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Math OER
 Study Skills 

The hardest thing being with a mathematician is that they always have problems.

- Tendai Chitewere

Study Skills
Time Management
Before the Term
During the Term
Study Groups
Learning Styles
Virtuous Variety
Brain Awareness
Writing Math
Writing Step-By-Step Answers
Note Taking
Critical Thinking Skills
Activities to Respect
Onward and Upward
Calculator Use
Non-Fiction Reading

There are several kinds of math study skills.

Math classes have their own best ways to take notes, do homework, cooperate in groups, and prepare for tests.

Any learning can benefit from time management and brain awareness, and for math these are especially important.

Finally, you will do better with certain attitudes about success, thinking, and getting or receiving help.

If you have never been taught all these study skills then of course you felt overwhelmed by previous math classes. You were actually trying to learn two classes at once: the math topics and these study skills.

Once you learn these study skills, your classmates will look at you and say, "You are such a good student!" What they mean is, "I am still doing a double curriculum but you're not. You are passing the class while doing half the work."

Please read all of this page at the start of the term. This page is organized with the highest priority study skills first. The study skills will naturally arise as tangents from our math topics, which will have links to this page to revisit them.

Please keep in mind that a college math class is not a spectator sport! Our pace is faster than a high school math class. You are responsible for not falling behind, which for most students requires doing work outside of class for 3 to 6 hours each week, preferably some work each day.

For a slightly different perspective on study skills, please refer to the LCC Core Learning Outcomes and the Four Cs of Foothill College. Learning math includes improving in communicating about math, using math facts and formulas, and deciding when and how math aids understanding personal and community issues.

Time Management

Before the Term Begins

This time, like all times, is a very good one, if we but know what to do with it.
    - Ralph Waldo Emerson

A math class is like a construction project. Before the real work begins there is organizing and planning to do.

Students are already behind if they instead treat a math class like a drag race, with a rushed start when they arrive at the clasroom.

How can you get ready for a math class during before the term begins?


support network

Organize your support network. Consider what help you will need from other people. Thank your support network people in advance as a "heads up" that you might be seeking their help during the term. Do you a backup babsitter? Do you have someone who can drive you to class if your vehicle breaks down?

A famous saying by Jim Rohn claims, "People are the average of the five other people they spend the most time with." That may not be entirely true. But consider if you already have a close friend who can help with math, and provide support for your math learning outside of the time you have scheduled as math study time. If you do not already have that person, could you find someone for the duration of the academic term?

Math students have more success and fun when they work in groups. You can be proactive and arrange a study group even before the term begins. Get the instructor's e-mail. Because your instructor can only share student personal information with explicit permission, write a study group help request with details about when, where, and what contact information you are asking be shared. One example coud be "Someone wants a weekly study group that will meet on Thursdays or Thursdays (but not both) after dinner for an hour. Please let the instructor know." Another example is "Hi! I'm Valerie. I want someone to be sitting by the phone on Saturdays at 1:00 pm so we can call each other if we are stuck. Please call me at this phone number if you are interested." Your instructor can forward your request to the entire class.


Schedule when to do homework for 20 to 30 minutes as soon as possible after class. Then schedule when to do the rest of your homework. The homework page describes the need for both "prompt homework" and "enough homework".

The hours each week set aside for homework are as important a part of your "class schedule" as the classes themselves. Do you need to adjust the hours you work? Do you need to arrange for more babysitting? Handle those details.

A typical LCC class will have 2 hours of homework for every 1 hour in class. So every credit is expected to need 3 hours each week. This allows estimating your school and job workload:

The School and Job Work Load Formula

Total Hours = (credits of classes × 3) + hours of work

If your answer is more than 40 then beware! Contact you teacher before class begins to check in, and maybe ask for advice about resources and time management.

Most instructors are willing to let students do homework in their office during office hours. This can be helpful for quite a few reasons. If that sounds like a plan that could fit your schedule, ask your instructor about this before other students claim the one or two chairs in most instructors' offices.

Look over the student responsibilities. Which need some work? Which need the kind of work that someone else can help with?

Many students have trouble asking questions during class. If you are like that, find a personal reward or wager that will motivate you to ask some questions. Perhaps you will buy yourself a certain treat on the weekend if you asked at least one question during every class that week. Perhaps you will pick a friendly classmate and agree that each week whomever asks more questions during class will buy the other a coffee. Plan your bribe and put it on your calendar!


If your life is busy or full of drama, set your mind to never give up. Students fail low-level math classes because they do not study enough—usually because they cannot make time to study enough in one term.

Past terms have shown that roughly one-fifth of Math 20 students, and one-tenth of Math 25 students, need to use time from two terms before the pass the class. For various personal reasons one term simply does not have enough hours. I have seen that students that take two terms usually invest comparable time overall to their classmates who pass in one term, but spread out the time over twenty weeks instead of ten. They are only less efficient in terms of financially paying for more credits.

When this does happen it is regrettable. The student must pay for more credits, and might delay graduation. It might also mess up financial aid plans. But as a "Plan B" it does indeed work for a lot of students. There is nothing shameful about a busy student needing two terms to finish a math class.

Start the term aiming for a good letter grade! Start the term aiming to finish in one term! But also remember that if real life gets demanding you can keep doing the best you can, change to pass/fail, and finish up in a future term.

Test Preparation

Review the foundational math from the previous class. Re-read the textbook chapters or your notes, but cover the steps of the example problems with a piece of paper so you check if you can do the example problems yourself.

From that previous class, look at chapter tests or cumulative tests. Try to get an old final exam (or practice final) and take it. Take an old test in an environment as much like your upcoming classroom as possible.

