The abstract goes to Concrete
The How always preceded The Why
No matter how hard Alice ran, she stayed in the same place, so has U.S. math achievement. "Faster! Faster! Now here you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!?" (The Queen to Alice, Through the Looking-Glass)
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| Running in Place Math Achievement Stagnation |
The constructivist approach of reform math hasn't worked. Scores on national and international tests have been low. Students are not required to practice arithmetic basics for mastery.
The only way to judge whether or not the instruction is working is through performance testing. Can the student do standard arithmetic efficiently? Zig Engelmann writes, "How successful is your version of the constructivist approach?"
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| Petrilli: Lost Decade of Educational Progress: 2007-2017 |
Michael J. Petrilli says the NAEP scores in math from 2007 to 2017 indicate a "lost decade of educational progress." Also, Bill Gates spent "hundreds of millions to improve teaching," but the initiative "failed to improve student achievement."
What is wrong with education? More money, various reforms, so-called innovative programs, and professional development haven't worked. We have been unable to improve the "teaching" in the classroom, that is, the teaching that substantially boosts real student achievement.
In schools of education, teachers are taught inferior (minimal guidance) instructional methods, unreliable theory, flawed equity and diversity guidelines, and a reform math curriculum that is not world-class. Moreover, K-8 teachers are weak in math and science content because they majored in education, which is not an academic subject. How can they teach subjects they don't know well? I am convinced that some elementary school teachers, today, do not know how to teach math or reading effectively.
The progressive scheme dumbs down math for equity. High achievers and low achievers are mixed in the same classroom or in the same math class (inclusion for fairness). Furthermore, all students get the same instruction (e.g., Common Core or its state rebrand) regardless of achievement level or ability (equalizing for fairness). Indeed, "equalizing downward by lowering those at the top," says Thomas Sowell, is "a crazy idea," It is a "fallacy of fairness" that hurts all children, especially children of color. In American schools, it has been taboo to sort kids by achievement for math. In contrast, Singapore sorts kids in math in the 1st. grade. All kids should be given opportunities in schooling but not always the same opportunities or the same instruction. Kids who are advanced in arithmetic, for example, need an entirely different math curriculum from other kids. Instead of advanced math, they are given grade-level material. It is called equity, but it is false equity.
Also, the priority should be given to learning the compact, standard algorithms first, but this is not the case in the reform math as taught in our schools. Starting in 1st grade, we don't stress the mastery of fundamentals. Memorization and drill-to-develop-skill have fallen out of favor in modern classrooms. Group work, discovery learning, and other fads are much more important than automating the fundamentals of arithmetic, progressives (aka liberals) say. They are wrong!
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| Help |
Let's say we needed to calculate 6 x 1584. In Mock Turtle's 3rd-grade "Uglification" class, students were taught a wide range of alternative methods or strategies to multiply: lattice, repeated addition, array, area, partial products, make a drawing, calculator, distributive. (Which method should I use?) But, learning all these alternative methods clutters the curriculum and the minds of children (cognitive load). Who multiplies using the area model? The standard algorithm has been pushed aside. Some kids never learn it. Progressive education strikes again. We spend a lot of money, time, and effort on progressive ideas and theories that don't work well. Why make arithmetic harder than it is?
The Real World: To move forward, students must know the compact, standard algorithms, not complicated alternatives.
Part I Abstraction
Why isn't math taught this way?
We think concrete goes to the abstract because that's what everyone says. But, what if the "experts" were wrong? The idea is that children develop number concepts only from concrete representations, then, later, as abstract ideas (Piaget's theory), but it doesn't work that way. Indeed, 5 is 5 and can be represented in hundreds of different ways such as 5 people, 5 pencils, 5 cubes, 5 dots, etc., but the number 5, itself, is an abstract idea. It is a concept in our mind. You don't need to go from concrete to abstract to understand the meaning of 5. The number 5 is already abstract, which is the starting point.
