Will I Make It to Calculus?
Reform math people teach kids alternative math rather than focusing on mastering traditional (standard) arithmetic content that is needed for success in Algebra in middle school and higher-level math. Knowing content enables problem-solving. Furthermore, the reformers insist on using minimal guidance constructivist methods during instruction, such as discovery/inquiry learning, problem-based learning, and others via activities in groups and discussions. These instructional methods are inferior to explicit teaching (explaining) with worked examples and conflict with the cognitive science of learning. Minimal Guidance Methods = Minimal Learning.
Fairness Crusades Downgrade Excellence
Unfortunately, the reform math people have dominated mathematics education in our schools for decades and believed that their ideas (pedagogy, ideology, & philosophy) are right regardless of evidence. Their "fairness" policies are flawed because they downgrade excellence. They think that "equalizing downward by lowering those at the top"* is good policy, but I think it has been a destructive policy. The "fairness crusades" have marginalized individual achievement, hard work, and excellence.
* The quote is from Thomas Sowell.
Common Core Is A Downgrade
1997 CALIFORNIA 3rd-Grade [World Class] MATH Standards:
In the 1997 world-class California math standards, students needed to be able to add, subtract, multiply, and divide using standard algorithms by the end of 3rd grade.
2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers (3,671 x 3 = __).
2.5 Solve division problems in which a multi-digit number is evenly divided by a one-digit number (135 ÷ 5 = __).
The 1997 California math standards re-established a sharp focus on standard arithmetic beginning in 1st grade. In 2008, the National Math Panel had determined that standard arithmetic was the best preparation for Algebra-1 by the 8th grade, not the trendy NCTM reform math. Common Core has embraced reform math. Sadly, the 1997 California standards were replaced by Common Core and reform math in 2010. It was a downgrade.
Reform Math Sidelines Standard Arithmetic
The floodgate for [NCTM] alternative math was opened again, this time when Common Core did not define standard algorithms. If we truly want children to understand and master math and gain number sense, then we should teach children essential content, not alternative math. It is traditional content in long-term memory that enables problem-solving, not reform math or calculator use. The hodgepodge of multiple, alternative strategies (algorithms) characterizes reform math. Reform math crowds the curriculum, increases cognitive load, confuses students, and sidelines the standard algorithms of core math.
Kids Are Novices, Not Experts
Dr. Wayne Bishop wrote that geniuses invented arithmetic, algebra, geometry, and calculus over thousands of years, so why would we ask novices to reinvent or "discover" math? Children are not pint-sized mathematicians; they are novices. It is a waste of instructional time for beginners to attempt to "uncover math procedures, properties, and proofs" or invent their own algorithms. Professor Wayne Bishop points out, "Professional mathematics educators continue to insist on discovery or constructivist pedagogy that forces students to invent their own algorithms rather than be taught the standard ones." The approach has not worked well.
Calculators Real-World Problems
According to Everyday Mathematics (EM), which is a typical reform math program, calculator use allows students to spend more time thinking mathematically and solving problems. But where's the evidence? "Calculators enable children to think about the problems themselves rather than focusing on carrying out algorithms without mistakes.” * This mindset aligns well with the National Council of Teachers of Mathematics (NCTM) reform math philosophy that started in 1989. The NCTM emphasized early calculator use, starting in Kindergarten. The alleged use of calculators, we are told, "can enhance children's understanding and mastery of arithmetic, promote good number sense, and improve problem-solving skills and attitudes toward mathematics" supported by a “preponderance of evidence.”** It’s not true! There is no preponderance of evidence. Also, there is a correlation between calculator use and the rise in so-called real-world problems. Real-world problems with messy numbers were used to promote the use of calculators (special interests) and advance the NCTM reform math agenda to marginalize the importance of standard paper-pencil arithmetic. Incidentally, according to G. Polya (How to Solve It), students should begin with routine math problems and their variants and incrementally build up to more complex problems. Also, the problems cited as examples in Polya's book require substantial background knowledge. The key to problem-solving is to know the content.***
* Teacher Reference Manuel Grades 1-3, Everyday Mathematics, 2007)
*** G. Polya: Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics." To be able to do mathematics means you know some content. You cannot solve a trig problem without knowing trig (in long term memory) and ample experience applying trig to solve problems. The same is true for arithmetic and algebra.
Switching from one reform math program to another reform math program is not going to turn around poor achievement. The fundamental difficulty is that students are not learning enough core arithmetic beginning in 1st grade. For decades, we have taught core [standard] arithmetic badly. Reform math with its alternative strategies (algorithms) and minimal guidance constructivist instructional methodologies (e.g., group work) crowd and needlessly complicate the curriculum with extras.
What We Used to Do
Over 150 years ago, American 2nd-grade students were taught multiplication and division in the public schools.* The young students memorized single-digit number facts and drilled to improve skill and performance using both multi-step abstract questions and word problems.
*Ray's Arithmetic Book, Grades1-2, the 1800s.
East Asian "rote learners."
The more "rote learners" of East Asia have dominated math for 20 years (TIMSS), not only in knowing and applying math but also in reasoning, i.e., problem-solving in math.* Isn't this exactly what we want our kids to do cognitively in mathematics: knowing, applying, and reasoning with content? It has been 20 years, and American math education is still on the wrong course. Sure, there has been an incremental improvement in U.S. math scores, but we are far from making the big gains that we could have achieved.
Lastly, it is up to the teacher to correct misconceptions on the spot with a counterexample. Math is not a matter of opinion.
Excuse typos and other errors.