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Learning Math – The Wrong Way or the Right Way

June 29th, 2014

Like everything we do, there is a wrong way and a right way to learn mathematics. My years of teaching math at the university level showed me that many students had developed unhelpful habits for learning math. These habits, unless corrected, can lead to failure in college math courses.

The Wrong Way to Learn Mathematics

  1. Memorize topics, but don’t bother trying to understand the logic behind them.
  2. If you have difficulty with a certain topic, skip it. Maybe it won’t be important later anyway.
  3. Use sloppy, messy writing for solutions to problems.
  4. Don’t do the practice problems.
  5. Copy someone else’s homework.
  6. Don’t learn basic definitions, or their consequences.
  7. Forget about learning and understanding mathematical proofs.

In short, bad habits (learning math the wrong way) are mostly the result of intellectual laziness. Mathematics is not an easy subject to learn. To learn it properly you have to understand every detail. The first way to check whether you do understand a topic is to do the exercises, or problems. But you also need to study and learn the logical basis of every topic. Ask yourself whether you could explain it to someone else.

To mention but one example, you probably “know” that the sum of the interior angles in any triangle is , but can you explain why this is true? The proof is not difficult, and I believe that every geometry student should learn and be able to recite the proof. Understanding the logic underlying each math topic helps greatly to reinforce your memory of that topic. It is also useful, and in fact necessary, for learning more advanced topics. Students who rely on memorization alone often remember math incorrectly – or forget things entirely.

Another, very important example is the equation a(b+c) = ab + ac, which is frequently used in Algebra and elsewhere. Misuse of this equation often leads to failure on college math exams. But a student who understands why the equation is valid is unlikely to use it incorrectly.

The Right Way to Learn Mathematics

  1. Always spend the effort to fully understand each topic, whether an algorithm (method of calculation), a proof, or (especially) a definition.
  2. Memorize everything you learn; invent your own mnemonic “tricks.” Understanding the logical basis of a given topic is a powerful aid to remembering it over the long term.
  3. Understand and memorize definitions. For example, what is the basic definition for adding fractions? How does it imply the standard method?
  4. Always do practice problems. It is the problems that cause you some difficulty that are the most helpful. Trying to understand and overcome the difficulty is an important aspect of learning math.
  5. Watch for “mental blocks” – things that you just don’t understand. Everyone encounters mental blocks once in a while. Take the time and effort to resolve such blocks.
  6. Work hard to understand and remember proofs. Gradually learn how to produce valid – and clearly written – proofs on your own. Figure out how the assumptions were used in the proof.
  7. Write down the solutions to practice problems clearly and succinctly, using plain English phrases such as “by substitution,” “therefore we have,” “by Pythagoras’s Theorem,” etc.
  8. Think about how a given topic being studied is related to other familiar topics. Look for instructive special cases.

Many people seem to think that learning math is like learning history. You memorize a bunch of unrelated facts (or problem-solving techniques) for the exam. But learning math in this way is pretty much a waste of time (the same is probably true for learning history). Mathematics is a tightly organized system of knowledge, based on strictly logical arguments which themselves make sense. Ignoring the underlying logic is a serious mistake, which leads to faulty memory, and eventually to increasing confusion and “math anxiety.”

Most people know whether they do or do not understand a particular topic in math. But what can a person who doubts his or her understanding of parts of math do about it? Should you use online resources as learning aids? Maybe, but many sites that I have looked at were pretty close to “wrong way” approaches – sets of routine problems requiring very little basic comprehension.

Two outstanding exceptions: The Khan Academy http://www.khanacademy.org/ and my book Math Overboard . The Khan Academy is mainly for people who are learning the math for the first time. Math Overboard (Basic Math for Adults) reviews all of school math, from kindergarten to Grade 12, with self-contained explanations of every topic. It is designed for people who need to re-learn parts of basic math. Frequent problems support learning. Math Overboard, Parts 1 and 2, are now available in printed and eBook versions. See  http://www.mathoverboard.com/ for printed versions, and an online book seller for eBooks.

