Quantitative Strategies

In the past, when some people get a problem wrong, they might have thought that they just don’t have the ability to study math--that they’re not math people. But when you talk to professional mathematicians, the people who are best at math, it turns out that they work a long time on the same problem--and they only spend their time on problems that they struggle with the most. And even though you might think they make up answers on their own, almost always mathematicians have to ask people for help.

—The Carnegie Foundation for The Advancement of Teaching

Learning Objectives

By the end of this section, you will be able to:

• Describe how personal attitudes toward quantitative courses can impact success
• Compare effective note-taking strategies for quantitative courses against those for other courses
• Identify strategies for reading quantitative texts
• Compare pre- and post-test-taking strategies for quantitative courses against those for other courses

Math Attitudes

Productive Persistence

Many students work hard in math classes—studying long hours, nights and weekends—yet many of them do so using ineffective strategies. Others simply withdraw effort soon after the course begins, or they make mistakes. To help you successfully complete your academic goals, we want you to both persist in your studying and attendance (tenacity) and to do so efficiently and effectively (good strategies). This is called productive persistence. The next few pages of this course will help you develop a plan for how you can implement the idea of productive persistence as an effective way of passing a math course.

Mindset

One aspect of productive persistence is something called "mindset." Work through the following activity to learn more.

Getting the Most From Your Math Notes

In gradeschool, you may have had a teacher who praised students for having neat, tidy papers. Learning can be messy, and if we restrict ourselves to neat, tidy papers, that is all we will have. Sometimes we need to try a homework problem over and over before we understand how to find a solution. In our hectic lives, it is important for us to allow ourselves time to reflect on the messy parts so we can tidy them up in our minds.

Preparation

Prepare for your classes as you would practice for an upcoming athletic event. Know the topics you are going to cover, and make a goal of being current with your assignments. Learning is not passive, so if you want to learn from your class time, you must prepare yourself to learn.

Consider these two scenarios:

Scenario 1

Greta is busy.  She has a job and works at night after spending hours at school every day.  The last thing she wants to do is prepare for class when she gets home from work at night. Despite her exhaustion, she takes 15-20 minutes before bed to check the syllabus from her math class to see what topic will be covered in lecture the next day.  She then finds the text material related to the lecture topic and quickly skims it, reading over the headings.

Scenario 2

Greta is busy. She has a job and works at night after spending hours at school every day.  The last thing she wants to do is prepare for class when she gets home from work at night. She decides to watch an episode of her favorite TV show before she goes to sleep at night.

How much value do you think Greta is gaining from her math lectures in each of these scenarios? Do you think 15–20 minutes really makes a difference?  If you haven't before, try spending 15–20 minutes skimming the material related to your lecture before you go—even if you just read the headings in your text. Maybe you take public transportation to campus. If so, you could skim your text or lecture notes on the bus. If you drive, maybe you can get to campus a few minutes early to do the same thing.

Preparing for class doesn't necessarily mean having read all the material related to that day's lecture. Some people gain more from reading after lecture. In general, most people do retain more from lectures if they focus their mind on the topic of the class before entering the class. Ask yourself these questions before your lecture: what did we talk about last time? What are we going to talk about this time? Am I current with my assignments?

Math concepts build on each other. Because of this, keeping up with homework and assignments will help you be prepared for the next session. If you are behind, you will lose a valuable opportunity to make important connections between last week's content and this week's.

Taking Notes

It is impossible to write down everything your teacher says during a lecture.  As you become a more skilled learner, you will learn how to glean what is important from a lecture. Here are some things that can help you organize your notes and your understanding of the content in them.

Consider these things before you start writing:

• Where are you going to keep your notes? 3 ring binder, folder, spiral notebook? Taking notes with a computer in your math class may prove difficult when you need to draw a graph or geometric object.
• How are you going to organize your notes on the page?
• What purpose do your notes serve?
• How do you plan to use your notes?

