Inquiry-based learning is at the heart of the controversial California Math Framework. Rather than teaching students through rote learning, which instructors believe gives a false sense of virtuosity that comes undone when students have to think for themselves and do difficult problems, students are instead taken through a process of guided discovery in which they are persuaded through example and given the given the freedom to discover solutions, construct arguments, and work with peers, until they instinctively understand mathematics. Applying the theory to practice is much harder than it is with rote teaching, but the potential is enormous. 

How Inquiry-Based Learning Works

With inquiry-based learning, an instructor has a clear goal in mind, and they create problems through which students will be guided toward the best solutions. For instance, teachers may use “back-pocket questions”, to scaffold and guide discovery. Students are asked a progression of questions as they grapple with the problem, questions that are designed around the student’s skill level. When students seem to stray, they are guided back toward the destination the teacher has in mind. When students are really struggling, the questions become more leading. 

A teacher needs to be very skillful in understanding her students and in knowing what kinds of questions to ask to help students get to a solution. This is what you will find if you search for "math tutor near me" in Google.  If the teacher is not skilled, the whole thing falls apart and leads to confusion. When it works, students have higher levels of achievement.

The Argument for Rote Learning

There is an argument for rote learning. Let’s take a simple multiplication problem: 43 x 8. With rote learning, the problem is easy: you know that 8 times 3 is 24, so you have the 4 and carry the 2, you know that 8 times 4 is 32, so you add 32 to 2 and the answer is 344. If you have really mastered your multiplication tables, this process will be virtually instantaneous. Critics have said that by downgrading rote memorization, students will lose this “math fact fluency”. However, as Danie Buck noted, students may suggest sub-optimal solutions: one student might suggest multiplying 40 times 8 and then adding 24, another student might suggest multiplying 10 times 8 and quadrupling that answer and then adding 24, and so forth. The idea is that this will give students a richer understanding of the problems, an understanding that is impossible with rote learning. The idea that students can learn with guided “discovery” has been around since at least 1961 when Jerome Bruner proposed it. Yet, as Bruck says, 

“This approach may sound good—but it doesn’t work. A vast body of empirical research has consistently shown that more structured methods—with clear objectives, clear explanations, clear corrections, and lots of practice—achieve superior results.”

By dismissing rote learning, students may suffer from poor math fact fluency. The issue of inquiry-based learning tends to be treated as a simple binary: do we want to promote “math fact fluency”, or a richer understanding of concepts? Yet, there is a middle way. 

The Middle Way Between Rote Learning and Inquiry-Based Learning

The student who can multiply accurately as fast as Scott Flansburg, the “Human Calculator”, while having a rich understanding of the concepts, is a better student than one who does mental mathematics but doesn’t understand the concepts. The “prince of mathematicians, “Carl Friedrich Gauss, is an example of a superior student: during class one day in the 1700s, his teacher asked the students to calculate the sum of the integers from 1 to 100. He quickly found a solution: 5,050. The 7-year old Gauss realized that there were fifty pairs of numbers when he added the first and last numbers of the series, the second and second-last, and so on, in the form: (1 + 100), (2 + 99), (3 + 98), …. He found that each pair summed up to 101, so really, the solution to the problem was 50 x 101.

To Bruner’s point, there is a lot of empirical evidence that combining rote learning and critical thinking does lead to greater “computational fluency”. 

Nevertheless, there are limits. Not every child is Gauss, and are better off with highly structured, goal-oriented classes where they are drilled to learn math facts before they can be given the freedom to explore. There is plenty of evidence that very young learners, especially struggling learners, desperately need rote learning and direct instruction rather than guided discovery. When students have their bases established, then they will have the tools and confidence to enjoy the benefits of inquiry-based learning. The California Math Framework errs because it is far too confident about the ability of early learners to efficiently and confidently discover optimal solutions. When inquiry-based learning has been used with older students, it has been a roaring success. When the Detroit Public Schools Community District introduced inquiry-based learning to some seventh and eighth grade students, those students did better on the Michigan Educational Assessment Program (MEAP) than other students in the district.