 

 

Understanding Basic Math – Test 3

May 13th, 2013

 

Euclidean Geometry

This is the third in a series of tests of understanding basic math. You need to be able to answer these questions accurately and confidently if you intend to take college-level math courses in Calculus, or Statistics. In answering the questions, please provide typical examples.

Angles and Triangles

  1. Define the term “angle.” When are two angles equal?
  2. Define the terms “straight angle”, “right angle”, and “acute angle.”
  3. Explain how the size of an angle is specified, in terms of degrees. (How much is 360° ?)
  4. Define the term “triangle.” Also define “right triangle”, “isosceles triangle”, and “equilateral triangle.”
  5. State the theorem about the sum of the interior angles of a triangle. Give the proof.
  6. Find the size of each angle in (a) an equiangular triangle, and (b) an isosceles right triangle.
  7. State Pythagoras’s Theorem and give the proof.
  8. Define “congruence” for triangles (and other geometric figures).
  9. One set of sufficient conditions for two triangles to be congruent is that the three sides of the first triangle are equal to the three sides of the second triangle. (a) Explain why this is true. (b) Is the analogous statement true for quadrilaterals?
  10. Give two other sets of sufficient conditions for congruence of triangles.
  11. Prove that the base angles of an isosceles triangle are equal. What is the converse?
  12. Prove that an equilateral triangle is also equiangular. State and prove the converse.

 

Circles, Arcs, and Sectors

  1. Define radian measure for angles. Give examples.
  2. 1° is how many radians? 1 radian is how many degrees?
  3. Define the terms “circle”, “arc”, and “sector.”
  4. State the formulas for the circumference of a circle, and for the area inside a circle. What do the formulas have to do with scaling?
  5. State and derive the formula for arc length.
  6. Ditto for the area inside a sector of a circle.
  7. IF ABC is a triangle inscribed in a circle (i.e., the vertices A, B and C all lie on the circle), with AB being a diameter of the circle, prove that ABC is a right triangle. Suggeston: draw the line OC from the center of the circle O to point C, and look for isosceles triangles.

Solving Right Triangles

  1. Given two sides of a right triangle, show how to find the third side.
  2. Define sin θ, cos θ, and tan θ for positive acute angles θ.
  3. Calculate these functions for θ = 30°, 45°, and 60°. Use geometry, not a calculator.
  4. Given two sides of a right triangle, show how to find the angles using trigonometry (and a calculator).
  5. Ditto, given one side and one angle of the triangle.

 

Understanding Basic Math – Test 2

May 4th, 2013

 

Algebra of Polynomials

This is the second in a series of tests of understanding basic math. You need to be able to answer these questions accurately and confidently if you intend to take college-level math courses in Calculus, or Statistics. In answering the questions, please provide typical examples.

Exponents

  1. Explain how the expression xn is defined, if n is a positive whole number (n = 1,2,3,…) and x is any real number.
  2. Prove that xn xm  = xn+m.
  3. Prove that (xy)n  = xn yn and that (xn)m  = xnm.
  4. Explain why x0 is defined as x0  = 1 (for x not equal to 0).
  5. How is xn defined if n is a negative integer?
  6. Prove that xn / xm  = xn-m.

Polynomials

  1. Define “polynomial.” Also define “degree” of a polynomial.
  2. Show how to add and multiply two polynomials.
  3. Expand (x + a)2 and explain.
  4. Explain division of polynomials with remainder. Illustrate with an example.
  5. Show how to solve a linear equation ax + b = 0.

Quadratic Polynomials

  1. What is a quadratic polynomial?
  2. Explain how to factor a quadratic polynomial by trial and error.
  3. State the quadratic formula, with examples.
  4. Explain the method called “completing the square.”
  5. Use completing the square to prove the quadratic formula.
  6. Define the discriminant of a quadratic polynomial.

The Binomial Theorem

  1. Expand (a + b)3.
  2. Describe Pascal’s triangle.
  3. What are the binomial coefficients C(n,k)? How are they related to Pascal’s triangle?
  4. State the Binomial Theorem, using ∑ notation.

