The Power of One in Math

The number 1 is such a simple number:

·         It’s the first number when you count positive integers.

·         It has no effect when multiplying or dividing by 1.

·         It also has no effect as an exponent.

·         Adding or subtracting 1 simply yields the neighboring integer.

Yet the number 1 can be very powerful in math.

One way to convert units is to multiply by 1. For example, to convert 4 ft. to inches, you could multiply by 1, writing 1 as a fraction: 12 in. / 1 ft. equals 1. Multiplying 4 ft. by 12 in. / 1 ft. yields 48 in. (Observe that the feet cancel out.)

Fractions can be added by finding a common denominator, and the denominators are made common by multiplying by 1. For example, to add 1/4 to 1/3, the least common denominator is 12. Multiply 1/4 by 3/3 (which is 1) to make 3/12 and multiply 1/3 by 4/4 (also 1) to make 4/12. Now add these together to get 7/12.

The number 1 is used to turn numbers into their reciprocals. For example, the reciprocal of 5 is 1/5. The reciprocal of x is 1/x. Any number times its reciprocal equals 1. For example, 2 times 1/2 makes 1.

We use this principle all the time in algebra. For example, consider 4 x = 12. In this case, isolate the unknown, x, by multiplying by the reciprocal of its coefficient. The coefficient of x is 4; the reciprocal of 4 is 1/4. Multiply both sides by 1/4 to make x = 12/4 = 3. (Note that dividing both sides by 4 is equivalent to multiplying both sides by 1/4.)

The equation x + y⁰ = 2 wouldn’t have a unique solution if not for the fact that y⁰ = 1. (This follows from the rule that y^a/y^b = y^a–b. Let b = a. Then y^a/y^a = 1 = y^a–a = y⁰.) Knowing this, the original equation becomes x + 1 = 2, or x = 1.

In trigonometry, the number 1 often comes in handy through trig identities. For example, the equation 1 – sin²x = 0.5 cos x would be difficult to solve because it includes two different trig functions, except that 1 can be written as sin²x + cos²x, which reduces this equation to cos²x = 0.5 cos x. We can then factor it as cos x (cos x – 0.5) = 0, which means that cos x = 0 or cos x = 0.5, for which the solutions are 90, 60, or 300 degrees.

In probability, the number 1 represents 100%. If you reach into a bag with 3 red marbles and 5 blue marbles, you have a 3/8 probability of selecting red and 5/8 probability of selecting blue. If you add these together, you get 3/8 + 5/8 = 8/8 = 1. That’s because there is 100% that you’ll select a marble of some color.

There are many derivations and proofs in higher levels of math that become eloquent when a term is multiplied by 1 and the 1 is written in terms of an identity (such as the summation expression for the completeness theorem in group theory – which itself is another example of the power of one, in establishing unity for a complete set).

Chris McMullen, author of the Improve Your Math Fluency series of workbooks

The Importance of Confidence in Mathematics – and How to Attain This

When learning mathematics, concepts often seem strange or intimidating to students. When this happens, many students proceed tentatively.

They sometimes give up when they were proceeding in the right direction; they just lacked the confidence to keep going.

They sometimes give up when they get the wrong answer, assuming that they solved the problem incorrectly, when all they did was make a simple mistake; they just lacked the confidence that the solution was correct to check it carefully. When you doubt your solution, you wonder if it’s worthwhile to check it over.

A student who lacks confidence in math is more likely to skip homework problems that seem too hard or rely on too much help from others. A more confident student will at least attempt the problems and is more likely to participate in the solution when receiving help. This small measure of confidence makes a world of difference.

The top students tend to be much more confident. They carry out their solutions fully because they’re sure they’re solving the problems correctly. Instead of doubting their solutions, when they get the wrong answer, they check their solutions carefully. Confidence plays a significant role.

The confident student is much more likely to find a solution to a challenging problem, whereas a tentative students is much more likely to give up easily.

Overconfidence can also be a problem. The overconfident student may not practice or study as much as necessary, or might not put enough thought and care into a solution.

How can students develop confidence in math?

It’s not easy for everyone to show confidence; some students have more trouble with this than others. But everyone can become confident.

It’s not just a matter of saying you’re confident. There’s much more to it than this.

Suppose you don’t know a word of Russian and suddenly wander down the streets of Moscow telling yourself you’re confident you can easily pick up on the language. Not gonna happen. (Maybe someone will speak English, but you’re not going to instantly pick up Russian no matter how confident you are.)

Students can become confident through experience.

The student who hasn’t practiced or studied enough definitely lacks confidence.

The top students know they have practiced plenty and studied hard, so they show confidence.

