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Math Tutoring Tips
Here are some tips for tutoring and mentoring students in math. These should be helpful for tutors who have not yet gained a lot of experience, and/or who have just started. Homeschooling parents and parents who want to delve into tutoring their kids should also find these tips helpful. Tips (1) to (11) are appropriate for all grade levels, and tips (12) to (14) are appropriate for more advanced students (9th grade and beyond). (Note that my book, "“hat Can I Do..?” is primarily aimed at parents who cannot, for a number of reasons, tutor their kids themselves: in that I book I show parents a multitude of different things that they can do, without having to solve a single math problem. Some of the tutoring tips below are more advanced and are appropriate for those parents who want to go beyond the methods I talk about in the book.)
(1) No matter what grade level a student is at, whether it is 1st grade, 12th grade, or college level, always encourage developing intution by means of setting estimation excercises. The excercises should be both oral and written. Fostering the ability to "sanity-check" an answer to any given problem is absolutely critical for success in math. The associated ability of a student to spot his/her own mistakes is also critically important. These skills should be developed alongside every topic in math, whenever possible. Making mistakes in math problems is part of the problem-solving process and developing an intution for correcting errors is an integral part of improving one's performance in math. I discuss these issues and specific techniques in considerable detail in my book, “What Can I Do..?”.
(2) Students should also be taught the mechanics of spotting mistakes and knowing when a mistake has been made. This is subtly different to the skill described in (1) above. The point made in (1) was about errors in understanding, but here I mean "clerical" errors or "slips." Enforce the habit of writing out problems strictly from left to right and top to bottom. Very often, students will start off with big writing, run out of space on the worksheet, and then jump to another part of the page, run out of space again, and jump to another part of the page. A muddled layout leads to muddled thinking and inevitably leads to mistakes, which are then extremely difficult to trace.
In algebra problems that take up several lines, a common source of error is to leave out a variable in one of the lines, or forget a superscript, or mix up positions of parentheses, or apply a manipulation which is mathematically illegal. It is impossible to avoid such mistakes. Rather, the skill that students should develop is to know the signs that indicate that something went wrong. For example, if an algebraic expression seems to become more and more complicated in each successive line rather than becoming simpler, it is a sign that something went wrong. When watching a student solve a problem, let him/her continue with the error rather than point it out straight away. Then, at the appropriate point, interject and ask the student where he/she thinks the solution is going and let him/her first attempt to find the mistake.
(3) When asking a student if he/she has had any problems with material covered in class, don't take the student's answer of "no" as final. In my experience it is extremely common for students to think that they have adequately understood and mastered a topic, when in fact important issues may remain. Assuming that everything is fine can be fatal later on. Never assume everything is fine until you have verified it for yourself. I suggest asking the student to show you his/her class notes, assignments, or relevant section in the text book to explain what he/she covered. Then, whilst browsing, casually ask the student several "feeler" questions as an instant "spot check." For example, if a student studied the remainder theorem you could casually drop in the question, "So, what would be the remainder when (x2+1) is divided by (x-1)?" If you are familiar with the remainder theorem you will see that the spot check has been carefully chosen so that the student should be able to quickly give you the correct answer without having to write anything down, if he/she has truly understood one of the key concepts. On your part, you have to be quick and "on the ball" in order to make up the question "on the spot" and deliver it in a manner that is not patronizing to the student. If the student has a problem with you not trusting his/her own assessment of understanding a topic or concept, you should explain that it is common and natural to be mistaken about whether a new topic or concept in math has been truly understood (because it's true).
(4) Contrary to popular opinion and widespread usage, the "pizza model" for teaching fractions has some severe limitations (so you should not overemphasize it). In particular, the pizza model does not lend itself to developing a good intuition for the decimal equivalent of fractions, and getting a sense of size for arbitrary fractions. In my book, “What Can I Do..?”, I discuss in detail, powerful techinques for developing intuition for fractions and the manipulation of fractions. A common type of question in tests and exams is to put a given group of fractions in numerical order. Dealing quickly with any fraction that may be "thrown" at a student is a skill that you should help students develop, and it ties in with the estimation and "sanity-checking" skills that I talked about in point (1). For example, a student should be able to mentally assess very quickly whether (7/47) is smaller or larger than (12/71).
