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:

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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 solving problems. You will not have erase mistakes (unless you wish to), and nobody else will see this work but you. Plus, you will wish to copy your work over (see below).

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Do you ask questions in class at least once per week? For many students, the answer is probably “no”.  Reasons for such an answer may include one or more of:
– I don’t want to let my peers or the teacher know I don’t understand something
– I am uncertain about what to ask… I just don’t get what the teacher is talking about
– I don’t wish to appear to be the teacher’s “pet”
– I am not being called on when I raise my hand
– Someone else asked a question first, and the teacher needed to move on
– The teacher has not answered my past questions – they just said “see me after class”

Preparation

A number of small preparatory steps may help get your questions answered in class, particularly if your class is a large one.  The need for such steps will vary greatly from one school to another, or one teacher to another, but they will not hurt your efforts to master the subject even if they are not necessary to get your questions answered during class time:

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

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

Solving Math Problems At The Board

Many students perceive their teachers to know more than they really do (Math teachers in particular). If a Math student who just observed a teacher solve a problem at the board is asked “What do you think was going through the teacher’s mind as they solved that problem?”, I suspect the average answer would be something very specific, like “This is a second degree polynomial in standard form, so solving this problem will require precisely six steps, the first of which is…”

In reality, the jumble of thoughts in a math teacher’s mind probably go something more like: “What looks easiest to simplify first? Oh… I see two places, no make that three, that I could start… but which should I choose? Does this look like it will take a lot of room to solve? If so, I had better organize my work a bit more…  OK – now that I have simplified things a bit, what options do I see from here?”

Teachers and mathematicians do not “see” the entire series of steps needed to solve a problem before they start work on it. Instead, they usually seek to take whatever step looks like it will simplify the problem the most, and have no clue (yet) what will follow that. Once that first step has been completed, the appearance of the result influences what is tried as a second step. It is an iterative process, one step at a time, with a re-evaluation of the situation being done after each step. There is much uncertainty in the process, even for a math teacher.

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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 send periodic spasms of word problems your way.  The SAT makes liberal use of them as well.  Since you cannot avoid them, you had might as well learn how to solve them once and for all.

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 a pulse during class when they were explained, you can probably do them without hesitation.

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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 piece of paper, and put aside for a day or two before being solved again.  When you sit down to solve it again, pay careful attention to how you feeling while you are working, to the pace of your work, and to your train of thought.  Are you feeling frustrated or hesitating anywhere during the problem?  Are you uncertain about which step to take next at any point?  Do you have any doubt that your answer is correct before you compare it to the known correct answer?  If you answered “yes” to any of these questions, then copy the problem onto a fresh piece of paper again, and put it aside for a day or two…  then repeat.

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 sense of when it is time to move on to the next question without having to look at a clock.

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