Random Thoughts on First Grade Arithmetic
Number lines are seldom found in primary school math materials. One exception (?) is the 11" by 16.5", 645-page 1st grade enVision Math textbook from Pearson 2011, but the number line is hardly used--just one lesson: "You can use a number line to find missing numbers: 2 _ 4 with "before" and "after" as vocabulary. A ten-frame is used extensively to model numbers and figure out single-digit number facts, not a number line. I think this is a mistake. Also, memorization and drill are not part of the enVision reform math program. Why not?
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| 11" by 16.5" enVision Math 1st-Grade textbook. It is 645 pages. |
The reform math idea is that students should calculate the single-digit number facts using a variety of methods, not memorize them, which is a shift from the Old School ideas of memorization and drill to develop skill in arithmetic. Frankly, the ten-frame method and counters are inferior to a number line, which shows basic arithmetic. In my self-contained 1st-grade class (the early 80s), I stopped using counters (manipulatives) after the first couple of weeks of school. The number line is essential mathematics, not the ten-frame, counters, or calculators. The 0-20 number line is an extension of the number line kids come to school with.
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Number lines show key math concepts. Students learn magnitudes, number relationships, patterns, whole number operations, fractions, decimals, etc. The "one more" idea shows children how numbers are built: add one.
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| First Grade Number, Line 0 to 20: Equal Units. I started with a 0-20 number line, then added a -10 to 10 number line. |
Equal Units
First-grade students come to school with a built-in logarithmic number line (unequal units), but in arithmetic, the number scale is linear (equal units) and needs to be taught that way. The distance between 4 and 5 is ONE unit or just plain 1. The distance between 10 and 11 is ONE unit, etc. Students can see this on a number line. To get the next number (a'), add 1 to a: a + 1 = a'. (FYI: a' is read a-prime, which is the next number) Thus, 2 + 1 = 3, 4 + 1 = 5, 18 + 1 = 19, 56 + 1 = 57, etc. Linear number lines should be used from day-one in teaching arithmetic to first graders, but I seldom see them in primary math textbooks. Counting starts at 1, but number lines and rulers start at 0. Magnitude is an important idea, too. Students can see that 4 is greater than 3 (4 > 3) or that 3 is less than 4 (3 < 4). These are inequalities, and the symbol always points to the smaller of the two numbers when they are compared. When comparing numbers m and n, only one of these is true: m = n, m > n, or m < n.
Okay, how do we get from 4 to 5?
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| ADD ONE to get the next integer. |
The number line shows how (and why). We add 1, which, on the number line, means you move one unit to the right of 4 to get to 5: 4 + 1 = 5, that is, to get 5, add 1 to 4. This is the successor rule for integers: a + 1 = a', in which a is an integer {4} and a' (a-prime) is its successor {5}. The idea that you get the next integer by adding one to the previous integer is very important, but, unfortunately, it is seldom taught this way in 1st grade. The successor rule works for all integers. For example, -6 + 1 = -5 or -1 + 1 = 0. You don't need to call it the "successor" rule. For whole numbers, you can call it the "add 1" rule. Furthermore, what seems obvious to adults, is not always clear to 5 and 6-year-olds. Remember, kids don't think like adults, which is a basic premise of cognitive science. Hence, presentation and explanation are important in teaching young children arithmetic as are memorization, "drill for skill," pattern recognition, place value, magnitude, etc.
Adding 3 and 4
Once children see the basic relationship of 3, 4 and 7, then they should memorize the fact (3 + 4 = 7) and solve missing addend problems such as 7 = x + 4. Also, note that subtraction is defined in terms of addition: 7 - 3 = 4 if and only if (iff) 3 + 4 = 7. And, 7 - 4 = 3 iff 4 + 3 = 7. It makes good mathematical sense that addition and subraction should be taught together.
The concepts in arithmetic are elementary, and the number line makes them accessible to 1st-grade students. Children are novices, not experts or little mathematicians. They need to memorize and practice to learn. Learning something is remembring it from long-term memory.