(5) Realize that skills in the mental manipulation of percentages (and the associated estimation skills) go far beyond the literal application of percenatges. These skills directly and intimately tie in with developing intuition and "sanity-checking" skills that I talked about in point (1) above. A solid foundation here has far-reaching and long-term benefits for a student's success in math in general. Yet it is common for even students in the higher grades (as well as adults) to be deficient in the manipulation of "reverse" percentages. For example, it is a common mistake to think that if a quantity is increased by a certain percentage, the result has to be decreased by the same percentage in order to recover the original value. If 100 is increased by 25% what percentage decrease of the result is required to recover the original value? It is not 25%. In my book, "“What Can I Do..?” there are some carefully designed oral drills to develop and enhance this fundamental mental skill.
(6) Do not overuse mnemonics. The reason is very simple. Whenever a student uses a mnemonic he/she moves one step further away from developing an intuition and deep understanding of the topic in question. It becomes a mere memory game. Pure memory is not helpful for success in math. The mnemonic divorces the student from the subject matter of the topic and discourages a deeper understanding. It is better to teach students how to derive results that are not fundamental (for example the cosine formula from Pythagoras' theorem) rather than simple memorization. None of the secondary results in a student's entire school career are very difficult to derive quickly. In the process of learning how to do derivations quickly from scratch, even if it is not required, students develop a deeper understanding of fundamental relationships and mathematical "truths," that will have longlasting and far-reaching benefits.
On a related note, do not ask students to memorize too many equations, keep it to an absolute minimum. I can tell you from over 25 years of experience, that memorizing equations does not help students solve math problems. A student can memorize all the relavant equations for a particular math problem and and still not be able to solve the problem. If a student is trained correctly, after having understood the concepts pertaining to a particular topic, and solved enough carefully selected problems, all of the apparatus will already be present in the student and there will be no need to memorize equations. Although at the end the student will have the relevant equations in memory, the process I have described is manifestly different to the process of memorizing an equation in the hope that it will help you to solve math problems. The truth is that it is the other way around. You don't memorize equations in order to solve math problems, rather, you solve math problems and the equations get memorized. Unfortunately, it is all too common to find this taught the wrong way around.
(7) Teach students how to use active techniques for studying, as opposed to purely passive techniques. Teach students how to study for exams. Use these techniques along with administering simulated exams to enhance and monitor a student's test-taking ability and peformance. I give some thorough discussions and details of these techniques, methods, and skills in my book, “What Can I Do..?”. In particular, there are methods appropriate for multiple-choice questions that are are powerful techniques for eliminating wrong-answer choices, that are not taught by the "big name" test-prep companies (you know who they are). The reason that the "big name" companies don't teach these methods is that the methods that they prefer can be taught by tutors with little or no experience, but their methods can require more time for problem-solving and can be more prone to a student making errors. Furthermore, the test-making organizations can often "slip up" because it is hard to make fool-proof multiple-choice questions. For example, in my book, “What Can I Do..?” I show you how a particular multiple-choice question in a standardized test in the USA (taken annually by more than 1.5 million students), could be done in 3 seconds, without even fully understanding the question. The test-makers slipped up big time because the average allocated time for that question is 90 seconds. This is not an isolated example: I see vast numbers of multiple-choice math questions that can be done much more quickly than the allocated time (thus leaving the student with more time for the harder questions). Obviously, you have to know what to look for, but I find that students are rarely trained in these techniques. Test-prep companies that charge thousands of dollars do not promote these techniques that utilize intuition-training skills (as opposed to simply plugging in numbers) and looking out for the test-makers' "boo boos." Taking advantage of the test-makers' mistakes is not cheating: the very ability to spot those mistakes distinguishes a student as having mastered his/her understanding of a topic much better than students who have to rely on "plugging in numbers" and "grinding and churning."