Make note of which problems you miss. Either study those topics on your own, get help at the Math Resource Center, or at the start of the new term be ready to ask your new instructor to do those review questions.

During the Term

Nothing is a waste of time if you use the experience wisely.
    - Rodin

The key is in not spending time, but in investing it.
    - Stephen R. Covey


Care for your support network. This helps you be prepared for a struggle, even if you cannot predict it. Even small problems can become huge crises for people without a support network.

Only a few decades ago nearly everyone had a huge support network. People had family nearby, all the neighbors as close friends, a religious congregation to call on for help, and a circle of friends you saw often because of a hobby.

The common trait shared by most students who fail a low-level math class is the lack of a support network. So even though school and family and work keeps you busy, make an effort to spend time with friends and be active in a social hobby or religious community. It might sound like an odd thing for a math instructor to say, but over the years I have known plenty of students who might have passed the class if they spent less time on homework and more time knitting, volunteering, or getting to know other single parents at the library/parks/gyms.

Remember the importance of studying in a group. Explaining a math topic to someone else helps you really learn it! Having a math topic explained to you by someone other than the instructor also helps.


Budget time for problems. Some people plan their week so full that they would always be busy even if no problems arise. When problems do appear, those people have no time to deal with them. So keep some cushion in your calendar. Maybe you will be fortunate and have some free time because no problems happened!

During the rush of school and work, adopt a long-term view. Most problems do not look so large when seen from a long-term view.


Spot the Main Character by Jaiden Animations

At the start of any class it can feel like the instructor is the main character of the story who makes the rules and dances in the spotlight at the front of the room, and you are only a side character. That is naturally. But it should change. Sometime during the term the story becomes your story. You are the main character. The story is about your progress and learning. The instructor is a side character, merely another resource you use. When that happens you are halfway to passing the class.

Respect your determination. Your success is from your effort, not some "one thing" that will make life easier. Over the years I have known many students who fixate too much on a new math tutor they just found, a new job with better hours that they are about to start, a tax rebate they are about to receive, etc. It is nice when our lives improve. But new things usually come with new stresses, and you still have to do all the homework and all the studying. Do not needlessly ride an emotional roller coaster.

Respect your investment of hard work in your scratch paper step-by-step answers from homework and tests. Fix the problems you missed so you have correct answers usable for future studying. These fixed problems are usually the most helpful to try explaining to someone else. Be the teacher in your study group, or to a family member or stuffed animal at home.

Press deeper for the why. The more you understand why the math topics work the less you have to memorize how they work. Understanding is better than routines. Simple routines are better than complex routines.

Test Preparation

Make your notes as helpful as possible. You want a "best parts of what I learned" organized in one place so that you only need to go to that one place before and during a test. Include completed practice tests. Make sure every topic has explanation, example problems, and comments about struggles (if any steps are tricky, if any homework problem stumped you) to help you remember where to do extra studying.

The day before the test write a cheat-sheet. Even if the test does not allow you to use a paper or index card of notes, the process of preparing the cheat-sheet is a great way to study. You will need to decide which math topics are the most important, most difficult, or most helped by seeing an example problem. You will memorize parts of the cheat-sheet, which will allow you to work faster during the test.

During the week before the test, pace yourself as you study. By the time you get to the day before the test you should be doing less studying than earlier in the week and more wellness care.

The day before the test get enough healthy food, exercise and sleep. Be good to your brain, so it can be good to you.

Do the quick problems first. If a problem turns out to be tricky or long, leave space on your scratch paper and keep going. Keep track of the problems you skip: either use marks by the problems or be careful with an answer key.

To get the most partial credit on problems, learn to "fake it" when you cannot remember a fact (for our class this is usually a measurement unit conversion). For example, perhaps you know how to do unit conversion problems but forgot how many feet are in one mile. Write a quick excuse and do the best you can. A student who writes, "I can't remember, so pretend 6,000 feet = 1 mile" and continues will get a more partial credit than a student who stops halfway through that problem.

Study Groups

It is all right to hold a conversation but you should let go of it now and then.
    - Richard Armour

No relationship was found between gains in content knowledge and study group use...We conclude that students require guidance in the successful use of study groups. Instructors can help students maximize study group success by making students aware of potential group composition problems, helping students choose group members who are compatible, and providing students materials on which to focus their study efforts.
    - Stephen M. Rybczynski and Elisabeth E. Schussler

Our math division at LaneCC has two study rooms (one upstairs, and one downstairs) with a big table and a nice chalkboard that students can reserve by going to the math office.

Although there is no "right way" to do homework in groups, there is a most efficient way. If you were to watch math graduate students working in a group, their routine is not very social, but it helps learning happen quickly and deeply. Most most work done by each student solo (but with company) intermingled with short discussions about the trickiest (and most interesting) parts of the homework.

The graduate students begin by doing homework problems at a table. They do not talk. They work independently until someone gets stuck.

homework in group picture one

When a student gets stuck, he or she moves from the table to the chalkboard and writes the steps to the difficult problem that he or she has done so far. The other students pay attention. If that problem is new to them they begin to work on it at the table. The student at the board waits for a response.

homework in group picture two

Soon a second student will get up and go to the board. This new student will help by giving a hint about the next step, or by pointing out where a mistake was made. The first student may not agree. The two students may debate whether a suggestions is useful. Students still at the table also offer comments.

homework in group picture three

Eventually the problem is solved. The students copy the correct answer to that difficult problem from the chalkboard. Someone erases the chalkboard. All the students go back to working by themselves at the table.

homework in group picture one

As I mentioned before, this routine is not very social. The students avoid talking off-topic until all the homework is completed. (As math graduate students they have lots of homework to do.) When they do discuss a problem, it is either a brief suggestion that offers enough help, or a discussion that sounds almost like an argument. Most problems are done independently.