The number line helps novices visualize what 5 is. It helps beginners with the idea of magnitude. The number 5 is a point on the number line that is one more than 4 and two less than 7, and so on.
The meaning of 5 is derived from its connections with other numbers.
What about 3 + 4? You don't need to show 3 + 4 is 7 in several concrete ways, or make a drawing, or write an explanation. If need be, the number line can verify that 3 + 4 is 7, which is obvious. But the number line is not a proof or a why. Also, showing different ways to represent 3 + 4 is not a verification of "deep understanding" or why 3 and 4 add to 7.
The number line is essential arithmetic, but it is seldom used in the primary grades, which is a huge mistake as children try to grasp magnitude and how numbers behave.
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| The Number Line Is Important Arithmetic |
Numbers are invented in our minds. They are abstract. Children start with the whole numbers as concepts and should learn connections between numbers on a number line, such as "add 1" (4 + 1 = 5), or 3 + 4 = 7, or 7 - 3 = 4, or 3 x 4 as the sum of three fours: 4 + 4 + 4 = 12.
So, why do we ask novices to explain the obvious several different ways? We are told that it shows understanding. Really? I don't buy it. If needed, the number line verifies it. That said, 3 + 4 = 7 should be memorized early in the 1st grade, but, unfortunately, the memorization of number facts has fallen out of favor in progressive classrooms.
In short, arithmetic is invented in our minds. In every word problem, no matter how simple or complicated, the student needs to pull out the numbers and figure out what to do with them (execute an operation) to find an answer. You see, arithmetic is about numbers that are abstract ideas. It's not about 5 pencils. It's about the 5 and how it relates to other numbers. We use straightforward arithmetic to solve word problems.
If arithmetic is abstract, then why isn't it taught that way?
Indeed, the equation 3 + 4 = 7 is the solution to hundreds of word problems. In 2nd grade, I would write an equation on the board and ask students to make up a story (word) problem. Some students surprised me by turning 3 + 4 = 7 into a missing addend problem (a subtraction problem) such as, "I have 7 apples and give 3 of them to Bill. How many apples do I have left? Or, 7 - 3 = 4. The problem isn't about apples; it is about the numbers and how they are related. The basic relationship between addition and subtraction is abstract.
The HOW always preceded the WHY.
Why isn't math taught this way?
Tobias Dantzig (Number: The Language of Science), a book highly praised by Albert Einstein, writes, "In the history of mathematics, the "how" always preceded the "why," the technique of the subject preceded its philosophy. This is particularly true of arithmetic. The strength of arithmetic lies in its absolute generality such as a + b = b + a. Its rules admit of no exceptions: they apply to all numbers. Every number has a successor [add one]. There is an infinity of numbers." Teaching novices arithmetic should be the "how," not the "why." Kids are novices, not little mathematicians.
Ian Stewart (Letters to a Young Mathematician) clarifies, "One of the most significant differences between school math and university math is proof. At school we learn how to solve equations or find areas of a triangle; at university, we learn why those methods work and prove that they do." We need to stop teaching kids as if they are little mathematicians, which they are not.
American teachers are hung up on the "why" of everything, but in arithmetic, it is proper to learn the "how" first and the "why" later, which is what happens in Asian nations. Learn the technique first and get it right (i,e., automate it). Learn the "how" first and don't worry about why it works.
Note: Standard arithmetic is simple and compact, but it is not easy to learn without the memorization of single-digit number facts and practice-practice-practice.
National Mathematics Advisory Panel (2008):
"Computational proficiency with whole number operations is dependent on sufficient and appropriate practice to develop automatic recall of addition and related subtraction facts and of multiplication and related division facts. It also requires fluency with the standard algorithms for addition, subtraction, multiplication, and division." I do not recall teaching arithmetic without some level of student understanding. Understanding develops slowly over the years. Also, automating the mechanics of arithmetic for fast access frees working memory space for problem-solving. Arithmetic is a tool for solving problems.
First-Grade Subtraction:
Go Vertical! Let's Borrow! Missing Addend Equations!
