Progressive educators believe students should not be taught standard algorithms -- no Euclid for you! -- until they discover mathematical relationships on their own, writes Lance Izumi of the Pacific Research Institute. This philosophy "has frustrated children and appalled parents."
Euclid’s method of finding the greatest common divisor of two integers was first described in 300 B.C., he writes. It's still used today, because it works.
Students should learn how and why these algorithms work because they "are our best representation of connections among mathematical concepts," write Alice Crary and W. Stephen Wilson in the New York Times.
InTraditional Math, Barry Garelick explains that “delaying teaching of the standard [addition] algorithm until the fourth grade and relying on place value ‘strategies’ and drawings to add numbers is thought to provide students with the conceptual understanding of adding and subtracting multi-digit numbers.” Reformers abhor "rote learning." However, he writes, alternative strategies are “inefficient cumbersome methods” that “confuse more than enlighten.”
Izumi interviewed math tutor Mike Malione for his 2024 book The Great Classroom Collapse. The tutor worked with students whose schools stressed conceptual understanding.
"When given the problem of ½ x ¾, children were not shown that they could get the right answer by multiplying 1 x 3 and 2 x 4 to get 3/8." Instead, the teacher would tell students "to draw a rectangle and split the rectangle into parts and then shade the parts to represent ½ and ¾."
“We’re going to draw a picture every time we’re given 10 problems with fractional multiplication, when you could do them in your head? We’re going to take time and draw 10 pictures and draw lines and shade? That’s insane.” -- Mike Malione, math tutor
Meanwhile, his students “don’t remember how to multiply two-digit numbers because they have never practiced it enough.”
In an analysis of Common Core for the Brookings Institution, education researcher Tom Loveless described a first-grade math program that "told parents not to teach the standard algorithm for addition to their children at home," because they were being taught to draw pictures of objects in groups of 10. That was "both time consuming and tedious," Loveless concluded.
Standard algorithms "are efficient and packed with mathematics,” writes Loveless. Once “students have mastered single-digit operations and the meaning of place value, the standard algorithms reveal to students that they can take procedures that they already know work well with one- and two-digit numbers, and by applying them over and over again, solve problems with large numbers.”