Kindergarten
Pre-Arithmetic Skills
When learning arithmetic, students should start with simple skills. For example, in Kindergarten, students should count and write numbers daily, repeatedly do simple arithmetic combinations on a number line, such as 2 + 7 = 9, and use an equal arm balance to measure masses of objects. Other pre-arithmetic skills include the commutative rule (2 + 3 = 3 + 2), adding zero (6 + 0 = 2), adding one (11 + 1 = 12. “The beginning exercises are simple and do not resemble later exercises (just as beginning piano exercises do not look much-advanced ones).
On the number line, start with simple sequences of calculations:
1 + 1 = 2, 1 + 2 = 3, 1 + 3 = 4, 1 + 4 = 5 Or 5 + 1 = 6, 6 + 1 = 7, 7 + 1 = 8, etc. OR 5 + 1 = 6, 5 + 2 = 7, 5 + 3 = 8, 5 + 4 = 9, etc. Do this rote training on the number line, again and again. When introducing larger numbers such as the teens, students should learn them as eleven: 10 +1; twelve: 10 + 2, etc. on the number line that goes from 0 to 20.
Also, K-students should learn to read using Zig Engelmann’s book (Teach Your Child to Read in 100 Easy Lessons).
Engelmann: "Only 10% of each lesson is new material. The remaining 90% of each lesson’s content is review and application of skills students have already learned but need practice with in order to master."
Teach Kids Algebra (LT)
I do not teach to the state test. I give lessons on pre-algebra skills, which puts stress on a student's cognitive effort. It's sensible but not always well received by some students who don’t want to think.
Also, the school should abandon reform math and return to teaching basics—memorizing number facts, practicing the standard algorithms for mastery, recognizing patterns, and solving word problems by writing and solving equations. The problem is that the teachers don’t know how to teach math basics. Their skills are weak. They don’t know how to teach for mastery. They are told to teach the test, which is a “ bits and pieces” curriculum.
In the reform math era, students are confused, and parents are baffled. Reform math stresses strategies, alternative algorithms, so-called mathematical practice standards, group work, and other extras, Students are novices, not little mathematicians. They don’t think like adults. Novices require explicit instruction and a lot of repetition, practice, and review to learn something like arithmetic.
We sent men to the moon using Newton’s Laws, a slide rule, and trig; today we send students to remedial math at community colleges using calculators and reform math. Even the state tests and the GED, AP, and SAT exams require calculators. Students are weak in arithmetic and don't know enough algebra. A whopping 87% of TUSD high school graduates who apply at Pima Community College (Tucson) are placed in remedial math classes. The math curriculum in Tucson and Arizona is not world-class. This is the case in most states. Common Core math is not world-class.
There is no generalized thinking skill independent of domain content knowledge (E. D. Hirsch). But, many educators believe that there is a generalized thinking skill that can be applied to anything. It simply isn't true. Problem-solving (i.e., critical thinking) is domain-specific. Thinking in math is different from thinking in science, and so on.
Also, “understanding" does not produce mastery; practice does. Even our best students are below international benchmarks. "It's not that Asian kids overachieve; it's that American kids underachieve!" We should focus on the mastery of fundamentals like Singapore, not “learning” for a state test. But, we don’t.
The way we teach math can block a child’s future.
For decades, we have taught arithmetic poorly.
The math I taught to 3rd graders in the early 70s was far different from what 3rd graders learn today. Today, students can’t calculate well, even though it is a key factor for problem-solving. You don't make drawings to do arithmetic. Who does that? It's another bad idea! Students should rely on memorized facts, fast algorithms, and pattern recognition to solve math problems.
In general, U.S. Children are not mastering basic arithmetic. They are predominantly taught a version of reform math that downplays memorization, standard algorithms, and traditional instructional methods such as drill-to-develop-skill. Also, “state standards” are based largely on Common Core, regardless of the rhetoric from state leaders. CC math is not world-class, which puts our kids behind.
