Early math is just as important as early literacy.  |
| Algebra-1 is a middle school subject for students who are prepared! Preparing for Algebra-1 starts in the 1st grade. In the early grades, students should memorize math facts and learn the efficient, standard algorithms from the get-go. Practice-practice-practice and explicit teaching will prepare students for Algebra in middle school. As it stands now, the math standards from Common Core are not world class, so students start behind and stay behind up the grades. Students won't be ready for Algebra by the middle school unless we upgrade the curriculum to world-class and boot popular constructivist instructional methods and theory (Deweyism). The theory is wrong. |
Discovery learning and other minimal guidance methods that had been promoted by Dewey have been ineffective. Kids need explicit teaching from teachers who know math. Starting in the 1st grade, students should memorize math facts and learn standard algorithms from the get-go. And, they need to practice-practice-practice so that fundamentals stick in long-term memory for problem-solving. Practice may not be much fun or even dull, but it is critical preparation for algebra and higher mathematics, so ignore grumbling students and push Deweyism to the side.
Social Process. Really?
John Dewey said that education is a social process: "Education is a process of living and not a preparation for future living." Dewey's idea that schooling is for socialization manifests itself in several ways in the classroom, including group activities, discovery learning, other minimal teacher guidance methods, collaborative learning, and so on. In short, the teacher no longer teaches.
Groups
Students sit in groups of 3 or 4, facing each other. They are encouraged to do group activities (socialize). In one group, I noticed that two students had words, but the teacher refused to separate them: "They must learn to get along with each other." But, at what cost? The fast learning of essential content is not the goal of the group work.
Note. Early Algebra is accessible to very young children through standard arithmetic. No manipulatives, no calculators, and no group work.
Early math is just as important--perhaps more important--as early literacy.
Ian Stewart, a mathematician, explains, "Without internalizing the basic operations of arithmetic, the whole of mathematics will be inaccessible to you." Stewart is referring to math facts and standard algorithms by hand. Maya Thiagarajan writes that the East Asians are obsessed with math and advocate early math: "A strong math foundation must be built in the first 10 years of a child's life."
In contrast, many U.S. students are deficient in math skills, which starts in the 1st grade and rises up the grades into adulthood. U.S. math standards are not world class! In short, many students stumble over simple arithmetic. Likewise, adults often boast about their poor math skills. Many elementary school teachers fear math, which rubs off on their students. K-5 teachers are weak in both math and science, especially chemistry and physics. Children are great imitators. They learn by imitation, repetition, and practice. The attitude of their teachers rubs off on them.
Elementary school science programs and textbooks lack math, something Nobel-Winning Physicist Richard Feynman pointed out as a significant flaw in the 70s. "None of the science books said anything about using arithmetic in science." He said that elementary school math textbooks were "universally lousy," too because they lacked sufficient applications and word problems. Not much has changed.
Contrary to the "constructivist" theory taught in schools of education, which lacks evidence, students should not be expected to figure out the basics of arithmetic on their own via group work, discovery activities, or other minimal guidance methods of instruction. Starting in the 1st grade, the fundamentals, which begin with math facts and standard algorithms, should be taught explicitly and learned through practice-practice-practice, so they stick in long-term memory for problem-solving. Without content knowledge in long-term memory, critical thinking (i.e., problem-solving in math) is empty.
Knowing something implies some level of understanding.
Educators should focus on knowledge, not understanding, which is ambiguous and difficult to measure. When we say a student's understanding is weak, what we mean is that his knowledge is inadequate and fragile.
Math is not a matter of opinion; it is a matter of fact.
The Abacus Problem --> The Calculator Problem
Richard Feynman writes, "With an abacus, you don't have to memorize a lot of arithmetic combinations: all you have to do is learn how to push the little beads up and down. You don't have to memorize 9 + 7 = 16; you just know that when you add 9, you push a ten's bead up and pull a one's bead down."
That's the problem with calculators. The NCTM pushed calculator use starting in Kindergarten. First-Grade textbooks like Scott Foresman - Addison Wesley (2001) had "Explore with a Calculator" activities starting in Chapter 1. I used number lines extensively in the first couple of weeks of school as 1st-grade students started to grasp simple combinations. But, I did not want students relying on the number line or counters (cubes) to calculate. The number line and counters were put aside after the first two weeks of school and replaced with flashcards. I wanted students to memorize the combinations.
Novices need to learn numbers, math facts, how numbers combine, and place value, that 12 is 1ten+2ones. The standard vertical algorithm, which lines up ones under ones and tens under tens, is efficient and the best model for place value.
In 1st-grade, the idea of "carry" is important.
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| Credit: WolframAlpha |
1. If you can't calculate it, then you don't know it. I developed this idea when I tutored high school students in precalculus. They would say that they understood the idea but had difficulty applying and calculating it. Thus, their knowledge and calculating skills were weak. Your knowledge of an idea, let's say of perimeter, is better if you gain lots of experience calculating perimeters of polygons.
2. I substitute the word knowledge for understanding, which is ambiguous and hard to quantify. Knowledge of something implies, in my opinion, that you have some level of understanding, which is difficult to measure. Kids need background knowledge. Math is mostly knowledge of content and calculating. Mathematician Ian Stewart states it this way: "Math requires a lot of basic knowledge and technique."
Reciting a definition of perimeter doesn't mean much. If you can't calculate perimeters, then your knowledge of perimeters is weak, hence the "implied understanding" of perimeters, is weak, too.
3. Practice produces improvement in performance, which is measurable, but, on the other hand, understanding is ambiguous, implied, and difficult to quantify. It is the reason I use the word knowledge, not understanding. Also, I often use the word ability rather than talent because practice does not create talent. Ability, I think, is something you are born with, but it can be developed only through excellent instruction and practice-practice-practice. We can measure improvement, but talent or understanding is ambiguous and difficult to quantify. We are not all equally creative, and we do not all have the same abilities. Not all children who have an ability have the same opportunities to develop and enhance it.
4. Mathematical Ability varies widely
In my opinion, all children are born with some mathematical ability, but math ability varies widely like any ability, such as the athletic ability or musical ability, and so on. Some kids are just better at math than others; however, this does not mean that the vast majority of students cannot learn arithmetic and algebra at an acceptable level with enough practice and good teaching.
5. Focus on Performance
My conclusion is that we should focus on ideas that are measurable, such as the student's performance in arithmetic, algebra, basketball, chess, violin, or piano, etc. If you do well on an arithmetic test, then the implication is that you understand the math at some level. Still, you would expect a 6th grader's understanding of perimeter is at a higher level than that of a 1st grader because the 6th grader presumably would have greater knowledge of and experience with perimeters than the 1st grader.
Note. Reading: New vocabulary should be introduced before the reading lesson.
To Be Revised: 1-4-19, 1-6-19, 1-12-19
©2019 LT/ThinkAlgebra
