Math: Pen vs Pencil

Many math students are given strict instructions by their teachers to do all their work in pencil. I disagree.

The advantage of doing work in pencil is that:

  • it is easier to erase, so students are less likely to be paralyzed by “I am not sure this is correct, so I don’t dare write it down”

The disadvantages of working in pencil are that: Continue reading Math: Pen vs Pencil

11 Ways To Do Better In Math

1) Reflect and Summarize
at the end of each class, at the end of each week, at the end of each month. Review your notes and/or think back over the material that has been covered, then decide which skills or ideas you think are most important. Summarize the material you are learning as concisely as you can, because summarizing helps you learn. Identify any skills or ideas that you are not confident about. Write your reflections and summarizations as part of your notes, with reminders about what needs more work.

2) Use scrap paper
Using scrap paper removes a source of anxiety when Continue reading 11 Ways To Do Better In Math

Studying to Understand vs Studying to Memorize

A number of historically “good” math students seem to reach a point during their High School years where their feeling of mastery starts to fade away. While teachers usually expect more from a student with each passing year, this alone does not explain the frustration these students experience. I believe it arises because a familiar study habit, memorization, is no longer enough to assure mastery.

My experience

I used to read a math or science textbook in pretty much the same way I read anything: as quickly as I could. In fact, for math I often skipped the reading entirely as I had been shown how to do the new types of problems in class, so all I had to do was sit down and follow the procedure I had been shown – no need for all the verbiage.

However, this approach stopped working when I got to college. If my notes from lecture did not help me figure out how to solve a problem, I had to rely on the text in the textbook for almost the first time. I learned that “believing I understood everything that happened in class” was a very different thing from being able to solve the problems assigned for homework.

After skimming through my math text, I often found that  Continue reading Studying to Understand vs Studying to Memorize

Ten Skills Every Student Should Learn

A recent eSchool News article by Meris Stansbury lists ten skills cited by its readers as being most important for today’s students to acquire:

  1. Read
  2. Type
  3. Write
  4. Communicate effectively, and with respect
  5. Question
  6. Be resourceful
  7. Be accountable
  8. Know how to learn
  9. Think critically
  10. Be happy

The list is interesting to ponder. I would not argue that any skills on the list should be dropped, however I suspect we could have endless debates about what order to list them in or how to best group them. I am happy to note that all of the skills are beneficial in studying just about any subject or discipline.

There are a few additional skills that I would advocate adding to, or being more explicit about in the above list:

Uncover the Hidden Game

The title of this posting is the title of a chapter in “Making Learning Whole”, by David Perkins (2009), which I mentioned in my previous posting.  I recommend it highly.

What is the “hidden game” in High School mathematics? What mindsets, approaches, techniques, etc. do those comfortable with the work asked of them rely upon, yet perhaps neglect to Continue reading Uncover the Hidden Game

Learn the Game of Learning

The title of this posting is the title of a chapter in “Making Learning Whole”, by David Perkins (2009). Of the books on education I have read to date, this is the first that resonated completely with me.  He describes the way I try to teach, and more – thus giving me much to reflect upon.  I recommend it highly.

The list of skills related to “the game of learning” I see as being most important for math and science students to acquire, and therefore worth devoting some time to teaching explicitly over the course of the school year (since they are also more generally applicable) are:

  • What is it you need to learn: a concept, a skill, or a fact? Concepts can often require thought, and time spent discussing them with others while being watchful for subtleties. Skills often require repetition and varying levels of difficulty. Facts can sometimes be obvious if they are based on an underlying concept; if the facts are not obvious, search for a way to link them to one or more concepts or themes, then practice retrieving them along with related information.
  • Frustration is a normal part of the learning process, one which can often lead to greater understanding and retention once you have worked your way through it. Expect to become Continue reading Learn the Game of Learning

Equivalence Deserves More Attention

Most students taking courses in Algebra or higher seem quite comfortable with the idea of “equivalent fractions”: improper or unsimplified fractions all of which evaluate to the same decimal value. An example would be

\dfrac{2}{3}=\dfrac{4}{6}=\dfrac{12}{18}=\dfrac{60}{90}=0.\overline{666}

To create such fractions, multiply whatever fraction you wish to start with by 1 (the multiplicative identity) in the form of a fraction whose numerator and denominator are the same:

\dfrac{2}{3}\cdot \dfrac{2}{2}=\dfrac{4}{6}=0.\overline{666}

\dfrac{2}{3}\cdot \dfrac{6}{6}=\dfrac{12}{18}=0.\overline{666}

The key concepts here are that
a) an infinite number of equivalent fractions can easily be created, and
b) while all these equivalent fractions sure look different, they all represent the same decimal value or simplified fraction.

