20 years of experience and investigation make me an irreplaceable asset here. Tutors are often either former humanities majors with a shaky grasp of math themselves, often even needing to review math questions before class, or they are current math/physics students.
This second type of tutor might know the subject well, but rarely has much experience effectively communicating on the elementary/middle or high-school level. Also, these tutors can be “math machos”. They usually know or care little about the incredible confusions to be found everywhere in the school math curriculum, confusions that are the result of the poorly conceived materials and have nothing to do with math itself. Since these scientifically minded tutors mostly come from an academic background and ended up loving math, they can scarcely comprehend the vast confusion that awaits normal mortals fighting their way through school-math’s dark forest of often contradictory terminology. I won’t lie, I was one of these “math machos” myself (magna cum laude in pure math at 21, see bio). It took nearly 20 years of tutoring and authoring math help books for me to comprehend what kind of experience math class is for most students. I never really paid attention to how badly math was taught when I was in school because I was ahead anyway. When I did notice something illogical or contradictory, it was just funny not threatening.
20 years later I am a walking encyclopedia of each and every confounding statement ever made in school-math classrooms and textbooks, with a huge toolbox to get students passed these roadblocks one-by-one and put them back on the correct path to actual mathematical understanding in no time!
My tutoring motto for many students is:
“Even if you do not and never will like math, you don’t have to hate it.”
If you think I’m exaggerating the rote memorization aspect and general instruction/material deficiencies of school-math, read this article about the STEM dropout rate. In essence, 60% of students who think they want to study a STEM field (especially in the Ivy leagues) end up dropping out because they are not able to think independently mathematically instead of just repeating patterns. Not to mention the widely discussed poor performance of the average US math students on international tests such as PISA etc.
As in a test prep situation, how to approach a school-math tutoring client is first and foremost dependent on the type of student and their current situation. Overall, it can be said that the most important step for most students is to let go of memorization and actually understand basic concepts again. School-math rarely works in this direction and I often unfortunately find myself undoing the confusions caused by math class, but of course this really depends on which category below a student falls into:
I. Excelling Students
This is obviously the most fun a good tutor can have. I always relish the chance to serve as a mentor to these young aspiring minds (see Danielle Quellers very flattering review.) Even though I did not continue with my academic career I still feel an incredible pull towards mathematics, the purest form of thought, the discipline closest to philosophy in the hard sciences. I do sometimes regret not having pursued mathematics further, and so these talented students are often opportunities for me to live vicariously.
There are so many fascinating math topics rarely mentioned in school classrooms, even at the lower grade levels. To name a few of my favorites: the four color theorem (an amazing example of a very hard problem a 5th grader can understand), the Möbius strip, Pascals last theorem, Gödel’s proof, Euler’s formula. Obviously some of these, such as Gödel’s proof, cannot actually be studied in any detail but their mere existence (carefully phrased, mind you) is astounding and inspiring. Then there are also plenty of topics that are only superficially mentioned in school, when they actually deserve much more reflection such as: pi (many students memorize a hundred digits of pi, but have no idea what it actually is), Pythagoras (the equivalent of the atomic bomb in those days, people were thrown overboard for even speaking about it), prime numbers (perhaps the greatest simple mystery of them all and the heart of modern encryption) or just simply the 2000 year story of the three milestones in human analytical thought: the symbol for 0, algebra and finally calculus ushering in the modern world. Calculus itself is hardly ever given the introduction and explanation the idea of a limit, this wonder of human thought, the closest thing to actual magic we have ever produced, rightly deserves.
I especially pride myself on emphasizing the underlying concepts (such as amount of information vs amount of unknowns) that permeate all of mathematical thought, rather than simply presenting students with further opportunities to tinker with tricky problem sets, of which they will always have enough when we cover AMC, Trinity, Math kangaroo, and Matholympiad.
II. solid students just looking to kick the tires
Often students are doing well in school in general, are hard working and smart and yet math continues to be the thorn in their sides. This can be because of widely varying scores or the realization that no matter how much work they put in to stabilize the math situation, they seem to always be just one misstep away from a catastrophe on the next exam. And then there are the upcoming standardized exams, where the whole of the student’s mathematical knowledge from 6-12th grade will be tested at once! In other words it seems as if all their hard work is really not paying off, the way it does in other subjects.