(8) Teach students how to mentally and quickly (a) evaluate equations for the value of 0 and 1, (b) assess the difference for negative and positive input values, (c) "reduce" equations into "extreme forms" (the behavior when a variable is very large or very small), and (d) recognizing if and when an equation exhibits "special" behavior for "special" inputs. These skills can be quickly learned with the right training and will significantly enhance performance in a large range of topics in math, as well as promoting the development of intuition that will result in wider benefits.
(9) Always emphasize the graphical (and whenever possible, the physical) representation of algebraic expressions and equations. Use quick sketches as you are explaining things and shift towards asking your student to do the sketch as you are explaining things. This also ties in with what I have said in point (8) above. Too often, students do not see algebraic expressions and equations as anything more than an abstract collection of letters and symbols. The goal of you and your student should be that the student will be able to look at any equation (up to grade level), and immediately grasp the essential features of its behavior, roughly visualize the form of its graph, and be able to discuss what kind of relationship that the function represents. If the form of the function is one that is commonly found in nature, or other familiar circumstances, a tutor should take every opportunity to point out such occurences.
(10) Tutors should continuously and consistently foster the idea that the "simple stuff" should not be taken for granted, and competence with the "simple stuff" should be maintained. I have seen too many cases of students doing themselves a disservice by neglecting the "simple," or "no-brainer" stuff, only to find out when it is too late. One reason is obvious, another is not so obvious. The obvious reason is that forgetting the basics can be a show stopper for a more advanced math problem. The not so obvious reason is that if the student continues to develop speed, accuracy, and proficiency in the "no-brainer" material, the student will continue to make more and more time in exams and tests for the harder problems, thereby increasing overall performance.
The principle of maintenance of the basics applies at all academic levels but obviously, what precisely is "basic" shifts continuously. For example, it could mean that an 8th-grader studying the factorization of polynomials should not neglect maintaining proficiency in mental skills with the basic multiplication tables. For a 12th-grade calculus student, "not forgetting the basics" might include not forgetting how to factorize polynomials. Having said that, I once worked with a 12-th grade calculus student who had forgotten her multiplication tables and it showed up when she got stuck on a calculus question at the very begining of the problem. This gap of at least nine grades in neglecting the "no-brainer" stuff is shocking, and unfortunately it is not uncommon. Tutors, parents, and students should never let this kind of thing happen because it is a senseless, unnecessary, and tragic waste of effort on the part of the student.
A tutor can help to avoid such catastrophies by routinely inserting "spot checks" into discourses and lessons involving more advanced material (see point (3) above). Another way a tutor can reduce the chances of "no-brainer" material causing "show stoppers" for more advanced problems is to make a habit of continuously pointing out relationships between different topics in math that the student has studied. This helps to keep older material at the forefront, and sends a powerful message to the brain that the stuff should not be "filed away" because it is likely to be needed.
(11) During tutorials, unless the lesson is on graphing-calculator skills, keep any calculators out of reach and only use a calculator whenever absolutely necessary. Develop a lightning-speed reflex to grab the calculator before the student can, if it is a situation in which a calculator should not be used. Abuse and misuse of a calculator diminishes students' mental agility, intuition, and sense of numerical size. This ultimately results in an overall degradation of performance in math in general. If you have read claims that "Research shows that calculators improve a student's performance in math.." or something to that effect, you have to ask yourself several questions. Is the claim backed up by showing you actual data or other evidence? Are any details given about how the study was conducted? In particular, how was performance measured and how was improvement defined? What period of time did the study cover and what demographical information is given about the participants? How many participants were there? Most importantly, how were other factors eliminated or otherwise accounted for? It is very difficult to eliminate other factors that could affect the outcome. You will have a hard time finding this information. I challenge you to find a convincing study that proves that calculators improve a student's overall performance in math. Now, I can't prove the converse to you, but I can tell you that whenever I restrict a student's use of a calculator and retrain him/her in basic numerical mental manipulation skills, the student's competence and performance in solving a wide range of math problems improves (as measured by speed and accuracy).