Roots of Polynomials

  1. Define the terms “root of a polynomial” and “factor of a polynomial.”
  2. Explain why any polynomial of odd degree must have at least one real root.
  3. State and prove the remainder theorem.
  4. State and prove the factor theorem.

 

Any Math Questions

April 3rd, 2013

Any Math Questions?

If you are reading Math Overboard (Basic Math for Adults), you’re invited to send me any questions about any topic in basic math. I’ll either direct you to the appropriate section of the book, or explain how to resolve your problem. Just send a reply to this blog.

Meanwhile, here are some errors and improvements to Part 1.

Errors and Improvements (Part 1) (See below for Part 2)

Page 8, Solution 1.4, Step 3. Second line should read “carrying over now to the hundreds position.”

Page 48. Solution 2.3 doesn’t explain how to borrow 1 from 0. To do this, first replace 0 by 9, and second, borrow 1 from the next digit (unless it’s also 0, in which case you repeat this step). For example, 203 – 75 becomes 19(13) – 75 = 128 (write this out to see what I mean).

Page 64, 3 lines below Box (2.15). The new sentence should start “Also, in Box (2.15),…”

Page 97, line 8. Replace “1 x a = 1” by “1 x a = a”

Page 101, line 6. Replace “45/99 = 5/9.” by “45/99 =5/11.”

Page 118, Solution3.8. This should read “A = (1/2) x 4.0 cm x 2.1 cm = 4.2 cm2.”

Page 130. Solution 3.14. Replace “and P = $5.00. This gives… square foot” by “and P = $5.00 per quart. This gives k = C/AP = .036 quarts per square foot”

Page 137, Problem 3.22. First sentence should include “conversion table on page 135”.

Page 140, Solution 3.23, line 2. Replace “5200 ft/mile” by “5280 ft/mile”, so that the answer is 6.8 mi/hr.

Page 181, line -7. Replace “discussed in Sec. 2.5.” by “listed on Page 97.”

Page 199, line -9. Shift the numbers 1331 to the right slightly, to line up properly in this table.

Page 206, Solution 4.54. Add: (b)  (u + v)p =   (Sigma x = 0 to p) (p! / x!(p-x)!) up – x vx .

Page 263, One line below Eq. (5.10). Replace “Eq. 5.8” by “Eq. 5.9”.

Page 394, Solution 3(b) should be (3x – 2y)/6.

Errors and Improvements (Part 2)

Page 454, Solution 8.12, line 2: Replace “Eq. 8.8” by “Eq. 8.1”

Page 460, Diagram: The smallest angle should be 15.6 degrees.

Page 479, line 4: Replace “Eq. 8.26” by “Eq. 8.27”

Page 483, Section 9.1, line 5: Replace “Section 4.7” by “Section 4.8”

Page 495, Problem 9.7, line 1: Break the line after “ln 1/2,” and start new line with “ln(1/3)”

Page 526, Example – Solution: Replace “We have … 22.5 degrees” by “We have tan 2 Theta = B/(A-C) = -1, so that 2 Theta = -45 degrees, or Theta = -22.5 degrees” Also, in the displayed equation, replace (4.7) by (2.24) and (.65) by (1.34). In the next sentence, replace “4.7 and 0.65” by “2.24 and 1.34”

Page 5.29, line -11: Replace “Section 4.8” by “Section 4.9”

Page 5.42, Solution 10.27, line -2: Replace “it can be” by “It can be”

Page 5.47, three lines above Problem 11.1: Replace “Section 4.6” by “Section 4.7”

Page 590. Solution 11.30, line -7: Replace “= .411” by “= .589”

Ditto, line -2: Replace “= .500 … .719” by “= .500 + .589 – .192 = .897”

Page 591, line -7: Replace “should result” by “should result in E occurring.”

