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

Mathematical English

You don’t have to love both math and English to appreciate mathematical English. It may appeal to people who are passionate about just one or the other. Mathematical English can also be used to help stimulate student interest in a subject that they don’t generally favor. Since word problems require using both sides of the brain at the same time, doing some basic mathematical English may be helpful practice for this, too.

Following are a few entertaining activities where mathematics and language intertwine:

(1) Victor Borge’s inflationary language: The concept here is to add one to any number used in ordinary speech or writing. For example, “forever” would become “fivever” and “tension” would become “elevension.” You can find video footage of this on YouTube, for example, or more information about Victor Borge at http://en.wikipedia.org/wiki/Victor_Borge.

(2) Mathematical word games. Anagrams involve permuting letters to form new words, as in “listen” and “silent.” In cryptograms, each letter is replaced by a different letter to make a secret code. Scrabble assigns a numerical value to every letter and players try to form words from the letters to make the highest possible score. Such games combine mathematical thinking or logic with language skills.

(3) Poetic verse that involves numbers. William Shakespeare used iambic pentameter when writing plays or sonnets, which follow a mathematical pattern (rhythm). Japanese Haiku is a short form of poetry following the mathematical pattern 5-7-5.

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

The Happiness of Chemistry

The noble gases in the rightmost group (column) of the periodic table – helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), and radon (Rn) – are all “happy” because they have filled outer shells. These elements are happy the way they are, and therefore seldom form bonds with other elements. Ah, but it’s a lonely existence for them.

The metals to the left of the zigzag staircase on the periodic table have one or more valence electrons, which makes them “unhappy.” They are eager to lose those electrons by forming ionic bonds with nonmetals. This way, their outer shells will be full, and they will be happy like the noble gases.

The nonmetals to the right of the zigzag staircase – except for the noble gases, of course – are equally unhappy. They wish to gain one or more electrons to fill their outer shells. Yes, all of the other elements are unhappy because they aren’t like the noble gases. So they bond in any way possible to try to fill their outer shells so that they, too, can feel like a noble gas.

Nonmetals can bond together with metals to form ionic bonds, or bond together with other nonmetals to form covalent bonds. (Sometimes, one or more atom groups are involved in the bond, in which case the bond consists of three or more atoms.)

When metal atoms and nonmetal atoms interact, the metal atoms donate their valence electrons to nonmetal atoms so that all of the atoms involved in the bonds have filled outer shells. The metal atoms become positively charged ions by losing electrons, while the nonmetal atoms become negatively charged ions by gaining electrons. These oppositely charged ions then attract one another.

For example, each sodium (Na) atom loses one electron and each chlorine (Cl) atom gains an electron when sodium and chlorine react together to form sodium chloride (NaCl). The positive Na⁺ ion attracts to the negative Cl^– atom in this ionic bond.

When two nonmetal atoms interact, they share valence electrons so that they both have the feeling of a filled outer shell. For example, carbon (C) bonds with four hydrogen (H) atoms in methane (CH₄). Carbon has 4 valence electrons, but feels the presence of 8 by sharing one electron with each hydrogen atom. Hydrogen has 1 valence electron, each but feels the presence of 2 by sharing one electron with carbon. Hydrogen needs 2 electrons to fill its outer shell, while carbon needs 8. This way, both carbon and hydrogen are happy through the covalent bond.

If only happiness were as simple to achieve in life as it is in chemistry. :-)

Chris McMullen, author of Understand Basic Chemistry Concepts and the Improve Your Math Fluency series of workbooks

Is Everything Really Just Nothing?

You might be inclined to wonder this with a little abuse of the transitive property.

The transitive property of mathematics states that if A = B and C = B, then it follows that A = C.

Let’s try to apply this to words and see where it leads.

Everything is something. Agree with that? (Really, it’s a whole lot of something’s.)

Let’s take everything to be A and something to be B. Then saying, “Everything is something,” is like saying A = B.

Nothing is something. Agree with this? (It’s just something that isn’t.)

Let’s call nothing C so that saying, “Nothing is something,” is the same as B = C.

If A = B and C = B, then C = A; that’s the transitive property.

Plug in the words: If everything = something and nothing = something, then everything = nothing.

Everything is nothing! What?!

Can you find the fallacy here?

Spoiler alert: The solution will be forthcoming. If you’re not ready to read the answer, don’t look below.

The problem isn’t with saying, “Everything is something,” or, “Nothing is something.”

The problem is that everything and nothing are two different something’s, not the same something.

In algebra, “Everything is something,” is like A = X and, “Nothing is something,” is like C = Y. The first something, X, isn’t the same as the second something, Y.

(Thinking of these “things” as numbers, you might want to write something like A = infinity and C = 0; both infinity and zero are something’s – as I said, that wasn’t the issue. The issue is more like the fact that infinity and zero are two different things.)

Really, in words, we should say, “Everything is something, but nothing is something else.”

Ain’t dat somepin’ else?

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