Then there are good students who worked and studied hard, but don’t show confidence; instead, they show much anxiety and make nervous mistakes. These students didn’t first convince themselves that because they’ve worked and studied so hard, they do know how to solve the problems. They had every reason to be confident. Maybe they have done poorly on tests in the past, and this prevented them from showing the needed confidence.

There is a way for students to overcome math exam anxiety: One step at a time. Try to solve one small problem by yourself. Then work your way up to exam conditions. Start with self-check exercises, try practice quizzes, make a practice exam. Put yourself in positions where you experience exam anxiety, where it starts out easy and becomes progressively more challenging. It takes a will to find the way.

Teachers or parents can help students with this through practice problems, practice quizzes, and practice tests. The idea is not to give a preview of the actual problems, but to recreate similar exam conditions in order to help students overcome their exam anxiety. After every practice test or actual test, spend time with the student discussing the student’s approach.

Chris McMullen, author of the Improve Your Math Fluency Series

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Issues with Multiple Choice Tests in Algebra

Suppose that an algebra problem asks a student to solve an equation for an unknown. The goal of the problem is to test whether or not the student knows how to apply a particular algebraic technique.

However, if the question is multiple choice, the student doesn’t actually need to know how to solve the equation in order to determine the correct answer. The student could simply plug each answer into the equation to see which one works.

Example: 3x + 20 = 12x + 2.

(A) x = 1 (B) x = 2 (C) x = 3 (D) x = 4 (E) x = 5

A student could simply try each answer. Plugging in x = 1, it’s easy to see that 23 doesn’t equal 14. Trying x = 2, 26 = 26. The student already has the answer, but hasn’t done any algebra.

If the correct answer is (E), or if the student makes a mistake in the calculation, it will take longer, but in principle, a student can get the correct answer without doing any algebra.

If there are fractions in the answers, that makes the calculation a little more cumbersome. Many students prefer to avoid fractions, so this may deter students from avoiding the algebra.

Suppose that the choices had been:

(A) x = 1/2 (B) x = 18/15 (C) x = 22/15 (D) x = 2 (E) x = 22/9

Some problems can be solved faster by actually doing the algebra. But more time-consuming problems, like using the quadratic or solving a system of equations, might be solved faster by just plugging the answers into the original equations. When algebra yields the fastest solution, this provides an incentive to solve the problem as intended.

If a student relies on plug and chug for every problem, the student risks not finishing the test. But students probably won’t try to solve every problem this way. Just the hard ones.

Fortunately, there are many problems that tests can ask that can’t be solved this way. For example, many problems ask students to simplify an expression, and there are also conceptual, strategic, and logical questions. This helps to ensure that students must grasp some algebra concepts in order to pass the class.

The main goal of an algebra course is for students to become fluent in solving for unknowns in a variety of types of equations. Multiple choice is convenient, especially in large classes. It’s worth considering whether or not students might find ways to succeed in the course without actually mastering the techniques.

One way is for the instructor to become acquainted with the students, perhaps by checking written solutions of homework or quizzes (or adding a few written problems to exams, where feasible), helping students solve problems on the board, or interacting with students during office hours. If students who ordinarily struggle solving equations are better able to figure out the right answers on multiple choice problems, this might be a signal (of course, this could also be the result of studying, tutoring, and improvement). If instead experience with their written solutions corresponds well with their ability to solve similar problems on multiple choice exams, then there may not be any reason to worry that the students who most need to improve their fluency might be finding an easy way out (which may very well be the case for many of the students – it might be the clever problem-solvers who are most likely to think of this).

Chris McMullen, author of the Improve Your Math Fluency series of workbooks

The Perception of Mathematics

Attitude, outlook, and support are just as vital to successful mathematics teaching as the content of the instruction and the expertise of the teacher.

It is much more difficult to teach mathematics to a student who has a negative perception of the subject or the instructor, or who has a fear of mathematics or a bad experience learning it in the past.

Students, and even some parents, believe that an excellent teacher will, figuratively speaking, inject the students with mathematics knowledge and fluency with a hypodermic needle.

Wouldn’t that be awesome, if we could all learn mathematics (or anything else in life) without any effort?

But the real world doesn’t work that way. At least, not yet. (Perhaps a future generation of students will simply connect jump drives to their brains for access to instant knowledge. That may be exceedingly convenient, but perhaps considerably dangerous if they don’t also learn the wisdom to accompany instant knowledge.)

Some of the most important life skills that mathematics can teach, in general (i.e. not just to those students who will make direct use of the subject), are those skills that most students like to avoid and feel won’t ever be useful.

Mathematics can help anyone learn to think more abstractly, apply logical deduction, visualize three-dimensionally, and develop useful problem-solving skills. These are skills and techniques that can benefit people in a variety of non-mathematical disciplines, not just in mathematics. For example, visualization can be highly useful in art.