Page 594, line -5: Replace “-n + k (k = 0,2, …, 2n)” by “-n + 2k (k = 0,1, …, n)”

Ditto, line -4: Replace “-n + k” by “-n + 2k”

Page 637, Solution 12.20, line -3: Replace “Yogi Berra” by “Any runner”

Page 653, line -5: Replace “wrong if Alpha = Beta, for” by “wrong, for if Alpha = Beta”

Page 683, line -7: Replace “Section 10.1” by “Section 9.1”

Page 717, line 1: Replace “x = 233.1 degrees” by ” x = 4.069″

Page 719 1(b): Replace “1/2 – i” by “-1/2 – i”

Ditto, 7: Replace “:Theta = 67 1/2 degrees” by “Theta = 67 1/2 degrees or -22 1/2 degrees”

Page 723, 7(b), line 3: Replace “t = 15/29” by “t = 23/29”

Ditto, line 4: Replace “therefore … /29)” by “therefore r(23/29) = (28/29,46/29)”

Page 727, line 1: Break line before “(repeating)”

Page 727, line -2: Replace “n(n-1)/3” by “n(n-1)/2”.

 

Understanding Basic Math – A Test

February 13th, 2013

 

Understanding Math

This is the first in a series of posts that allows you to test whether you understand basic math, by which I mean the math that you learned in school, from kindergarten to Grade 12. I will assume that you are either a parent, or a student (school or college-level), and that you realize that in fact you don’t fully understand basic math. No doubt you have already made an effort to upgrade your math skills and understanding, but have found the available material (books or the web) not too helpful for some reason. My recently published book Math Overboard! (Basic Math for Adults), which reviews all of basic math (K-12), attempts to explain fully and understandably all of basic math. By skimming through several of the tests in this series you can find out what you need to study, and what Chapters of Math Overboard! are relevant.

But what does “understanding” math actually mean? I once asked a class of Honors Calculus students to write a short essay on this question, and was surprised to find that they didn’t have a clue. I soon found that I wasn’t very good at answering the question myself. Most mathematicians will say that to understand a theorem you have to understand the proof. That is true, but it’s only part of the story.  First, there are many topics in math other than theorems as such. Algorithms are one example. Applications are another. You have to know (memorize) these things, and also to understand why they are valid. For example, you need to know the long-division algorithm, and why it is valid. How are long division with remainders, and long division with decimal expansions, related? And why is it true that the decimal expansion of a fraction of whole numbers either terminates or repeats?

I am not going to provide you with the answers to the test questions, as this would take too much space. You will need to read Math Overboard! or some other book (or use the web). By reading through the tests you can find out what you need to study.

Arithmetic: Base-Ten Numbers

  1. Explain what base-ten notation, as in 2,316 for example, means, in terms of the powers of ten.
  2. Explain how the addition (or sum) of two base-ten numbers is defined, in terms of counting.
  3. Of course you know that 59 + 37 = 37 + 59 (without actually doing the additions), but how do you explain that a + b = b + a in general?
  4. Describe the addition algorithm for base-ten numbers (which are the same as whole numbers). Explain why the algorithm is valid. (An algorithm is a method of calculation that is routine and repetitive.)
  5. Repeat questions 3-5 for the multiplication of base-ten  numbers.
  6. Another basic law of arithmetic is that a + (b + c) = (a + b) + c. What is the meaning of the brackets (or parentheses)? How do we know that this law is true?
  7. Repeat question 6 for multiplication.
  8. Yet another all-important law of arithmetic is that a(b + c) = ab + ac. Give some examples. Then explain why this law is valid.
  9. What does the expression a < b mean? Give examples.
  10. State and explain the algorithm for deciding which of two given whole numbers is the larger.

Decimal numbers

  1. Explain what decimal number notation, as in 17.62, means.
  2. What is meant by the number line for decimal numbers? How is a given decimal number located on the number line?
  3. Explain the addition algorithm for decimal numbers.
  4. Show how to multiply two decimal numbers (Math Overboard! doesn’t explain why this method is correct, so perhaps you can explain that.)
  5. List the Laws of Arithmetic for decimal numbers. How do we know that these laws are valid?

Subtraction

  1. If a < b, how is the difference b – a defined?
  2. State and explain the subtraction algorithm for decimal numbers a < b.
  3. Explain both addition and subtraction in terms of the number line.

