Philosophy of Teaching

If what has been done in class will not promote future success, then class was a waste of time!

For students to be successful in Cambridge Math class itself needs to operate differently than in a traditional United States math class.  The students need to take on a much greater responsibility and the teacher needs to let go.  It is impossible for a teacher to cover every possible iteration of every topic to be taught in two years of Cambridge math. Further, there would be an even lower chance of student retention, and no chance at all of a student being able to apply what they've been taught in a new way.

First Idea:  Math is Not Steps

It is typically the situation where students believe that math is about the steps.  They'll ignore all of the information about the concept and the reasons behind things, but will jump all over the process for a particular problem.  Cambridge tests are too well written to allow this type of superficial learning to be rewarded.

This is readily apparent when I teach the first topic, square roots.  Students jump all over process, and avoid concepts and ideas because they see no value in them.  It is only after they learn that the steps are determined by the concepts and ideas that they find value in them.

As a teacher it is my job, and your role if you so choose, to break students of this habit.  Through constant reminding, withholding steps, asking students to describe what parts of equations mean, and coaching, I get students to write out their ideas, not in complete sentences, but in short hand.  A prime example is in the picture below.

The most important writing on this problem is "Sector - Δ," because that's what drives all of the numbers and calculation after.  Vigilance in this regard, focusing on the why we do something more than what or how we do it, is one of the major keys to success.  And while the teacher sets the group norm by rewarding students for doing this and redirecting the reluctant, it is also the job of the student to always push for the deeper understanding and not to latch onto quick fixes.

Second Idea: Lack of Information is NOT a Problem

The amount of information that a student has access to is incredible.  Whether a struggle student seeks the information they need or not is entirely up to them.  It matters not if you provide it for them during remediation because the problem is failure to integrate information, not lack of access to it.

Students need to integrate the things their learning into their body of mathematical understanding.  They need to apply new concepts to discover process, not just be taught procedure.

An example of this would be when teaching exponents and negative exponents come up.  I teach that exponents are repeated multiplication, written in short hand.  I also teach them that a negative sign means opposite.  So, negative exponents are the opposite of repeatedly multiplication -- they're repeated division.  All of the properties and "rules" that come after are consequences of this!

So when we summarize the properties, some people call them laws, of exponents, I make sure the students can explain why things work the way they do.

Third Idea:  Kids are Smart ... Let them Solve Problems

Perseverance, confidence and persistence are three great qualities.  I believe students develop these in Cambridge Math.  However the staying power of freshmen is notoriously lacking.  Often, when challenged with a problem they'll likely give up, saying, "I don't get it," and engage in something more entertaining.

The phrase, "I don't get it," has no place in any math classroom because it is a blanket statement, it's emotive and a complaint.  There is no way to respond to the statement in a constructive way beyond suggesting that they need to be specific about what is causing the confusion.  Not only will this force them to reflect and think instead of just react, it develops a huge component of problem solving ... thinking!

As a teacher, set up problems that are nuanced, layered and complex, but approachable.  Encourage students to stay engaged, explaining that if the first time they really try something is on test-day, they're going to fail.

Also, don't let the students off the hook by confirming complex answers as right or wrong too soon.  In fact, get them used to defending and understanding their solutions to the point where they no longer look to you for confirmation.  They need to understand that math is right or wrong because of situation, application and process, not because a person or book says so!

Fourth Idea:  If You Can't Say It, You Don't Know It

Memory and language are inseparable.  Memories are stories and without the language of the story, there is no memory.  A student that cannot articulate what they know will have little to no chance of retention.  So carefully stating what is understood or intended, with accurate vocabulary and limited pronouns, is a great exercise that promotes retention.  Students that understand this are far more likely to engage in the act of deliberately expressing their ideas or understanding.

Fifth Idea:  Cover all the materials with time to spare, then practice

Cambridge math covers a ton of material from all of Algebra 1 and Geometry, most of Algebra 2, a huge chunk of Trigonometry, Probability and Statistics as they're taught here in the US, in addition to many other topics we don't typically cover.  Then, when the students take their end of course examinations, they're likely to see a problem that involves many of the topics all at once, and without prompt or warning that one of the topics is to be used.

In response to the nature of Cambridge exams I teach all of the subjects completely in the first 6.5 quarters of the program.  That leaves me about 2 months of time to practice, review and shore up misconceptions and hit small things that always appear on the examinations.