The standard algorithm of subtraction is a key math skill that can be taught in the 1st grade. The vertical format is a place value system. Start with problems that don't involve borrowing, such as 47 - 25, but use the vertical format to stress place value.
Teach borrowing. Borrowing shifts a ten back into the ones place, that is, one ten equals 10 ones. The 16 can mean 16 ones. It is simple place value and reverse engineering. Students can compose missing addend equations in their heads and recall memorized addition facts to solve them. What number plus 9 is 16? {7 + 9 = 16} Or, x + 9 = 16 . It is a missing addend equation. Children can use memorized addition facts to do subtraction:
addend1 + addend2 = sum. It is important to relate subtraction to addition.
Like it or not, Arithmetic is based on Rules
I often hear teachers say that they don't want kids memorizing a bunch of rules. But, arithmetic and algebra are governed by rules, and students must know the rules and be able to apply them to do correct mathematics. For example, the commutative rule should be learned in the first full week of 1st grade: 2 + 3 and 3 + 2 give the same answer. Arithmetic is based on rules. There are not that many, but they should be learned as early as possible, such as the "zero rule" for addition: 3 + 0 = 3. The properties of numbers that children learn are rules. They are also mini procedures.
Learn the procedure first (i.e., the "how," such the procedure for solving 1/2 of 1/2, which is a multiplication of fractions 1/2 x 1/2 = 1/4.)
The stress on understanding "why" something works is misplaced. For example, Isaac Newton invented a calculating method (calculus). He knew it worked because the results of his calculus agreed with the experimental results, but he didn't know why his calculating method worked. The "why" wouldn't come for another 200 years with "limits." But, that didn't dissuade Newton from using his calculating method to figure out physics. It always worked. He worked out the how, but not the why. Still, he was a brilliant mathematician and physicist.
So, when we teach kids (novices) an operation, such as addition in 1st grade, we should first make sure they can do the procedure well (the "how"), which requires practice-practice-practice. Focus on the how not the why.
Teachers should teach the procedures (aka algorithms) that are the most efficient and compact and work all the time. These are the standard algorithms.
Understanding comes out of doing. Why are parents concerned about their children's math education? Kids can't do simple arithmetic.
Students Learn Math Inductively
Mathematics uses deductive reasoning and logic to create new math. But, students--who are novices, not experts--don't learn math that way. They learn math inductively. That is, the teacher explains several examples and then tells students that the method always works. It is impossible to test every possible case or present a formal proof. The teacher should also explain the counterexample.
Students don't need to draw pictures or count objects to understand math.
enVision Problem Solving
Kara found 5 shells. Then she found 3 more. How many shells did Kara find? Draw a picture. Write the number.
The problem-solving idea of enVision Math, a popular reform math program, is for the student to draw little pictures then count the objects depicted. The memorizing of critical single-digit number facts is not part of the curriculum in 1st-grade state standards. In my opinion, 5 + 3 = 8 should be memorized in 1st grade. It can be verified on a number line if need be. Moreover, the answer to the question is "Kara found 8 shells," not the number 8.
Teachers teach math for test-based proficiency, not mastery, and that is what is wrong with math instruction today. Proficiency in state math tests should not imply that the student has mastered grade-level math content. Under Common Core, state standards, test-based accountability, and "teach to the test items" mentality, our curriculum has become fragmented, and our instructional methods have been inferior.
Let me repeat. Educators focus on low achievers. It is called "equalizing downward by lowering those at the top," which is a "fallacy of fairness," says, Thomas Sowell. Instead, teachers should focus on equalizing upward to challenge high achievers by tracking them in at least one of their best subjects, such as mathematics, but not all their subjects. It is tracking-by-subject, but there should always be some flexibility in grouping along with way.
To Be Revised
Updated: 7-7-18, 7-10-18, 7-11-18, 7-13-18, 7-16-18, 7-17-18, 8-31-18
Model Credit: Gabby, Hannah, Shayna
The NAEP chart is from an article by Michael J. Petrilli (Fordham)