Negative Numbers

  1. Describe the system of integers. Show how to subtract two whole numbers b – a, in the event that a > b.
  2. Explain the subtraction algorithm for two decimal numbers b – a, in the event that a > b.
  3. Define multiplication for integers, and for decimal numbers (positive and negative).

Division and Fractions

  1. If a, b, and c are whole numbers, what does the equation a / b = c mean? Discuss examples. (Be careful: it may not mean anything if we insist that all three symbols refer to whole numbers.)
  2. Now explain the concept of division with remainders.
  3. State and explain the algorithm for division with remainders. Discuss examples.
  4. What is meant by a fraction m/n? Explain how to locate a fraction on the number line.
  5. Explain the cancellation law am / an = m/n.
  6. Explain addition of fractions, in the case that both fractions have the same denominator.
  7. How do you add fractions having different denominators? What is the quickest way to check such an addition?
  8. Explain how to obtain the decimal representation of a fraction.
  9. How are two fractions multiplied?
  10. What is the definition of division for fractions? What is the algorithm, and why is it valid?

If you can answer these questions confidently, you already understand the number system, at least as far as rational numbers (fractions, positive or negative) are concerned. If not, you need to brush up on basic arithmetic.

The Disconnect Between School and College Math

January 9th, 2013


Why are many college students unprepared for math?

This article is about what I call the “disconnect” between mathematics instruction in the schools and in our colleges and universities. In a nutshell, the teaching of math in the schools generally downplays theoretical aspects (such as proofs), whereas university math courses do stress theory. According to Dr. Kim Maltman of York University, “…we have found that our first-year science majors [are] in general rather poorly prepared for first-year university mathematics. The result is very high drop-plus-fail rates in our first-year math courses….” (see Math Remediation…). This statement probably applies to almost every post-secondary institution in North America.

The leap from school math without theory to proof-based college math courses is usually abrupt, and takes many students by surprise. As one student said to me, “I can’t hope to understand math. My only chance of passing your course is to memorize the problem solving techniques needed on the exam.” She didn’t pass.

The best students eventually get through the disconnect, and learn to master theory. This works well for students in math, physics, and engineering, but not for many of those going into biology, economics, or psychology. Many students find their educational hopes dashed because of their failure in basic math courses.

Knowing Theory is Necessary for Understanding Mathematics

The trouble with studying mathematics without learning basic theory is that the student then fails to understand the math that he or she has studied. He or she may be able to solve routine exam problems, but even that ability begins to fade as memory falters later on. Such rote “learning” is all but useless in more advanced courses, and in later life generally. Among people I meet who are not scientists or engineers, well over half tell me that they never really understood math in school, and have now forgotten most of it. Why is so much time and effort spent teaching math in school if this is the result?

Teaching – and learning – mathematics with a secure understanding of each topic is much more difficult than just memorizing mysterious formulas and methods for solving routine problems. Understanding math as you learn it has the following important consequences:

  1. The effort of understanding the logic helps immensely in imprinting a given topic permanently in your memory.
  2. Understanding a given topic helps greatly later when learning more advanced topics.
  3. Understanding each topic fortifies your confidence in using that topic later to solve problems, both routine and novel.
  4. Learning basic theory (proofs) in mathematics places the learner in a long historical-cultural stream extending from the ancient mathematicians down to the present day. Our current knowledge of mathematics is entirely dependent on the existence of valid, tested proofs for every known result.

The lazy approach, whether in learning or teaching math, is to ignore the theoretical, logical foundation and simply concentrate on memorizing techniques for solving routine problems. This approach may get the student through exams, but the long-term consequence will be an increasing level of confusion and uncertainty about math.

Of course, even if you do understand things, you still need to memorize many techniques and formulas. You also need plenty of practice in problem solving, in order to firm up both memory and understanding.

But What Can be Done, Today?

Perhaps some day in the distant future the teaching of math in the schools will emphasize memorization and comprehension equally. Meanwhile what can a student do to re-learn school math with a proper level of understanding, as required for college studies? Read through the school textbooks again? – not. Search the web? Hire a private tutor? Take a remedial course?