Before and During Class

As with any kind of class, you may need some time to figure out how to best organize the information you want to record. There are many popular styles of note-taking and if you have one you prefer, there is no reason to change.  If you want to explore more ideas that have worked well for other students in math courses, consider these:

• Split your page into two columns, one for descriptions/ definitions and one for examples
• Cornell notes—good old "C Notes"—can be helpful, but require some deeper thinking than you may be able to do on the fly during a lecture. Consider saving C notes for use while reading your textbook, instead.
• If you don't understand something, write it down anyway and mark clearly that you have a question about it. If you don't have time in class to ask about it, get help with it later. Writing down what you don't understand may be the most important part of taking notes!

Homework Journal

There's a good chance you are going to be assigned online homework in at least one of your math or science classes while you are in college. Often, students fall into the habit of working through online homework without keeping track of their work.  If there's one thing you take from this page, it should be this: Keep an online homework journal!

Here are some important things to include in your homework journal:

• The title of the homework set—even if it is just the number, such as 2.3. This will help you use your work as a reference for study.
• Write down each problem you do—if there are easy ones, just write the answers to them if you don't think you will forget the steps.
• If you make a mistake—just circle it and note that it was wrong. If it takes a ton of tries, that's okay—it's your homework journal, and you're not turning it in for a grade.

Allowing yourself to make mistakes in your homework journal gives you freedom to learn and not be worrying about a perfect paper.

Summarizing/ Reviewing

Summarizing and reviewing what you have done will help to solidify the ideas that are now swirling around in your head. You go to lecture, take a bunch of notes, do a ton of homework problems, and then what?  Your brain needs time to make connections between practice, what you have in your notes, and what you may have read in the text.

After Class / Homework

What's the point of taking notes or doing homework if you never look at them again?

• The sooner you can get your questions answered the better you will understand the answers. Remember those questions you circled in your lecture notes? Make sure you resolve them as quickly as you can.
• As you do your homework, keep your notes open. Ask yourself, "is this question like an example from class?"
• After you do your homework, try to place the practice problems into the corresponding readings in your text—what are the key terms or definitions related to those problems?

Recall from the Active Learning section that effective reading requires more engagement than just reading the words on the page. Reading a quantitative math text effectively uses the same skills as reading any academic text effectively. It's still a good idea to do things like circle key words, write notes, and reflect. You can still employ the same steps that were presented previously:

• Preview: You can gain insight from an academic text before you even begin the reading assignment. In the section about preparing for lecture, you were encouraged to preview the material associated with the day's lesson. In this way, previewing serves you in two ways.
• Read: While you read a math text, you should have a pen or pencil in hand. Circle or highlight key concepts, definitions, or examples. Write questions or comments in the margins or in a notebook.
• Summarize: After you read a math text, it's worth taking the time to write a short summary—even if your instructor doesn't require it. The exercise of jotting down a few definitions or examples can help to solidify new ideas and help you when you do homework or study for a test.
• Review: It always helps to revisit what you've read for a quick refresher. It may not be practical to thoroughly reread assignments from start to finish, but before class discussions or tests, it's a good idea to skim through them to identify the main points, reread any notes at the ends of chapters, and review any summaries you've written.

Get to Know the Conventions

Math texts may be organized in a way that is new to you. They are full of symbols and notation, and not as much text as other subjects. A few important features make up a math text. These include

• definitions
• examples
• descriptions of notation
• text
• graphs
• tables

You may be tempted to skip over examples or boxes with definitions in them when you are reading a math text and just get to the "regular" text part. BEWARE! Most of the important information in a math text is in the definitions, examples, and notation. Notation is very important to most college math instructors, so take the time to pay attention to how mathematical ideas and processes are written.

Look up and Keep Track of Unfamiliar Notation and Definitions

If you don't understand a definition or how it is applied, make note of your confusion.  You can circle an example or the definition, or write it in your notes. Ask for help to clarify your confusion. Try rewriting mathematical expressions or equations as words if you are confused by them. Remember that being confused is probably the most important part of learning: it means that you know where to focus your learning strategies!