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Also, the standard algorithm is not found in the enVision 1st-grade text. The textbook is too big to take home, so students tear out the lessons. Moreover, there are numerous calculator activities called Going Digital to get kids on calculators. Using calculators starting in kindergarten was an imprudent guideline introduced by the National Council of Teachers of Mathematics (NCTM 1989). You would think the NCTM would know better.
W. Stephen Wilson, a mathematics professor at Johns Hopkins University, sets the record straight by pointing out that using calculators is "absolutely unnecessary" in arithmetic and algebra. He writes, "The concepts and skills we teach are so fundamental that technology is not needed to either elucidate them or enhance them."
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| enVision Math - 1st Grade |
You see, where reform math programs such as enVision Math are headed, which is to do away with memorization of facts for auto recall and the standard algorithms (Old School). Calculators (tech) are substituted for paper-pencil standard arithmetic. It is already happening. The reform idea is that computations can or should be done on calculators, even easy ones. Therefore, the student can focus on critical thinking or problem-solving. But, the math doesn't work that way. You cannot apply something you don't know well in long-term memory. Students should master the fundamentals, but many don't.
The content in 1st-grade enVision Math is significantly below the benchmarks of some Asian nations. American kids start behind and stay behind their Asian peers. Often, in reform math, kids are asked to draw something to get an answer or as proof. A drawing is not math. Counting is not a property of numbers. Using a calculator is pressing keys; it's not math.
Arithmetic is mostly addition, subtraction, multiplication, division, fractions-decimals-percentages, and ratio/proportion. In the early grades, parts of arithmetic are unexciting. Also, it is not much fun memorizing the addition and multiplication tables or practicing the standard algorithms. Reform math and minimal guidance constructivist methods seem to dominate K-8 classrooms. Consequently, many students do not master traditional arithmetic or standard algorithms. They are ill-prepared for Algebra. Indeed, basic arithmetic has not been taught well for years. The problem starts in 1st grade and jumps up the grades.
Kids are novices and need to memorize and drill for developing skills. Learning arithmetic means remembering arithmetic from long-term memory. Learning arithmetic is work, but it is essential because it forms the cognitive architecture of mathematics in your long-term memory and helps students develop number sense. But, later, students will likely use calculators for more complex calculations, especially in chemistry and physics. I used a slide rule. Today, unfortunately, calculators often cover up weak arithmetic skills even in elementary school.
Liberal reformers (the progressives) tossed out the Old School stuff even if it worked well when it was taught well. Tossing out the good things of the past is hardly a reasoned strategy.
The doing of mathematics and the understanding of mathematics are rooted in the symbolic language of mathematics (i.e., abstraction). But, teachers don’t know symbolics or properties (rules) of numbers. They do not know that equality is reflexive, symmetric, and transitive (Peano axioms), much less the ideas that subtraction is defined in terms of addition and division is defined in terms of multiplication. In algebra, subtractions are changed to additions, and divisions are changed to multiplications. I am not sure that teachers know that the sum of identical addends is called multiplication. Counting is not a property of numbers.
Note: The two primary operations in the set of real numbers are addition and multiplication, which make a Field. A number line is all that is needed to explain addition, subtraction, multiplication, and division of whole numbers and regular fractions.
The idea that elementary students must know the why of everything and show proof rather than the "how" is nonsense. Indeed, students need to practice procedures until they are automatic, which is what kids in many other nations do, including the East Asian countries that trounce American students in factual and procedural knowledge and creative problem solving on international tests (TIMSS, PISA). Memorization and repetition are keys to learning because learning is remembering from long-term memory.
Asian children are taught mechanics first with the explanation later, and it works! We do it backward with understanding first, and it doesn't work. In short, the progressive math reforms have not stressed the mastery of standard arithmetic in long-term memory.
As I had said many times: There is no substitute for knowledge in long-term memory and the practice that gets it there.
The reality is that the more "rote learners" of the East Asian nations have excelled in factual and procedural knowledge and creative problem-solving (TIMSS, PISA), leaving most American students in the dust.