So you may think this routine sounds like it is not much fun. In part, that is true! The group would have more fun if it was more social. But those math graduate students find learning math to be fun, even if socializing is postponed until later.

Learning Styles

Virtuous Variety

If the human brain were so simple that we could understand it, we would be so simple that we couldn't.
    - Emerson M. Pugh

A nice Scientific American article describes what scientific studies have shown are important learning methods.

Students are more successful when they space out their study sessions over time, experience the material in multiple modalities, test themselves on the material as part of their study practices, and elaborate on material to make meaningful connections rather than engaging in activities that involve simple repetition of information (e.g., making flashcards or recopying notes).

The Yale Poorvu Center has its own advice.

Students will develop their own preferences for reviewing content, but these practices differ from deeper cognitive processes like "chunking," building on prior knowledge, making conceptual connections, and transferring knowledge.

Instructors can incorporate active learning, group work, and inclusive teaching strategies to invite students to engage their full faculties and experience peer learning. Multiple modalities can assist all students regardless of proposed learning style.

Learning outcomes improve when instructors help students think about how they drew connections, digested content, or arrived at conclusions.

In other words, a student who is focused and thinking is learning—but staying focused and thinking is hard! A variety of learning activities helps everyone stay on task.

Learning involves experience, practice, discussion, organizing the material both alone and with groups, making connections, attempting application, and then reflecting about the entire process. That is not a to-do list for students. It is an outline for worthwhile lesson design.

You might have heard about various "learning styles". Those are mostly a myth, as those two articles explain. But we should talk about them and their limited usefulness.

Reflect About Repetition

receptive styles

Many college students are familiar with the traditional trio of receptive styles: visual, auditory, and kinethetic.

These are receptive styles. They provide variety and help maintain interest.

Some skills do require a lot of repetition to learn. The three receptive styles might indicate what type of repetition most helps you most: flashcards, recitation, or practice.

But the topic and situation might matter more, even for repetitive memorization. For example, in our math class I have known some students whose biggest help with learning to do food preparation problems came from explaining the process to a classmate, yet these same students needed to repeatedly do practice tests to master the problems about saving for retirement.

Reflect About Homework

Skip Downing, in his book On Course, proposes a set of four "learning styles". These focus on what to do outside of the class when what you learned in class only introduced a topic and you need to learn more. What questions do you ask yourself? What types of researching, memorization, and discussion do you enjoy most?

Pages 213 and 214 of his book describe these four learning styles in more detail. You can read a PDF copy of those two pages.

As before, Downing's tips are worthwhile if they help keep you focused and thinking when doing homework. They work if you use them to think more about your own learning. They are not intended to limit you or box you in to a single category.

Brain Awareness

Where there is personality, there is discord.
    - Terry Pratchett

Internal Perception

Our body sends us subtle signals to recommend certain actions. Noticing these signals tells us when we are hungry or thirsty, hot or cold, nauseous or creeped out. Awareness of those examples and our body's many other metabolic and "gut feelings" is called interoception.

Our body also has receptors in our muscles to provide body awareness. If we lift our arm above our head we can sense this without checking in a mirror. If we slowly cross our hands or our feet we can tell when they are above each other. This awareness is called proprioception.

Most people ignore some of their body's interoceptive and proprioceptive signals. We often need to focus on these sensations to decide if our hunger means "It would feel nice to snack" or "I should eat now or I'll get hangry". We often need to focus to feel our feet in our shoes, or how the weight of our head is pressing down on our neck.

(We can watch our kids learn interoception and proprioception. People with sensory processing issues develop less of a constant background amount, and must focus to sense what others naturally notice.)

Anxiety disrupts and confuses our interoception and proprioception. An anxious student often "hears" false cues about hunger and breathing, and interprets signals meant to inspire activity and engagement (increased heart rate, tensed shoulder muscles, etc.) to instead signify danger. During a test, watch your classmates who normally sit quiety start touching their faces, drumming their fingers on their desk, and in other ways subconsciously try to increase their interoceptive and proprioceptive self-communication.

Anxiety thus robs us of agency. We become disconnected from our bodies, and unable to connect to our feelings. Our internal cues to the mildness or severity of a frustration go missing, so any frustration seems intolerable. Our gut feelings of safety or danger become confusingly entwined. Bodies that are too often exposed to this state recognize its unhelpfulness and replace it with a numb fog that is less disruptive but not any more helpful. Lacking words for our feelings is called alexithymia.

Alexithymia leads to personal baggage. This happens even when we pass through situations successfully. Accomplishments feel hollow when made while disconnected from our bodies and agency—especially if in that wordless numb fog. We know we survived but cannot own the victories.

Almost everyone feels some amount of math anxiety. What can we do about it?

Some Bad Science

In the 1960s, Michael Gazzaniga and Roger Sperry studied patients in whom surgeons had cut certain connections between the two halves of the brain to try to reduce epileptic seizures. In those patients, the halves would develop specializations in different tasks. This led to an incorrect popular belief that the left and right halves of brain worked quite independently.

Modern fMRI research shows the whole brain's tremendous networking complexity. For normal brains, in which surgeons have not cut any connections, the two halves work closely together.

Some Good Science

Even though the brain works as a whole to do its things, some types of things do go together. Certain types of brain activity happen quite naturally together.

We know that our muscles have some patterns that fit together easily and others that resist each other. It is much easier to circle both arms the same direction than in opposite directions. You can bend forward more deeply after massaging the soles of your feet, but less deeply after sitting on the front edge of a chair.