A better solution – if you’re willing to work hard on your own – may be to obtain a copy of Math Overboard! (Basic Math for Adults). This recently published book reviews all of school mathematics, from kindergarten to Grade 12, with the aim of instilling detailed understanding of every topic. Nothing is left out; everything is included. Part 1 is now selling from the website for $24.00 net. Part 2 (expected in June 2013) completes the book, and will sell for about $20.00.

Math Remediation and Tutoring

December 31st, 2012

Math Remediation

The following Abstract refers to a talk presented recently at my home university (British Columbia) by York University math professor Dr. Kim Maltman:

Abstract. At York, we have found first-year science majors coming to us from the
Ontario high school system in general rather poorly prepared for first-year
university mathematics. The result is very high drop-plus-fail rates in
our first-year math courses and a resulting high attrition rate in the
early years of our degree programs. A major source of the problem appears
to be the widespread use in the schools of an approach heavily emphasizing the
memorization of solution problem templates, an approach which leaves a
majority of our incoming science majors with deficiencies in very basic
algebra, trigonometry, and, even more problematic, their intuitive
understanding of the basic operations of arithmetic. In this discussion,
I will outline an approach I have developed involving 4-day, 4-hour-per-day
intensive remediation sessions focused on changing the way such students
approach mathematics. The program was begun in 2005 and significantly
expanded in 2009, now handling between 15 and 20% of the incoming
class each year. I will present statistics outlining the significant impact
we have seen on student performance.

It seems that the phenomenon of “high drop-plus-fail rates” is almost universal in North American colleges and universities. I don’t know how other universities are dealing with the problem, but at UBC until recently little was done, other than to provide students with a math-help facility where they could obtain help with their calculus courses. Many students still had difficulties, as attested to by the large number of posters advertising Math Tutoring. Besides this, almost every shopping mall in the area has an office advertising Help classes in Math.

Why is the phenomenon of high dropout rates so common and persistent? I think the Abstract has got it right – “…heavily emphasizing the memorization of solution problem templates,…which leaves incoming science majors with deficiencies in … algebra, trigonometry, [and] their intuitive understanding of … basic operations….” Just so.

But how to deal with the problem? Two possibilities are:

  1. Re-design the school curriculum to place more emphasis on understanding math.
  2. Provide assistance for poorly prepared college entrants.

The first alternative probably cannot be achieved easily, if at all. Remediation sessions are one approach to the second possibility, and it’s certainly impressive if a total of 16 hours’ class time can reverse 12 years of “math-is-memorization” instruction. I don’t know how  many universities are providing this service at the moment.

A second approach is streaming, with poorly prepared students being forced to take a course such as “Precalculus.”  But unless this course includes remediation for weak training and poor habits in arithmetic, algebra and geometry, I doubt if it will succeed in rescuing many students.

Tutoring

Private tutoring is yet another approach, but whether tutoring is likely to result in the necessary changes in the way that students perceive math is doubtful. Commercial tutoring may be better, but expensive.

Rather than hiring a tutor, a struggling student might try to re-learn math by searching the web. But what should they re-learn? Proofs? Set theory? Long division? Inverse functions?

Math Overboard!

These were the issues that prompted me to write Math Overboard! (Basic Math for Adults), which is a complete review of school math, emphasizing understanding and algorithmic, problem-solving skills. Math Overboard! is a self-study resource that helps motivated students to upgrade their comprehension of basic math to a level suitable for college math courses.

Here is what Prof. Marc Mangel (UC Santa Cruz) says:

I have now had a chance to read through MATH OVERBOARD and like it very much; I look forward to Part II. I will recommend it to grad students in biology as a reference book and once I am back on campus … I will hawk it to my colleagues who teach pre-calc and calculus. I think that for the calculus classes it would be wonderful resource book, if the students would use it.

 

Understanding Mathematics

December 10th, 2012


Why Should You Try to Understand Math?

 

Many students in America graduate from high school with a limited understanding of the math that they studied in school. If they go on to college or university, these students often have difficulty with the math courses there, for three reasons:

  1. College math courses are founded on basic school-level math.
  2. A lack of understanding of basic math usually leads to faulty memory, confusion, and errors.
  3. A habit of studying math without trying hard to understand it backfires in college courses.