Look for Main Ideas and Themes

Rather than presenting ideas with a thesis statement, then supporting them with examples and discussion, a math text will present a mathematical definition or classification, and support it with examples. As a college student, you are not expected to understand every single word or idea presented in a reading, especially if you haven't discussed it in class yet. However, you will get more out of class or homework practice if you can identify the main concepts in a reading.

Pay Attention to Visual Information

Math texts present a numerous graphs, tables, charts, and images. These items contain valuable information to help you more deeply grasp a topic. Graphs can show a visual representation of a mathematical rule or equation. Tables can help you see trends or describe relationships.

Data-rich graphics can take longer to "read" than the text around them because they present a lot of information in a condensed form.  Give yourself plenty of time to study these items, as they often provide new and lasting insights that are easy to recall later (like in the middle of an exam on that topic!).

Math Exams

Recall the four sensory modalities in Fleming's Learning Styles model:

1. Visual learning
2. Auditory learning
4. Kinesthetic learning

You can use the understanding you have of your own learning style to your advantage as you prepare for a math exam.

Visual/Spatial

Visual learners make sense of the world through pictures and images. They learn well when lesson plans incorporate photos, videos, or visual maps. A visual learner may describe their understanding as images or pictures rather than with words, and are drawn to design and spatially focused.

If you learn with pictures and images, you may benefit from creating drawings related to each concept. Diagrams and visual cues can help visual learners make important connections between concepts. Mind maps may also be a helpful tool for further solidifying your understanding of how mathematical concepts are connected. Additionally, if you are visual, you could color code your notes and study materials.

Auditory Learners

Aural learners are best equipped to understand and store information absorbed via sound and music. Their ears are particularly adept at deconstructing and parsing heavy mixes of tones. They will often do better with books on tape versus printed versions.

Many aural learners enjoy listening to music. Playing pleasant music in the background while learning mathematics can evoke positive emotions and stir up a bit of energy while you are working. Just make sure that the music is not overly distracting or played too loud. In addition, musical minded people can try to organize formulas and operations into musical patterns or rhymes. Coming up with a rhyme or melody to remember the quadratic equation may be more effective than simply attempting to remember the visual image of the formula. It may also help to find someone who will listen to you explain what you are learning or studying.

Read/ Write learners may prefer learning with words -  both with speech and writing. If you are a read/ write learner you may soak up knowledge through various mediums centered around language. If you prefer learning this way, try sitting down after class to rewrite your notes. Translating a mathematical expression, graph, or equation into words may also help. If you learn from reading and writing, keeping a homework journal may be very beneficial to you.

Physical/Kinesthetic

Physical learners learn best by touch and movement. A lot of superb athletes tend to fall into this category as physical processes and activities seem to sync well with their learning and memorizing capabilities.

For physical learners, studying may seem like torture, but if you are committed to it, there are ways. If you tend to learn best when active, it is important for you to stay in motion while studying. This could involve squeezing a stress ball while working, or simply taking a break every 20 or 30 minutes to walk around the room. Hands on models are terrific as well when applicable. If there is a tangible learning device that you can actually touch and interact with, all the better.

Group Study

Some learners prefer to learn in groups surrounded by other people. If you thrive in social environments, you may benefit from working in teams to complete homework assignments and prep for exams. This kind of activity can can often charge your energy levels and can create a supportive network of caring classmates. It is imperative that at least one person in the group be willing and able to keep the group on track and focused; otherwise your study session will turn into social hour.

If you plan to study in a group, try preparing an agenda before you get together. Your agenda could include a summary of the topics you will review, and a few problems for each topic that you want to try to work out together. Beware of the "attention seeker" when you are part of a study group. Try to share the work of solving problems or answering each other's questions. All of you will get much more our of the experience.