Our brains work the same way. Some patterns of thinking fit together easily and other patterns of thinking do not.

Most famously, there are maybe grooves of thinking that fit together, and there are now grooves of thinking that fit together. Mixing can work but does not flow as naturally.

The Maybe GrooveThe Now Groove
uses large-motor muscles consciouslyuses metabolic and proprioceptive muscles unconsciously
ponders what might beattentive to what is now
abstract thoughts about relational informationpictoral and emotional thoughts about sensory information
focuses on complex possibilities—scenarios, comparisons, "if..." fictionsfocuses simply on actual, individual stimuli
judgmentally categorizes things as good/bad or wanted/unwanted curiously observes things as they are
loves to create rules and restrictionsloves to openly accept and experience
selfishly prioritizes grasping pleasure and avoiding fearcompassionately prioritizes a mellow and joyful "energy in the room"
analytic external point of view—Cartesian dissection and Systems Theory networkssensitive internal point of view—how do my components feel, and how do I feel as a component?
rapid pace, stimulatedmellow pace, relaxed

When our brain is in the Maybe Groove we are not relaxed. We worry, plan, and make up stories—usually pondering the past or future instead of dwelling in the present moment. We think abstractly. We try discern differences, see patterns, make judgments, analyze relationships, and create rules.

When our brain is in the Now Groove we are mellow and relaxed. We sense the present moment without adding commentary. If the present moment is happy, we are happy. We try to feel togetherness. We acknowledge community without judging or evaluating relationships.

Those two brain grooves are important for a math class!

Lego Movie not listening meme

Most of what we do in a math class is in the Maybe Groove. We study patterns, look at differences, and judge whether a process or problem is correct or wrong. As we spend time in the Maybe Groove, our thoughts naturally become more busy and worried. We get less relaxed. Our thoughts get stuck in comparisons, judgments, and nit-picking. Distracted by our own minds, we have trouble paying attention to what is happening around us.

We can be aware of this and purposefully add some Now Groove style thinking. We can be happier math students if we keep those two styles of thinking balanced.

combo move But it takes a little effort. Just as we must concentrate a bit to circle our arms in opposite directions, we must be aware and intentional to put some extra Now Groove into a very Maybe Groove mental situation. It is a bit like doing a combo move in a video game. At first the extra effort takes practice. But it can become a habit that you naturally do when it is needed.

We can do this by focusing on metabolic and proprioceptive muscle use. Pay attention to something metabolic: breathing, heartbeat, diaphragm, stomach, etc. Also pay attention to something proprioceptive: feel where our arms and legs are, how our scalp feels, how you press against your chair. Sit or stand still to stop large-motor muscle use.

We can do this by focusing on immediate sensory information. What colors can we see that we were too busy to notice a moment ago? What sounds and smells do we notice now that we are paying attention? It helps to acknowledge this changing stream of sensory input wordlessly.

If we can keep that focus for a few minutes then more of our thoughts will flow in the Now Groove.

This should sound familiar. Many, many religions and philosophies have developed their own ways of doing this, and expanding on this. These traditions are called praying, meditating, communing, mindfulness, self-soothing, and many other names.

But for us the importance is just to recognize that our thoughts fit more or less easily into two grooves, just like our muscle movements can fit more or less easily together.

Writing Math

Writing Step-By-Step Answers

It's not that I'm so smart, it's just that I stay with problems longer.
    - Albert Einstein

What qualities help make writing step-by-step answers helpful to the student and instructor?

To add clarity we'll consider a practical example: a PDF file of handwritten step-by-step answers for an old Math 20 midterm.

(If PDF files do not work for you, you may also click on the images below to see larger versions. You may need to use your web browser's zoom, often by holding down the Ctrl key and tapping the + key).

sample page one   sample page one   sample page one   sample page two

1. Paper is ruled with lines tall enough for factions. Work is written using pencil in a reasonable size.

Some paper has such closely spaced lines that each fraction must be written spread vertically on two lines. This needlessly invites confusion about which lines start a new step.

Writing too big wastes paper. Writing too tiny annoys the person grading your work.

Use a pencil so that you can erase to aid formatting. For example, if you write fraction addition too compactly you would want to erase a bit so you can create room for the little numbers that show how you are multiplying each fraction. (Only erase for reaons of arranging your work on the page. Do not erase actual math errors—cross those out neatly instead. It is important to be aware of which types of problems are difficult and which types of careless mistakes happen most often. Do not erase to hide this information!)

In the sample Midterm step-by-step answers, I used fractions in five of the problems. In all of these problems the fractions are written legibly and take up about as much room as would a whole number.

In the sample Midterm step-by-step answers, I switch to writing every other line when the equations are full of fractions. Problem #10 would be less legible if written on only two lines of paper.

2. Fractions are written vertically, not diagonally.

Unfortunately, when writing text on a website it often works best to write fractions diaglonally like ab. But this makes fraction canceling and arithmetic less visually intuitive. Since you are writing on paper instead of typing, all of your fractions should be nicely vertical.

In the sample Midterm step-by-step answers, I always write fractions vertically.

3. Dots for multiplication and decimal points are not confusing.

Most students in a "learning math skills" class use · instead of × to write multiplication. This is a good habit to prepare for later algebra classes.

However, using · in a class that uses decimals frequently can be tricky. Make sure decimals are at the bottom of the line and multiplication dots are higher. Use horizontal spacing of work to increase clarity when decimals are multiplied. In some problems it may even be necessary to take the time to draw multiplication dots larger than decimal dots.

4. Steps are clear and small enough to answer "What was I thinking here?"

The purpose of step-by-step work is to create clarity about where errors happen. Hopefully you will notice the errors as you do the problem, and fix them. If not then whomever is grading your work still gets the benefit of knowing exactly where you were careless or confused.