These considerations lead to the following questions:

  1. What constitutes “understanding math,” anyway?
  2. What is the proper emphasis on understanding basic math?
  3. How can students re-learn their school math so that they do understand it?

 

What does Understanding Math Mean?

The phrase “understanding math” refers to the understanding of each specific topic in math. Do you understand quadratic equations? Logarithms? Inverse functions? and so on.

Understanding a specific topic implies:

  1. Knowing exactly what the topic (result, theorem, algorithm) is, including knowing the precise definitions of all relevant terms.
  2. Knowing why the result, theorem, or algorithm is true or valid.
  3. Knowing how to use the topic.
  4. Knowing how the topic is related to other topics in math.

Understanding a certain topic in math is more difficult than just memorizing it. Failure to understand a topic may have been the result of laziness on the part of the student, or the result of the teacher’s lack of insistence on understanding. SAT tests don’t usually test understanding directly, so there may be little incentive to learn understanding.

The Distributive Law

Here’s an example, the distributive law:

a (b + c) = ab + ac

This equation is absolutely fundamental in all of mathematics, from algebra to trigonometry and beyond. It’s also one of the most frequent sources of error made by weak students.

Are you familiar with the above equation? What does it say in words? Can you give some examples? What do the letters a, b, and c represent? And especially, why is the equation true?

Does it matter why the equation is true, in general? If you don’t know why it’s true, how would you know whether the similar-looking equation a/(b + c) = a/b + a/c is true? (It isn’t.) It’s quite a feat of memory to remember that the first equation is true while the second is not, without understanding why in either case. But a student who remembers (even if vaguely) why the first equation is true, is also likely to realize that the same logic does not apply to the second equation, which is probably therefore incorrect. This student could also probably prove that the second equation is wrong, by supplying a counterexample.

Knowing why the formulas, theorems and algorithms of basic math are true is an important aspect of learning the subject.  Without such knowledge, math becomes just a jumble of unrelated, mysterious factoids.

Too Much Emphasis on Understanding?

Is it possible to over-emphasize understanding of basic math? Yes, if this means that problem-solving skills are downplayed. Students need lots of practice (“drills”) to perform routine operations quickly and accurately, without consciously thinking of the underlying theory. But the fact that you do subconsciously remember the basic logic is strongly confidence-building. A sound mathematical education combines theory (understanding) and practice in balanced proportions.

Math Overboard!

My recently published book Math Overboard! (Basic Math for Adults) reviews all of school mathematics, from kindergarten to Grade 12, emphasizing both understanding and skills. To my knowledge no other available book accomplishes this outcome. Please visit Math Overboard!

The Math Education Crisis

November 23rd, 2012


 Mathematical Mis-education

Mathematicians and other scientists are upset about the current state of math education in American schools. Here are a few nuggets taken from the website http://wisemath.org/.

“We support a balanced approach between understanding and skills. Unfortunately, in the shift towards ensuring that children understand math concepts, which we support, several important elements of mathematics have been neglected, or completely eliminated, from curricula and math classrooms.”

“The most recent version of the WNCP (Western and Northern Canadian Protocol) math curriculum omits all standard algorithms for addition, subtraction, multiplication and division.”

“Martin Scharlemann, while chairman of the Department of Mathematics at the University of California at Santa Barbara, wrote an open letter deeply critical of the K-6 curriculum MathLand, identified as “promising” by the U. S. Department of Education. In his letter, Professor Scharlemann explains that the standard multiplication algorithm for numbers is not explained in MathLand. Specifically he states, “Astonishing but true — MathLand does not even mention to its students the standard method of doing multiplication.” ”

“Post-secondary instructors are also frustrated by the weak math skills of many new graduates and are troubled by the fact that many math teachers are not receiving adequate training in math before entering classrooms in Canada.”