Positive Mental Habits for Testing

• Define and repeat a positive mantra like “I have all the tools I need to do well on this test,” or “I am capable of answering all the questions on this test.”
• Avoid over-thinking—your gut reaction is usually right. Math teachers grade innumerable tests where students erased the right answer.
• No amount of positive self-talk can make up for not being prepared. Give yourself lots of time and practice so you don’t feel stressed.
• Stop studying at least 30 minutes before your test, and let your brain rest.
• Before your exam try to laugh or get some light exercise; read a funny book or comic.
• If others are commiserating before the exam about how hard it will be, don’t join in. Remove yourself from self-doubt and negative talk.
• Before the exam—visualize yourself taking the test and knowing how to answer the questions and feeling good about your performance. Repeat this activity as much as possible. It works!
• Make mini exams for yourself. Take homework problems that were not assigned and give yourself an allotted amount of time to do them. Getting good at something happens from practicing the mechanics of the motions frequently.

Math Test-Taking Strategies

Just as it is important to think about how you spend your study time (in addition to actually doing the studying), it is important to think about what strategies you will use when you take a test (in addition to actually doing the problems on the test). Good test-taking strategy can make a big difference in your grade.

• First, look over the entire test to identify problems you definitely know how to do right away, and those you expect to have to think about.
• Do the problems in the order that suits you. Start with the problems that you know for sure you can do. This builds confidence and means you don't miss any sure points just because you run out of time. Then try the problems you think you can figure out, with more time. Finally, try the ones you are least sure about.
• Time is of the essence—work as quickly and continuously as you can while still writing legibly and showing all your work. If you get stuck on a problem, move on to another one. You can come back later, and something you do on another problem might help inspire you when you return to one you skipped.
• Work by the clock. On a 50 minute, 100 point test, you have about 5 minutes for a 10 point question. Starting with the easy questions will probably put you ahead of the clock. When you work on a harder problem, spend the allotted time (e.g., 5 minutes) on that question, and if you have not almost finished it, go on to another problem. Do not spend 20 minutes on a problem which will yield few or no points when there are other problems still to try.
• Show all your work: make it as easy as possible for the Instructor to see how much you do know. Try to write a well-reasoned solution. If your answer is incorrect, the Instructor will assign partial credit based on the work you show.
• Never waste time erasing! Just draw a line through the work you want ignored and move on. Not only does erasing waste precious time, but you may discover later that you erased something useful (and maybe worth partial credit if you cannot complete the problem).
• For a multiple-step problem with limited space on the page—you can put your answer on another sheet to avoid needing to erase. Outline your answer and indicate where the solution can be found.
• Don't give up on a several-part problem just because you can't do the first part. Attempt the other part(s). If the actual solution depends on the first part, at least explain how you would move through each step. Read the questions carefully, and do all parts of each problem.
• If you finish early, check every problem—that means rework everything from scratch. Does each answer make sense given the context of the problem?

Your instructor passed back your first math exam and you are devastated.  You thought you did so well preparing, you kept up with assignments, and came to every class. Why did you get a C−? You may feel like you have failed, but this is an opportunity take charge of your education! You have been given several opportunities by getting a C− on your exam.

Don't throw away any tests, even if you are upset about your grade. It is very important to take an inventory of the errors you made on each question.   Use the following inventory to assess your mistakes.

The Six Types of Test-Taking Errors

1. Carelessness. You lost focus on the question and made a silly error (like changing a sign or inventing a new rule of algebra).
2. Directions. You skipped over or misunderstood directions and as a result you did the problem incorrectly.
3. Concept. You did not understand the properties or principles required to work the problem.
4. Application. You understood the concepts involved, but did not apply them correctly in the context of the specific problem presented.
5. Nerves. You made errors in judgment due to the pressure of the test-taking environment. These include not completing the problem to the last step, changing a correct answer to an incorrect answer, getting stuck on one problem and spending too much time on it, rushing through the easiest parts of the test and making careless mistakes, leaving answers blank (no partial credit), or leaving early and not checking all of your answers.
6. Preparation. You studied the wrong material, or did not spend enough time studying the relevant topics.

Identifying what kinds of error you made will help you focus your strategy for study and preparation for the next exam.  You can develop a plan for how to prevent similar types of errors, once you know where to concentrate your efforts.