5. Proper math grammar is used when moving from step to step, and when canceling.

When solving problems you have two big tools. You can do the same thing to the top and bottom of a fraction (or ratio). You can do the same thing to both sides of an equation.

There are correct and incorrect ways to write reducing or un-reducing a fraction. See "Writing Un-reducing" and "Writing Reducing" in the fractions lecture notes.

It is a good habit to use distinct cross-out lines each time fraction canceling happens in the problem. If all the cross-out lines are single diagonal lines going the same direction then when reviewing the problem it is very slow to double-check how you did more than one bit of canceling. Vary how you write cross-out lines: single versus doube, diagonal right versus diagonal left, straight versus wavy.

In the sample Midterm step-by-step answers, I had two bits of canceling in one step within problem #5. I wrote my cross-out lines diagonally but going opposite directions.

6. Rates that should have labels do have labels. Proportions have equal signs.

These are two aspects of proper math grammar that are significant enough in Math 20 and Math 25 to deserve extra attention.

Labeling rates allows double-checking that canceling or cross-multiplication is legal. Math 20 includes many "trick questions" in which the initially provided measurement units do not match properly, and must be adjusted before a proportion can be solved.

Including an equal sign in the middle of a proportion is important, especially if more math happens to either side.

7. There are enough equal signs for flow. Equal signs really mean everything they connect is equal.

A single line can include more than one equation if all items are equal. For example, there is nothing wrong with writing

y = 7 + 8 + 9 = 15 + 9 = 24.

However, it is wrong to chain together equal signs when all items are not equal. This error most commonly happens when a new step is begun not with a new line but with an addition operation. For example, the previous addition might be sloppily and incorrectly written as

y = 7 + 8 = 15 + 9 = 24.

My colleague Eric Hogle called this the "read it over the phone test". If you pretend to read your work to someone over the phone you can more easily notice when you are trying to start a new step in the middle of an equation.

8. Problems are clearly numbered, spaced apart sufficiently, and easy to find.

We do want to avoid wasting paper. But if we cannot clearly find and read each problem the entire assignment is somewhat of a waste.

In the sample Midterm step-by-step answers, I could have started writing problems #2 or #14 one line higher. The work would have fit. But it would be slightly less legible and would not have saved any paper.

9. Enough of the original problem is included to allow double-checking for copying errors.

For many students the most common careless error is copying the original problem incorrectly. This can happen in your head, so do not jump write to solving the problem without writing enough of the original problem. A key example of this criterion is writing a percent sentence (or extracting which values are "part" and "whole") before changing it into an equation.

In the sample Midterm step-by-step answers, I made sure to write the original, unestimated amount for problem #1. I also wrote the task for problem #2 to aid later double-checking that I rounded correctly.

10. Appropriate parts of the problem line up vertically. Equations are solved with the Vertical Format.

Double-checking work (or grading it for partial credit) is much more slow and succeptible to errors if steps do not line up appropriately.

In the sample Midterm step-by-step answers, I lined up my equal symbols and letters for problems #7, #8, #15, and #16. Also, problem #17 involved an expression with four terms and I kept each term in its own horizontal column while simplifying.

11. Work written horizontally to emphasize terms and to leave room for fraction reducing/unreducing.

Leave more room around + and − symbols than × and ÷ symbols. This helps your eye and brain naturally see terms, and prioritize multiplication and division.

When adding or subtracting fractions remember to leave room before the + or − symbol and after the second fraction for those little numbers that show how you are multiplying each fraction.

In the sample Midterm step-by-step answers, I put a lot of space after the fractions in problems #9 and #10 before writing an equal symbol. I know I will be reducing them, and leave room for writing how I divide the numerator and denominator.

12. "Off on a tangent" work is to one side, set apart with a brace or cloud, and points to where it fits in.

Fraction arithmetic and long division are the most common "off on a tangent" work. When solving an equation, do not try to squeeze fraction arithmetic or long division into the Vertical Format of solving the equation. Do that work off to the side where it can be written large enough and neatly enough to make double-checking easy.

In the sample Midterm step-by-step answers, I had "off on a tangent" work in a cloud as part of problems #7, #8, #12, and #15. In each case I drew an arrow to show where that work related to the main problem.

13. Abbreviations for unit labels are proper. (meters = m, miles = mi, minutes = min, etc.)

This is polite. If you use non-standard abbreviations it makes extra mental work for the person reading your paper, whether a study partner trying to work with you or an instructor grading a test.

14. The answer is easy to find in a box (or otherwise made distinct).

Whomever is grading your work for partial credit will appreciate this small politeness.

In the sample Midterm step-by-step answers, I put a box around my answers.

15. Answers that should have labels are correctly labeled.

Do not be forgetful or sloppy if the problem is a word problem or another type whose answer needs a label. Be professional.

16. Fraction answers are reduced.

I am not sure who invented this rule for students learning math skills. We might not like it. But reducing fraction answers is required.

To conclude, here are two examples of student homework with nicely written step-by-step answers.

sample page one   sample page two

Note Taking

He listens well who takes notes.
    - Dante

The best way to waste your life, is by taking notes. The easiest way to avoid living is to just watch.
    - Chuck Palahniuk

Four Sources to Merge

After a math lecture you have four sources of information about the material:

Most students find it helpful, as soon as possible after each class, to combine these sources to make their notes "completed". Add to what you wrote during class your thoughts you did not have time to write. Then augment your notes with helpful things you notice when looking at the textbook pages and the instructor's public lecture notes.

