The New Math of 1960 Flopped, and Today's Math Fights Keep Repeating It

It is 1965. A father comes home from work, tired, sits at the kitchen table, and opens his second-grader's math homework to help with the subtraction. The page asks the child to explain subtraction in terms of sets and to distinguish between a number and a numeral. Dad, who has balanced a household budget for fifteen years, stares at it. He understands subtraction. He does not understand this. He says, with great dignity, "Ask your mother."

Mom, a hypothetical but entirely recognizable woman, is already at the stove, reading the same worksheet. This is the great American drama of mathematics education, folks, and it has run for more than sixty years with the same cast, the same plot and the same ending, which is that a group of experts discovers that arithmetic has been taught wrong for centuries, fixes it, and finds that the children cannot do arithmetic.

The claim, and what the words mean

Thesis: the New Math of the late 1950s and 1960s failed for reasons that recur in later reform movements, and an honest math program teaches facts, standard algorithms and practice, with understanding built on top of them rather than substituted for them.

Terms first.

• New Math refers to the wave of mathematics curricula, begun in the late 1950s, that emphasized abstract structure, set theory and the logic of number systems in elementary and secondary schools.
• A standard algorithm is the conventional, efficient, step-by-step procedure for an operation, such as carrying in addition or long division. It works for every number, which is the point.
• Procedural fluency is the ability to carry out procedures accurately and efficiently. Conceptual understanding is grasping why the procedures work. Automatic recall is knowing basic facts, such as the times table, without having to work them out.

I am going to argue that those three belong together, and that every reform movement in this story went wrong by treating them as rivals.

One: Sputnik and the panic

The story begins with a launch. In October 1957 the Soviet Union put Sputnik, the first artificial satellite, into orbit, and America concluded that it had been beaten in the classroom as well as the laboratory. Congress passed the National Defense Education Act in 1958, and the National Science Foundation put money into rewriting school science and mathematics.

That same year, the School Mathematics Study Group, usually called the SMSG, was launched under the direction of the Yale mathematician Edward Begle, with funding from the National Science Foundation. Its purpose was to produce a modern curriculum, written in large part by university mathematicians, that would give American children the kind of mathematical thinking that research mathematicians valued. The SMSG produced textbooks for grades from elementary through high school, and they were distributed widely.

I want to be fair to those mathematicians. Their goal was honorable, and some of what they wanted to do was sensible. Mathematics is not just a bag of tricks, and a child who sees why the procedures work is better off than one who does not. The Cold War gave the project urgency, and the money gave it reach.

Two: what went wrong

The trouble began when a curriculum designed by research mathematicians met an actual second grader and an actual second-grade teacher.

1. The authors wrote for people like themselves. An abstract treatment of number as an element of a structure appeals to someone who already knows arithmetic cold. It is not how a seven-year-old gets there. The child who has not mastered the facts has nothing to build the abstraction on.
2. Teachers were not ready. Elementary teachers were generalists, many with limited mathematical preparation, and were handed new material and a short training period. A teacher who is herself unsure of the new terminology cannot teach it with confidence.
3. Parents could not help. This is the most underrated cost. Homework is how a school extends its reach into the home. When the vocabulary changes, the home can no longer help. The satirist Tom Lehrer captured the national bewilderment in a 1965 song called "New Math," which walked through a subtraction problem in a base other than ten and treated the whole business as a joke at the expense of anybody who had ever carried a one. The audience laughed because they recognized the situation.
4. Computation slipped. Critics argued that in the rush to teach structure, schools neglected practice. Children could talk about sets but make errors in multiplication. Reformers had the concepts right on the page and the arithmetic wrong in the notebook.

The mathematician Morris Kline, of New York University, made the case in a book that has become the classic indictment. Why Johnny Can't Add: The Failure of the New Math appeared in 1973. Kline argued, with the authority of a working mathematician, that the reformers had mistaken the polished structure of mathematics as finally written down for the way a person learns it, and that the result was formalism without meaning. By the time his book appeared, the New Math was already in retreat. The country had tried it, and the country went, in the phrase of the time, back to basics.

Three: the pattern returns

If the story ended in 1973, it would be a curiosity. It does not end there, because the same structure reappeared.

1989. The National Council of Teachers of Mathematics, the main professional body for mathematics teachers, published its Curriculum and Evaluation Standards for School Mathematics. The document, widely influential, emphasized problem solving, communication, reasoning and conceptual understanding, and its adoption was followed by a new generation of curricula, often described as "reform" or "inquiry-based" mathematics. In many classrooms, those curricula leaned on having children discover their own methods for operations, drew back from teaching the standard algorithms early, and used a great deal of group work and manipulatives.

Again, the goals were admirable. Again, the practice drifted. Parents saw their children being asked to invent methods for multi-digit multiplication, and heard that the traditional method was optional or even discouraged. The homework table recovered its old drama.

California in the 1990s. California was the great battleground. Its 1992 framework embraced the reform approach, and the results on state and national assessments were disappointing enough that the state reversed course. In December 1997 the State Board of Education adopted new mathematics content standards, which were widely praised for their clarity and for their emphasis on arithmetic fluency, the standard algorithms and the core content of algebra. The "math wars" that surrounded those years, with parents and mathematicians on one side and education schools and curriculum publishers on the other, were an echo of the New Math fight, with the roles recast.

1999. A group of mathematicians and scientists, including Nobel laureates, signed an open letter to the Secretary of Education, published in the Washington Post, objecting to the Department of Education's endorsement of a list of ten programs it called exemplary or promising. The signers worried that several of the programs de-emphasized basic skills. The episode is worth remembering for who was in the dispute: working mathematicians complaining that the government's expert panel had gone too far toward the same side that Kline had criticized a generation earlier.