Fundamental concepts are often a wobbly base on which all of a student’s mathematical understanding is built, constantly threatening to tip over. The reason these basic concepts are so unstable is that they were never explained in a correct and memorable way that actually made sense. This is where my vast experience in school-math’s obstacle-course really comes into play. Many basic topics that should be simple are often buried under mountains of unnecessary confusion because of the way school-math taught them (see the pdfs below for two astounding examples of what I mean).
So what to do? After decades of tutoring I have come to a very optimistic conclusion. I truly believe that the vast majority of students are far smarter than the school system gives them credit for. I have always found that if you can remove the blinders, take away the fear and have students look at math with a fresh attitude as if they had never seen it before, if you can get them to just use their natural common sense again, then miracles will happen much faster than you would ever believe! But this can only be accomplished if the tutor knows exactly which confusions are swirling around in the students head for each topic and can clear them up one by one, which, after 20 years of investigation, I surely do!
This quite often involves kicking the tires on some very basic math concepts, even going back to things like the distributive law from 7th grade (A law that is never well explained in school, see below). This process may sound terrifying and overly ambitious, certainly it would take forever? Wrong! It is astounding how quickly most students rebound and dramatically improve once the “leaks in their mathematical life raft” have been found and fixed. This is, in my quite extensive experience, surprisingly true of even the most troubled math students!
It is as if they had been longing to be saved the whole time. They have just been guessing at math, walking around in a room full of obstacles blindfolded and randomly bumping into things. This has kept many students in a subliminal state of silent panic for years, even if their grades hardly reflect it. In school math classes students usually spend the week memorizing a bunch of patterns for the test on Friday where all they have to do is guess which pattern to apply to which question. Sadly, the questions are often so badly phrased that thinking about them will actually make things harder rather than easier, and so many students have learned to memorize rather than think about math. In the vast majority of cases it turns out that students are actually longing to finally understand what it is they are doing. The amazing thing is that this recovery of basic common sense is actually possible and manageable, even given a limited time frame of a couple months. I have done it successfully hundreds of times!
III. Students with math anxiety
Some students have an almost physical reaction to mathematics. They have had so many terrible experiences that they have developed a host of defense mechanisms to make math less painful. A common example of this is just declaring that they can’t do any math at all. Instead of trying again for every exercise, they just declared themselves a failure and give up. In this way they avoid failing over and over. Math to them is just like Charlie Brown and the football. Why in the world would they wind themselves up every time just to have it pulled away again at the last instant? How this has become the reality of school-math for far too many, is a whole other story and the subject of a math-help book I am writing. More often than not, this math anxiety, or math trauma as I call it, is the result of terrible material and instruction (see below). There are actually quite a few very smart kids in this category. Smart enough to notice the contradictions and ambiguous phrasings that are unfortunately everywhere in school math, these, otherwise gifted students, have a very hard time swallowing the bitter pill of nonsensical memorization. They are often, quite frankly, insulted by it.
Even here, the most important step is too let go of memorization and understand basic concepts. Even if the test is tomorrow, the first step must be to make sure the student knows whether what he/she is doing makes any sense at all! Otherwise valuable time will be lost during the test doing exercises that result in absolutely zero points. Once random guessing starts, students will miss even the easiest questions, questions they would get right under any other circumstances.
On the other hand, I will not deny that sometimes tests will really just be about memorizing a handful of patterns. If it really is the day before the test and the student is desperate enough, blind memorization may actually be the best short term option. After many years of experience I know when it is time to give up all pretense of actually teaching concepts and just make the best of the situation at hand.
In fact, “making the best of it” is what I am always doing! It is a fact that, even for these very math troubled students, the only long term solution is still understanding a bare minimum. They must at least know when what they are doing makes no sense at all. If not, they will be completely lost and spend half the exam doing nothing at all. They must at least maximize the points they could actually get from the things they can do .
Again, my tutoring motto often is:
“Even if you do not and never will like math, you don’t have to hate it.”
Students who really are suffering from math anxiety or math trauma must be approached with more care and a softer touch, but in essence the recipe for improvement must be the same as for case II above (the solid students who doesn’t love math). Blind memorization is by far the most painful and fruitless road to continue down, regardless of where the student is currently. Again, as above, I am consistently surprised by how fast students, no matter how mathematically desperate they may be, respond to meaningful explanations and exercises. It is as if they had been waiting all along for someone to finally reach out a hand and guide them out of the labyrinth of meaningless memorization they have been lost in for years. Even if some students initially do protest too much that they simply cannot under any circumstances ever do math correctly!