Making notes "completed" is a great aid to learning the math topics. It is especially important to try this early in the term. Early practice will help you become quicker at "completing" your notes, saving you time later in the term.

Not only does making notes "completed" help you review and better learn the math topics, but when it is time to study for a test all four types of information are now optimally accessible in one place.

Someone who reads "completed" notes should be easily able to see how you processed the notes after class ended. Make your after-class additions distinct!

Four Formatting Characteristics

Equally important for creating readable and helpful notes is the formatting. Notice how the lecture notes on this website are examples of good note taking formatting.

Your in-class notes should also have these four characteristics. (Then your "completed" notes automatically will.) It makes a big difference when studying!

How can we include those four characteristics when using pencil and paper?

Try Columns

Many students like using two or three columns. Let's look at examples of each. You are not required to use columns as your method of including those four characteristics of note formatting. The following examples are only suggestions.

To use two columns, simply reserve the right-hand margin for "completing" the notes you write in class. This means you need to find some way to keep narrative and examples from blurring. You also need to label when a topic changes, and make definitions stand out.

Here is an example.

two column example

Some students like using three columns. This method puts narrative in one column and examples in the second column. This helps a student realize when a topic is missing an example. The third column is for "completing" the notes. Topic changes and definitions still need to be clearly marked.

Here is an example.

three column example

To summarize, you have two tasks when creating quality notes. First, be sure to "complete" your notes by spending time with them after class, adding whatever is needed to make them a maximally useful aid for studying in that math class (and perhaps future math classes too). Second, in class write nicely formatted notes using any system that includes those four formatting characteristics.

Here are three great examples of student notes. As before, click on the small version to go to the full size version.

sample page one   sample page two   sample page three


All I need is my brains, my eyes and my personality, for better or for worse.
    - William Albert Allard

What is success in our math class?

As mentioned above, success is not about passing the class in ten weeks if your real-life circumstance require you to take twenty weeks. Success is instead about three things.

First, success is about growing your critical thinking skills. Everything we do is about critical thinking: understanding a lecture, asking a question, participating in a discussion, studying at home, analyzing a math book, and organizing your notes and brain before a test.

Second, success is learning to stand on your own two feet in a math class. Successful students motivate themselves to do necessary work, manage their time sufficiently, have experienced the benefits of group work, know where and how to ask for help outside of class, accept responsibility and accountability for their choices, and leave the class with solid study habits.

Third, success is seeing yourself as belonging. Successful students are no longer afraid of the math topics, have accumulated enough learning to view themselves as effective students and learners, have experienced enough small victories to understand how learning is fun, and no longer doubt whether they belong at LCC and in a math class.

Let's look more carefully at these three elements of success.

Critical Thinking Skills

Critical thinking is thinking about your thinking while you're thinking in order to make your thinking better.
    - Richard W. Paul

College culture values Critical Thinking Skills. What are these?

Students who arrive from college directly from high school can be surprised by this change. High school culture mostly cares about learning objective, concrete, memorizable facts and procedures. But even when college assignments require a lot of that type of learning, college culture do not value it as much as critical thinking.

This change can be scary! Most students who graduate high school are experts at extracting memorizable facts and procedures from a teacher's lectures and demonstrations, and from textbooks. But when students arrive at college they are surprised to discover that their expertise is valued little, and is only a minor part of what will be tested. Eek!

The good news is that your college instructors will not immediately expect you to be good at those skills. Instructors of lower-level college classes realize that high schools often do not teach critical thinking skills. These instructors know they need to teach these as part of the class topics.

Initially, your job is be patient with teaching that is not about memorizable facts and procedures.

Some discussions, activities, and assignments will seem to be pointless or all about opinions. These are almost certainly trying to teach you critical thinking skills. You might not realize how or why. But that is almost always what is happening.

However, nobody learns from pointless work or opinions. When tasks seem to be like that, ask which skills the task is trying to teach you. That can change the task from seeming pointless or opinion-ish into something you can learn from.

The great news is that critical thinking skills are a lot more interesting than memorizable facts and procedures. You will soon learn to recognize when and where these skills are being used. Then discussions and essays will become more fun than similar high school tasks.


You have good reason to believe that you can trust yourself. Not because you've always made the right choices, but because you survived the bad ones.
    - Sandra King

A single word—a key philosophical term—has for centuries encompassed both "learning to stand on your own two feet" and "seeing yourself as belonging". That word is dignity.

A person with dignity stands on his or her own two feet. No tyrant or issue is conquering them completely to make them bend their knee. Adults can "stand with dignity" in a way children cannot, because adults are responsible and accountable in a way children are not.

Yet people with dignity are standing as part of something bigger than themselves. Dignity requires solidarity. Dignity requires a virtuous community. And although our culture's stories might emphasize when dignified people have stood together in defiance or protest, dignity does apply equally well when untroubled people stand in celebration of the values and virtues that unite them.

(However, when Robin Hood swings in on a rope, and stands with a jaunty knee, hands on hips, and devilish smirk, we all agree that pose is too egocentric to be called dignified despite his community or its virtues. Dignity is about belonging, not standing out.)

Is there some obvious philosophical link between these two aspects of dignity? Does responsible and capable accountability somehow have an inherent connection with belonging to a virtuous community? So far a link eludes me. But dignity means both, and always has.

To show dignity someone must show courage. This is easy in our math class. Our class has no trigger warnings. Can you imagine if the instructor said, the first day of class, "I am about to talk about fractions. Anyone who has had bad experiences with fractions, or suffered emotional harm because of fractions, might want to leave the classroom." Everyone would leave, including the instructor!

kicking the snot out of inner demons Of course we have emotional baggage. Math victories happen as we recognize just which inner demons haunt our math ability, acknowledge how they do that, and then kick the snot out of them.

