for box, so that (☐ + ☐) - 6 = 2.

My first-grade students worked equations like this but did not use algebra techniques. They used “guess and check,” reasoning, and knowledge; e.g., memorized facts and procedures, the order of operations (parentheses), conventions, equality axioms, and rule for substitution (variables). It takes time, repetition, and practice to reach this level. I started with very simple equations (e.g., 3 + ☐ = 7) and advanced step-by-step to more complex equations. Solving equations is one approach for students to practice math facts, algorithms, and number properties and to apply pre-algebra ideas (e.g., variable) and reasoning. Incidentally, box = 4 because (4 + 4) - 6 = 2.  When using x (instead of box) as the variable, the equation is (x + x) - 6 = 2 or 2x - 6 = 2. 
(See Endnotes 1, 2)
Draft 1 TBR Updated on 9-27-2010
Endnote 1
Find the number for box, so that (☐ + ☐) - 6 = 2.
If the right side is 2, then the left side must also equal 2 to make a true statement. Also, because the box is the same variable (☐ + ☐), then the number that replaces box must be the same number (rule for substitution). The purpose of solving an equation is to make a true statement (left side = right side). 
What is important for first-grade students is to apply the idea of equality(THINK LIKE A BALANCE) This means the left side of the equation must be equivalent in value to the right side. In this equation, the left side = right side (2 = 2), if and only if box = 4, which is a unique solution; there are no other solutions. 
Introducing layers of difficulty through a sequence of equations is important, too. Here is another first-grade equation. In this example, the same variable is on both sides. All boxes must be the same number. 
(☐ + ☐) - 2 = ☐ + 5 
There are prerequisite equations to this equation, of course. Building to more complex equations takes time, persistence, repetition, and practice. My first-grade students did not use algebra equation solving techniques; they used “guess and check” and reasoning, along with memorized facts, procedures, axioms, rules, and conventions. Students were asked to think like a balance. Learning is based on repetition and practice. Knowledge in long-term memory is key to problem-solving. 

In (☐ + ☐) - 2 = ☐ + 5, box = 7 because 
      (7 + 7) - 2 = 7 + 5 
                   12 = 12. 

Note. In pre-algebra and Algebra 1, the equation is written 2x - 2 = x + 5In short, an equation approach to arithmetic prepares students for real algebra they encounter in pre-algebra and algebra classes. Even though I did this in 1st grade as a means to practice number facts, properties, and procedures, elementary teachers in higher grades should find the ideas useful. 9-27-2010. (On 9-27-1905, Einstein introduced the mass-energy equivalence: E = mc².)

The properties of equality (equality is reflexive, symmetric, and transitive) and other properties are simple enough to teach to 1st grade students through practice and repetition, but this approach is seldom used in the United States, even though the properties of numbers provide a conceptual basis of “number” and help students understand how numbers work. For example, 12 = 10 + 2 is a true statement because the left side (12) is equivalent in value to the right side (10 + 2). Both “12” and “10 + 2” name the same point on a number line {12}. Therefore, they are equivalent to each other by the transitive property of equality, which roughly states that things equal to the same thing are equal to each other. If a = and b = c, then a = b. Also, read First Grade Notes
The equal-arm balances my 1st graders used in the early 1980s came from a 1960s Science--A Process Approach (SAPA) module that I found in storage at district elementary school. (SAPA Equal-Arm Balance)