What is Gentle Knowledge?

Gentle Knowledge is a mathematics program for school-aged kids. We provide a complete mathematics education for homeschooled kids, as well as for kids who attend school. For students who want to go beyond a standard mathematics curriculum, we provide various enrichment opportunities. In addition to mathematics, we offer computer science and programming classes.

How is Gentle Knowledge different from other math programs?

The single biggest differences between what we do and what most other programs do is our focus on problem solving. The standard mathematics curriculum is devoted almost exclusively to an ad hoc collection of mechanical procedures that students need to memorize in order to solve standard exercises. Most textbooks and math programs teach students by presenting a step-by-step solution to a prototypical exercise and then asking students to repeat those steps in dozens of similar exercises. This type of rote learning has some short-term benefits, but it does not teach students how to think. In contrast, our approach puts thought-provoking problems center stage. These problems are nonstandard because they don’t follow a predefined template, and because they require students to uncover and explore new mathematical topics by thinking, not by mindlessly plugging numbers into formulas. Beyond covering all of the same topics covered in the standard mathematics curriculum, we introduce interesting and nonstandard concepts that are usually left out, all while students are thinking, not simply following directions.

Can you give an example that illustrates the difference between problems and exercises?

A typical textbook exercise involving averages is:

Find the average of the numbers 12, 35, 67, 80, and 95.

This exercises requires nothing except the straightforward application of the textbook definition of the average of a set of numbers (adding the 5 numbers and then dividing the sum by 5). In contrast, a sample Gentle Knowledge problem involving averages is:

Some number of scientists moved from country A to country B. As a result of this move, could the average IQ have increased in both countries?

At first glance, it seems that the answer is no. Students’ everyday experience with conservation laws (e.g. if water is poured from one glass into another, one glass gains water and the other loses water) tells them that the same principle should apply to averages. After some thought, students see that if the average IQ in country A (call it AIQ) is greater than the average IQ in country B (call it BIQ), then, if the IQs of the scientists who move from A to B are between BIQ and AIQ, the average IQs of both countries will increase. This careful analysis not only reveals a lot more about averages than the textbook exercise, it makes students feel a lot more accomplished because they have to think, not just do trivial arithmetic.

How will my child learn basic math skills if you focus so much on problem solving?

We do not believe that achieving computational fluency (the ability to do basic arithmetic and algebra) and learning how to solve difficult problems are mutually exclusive goals. Fundamental math skills are a vital first step, and students need to practice them even while working on interesting and challenging problems. For this reason, we assign practice exercises as part of every homework assignment.

I am worried about college admissions and standardized tests like the SATs and AP exams. Will you prepare my child for these future challenges?

The short answer is yes. We are not a test preparation service, in that we do not “teach to the test,” but we do something better. In our experience the most difficult part of a test like the SAT is the set of problems that do not exactly follow a standard template. By focusing on nonstandard problems that require thinking, we prepare students for the hardest problems on any standardized tests. Beyond classwork and homework, we offer students opportunities to work on independent projects that can play a major role in the college admissions process. We help motivated students find projects that interest them, and we either supervise them directly, or we connect them with mentors working at top universities and high tech companies. We also write recommendation letters and provide college counseling services.

What can you offer children who are ahead of their peers, and how are you different from other advanced math programs?

Unlike math programs that specialize in working with only the most mathematically mature students, we are open to everyone. Most students will find something that is challenging in what we do. For extra ambitious students who want to prepare for math olympiads, or who have math backgrounds that are exceptionally advanced, we offer specialized classes and private tutoring.

What does computer science/programming have to do with math?

Many computer science concepts are part of mathematics, and programming is, in many ways, the new literacy. Writing computer programs requires serious problem solving, gives students their first real taste of engineering, and most importantly, provides endless opportunities for rewarding mathematical exploration in a new medium.

Do you assign homework and give tests, and do you have any other requirements?

We don’t give tests, but we do assign homework after every class. Although we don’t expect students to solve all of the problems, we expect them to devote some serious thought to each problem and to do all of the exercises. Even partial solutions or rough ideas are an important part of the learning process. Our goal is to gently nudge students to the limits of their abilities, and we would rather challenge them, than give them a false sense of accomplishment.

Who are the people behind Gentle Knowledge?

Gentle Knowledge instructors are mathematicians and educators, all of whom have graduate degrees in mathematics and years of experience teaching students at all levels. Several of our team members are celebrities in the world of math education and have made major contributions to both mathematics and education. Learn more about them here.