The main reason for our educational problems is "the teaching" in the classroom, but teachers, educators, and administrators don’t think that way. Educationists give excuses such as poor parenting, societal problems (e.g., poverty, drugs), and not enough money. The same excuses were given 50 years ago. "If we only had more money…."
Even the best students complain that the reform math they are taught is confusing and overly complicated. Reform math is a hodgepodge of strategies and alternatives, not traditional arithmetic. Parents are baffled and can’t help their kids. Reform math is a hodgepodge of strategies and alternatives, not traditional arithmetic. The many alternative algorithms and so-called strategies take precedence over the standard algorithms.
(Note. I call today's math "reform math," which stems from the 1947 NEA Yearbook and the 1989 NCTM standards. Reform math is promoted in Common Core and state standards and taught in schools of education.) Even in GATE classrooms, students are taught grade-level math, which is actually below grade-level at the international level. Our kids are behind, but no-one takes notice. We keep doing the same things, again and again, hoping for different outcomes that never happen.)
This is the status of many 4th graders. They struggle over the content they should have mastered in 2nd and 3rd grade but had not. Arithmetic isn’t taught for mastery. Memorization and practice-practice-practice and review-review-review have fallen out of favor in the progressive schools across America. Teachers use inefficient minimal-guidance methods and test prep. They are no longer the academic leaders in the classroom; they are facilitators, which is a radical change.
My 3rd Grade: 1971-1972
In contrast, my 3rd graders (1971-1972), 28 of them, memorized the x-facts and practiced both the multiplication and long-division standard algorithms. They also learned fractions and parts of measurement and geometry. In addition to math, my 3rd graders had lessons in reading/phonics, writing/grammar, science, history, cursive, and so on. All students were expected to use cursive writing starting no later than December for spelling and writing assignments. The 2nd semester was long-division time. Incidentally, addition, subtraction across zeros, and place value were reviewed and extended in the first couple weeks of school. No manipulatives were used. Discipline problems in my classroom did not exist. It was a fun time, but hard work.
My algebra program (Teach Kids Algebra - TKA) has been in decline lately because it is linked to basic arithmetic. Students are not learning enough arithmetic. For example, they don't master multiplication in the 2nd and 3rd grade. The problem will persist as long as teachers focus on strategies and alternative algorithms (aka reform math), and use inefficient minimal-guidance methods (e.g., group work) and test prep instead of the explicit teaching of traditional arithmetic from the get-go (1st grade on up).
Students need to memorize stuff and practice the standard algorithms starting in the 1st grade. Continual review is needed, too.
Memorization is good for kids.
Facts in long-term memory boost thinking and problem-solving in working memory.
Note. I taught and supervised the Talented & Gifted programs (TAG) at five elementary schools when I lived in Delaware. The TAG program was for the academically advanced and had stringent qualifications for admittance.
The GATE program is mostly an enrichment program for bright students. As implemented at R/N, it is not for advanced math students or acceleration in math. Although some would disagree, GATE is not designed to improve an elementary student's achievement in specific academic areas such as math, reading, or science. Even the students in self-contained GATE classes get grade-level math for equity. Equity, which is a "fallacy of fairness," cannot produce equal outcomes (Thomas Sowell). “Equalizing downward by lowering those at the top is a crazy idea.”
In contrast to GATE, my algebra program is focused on content. It does not pretend to teach children to be more creative nor does it stifle curiosity. It is not developmentally inappropriate as many believe. However, TKA does require children to use their cognitive abilities, but some bright kids don’t like that. Math is harder than other subjects because it is abstract. I have observed that some very intelligent students are weak in standard arithmetic.
Another myth is the right brain/left brain. We now know that any cognitive activity goes through both sides of the brain, not favoring one side over the other. Also, there is no evidence for learning styles (Willingham). There are many common practices and theories in education that are not supported by scientific evidence.
Note. Kids aren't reading books this summer. They would rather spend much of their free time playing games, texting, or doing social media (Instagram, etc.) on their smartphones. Kids, today, are glued to screens.
LT