Turning to algebra, the very similar concept of “equivalent equations” is helpful in Continue reading Equivalence Deserves More Attention

On the origin of algebra problems

As you are solving a math problem, have you ever wondered how textbook authors come up with algebra problems that have simple answers?

Just as you use inverse functions to solve an algebra problem, problem authors reverse the solution process to create a problem: they start with the answer.  From there they use the same principles that are used to solve an algebra problem, except instead of simplifying the equation at each step, they seek to make it more complicated by substituting a series of equivalent expressions and/or performing a series of operations to both sides of the equation.

Suppose you want the answer to be

x=3

Pick something to do to both sides, say multiply by 2:

2x=6

Do something else to both sides, say add 4:

2x+4=10

I could add 4x to both sides… why 4x?  Because I felt like it:

4x+2x+4=4x+10

I could rewrite the 4 on the left side as 2+2:

4x+2x+2+2=4x+10

Now to add another step to the solution process, I’ll factor part of the left side:

4x+2(x+1)+2=4x+10

And factor part of the right side too:

4x+2(x+1)+2=2(2x+5)

And voila!  An algebra problem which is guaranteed to have x = 3 as a solution.  Try creating several problems this way yourself.  Now that you know how to create your own algebra problems, and once you get some practice creating them, you may find it easier to solve algebra problems.

Word Problems… !#$%@;*!!

Word problems can be… frustrating.  Most of their reputation arises from their use of words to describe a quantitative problem.  And if the problem’s author did not choose their words very carefully, you’ve got Trouble (with a capital T). So why are so many word problems assigned? Because they are more similar to the quantitative problems you might encounter in life than many of the practice problems in your textbook: you have to supply some insight and organization in order to arrive at a solution.

Just about every level of mathematics, not to mention chemistry and physics, seems to Continue reading Word Problems… !#$%@;*!!

Problems fall into four categories

Math and science problems fall into four categories: Easy, Medium, Ugly, and Hard.

Easy Problems are ones you can solve with no difficulty in a short time.  An example from Algebra I might be:

3x+2=8

The problems that come at the beginning of each group of problems in a textbook are usually Easy Problems. If you had Continue reading Problems fall into four categories

Lost points on a problem? What to do…

People don’t like losing points for errors in their solution of a problem.  So, what can you do to insure you won’t lose points again the next time you are given a similar problem?  Most folks seem to look through the corrections, then perhaps ask the teacher to solve the problem for the entire class, and perhaps even make a note to study that type of problem before the next test or quiz.  Few do more than that… and the next time they are given a similar problem the probability they will lose points unfortunately remains greater than zero.

The solution I advocate to this dilemma involves a combination of repetition and self-awareness.

Every problem you lose points on, whether on homework, a quiz, or a test, should be copied neatly onto a new Continue reading Lost points on a problem? What to do…

Getting the most out of standardized test (SAT, ACT) practice books

1) For the SAT, I prefer the study guide and practice tests published by The College Board (it can be found at Amazon: The Official SAT Study Guide (2016 Edition)) as it contains good scoring guides and is published by the authors of the test. The College Board is now also providing on-line practice test resources, as well as an app for your phone which provides daily practice problems.

2) Most formulas needed to answer SAT questions will be given at the start of the test section.  Overall, it is more important to know how to use a formula than it is to memorize it.  In studying for the SAT, focus on understanding the mathematics instead of memorizing facts or formulas.

3) Always time yourself when taking a practice test.  This will help you develop a Continue reading Getting the most out of standardized test (SAT, ACT) practice books

Improve Your Math Grade: Use Scrap Paper

Never solve math homework problems on the piece of paper you intend to hand in (unless it is a problem that is very, very easy for you).

I used to do my work on the same piece of paper I intended to hand in, and when I encountered a problem I was uncertain about… I froze. I did not dare write anything incorrect on the page I was going to hand in. If I did, I might have to recopy all my work in order to end up with a neat-looking page to hand in. So, I would just skip the problem, telling myself I would ask the teacher about it in class next time. The result was that I greatly slowed my learning how to do that type of problem.

I now know that if I figure out how to do a problem all by myself, I will remember how to solve it for years. But, if I ask someone else how to do a problem I was struggling with, I will probably Continue reading Improve Your Math Grade: Use Scrap Paper