As for graphing calculators, there is an entire "Pandora's box" of problems that students face with graphing calculators and in my book “What Can I Do..?” I devote an entire chapter to discussing these problems and how they can by tackled, even by parents who have no knowledge of the actual math that is involved.
(12) When teaching students about the square root of negative numbers, imaginary numbers, and complex numbers, do not overplay the apparent "mysteriousness" of imaginary numbers. Too many math teachers do portray imaginary numbers with some kind "mystical" reverence. The result is that students who are already apprehensive of math will have their confidence eroded and they will be unnecessarily confused. The truth is that there is absolutely nothing mysterious about imaginary numbers: they are simply a device for manipulating at once, pairs of quantities that are in some physically meaningful way related to each other. The rules of manipulation are, by design, such that many types of real-world problems can be solved using complex numbers. If humans had not invented complex numbers, they would have invented some other mathematical device to solve the same problems. Unfortunately, I have come across plenty of professional mathematicians and physicists who believe that there is something "deep" about the square root of negative numbers and they impart this belief onto their students. Complex numbers are simply human inventions that provide a particular, but not unique, way of solving certain classes of physical problems.
The above tips are very general; here are a couple of specific ones for students in 9th and higher grades.
(13) Teach students to how to quickly compute (mentally) the six trigonometric function values for the special angles of 0, 30, 45, 60, 90, 180, and 270 degrees. Again, don't start with memorization: practicing the mental skill of computing these quickly will make explicit memorization unnecessary. When this has been mastered, add in the ability to work out the correct sign for any trigonometric function in the 0 to 360 degree interval.
In addition, students should learn how to quickly (mentally) compute the common and natural logarithms (and their inverses) for a suite of very simple arguments. For example, log(1), log(rational powers of 10), ln(1), ln(rational powers of e), and the corresponding inverses. Students should be familiar with the values of log(2) ≅ 0.3 and ln(2) ≅ 0.7, and thereby mentally estimate things like log(40) [=2log(2)+log(10) ≅ 1.6], and log(0.08) [=3log(2) − log(100) ≅ 0.9 − 2.0 = − 1.1]. Students should then also be able to mentally estimate log(3 to 9) and ln(3 to 9). For example, log(5)=log(10) − log(2) ≅ 0.7; log(3) ≅ log[√10] =0.5log(10)=0.5. Don't underestimate the far-reaching beneficial effects of these proficiencies in trigonometry and logarithms in dramatically enhancing a student's speed and accuracy in problems in many different math topics from 9th grade and well beyond.
(14) Don't make calculus seem more mysterious and/or more difficult than it is. This is because propagating the false myth leads to the erosion of confidence in the student. The fact is that many students find precalculus harder than calculus, once they have been properly trained in calculus. However, before a student embarks on studying calculus, or if a student is unsure whether to take a calculus course, you should reassure the student that if he/she is comfortable with other advanced topics in math then he/she should not encounter serious problems. In fact, most high-school level calculus problems are very straightforward. The hardest part is learning to recognize what method should be used to solve a given integral that may be "thrown" at the student. Students find this somewhat bewildering because there seems to be no clear-cut "rule" that serves as a "crutch." However, this can be easily and fairly quickly overcome with proper training from a tutor. Even though there is no single "rule" there are only a very limited number of possiblities, especially at this academic level, and the tutor should make up some oral drills covering each possiblity, in about a dozen different examples. Then, after going through with the student how each class of problem is solved, the tutor should randomly go through the example integrals, asking the student only the method that the student would use to solve each integral, without actually solving it. In this way, the student can very quickly learn how to recognize what approach should be taken to solve any integral that might come up on a high-school exam or assignment.
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