It is generally agreed in the modern world, that all students need to learn math. And the fact that our schools require everyone to study it, assumes that every pupil is capable of learning it. Unfortunately, standardized tests show us that vast numbers of pupils are not learning it. What is the problem here? Are the math teachers not doing their job? Most teachers would sincerely and passionately assert, “I taught it to them, but they didn’t learn it!”

The fact that so many middle and high school math students are doing so poorly is a source of great pressure and frustration for everyone involved. Employers—who need well-educated employees to be competitive in a global economy—blame schools, administrators, and teachers for not getting better results. Teachers—who work very long hours in situations of intense pressure—often blame uncooperative, unmotivated, careless, lazy students, and their parents, and society at large for preventing them from doing a better job. And many low-achieving students—forced to deal daily with things that they truly don’t understand—are understandably frustrated, too. Who cannot sympathize with the complaints of any of these aggrieved parties? But after all the blame is spread around, not much changes. The number of under-performing math students is still staggeringly high.

Many students struggle sincerely with their classwork and homework and tests, and often they can master no more than 60% of the material presented. That actually represents a significant amount of learning; but for all their trouble, those students are rewarded with nothing but an “F.” Frustrated teachers and students—who have had more than enough angry scolding from concerned citizens—may justifiably feel that, since they are already doing their honest best and working as hard as they can, it just doesn’t get any better than this. They realize that their best is just not good enough, but are convinced that that’s just the way it is.

However, slumbering at the back of everyone’s mind is the vague, hopeful feeling that there is, there must be, a better way of enabling more students to succeed—if only they knew what it was. The fact that something needs doing implies that there must be a way of getting it done. And that is true. But the achievement of different results requires the use of different methods.

Current math teaching methods and materials are actually quite effective, but only with a limited number of learners—students whose minds work in much the same ways as their teachers’. Many of today’s math teachers and textbook authors went to school at a time when only an elite group of pupils was expected to study college prep math. But today, all students—regardless of their personal background (socio-economic level, home language, parents’ level of education, previous mastery of prerequisite math concepts and skills, etc.)—are expected to master algebra. The problem is that today’s pupils are being taught mainly with methods and materials that were designed for use with the elite students of an earlier period, whose personal background and level of preparation was markedly different.

For more universal access to success, pupils and instructors need a truly different model for teaching and learning—a method that takes into account the mindset of the great mass of students whose minds, for the most part, do not work in the same ways as their teachers’. That is the aim of the new books based on the guided discovery method: to provide a new model for teaching and learning about math. Fortunately, the new model is simple and pleasant, for neither teachers nor students have time or energy to take on a complicated new methodology.

Some of the main features of this new approach are: exploration of physical models as a starting point for taking on new abstract ideas; engaging students’ power to notice relationships when working with concrete, numerical examples; using students’ amount sense to inform their sense of procedure; replacing book/teacher-centered explanations with pupil-centered development of reasoning; providing frequent links to underlying concepts and skills; and developing fluency in a context-derived way.

Many (but not all) college prep students in years gone by were capable of memorizing and successfully applying rules, short-cuts and formulas for concepts that they truly did not understand. Students and teachers alike tolerated the ambivalence of plunging ahead without real comprehension, confidently assuming that understanding would come along in its own sweet time, made possible by the enriched perspectives supposedly provided by successful practice (“Don’t worry, you’ll understand it later”). Sometimes that understanding came, sometimes it didn’t. Some students never did understand what they were doing, but got good grades anyway; they relied on the expediency of rote mimicry to eke out reasonable academic success. The goal they achieved was getting good grades; they did not achieve the goal of increased comprehension.

This approach simply does not work with many math under-achievers. For them, real sustainable progress only flows from real understanding. And the development of real understanding generally requires a slower pace, giving proper time to the thinking of deep new thoughts. This may seem grossly inefficient to many teachers, considering the tremendous pressure they are under to produce quick results. But this worry is misplaced. Just as we must spend money at a good sale in order to save money in the long run, so we must spend time to save time. Students who work their way carefully through guided discovery lessons progressively display increased confidence, accuracy, and speed. And each new concept, solidly understood, provides a sure basis for taking on the next new concept. The students also begin to feel a greater connection between their formal math education and their natural sense of curiosity.

Books based on guided discovery are purposely written in a simple style so that students can just read the directions and follow them—and then what should happen in their minds will happen. However, there are always students who have trouble reading or following directions. When teachers or parents are called upon to assist these students, they will get the best results not by telling them what to do and explaining why and how to do it (that approach has already proven not to work with many low-achievers)—but by asking the students leading questions, guiding them to figure things out for themselves. One of the main reasons that students are required to study high school algebra, after all, is for them to learn to think and reason more effectively. Guided discovery, when properly practiced, makes this achievement possible.

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