Yet many students are just focused on getting the answer and getting the course over with so that they can promptly forget everything they’ve learned.

Many students don’t like to sit through an abstract lecture in mathematics. If they would try their best to follow along, it would help them develop abstract and symbolic reasoning skills that could greatly benefit them later in life when critically thinking about totally different subjects.

Many students don’t embrace the challenge of solving new types of problems, especially word problems and proofs. They want to study a small number of examples and encounter only problems that closely resemble those.

But life doesn’t just throw a small number of simple problems at people. Life throws curveballs. The ability to apply concepts in different situations is a vital skill in any field. A strong repertoire of problem-solving skills is highly practical. The logical thinking involved in proofs helps to develop sound reasoning.

Another problem teachers encounter is when the struggling students refuse to rise to a challenge. For example, everyone becomes a better problem-solver when the teacher at first skips a couple of steps, asking students to try to fill in the steps on their own, and then fills in the steps a few moments later. This helps students learn to look a couple of steps ahead to see where the solution is going, instead of just being lost in the algebra.

But when the struggling students think to themselves that the teacher will simply go over the steps in a little while, so it’s not worth paying attention, they lose out on this opportunity to improve a valuable skill.

Most students who don’t go into a highly mathematical career will forget the quadratic equation, forget how to factor, forget how to integrate, forget properties of logarithms, etc. But if they can be motivated to learn how to think more abstractly than they would like, apply logic more often than they would like, learn to solve a variety of problems, and learn to apply concepts in different situations, they may very well benefit from these skills years down the road without ever realizing what a wonderful advantage they have indirectly derived from studying mathematics.

So much of this boils down to perception. Students do better in mathematics when they can be led to believe that the material is worth studying, when they feel that mathematics is fun, when they approach the subject with confidence, when they overcome their fears, when they have a positive outlook, and when other teachers and parents support the learning process.

This perception can be fostered through marketing. You can’t just tell people what to think. That’s not what marketing is. But marketing is a more indirect means of spreading a positive perception and outlook.

You can explain ways that mathematics can be useful to all students and hope that some of the students pay attention and believe this to some extent.

You can try to foster a positive outlook. You can try to motivate students and engage them in learning. You can make mathematics seem more fun and less daunting.

You can support other teachers and hope that they return the favor.

You can foster a positive image as a teacher.

You can show your passion for the subject and hope that students, parents, other teachers, and administrators sense this. Students often respond positively when they come to believe that a teacher truly cares about their success and really enjoys the subject.

Marketing a positive perception and learning ambiance is just as important as the content of the instruction and the expertise of the teacher.

Chris McMullen, author of the Improve Your Math Fluency series of workbooks

Does Practice Make Perfect in Math?

The saying is, “Practice makes perfect.” However, this is not necessarily true.

It’s more precise to say, “Practice makes permanent.” Only when the practice is correct does it make perfect. Practicing bad habits simply reinforces those bad habits.

This is true regarding sports, academics, and anything else. But let’s focus on math drills.

The spirit of math drills is to provide ample practice with the hope that the technique will become both permanent and correct.

This works when there is frequent feedback. Students need to have their answers checked frequently so that they can see if they are solving the problems correctly.

There is some controversy over traditional math drills. Some still embrace them, while others hate them.

The problem that teachers have with employing traditional math drills in school is that some students may become bored.

The weak students get frustrated by not being able to solve the problems, or not really wanting to learn how to solve them. Having to do things they really don’t like over and over doesn’t improve their learning.

The top students catch on quickly and get bored as soon as they’ve mastered the material.

But this doesn’t mean that math drills don’t work. There are many people who have become extremely fluent in mathematics through ample practice this way.

Teachers often strive to engage all of the students on the same activity. They want students with different learning styles, capacities, background, and interests to all get interested in the material and to all learn. To some extent they may be able to apply differentiated instruction.

Nonetheless, math fluency is severely lacking and needed with many students. Many parents turn to traditional math drills to help improve their kids’ fluency, remembering that these techniques worked well for them.

Some parents use such drills to help their gifted children advance ahead of the class. These students are bored in school, yet are engaged by the same drills that others call “boring.”

Other parents use these drills to help weaker students become more fluent through practice.

There are also still many teachers who find effective ways to use traditional drill sheets. Variety helps, both in the nature of the drill and by combining drills with other methods of instruction. Giving sheets of differing difficulty helps to improve the levels of all students.

Some home-school teachers also make extensive use of drill books.

Becoming too fluent in mathematics or learning too much math is never a problem, but lack of fluency can be a hurdle.

Chris McMullen, author of the Improve Your Math Fluency series of workbooks