Symbolic Arithmetic
Standard arithmetic is the basis for higher-level mathematics, such as classical algebra. Algebra is symbolic arithmetic. Newton called elementary algebra the universal arithmetic because the calculation of numbers (arithmetic) and of symbols for unknowns (algebra) were the same. The rules that govern the calculations of arithmetic and algebra are called the field axioms. Simply, elementary algebra obeys the rules of arithmetic. A focus should be on learning of field axioms (rules) that govern arithmetic and elementary algebra calculations. It impacts 1st-grade arithmetic and algebra.
Should we teach algebra concepts in 1st grade?
YES! 3 + 7 - x = 5 - 2
(x = 7 to make a true statement. The inverse idea is often used in solving equations.
Left side = Right side: 3 = 3)
John Stillwell (Elements of Mathematics From Euclid to Godel) said that the point of doing arithmetic is not to do millions of calculations but to learn the axioms that govern them along with efficient calculation methods using single-digit number facts. Also, He said that the algorithms (step-by-step recipes) to calculate numbers should be fast and efficient. For novices, one algorithm per operation is sufficient to start. The operations can be understood using a number line, and when the numbers are larger, the standard algorithms should be used. This idea goes against the "many methods" of reform math that clutter the math curriculum and create cognitive load. Also, children should learn the mechanics first with an explanation later. Learning the mechanics of an algorithm requires practice-practice-practice and auto recall of single-digit number facts.
In the first week of school, first-grade students should learn at least two field axioms (rules) via the number line: a + 0 = a and a + b = b + a. Also, students should learn an important "common notion" from Euclid: Things that are equal to the same thing are also equal (i.e., the transitive axiom of equality).
The meaning of the equal sign is important and often overlooked. If 2 + 3 = 5 and 12 - 7 = 5, then it follows that 2 + 3 = 12 - 7 by the transitive axiom (rule) of equality. The left side of the equal sign {5} is equivalent in value to the right side {5}. Simply, students should "Think Like a Balance." Mathematics is built on true statements.
True/False Statements
3 + 4 = 7 TRUE because both sides are 7 (7 = 7)
3 + 4 = 5 + 6 FALSE because 7 ≠ 11
Here are the nine "field axioms."
Elementary Algebra (i.e., symbolic arithmetic ) obeys the rules of arithmetic.
I have given an example in arithmetic and the grade level of introduction.
1. a + 0 = a
5 + 0 = 5 (1st, zero property of addition)
2. a ⋅ 0 = 0
5 x 0 = 0 (2nd, zero property of multiplication)
3. a + b = b + a
2 + 3 = 3 + 2 (1st, commutative property of addition)
4. ab = ba
2 x 3 = 3 x 2 (2nd/3rd, commutiative property of multiplicaiton)
5. a + (b + c) = (a + b ) + c
3 + (5 + 2) = (3 + 5) + 2 (1st, associative property of addition)
6. a(bc) = (ab)c
4 x (3 x 2) = (4 x 3) x 2 (2nd/3rd, associative property of multipication)
7. a + -a = 0
2 + -2 = 0 (1st/2nd, the addition of opposites is zero; the inverse property of addition. Note 2 + -2 the same as 2 - 2.)
8. a ⋅ a^-1 = 1 (for a ≠ 0)
3 x 1/3 = 1 (2nd/3rd, the product of reciprocals is one; the inverse property of multiplicaiton)
9. a(b +c) = ab + ac
3 (4 + 7) = 2 x 4 + 3 x 7 (2nd/3rd, distributive property)
Note: Subtraction can be changed to addition, and division can be changed to multiplication. Thus 5 - 3 = 5 + -3, and 6 ÷ 5 = 6 x 1/5, or 6 times the unit fraction 1/5, which is 1/5 + 1/5 + 1/5 + 1/5 + 1/5 + 1/5 or 6/5 by repeated addition. Why is this important? Both Addition and Multiplication are commutative. (Subtraction and Divison are not commutative) You can add or multiply numbers in any order.
Rules in math are essential.
I often hear negative remarks about rules in math. Math is governed by rules such as the properties of numbers and equality. The rules should be learned in the early grades. Simply, young students must know the rules and be able to apply them.
Last update: 11-19-17, 11-21-17, 11-23-17, 12-25-17
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