Four: the best case for the reformers

I should state the opposing case in a form its defenders would recognize.

They would say, first, that rote memorization without understanding is brittle. A child who knows the algorithm but not why it works cannot adapt when a problem changes. Second, the old way of teaching was not a golden age; large numbers of American students failed at arithmetic then too, and the idea that all children were once drilled to excellence is a myth. Third, the real world demands reasoning and problem solving, not just calculation, and a curriculum that prepares children only to execute procedures leaves them unprepared. Fourth, new curricula have been revised and the worst excesses corrected.

All four have something to them. Understanding matters. The past was not perfect. Problem solving matters. And reformers do adjust.

Five: the answer

Still, the record leaves a conclusion that is hard to avoid, which I will give in numbered points.

1. Understanding rests on fluency; it does not replace it. A child who has the basic number facts in memory is free to think about the problem. A child who must count on her fingers for every sum spends her working memory on the arithmetic and has nothing left for the idea. Fluency is not the enemy of reasoning. It is the thing that makes reasoning possible.
2. Standard algorithms are not arbitrary. They are the compact products of centuries of human ingenuity, and they work on every case. Teaching them, with an explanation of why they work, gives a child a tool that scales from two-digit numbers to two hundred digits. A method that a child invents is a fine exercise. It is not a replacement for the best method humanity has found.
3. Practice is not a dirty word. Every skill that anyone has ever mastered, from piano to free throws, takes repeated practice. A curriculum that apologizes for practice is apologizing for how learning works.
4. The national panel agreed. In 2008 the National Mathematics Advisory Panel, appointed by the President, issued its final report. It emphasized that students need a strong grounding in whole numbers, fractions and the topics leading to algebra, and said that conceptual understanding, computational fluency and automatic recall of facts support one another and should not be set against each other. When a body convened to settle a fight concludes that the fight was a false choice, the fight deserves a moment of silence.
5. The cost falls on children without help at home. A family with money hires a tutor and the child is fine. A family without one depends on the school. That is the same lesson as Ending Gifted Programs in the Name of Equity Hurts the Kids It Claims to Help: when schools fail to teach, the wealthy purchase a substitute and the poor go without.

A worked example of fluency and understanding together

Take 47 times 38. The standard algorithm handles it in two lines. Multiply 47 by 8 to get 376. Multiply 47 by 30 to get 1,410. Add: 376 plus 1,410 is 1,786. That is the whole procedure.

Now notice what it contains. The two lines are partial products, one for the ones digit and one for the tens digit, and the sum reassembles the pieces by the distributive law: 47 times 38 equals 47 times 8 plus 47 times 30. The algorithm is the concept, packed into a form you can do half asleep. A child who learns it with the explanation, and then practices until it is automatic, owns both the understanding and the speed.

A child who is told to invent her own method may produce something lovely and slow. She may draw a grid, or break the numbers into friendly chunks, and I have no objection to showing that once, because it illuminates why the algorithm works. My objection starts when the inventive method replaces the efficient one and the child is still working on it in seventh grade while her classmates in the next county are on to fractions and equations. Mathematics is cumulative. Every skill is a step on the stair to the next, and a child who is slow on the lowest step is late to all of them.

Six: what an honest math program looks like

If you want a test for a math program, ask these questions of any curriculum your district adopts.

• Does it teach the number facts to automaticity, with regular practice, by the end of the early grades?
• Does it teach the standard algorithms, with an explanation of why they work, and expect students to use them?
• Does it build in a coherent sequence, each topic resting on the last, rather than spiraling through a topic briefly and leaving?
• Does it give enough practice, with feedback, that students can do the work independently?
• Can a parent follow the homework? If not, the curriculum has cut the home out of the loop.
• Does it publish its results? Programs that work can show it.

What the Nation's Report Card Measures and What Proficient Really Means explains how to read the national assessment, which is the best long-run yardstick we have of what children know in math.

The Founding angle and the larger lesson

Here is the constitutional point, and it is simple. The curriculum of the school is a decision for the people, through their states, districts and families, and not for a panel of experts however distinguished. The New Math was funded from Washington and written by a national panel, and it reached districts across the country at once. Federal money helped shape local curriculum and the error was national in scale. The same dynamic appears in Common Core Was Voluntary on Paper, Here Is How Washington Made It Stick. A decentralized system makes mistakes too, but it makes them in a thousand places and corrects them in a thousand places. A national mistake has no competitor to show it up.

I also want to offer a small, cheerful observation. Every one of these cycles was ended not by experts but by parents who sat at the kitchen table, looked at the page, and said that this did not look right. The parents were right in 1965, right in 1997, and they are right when they ask about the homework in your child's backpack. They did not need a degree in mathematics. They needed the nerve to say what they saw. That is the most hopeful part of the whole story.

The case in brief

1. The New Math, launched after Sputnik by the School Mathematics Study Group (1958), taught abstract structure to young children and left parents, teachers and arithmetic behind, as Morris Kline argued in 1973.
2. The pattern returned with the NCTM standards (1989) and California's reform and its 1997 standards, and the 1999 mathematicians' letter showed the same fault line.
3. Conceptual understanding, procedural fluency and automatic recall support one another, a point the 2008 National Mathematics Advisory Panel made explicitly.
4. An honest program teaches facts, standard algorithms and practice, in a coherent sequence a parent can follow.
5. Curriculum decisions belong close to families, where mistakes are small and correctable.

What you can do: ask your district for the name of its elementary math program and look at one week of homework, then ask your principal whether the program teaches the standard algorithms and number facts to fluency.