It is inconceivable to me that anyone would think that you can understand Arithmetic, let alone Algebra, without mastering the basic algorithms for addition, multiplication, subtraction and division. Yes, you can buy a $10 calculator that will “find the answer” to any given numerical calculation, but this does NOT imply that learning these algorithms is not necessary.

The 7-11 checkout clerk tells you your purchases add up to $8.63, so you hand him a $10 bill and he gives you $1.24 change. When you get outside you ask yourself how come a coke ($1.49) and some chips ($2.49) can add up to over $8.00. So you reach for your calculator – oops, it’s at home. So you return to the store to complain, but the clerk explains that there was tax of 87c. Now what? Shrug it off? Too bad you never learned how to add or multiply, and after that how to do quick approximate sums in your head.

Then you go to a political rally, where the candidate tells you that his opponent’s tax policies will cost taxpayers $750 billion. Is this realistic? And is it dollars per year, or over a 4-year period? And how much is that per average taxpayer? Would you dare take out your calculator there among all the screaming audience? And how do you enter the number 750 billion into the calculator, anyway?

Math Overboard!

If that was your experience in school, you might want to re-learn your basic math from scratch. I would like to recommend my recently published book Math Overboard! (Basic Math for Adults). Covering all of school math, from kindergarten to Grade 12, Math Overboard! stresses the importance of understanding math in detail, as you learn it, or in this case, re-learn it. Frequent Problems test your understanding as well as your skills. It’s not an easy book, but it’s your best hope to really learn what math is all about.

For further information, please visit Math Overboard!

Re-learning Math with Math Overboard!

November 21st, 2012

Re-learning Math

Millions of people could benefit from re-learning the math that they were taught in school. For example, you may wish to prepare for college courses in Science, Economics, or other fields. Or you may just feel frustrated that you never really understood math at school.

Two possible methods for re-learning math are:

  1. Search the web using keyword phrases like “basic math,” “understanding math,” and so on.
  2. Obtain a book.

But where should you start? And what book? There are thousands of web sites and many books.

The Khan Academy

Web searching will probably lead you to the Khan Academy, a fantastic site (sponsored in part by the Bill and Melinda Gates Foundation). Under “Mathematics,” the Khan Academy has over 1,200 excellent, free videos covering all topics from school math. The videos are great, but you might have to view the whole lot of them, and go through the associated Problem sets,  to re-learn math. The videos are aimed at beginning students who have never seen the math before, so they’re very time-consuming. Is there some way to choose just those videos you need?

Math Books

A math book may be a better way to go. But should you buy 12 books, one for each grade? Or a separate Algebra text, Geometry text, Trigonometry text, and so on? Most of these are school texts, written for kids. Is there a single book that reviews all of Basic Math, addressed to adults, not children?

Math Overboard!

My recent book Math Overboard! (Basic Math for Adults) is designed precisely for this purpose. It covers andexplains all of school math, from Kindergarten to Grade 12, in an understandable fashion. For example, why can’t you divide by zero? Why is it true that a(b+c) = ab + ac? Why is the sum of the angles in a triangle equal to 180 degrees? Why is the quadratic equation valid? What is a logarithm? What is a probability? And so on. The book is readable by students, parents, and anyone interested in re-learning, or improving their understanding of basic math.

Math Overboard! consists of two volumes:

Part 1: Arithmetic, Algebra, Geometry, Functions and Graphs. (Published November, 2012; 444 pages. Price if ordered from the website (includes 20% discount from retail price), $24.00.)

Part 2: Trigonometry, Exponential and Logarithmic Functions, Complex Numbers, Statistics and Probability, Advanced Topics. (Expected publication date June, 2013.)

Your Re-learning Strategy

Re-learning math is not going to be easy. It will be necessary to concentrate, and to really try hard to understand why formulas and theorems are valid. Work hard on solving the given Problems (before looking up the solutions!).

  1. Combine your intensive study of Math Overboard! with occasional videos from the Khan Academy, when you need more elaborate help with unfamiliar or confusing topics.
  2. Follow the advice in Math Overboard! for learning math, and avoiding errors.
  3. Retain Math Overboard! as an indispensable reference for use in later math courses.

 

For further information, please visit Math Overboard!