Group Discovery ???

One of the marks of "new math" is discovery learning. But it's more than just discovery learning; it's "group discovery learning". Students are expected to come to knowledge of mathematical facts and procedures through "communal discovery".

You've heard it before. Only what your child discovers himself will he really grasp and remember. He must "own it" to be able to "use it and truly know it".

Now, let me see. If we want everyone to make the discovery himself, why would we put them in a group? That is exactly the opposite way you have every person discovery on his own. The whole group discusses and comes to an action, tried and agreed upon by the group. (Of course, someone in the group had to make the suggestion, to begin with, but never mind that.)

Now, remember through all of this, the purpose is for each student to make the discovery for himself -- so he can "own" that knowledge.

If you're in a group, how can all students come to "discovery" -- of a math procedure -- at the exact same moment? Someone in the group will come to the discovery first. Must he or she remain absolutely quiet? Must he hide his procedure? Remember this is a group discovery.

So, if someone else in the group makes the discovery, I guess no one else ever gets to "own" anything! Be First or Lose Your Shot at It.

By the way, the teacher is very quietly staying to the side, offering no guidance. And once the answer is agreed upon, the teacher is not allowed to check to see that the procedures were used correctly. Later, no drill will be given for students to use for practice. And there is no memorization of facts -- the facts can be 'rediscovered" on the spot, whenever they are needed.

In a traditional classroom, a teacher gives students an opportunity for small discoveries throughout the lesson, but he or she carefully directs the lesson and teaches precise methods, so that all students learn the most efficient procedures. Then the students practice and practice the steps under the supervision of the teacher. Following the guided practice is the independent practice, during which kids do more practice. Teachers are available to guide students and reteach if necessary. It is during this practice that some students "discover" and learn. And as the procedures are practiced over the next days and weeks, still other students may finally make the discovery on their own. But that "discovery" is just as real as the "discovery" made earlier by other students. In the meantime, the student has been able to successfully get the proper answer on the work because the teacher had taught just that.

I disagree that there is no discovery in a traditional classroom. There is. I see it every day.

And I don't think that it is possible for all students in a new math room to "discover" every single fact and procedure without input from the teacher.

Perils of Discovery Learning, Part II

Dr. Wickelgren, author of Math Coach, A parent's Guide to Helping Children Succeed in Math, explains that it is not possible for children to "discover" much of what is important for children to learn in elementary grades. How, he asks, can a student deduce, through discovery, the meaning of "3 to the 4th power" or that there are 5,280 feet in a mile. There are many things which must be taught by the teacher and then memorized by the student. Now you may avow that these are things that a student could conceivebly discover, so let Dr. Wickelgren describe how a typical "discovery" session works.

"Using the discovery method, students are given very little of the information necessary to solve problems. Solving problems thus requires gigantic leaps of intuition that virtuallly no students possess, so they flounder. A class typically spends half an hour -- and sometimes as long as two and a half hours -- on a single problem."

Because the "discovery" time is largely unsupervised by the teacher, it proves to be a very inefficient use of students' time.

Dr. Wickelgren tells of a student who "complained to me that his class spent an entire week on one problem before the teacher told the students how to solve it."

And remember that in addition to the inefficiency of this procedure, students are also asked to spend time discussing what to do, and then to spend more time writing about how they arrived at their solution. And what if their solution isn't even correct??!! Remember, the teacher is not involved, is not guiding students back on track when they wander off on a rabbit trail.

As a result of all of the wasted time, very few problems actually are solved by the students. Teachers are giving little or no information, and student proficiency takes a nose-dive.

Contrast this scenerio to the traditional classroom, where children are taught by direct instruction. Students are given most, not all, but most of the information needed for solutions. Smaller amount of insight is needed and thus some students, though not all, solve the problem quickly. Several of these problem-solving activities are given throughout the course of the lesson, affording students additional opportunities to use small bits of insight to solve other problems. (And a teacher steps in to help, giving additional information when needed, if the problem takes too long.) Many students gain confidence because of the successes they make each lesson. And they see the point in what they are doing. The goal is short ranged.

Solutions are found quickly, not hours later, and students do not lose interest because the problem's answer is so long in coming. I'd rather have students engaged for several small problems they have hopes of solving than disengaged for an hour because they see no point in what they are doing.