For example, consider the following two published test questions. The first comes from California’s old standards, the second from the Common Core.

1. Which of the following best describes the triangles shown below?

A both similar and congruent

B similar but not congruent

C congruent but not similar

D neither similar nor congruent

California Standards Test, released test questions, geometry, 2009

2. Triangle ABC undergoes a series of some of the following transformations to become triangle DEF:

Rotation

Reflection

Translation

Dilation

Is DEF always, sometimes, or never congruent to ABC? Provide justification to support your conclusion.

Common Core Smarter Balanced Grade 8 Sample Item, 2013

The second question, from one of the Common Core assessment teams, does not simply test a mathematical definition, as the first does. It requires that students visualize a triangle, use transformational geometry, consider whether different cases satisfy the mathematical definition, and then justify their thinking. It combines different areas of geometry and asks students to problem solve and justify. It does not offer four multiple-choice options. Common Core mathematics is more challenging than the mathematics it will replace. It is also more interesting for students and many times closer to the mathematics that is needed in 21st-century life and work.

An important requirement in the Common Core is the need for students to discuss ideas and justify their thinking. There is a good reason for this: Justification and reasoning are two of the acts that lie at the heart of mathematics. They are, in many ways, the essence of what mathematics is. Scientists work to prove or disprove new theories by finding many cases that work or counter-examples that do not. Mathematicians, by contrast prove the validity of their propositions through justification and reasoning.

Mathematicians are not the only people who need to engage in justification and reasoning. The young people who are successful in today’s workforce are those who can discuss and reason about productive mathematical pathways, and who can be wrong, but can trace back to errors and work to correct them. In our new technological world, employers do not need people who can calculate correctly or fast, they need people who can reason about approaches, estimate and verify results, produce and interpret different powerful representations, and connect with other people’s mathematical ideas.

Another problem addressed by the Common Core is the American idea that those who are good at math are those who are fast. Speed is revered in math classes across the U.S., and students as young as five years old are given timed tests—even though these have been shown to create math anxiety in young children. Parents use flash cards and other devices to promote speed, not knowing that they are probably damaging their children’s mathematical development. At the same time mathematicians point out that speed in math is irrelevant. One of the world’s top mathematicians, Laurent Schwartz, reflected in his memoir that he was made to feel unintelligent in school because he was the slowest math thinker in his class. But he points out that what is important in mathematics “is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn't really relevant.” It is fortunate for Schwartz, and all of us, that he did not grow up in the speed- and test-driven classrooms of the last decade that have successfully dissuaded any child that thinks deeply or slowly from pursuing mathematics or even thinking of themselves as capable.