At the end of each term, ask yourself how your LCC experiences affected your dignity.

Hopefully your dignity was helped by your learning experiences, by instructors and staff behaved, by how you treated yourself, by how college systems and procedures treated you, and by your participation in college community.

Be bold. Beware of tricking yourself in a common but harmful way: many choices that appear to minimize short-term discomfort often actually cause long-term discomfort. Turn in all assignments, even if some will earn a bad grade. Ask questions in class, even if they reveal your confusions. Visit office hours, even if that reveals your weaknesses to the instructor. And so on. All of that stuff helps pass the class and bring long-term life success.

Never quit because of anxiety. In fact, ignore anxiety as much as possible because a wise person makes use of good advice and anxiety always gives terrible advice.

Activities to Respect

Great things are done by a series of small things brought together.
    - Vincent Van Gogh

We can wrap up our thoughts on success, critical thinking, and dignity by pondering what math class activities earn our respect.

silly Dora the Explorer saying

• Right answers that show me what I know
• Wrong answers that show me what I know
• Questions and guesses that no matter what the answer show me what I know

• Asking questions to resolve confusion
• Asking questions to calm uncertainty
• Asking questions to explore what happens

• When a math topic is scary
• When a math topic is no longer scary
• When it doesn't matter if a math topic is scary, I am doing it anyways

• Math is a bunch of problems that need answers (I start by hunting for the answer)
• Math is a bunch of problems that need new tools (I start by hunting for the right tool)
• Math is a bunch of tools to answer other problems

• When I can first see something in class, and then learn by myself
• When I can first see something in class, and then learn in a group
• When I can first see something by myself, and then learn by myself
• When I can first see something by myself, and then learn in a group

• Trying to fix an ignorance arising from never before having been taught that topic at all
• Trying to fix an ignorance arising from never before having been taught that topic skillfully
• Trying to fix an ignorance no matter what its source
• Establishing accountability for learning and growth



Those who have not learned to do for themselves and have to depend solely on others never obtain any more rights or privileges in the end than they had in the beginning.
    - Carter Godwin Woodson

I know what privilege sounds like.

Privilege can look like many things. But in my experience it almost always sounds like a certain phrase thought or said aloud: "But they always have a choice!"

Many very privileged people think everyone always has a choice. Can't you just choose to shrug off the negative messages? Can't you just choose to spend time with a healthy group of friends? Can't you just study more? Can't you just choose to avoid alcohol?

Many very privileged people understand that some people do not always have a choice. (The negative messages will eventually penetrate. There is no functional crowd to hang out with. Worrying about how to afford food tomorrow totally destroys the ability to do homework. Drinking is what men in that place must do.) Yet they fail to see the deeper truth.

The deeper truth is twofold. No one always has a choice. Moreover, for people who are privileged, the lack of choice is often a lack of the option to fail.

Someone was there (of course!) to encourage you so the negative messages did not sink in. Someone was there (of course!) to steer you away from the unhealthy friends to the more functional crowd. Someone was there (of course!) to make you do your homework whether you wanted to or not. Someone was there (of course!) to keep your drinking from getting out of hand, or if it was already out of hand to make you deal with the addiction properly.


What a child can do in cooperation today, he can do alone tomorrow.
    - Lev Vygotsky

Privilege is a great thing! It is nice when someone "has got your back".

Privilege is not wimpy or cheating. A soldier who goes into battle alone is especially foolish, not brave. A student who avoids office hours and tutors is fleeing help, not admirably independent.

LaneCC Student Resources

Students should have privilege. This is why privilege is given out freely at LaneCC like candy on Halloween.

You can visit an Early Outreach Specialist to quickly learn about college resources. You will be amazed how many resources there are, and how many people are ready to say "I got your back".

Your instructors, tutors, academic advisors, counselors, TRiO staff, and many other people all want to be those caring allies who will encourage, advise, and nag you, so you cannot choose to fail.

The image to the left leads to my own compilation of LaneCC student resources. But I am always a bit out-of-date.

A famous psychologist named Vygotsky used the word scaffolding to describe the support people need to achieve their next milestone or plateau. It is an old term very rooted in educational theory.

Good teaching is half scaffolding and half morale management. Keep an eye out for both in this website and during class.

Onward and Upward

You are defeated not when you fall down, but when you fail to rise up.
    - Anonymous

flawed illustration of equality versus equity

A famous illustration was comissioned in 2016 by The Interaction Institute for Social Change to illustrate the problem with providing people with equal support.

Unfortunately, this illustration has such big problems I am drawing a big red X over it.

Here is my improved version, thanks to the help of my old-school Lego friends.

Lego equity 1   Lego equity 2   Lego equity 3

Much better!

• The three Lego people are the same. Math students, and people in most other situations, are much more similar than they realize. The images should emphasize our similarities, not our differences.

• The starting support structures are different. Math students arrive at the first day of class with very different personal histories and skills. We each have our own helpful scaffolding.

• The starting support structures are all incomplete. The brown support has solid blocks but they are barely assembled in a stable way. The red support hass small bricks, and is quite unstable. The yellow support is just three unconnected things that look ready to roll over.

• The people work together to improve their support structures. These study skill tips have already emphasized class participation and homework groups. Even the college Early Outreach Specialists cannot build your support structure for you. The work takes many hands, including your own.

• More blocks appear as the support is improved. Our original support structure might need some patching up or reorganizing. But no person has to give up pieces of his or her original support. We can share ideas and habits without diminishing ourselves.

• The end result is greatly improved but still incomplete. All three people are higher than before. But two people end up balancing on a thin beam made of two red bricks. Two people still have holes in their structures. We never finish learning. There are always habits, skills, and people that could help us more.

Life is not about some outside influence handing you a pre-built support structure. The closest anyone gets is an upbringing full of hard work where people who love them try to deny them the option to fail. (And we all know stories of people whose rebellion or terrible choices undermimed their well-meaning governance.)

Life is not about climbing as high as someone else. They surely would say that what they are standing on is also in many ways incomplete and unstable. They are not so different from you. They also need community more than competition.

Remember that success is about critical thinking. The Lego people have success because they had learned where, how, and with whom to build better support whenever their current structures were insufficient.

Remember the aspects of dignity. The Lego people got better structures to stand on, developed stability standing on their own two feet, and had belonging in a cooperative community.

Calculator Use


Computer science is no more about computers than astronomy is about telescopes.
    - Edsger Dijkstra

They are useless. They can only give you answers.
    - Pablo Picasso, about the calculating machines of his day

demotivational poster

Enough inspiring life advice! This is a study skills webpage, not a motivational poster.

Let's talk about using a calculator as part of a math class.

Using a calculator when learning foundational skills can hinder developing a conceptual understanding of math topics. The concepts of ratio, proportion, percent, and unit conversion are closely linked to foundational concepts about fractions and decimals. Those concepts should be taught with friendly numbers and no calculators. Students should be learning that in those types of problems the numbers are as much structures as values.

However, after a foundational skill is learned, it should be practiced with realistic and messy numbers. A calculator is needed.

Just remember to keep thinking of the numbers as part of a unified concept dealing with structure and meaning. Do not diminish your math ability into a collection of procedures.


I really hate this damn machine,
I wish that they would sell it.
It never does quite what I want
But only what I tell it.

    - Poem about computers on my grandfather's refrigerator

Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.
    - George Orwell

When Numbers are Too Big

What can we do if a problem involves numbers too big to fit into the calculator?

Most calculators cannot handle a fourteen-digit number. How can we answer the following question?

In September 2019, the U. S. national debt is about $22,711,880,000,000. The U. S. population at that time is about 329,754,250 people. How much was each person's share of the national debt?

We can round both numbers to the same place value. For this problem, let's use millions.

In September 2019, the U. S. national debt is about $22,711,880 million. The U. S. population at that time is about 330 million people. How much was each person's share of the national debt?

Now the word "million" can be treated as a label that does not actually go into the calculator.

$22,711,880 ÷ 330 ≈ $69,824 per person

Previous Answer Key

The previous answer key is very useful. Please take a moment to make sure you know how to use the version on your calculator.

In September 2019, the national debt per person in Japan was about $99,800. Express this as a percentage of the U. S. national debt per person.

We want to do $99,800 ÷ $68,824 without having to retype the $68,824.

Try it with your previous answer key.

$99,800 ÷ previous answer key ≈ 1.45 = 145%

(Older calculators only have M+ and M- keys. These can be used to mimic the previous answer key in a slower manner. But most Math 20 students find it worth the small expense to get a calculator with a previous answer key.)

Exponent Key

The exponent key is used in two of our formulas about saving for retirement. Please take a moment to make sure you know how to use the version on your calculator. Try 23 = 8.

Notice that on some calculators this key looks like yx and on others it looks like the caret symbol ^. But the display shows the caret symbol for both kinds of calculators!

(There is also a calculator key labeled EXP. This is for Scientific Notation. Do not confuse it with exponents!)

Pi Key

The pi key is used in Math 20 for geometry with circles in Chapter 9. Unlike the other keys, it does not hook two numbers together. Instead, it stands for one number.

Using the pi key simply saves time typing π = 3.14159265358979...

Please take a moment to make sure you know how to use the version on your calculator. Try 1 × π = 3.14159265358979...

Percent Key

The percent key is never used. Ignore it!

Recall that percent sentences appear three different patterns:

The percent key works differently in each case! It is much simpler and safer to always use decimals with the calculator. After all, it is so easy to change between percent format and decimal format by scotting the decimal point twice with RIP LOP.

Square Root Key

The square root key is wonderful. Math students today do not need to learn the tediously lengthy ways that people once used to estimate square roots to a decent number of decimal places.

On some calculators this key automatically does "equals" and shows the answer. On other calculators you still need to press "equals" or "enter" to see the answer.

You might never use this key. But you should read about the simplest of many square root algorithms just to appreciate what people needed to do before calculators were invented!

Non-Fiction Reading

Young people should not donate money to charity, but should buy non-fiction books instead.
    - a businessman's purposeful exaggeration

(I cannot find the article I once read that had the above quotation. If you find it please let me know!)

That quotation is wrong. Giving is good for us. Many studies have proved both short- and long-term benefits from having generosity in personality and habit.

But the article from which I summarized the quotation did make a valid point.

Individuals in the U. S. donate more than $286 billion to charity each year. The average household in the U. S. that gives money to charity donates about $2,500 per year.

Bequests from people's wills are not included above. These add another $35 billion each year.

(Then U. S. foundations and businesses donate another $87 billion each year. And all these numbers are increasing.)

So as a student whose career has not yet matured, donating $20 to charity can be great for your personality and habits. But that amount is almost always insignificant for the charity.

The point of that lost article was that for that same $20 you could buy a non-fiction book that could help your career potential. Reading the right books now could make a huge impact on your finances in the future. Then you can be generous in amounts that noticeably change the world.

So do both. Respect your generosity and your potential.

By the way, the strong urge to donate to charity is very much a part of U. S. culture. Canadian individuals on average only donate half as much. Continental Europeans average one-seventh as much. It is hard to find more data for other countries.