# 46 Quantitative Strategies

April Ring

# Introduction

## What you’ll learn to do: explore factors that can impact academic performance in quantitative courses

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

By the end of this section, you will be able to compare effective note-taking strategies for quantitative courses against those for other courses. You will be able to identify strategies for reading quantitative texts. You will also learn how to identify math study strategies for before, during, and after taking quantitative classes.

# Math Study Skills

### Learning Outcomes

- Identify math study strategies for before, during, and after taking quantitative classes

Simply put, math is a cumulative subject. You cannot learn the next level or topic in math without first building a strong understanding of the preceding topic. Think back to when you first learned addition. Would that have been possible if you didn’t know how to count? What if you didn’t know your numbers up to 100? In order to add numbers together, you had to have a solid understanding of counting.

And then when you first learned multiplication, it probably didn’t quite click until you were taught, or realized, that it’s really just repeated addition—you needed to learn how to add before you could learn how to multiply.

For example, if you haven’t memorized [latex]5(12)[/latex], you can first multiply [latex]5(10)[/latex] (easy!) and then add two more [latex]5[/latex]s. Both of these examples serve to demonstrate why having good math study skills is so important, and how growing your knowledge in mathematics takes dedicated time and effort.

Though math study skills are helpful when learning any topic and can be applied in any field, they are especially useful where objectivity, logical reasoning, and methodological approaches to problem solving are crucial.

Math study skills can be broken down into three components: what you do before class, what you do during class, and what you do after class. These three components are then broken down into further steps:

- Prerequisite understanding
- Preview the upcoming topics
- Actively participate
- Ask questions
- Annotate your notes
- Confidence in the new material
- Chase a deeper understanding
- Carry out practice, practice, practice
- Connect ideas

## Before Class

### Prerequisite understanding

The cumulative nature of math means that you need to revisit previous concepts often enough that you don’t forget them. Taking College Algebra, passing, and then immediately trying to push aside all the math equations you had memorized will do you no good in Precalculus when you have to build upon that previously learned material. The same thing applies throughout the duration of a course. Completing the first assessment does not mean you can promptly forget what you were just tested on. Learning everything from Module 1 and then moving on and forgetting that content will be a huge disservice to yourself when you get to Module 5 and you have to use those concepts again.

This is why prerequisite understanding is so important. Before every lecture or new section, review any previous content that will be key to mastering the new material. Your knowledge should be like a rubber band ball and keep layering on top of itself. Spend some time prior to each class reviewing material from the class before that will serve as the foundation for the upcoming material and older concepts that you may be rusty in using, but will need to be used to master the upcoming material (TIP: ask your professor for a list of these prerequisite skills if you are unsure).

### Preview the upcoming topics

There are two friends who have never been snowboarding, heard of snowboarding, or seen anyone snowboard before. Their roommate Ashlee is taking them snowboarding for the first time tomorrow and she hopes it will go well. One of the friends, Janaan, decides to watch some YouTube videos of people snowboarding, and goes to look at the gear that is packed in the truck. The other friend, Kyle, decides that he’s just going to wing it the next day on the slopes. Who do you think is better equipped to learn this new skill of snowboarding?

In math, the same concept applies. By looking ahead at the content that you will be taught or are about to learn, you give yourself a tiny head start and a framework for where to start storing newly learned information. This preview doesn’t mean that you need to complete all the homework before it’s even assigned or study the section in great detail (though it can’t hurt!). It simply means that you familiarize yourself with the vocabulary, new equations/theorems, and the type of questions that will be asked within the next lesson.

The first step in previewing the material is to simply learn the key terms. It’s hard to learn a new concept in any subject if it’s being taught to you in a foreign language. If you don’t have the basic vocabulary memorized for the course, you will struggle to follow your instructor’s lesson or the textbook prompts. At the bare minimum, learn the terms and basic theorems and rules that will be covered in class. One step further to be really prepared to get the most out of class is to do the following: skim the chapter text once through, note anything that seems especially important or difficult, and develop a rough understanding of what you should learn in more detail the next day.

## During Class

### Actively participate

Learning is not sitting and listening to someone explain something to you. In fact, sitting and listening is one of the most ineffective ways to retain information. You need to actively engage with the speaker by taking notes, responding to questions even if you’re not sure of the answer, and staying focused on the task at hand. If you are sitting through a lecture, make sure you are taking the time to reflect on what’s being said. If you are tasked with group work, make sure you are completing the activity in a way that you are both helping and learning from others. When you actively participate in class, it automatically leads to behaviors that help you become a better student.

### Ask questions

This step at first seems to be an obvious and simplistic piece of advice. But we are not referring to things you can look up on your own like definitions of words, or when the midterm is, we are referring to conceptual questions. The art of asking a good conceptual question is a very important math study skill. A good question is not, “I don’t get it, why is that the answer?” A good question is, “I see how we combined like-terms to get to the third step, but why did the negative sign in front of the [latex]3[/latex] change to a positive after we distributed and multiplied?” The act of identifying your uncertainty, thinking of what specifically you are struggling to grasp, formulating how to ask that question, and receiving the answer as feedback to process is a powerful tool.

Another aspect of cumulative course information in math is that waiting to ask for help and falling behind can make or break you in that class. It’s possible in some other subjects to miss a chunk of content and then come back strong and redeem yourself in that course. In math, that is almost impossible because the next concept depends on your understanding of the first ones.

Becoming active in your learning journey means developing the confidence to ask for help. Speak up when you don’t understand something; you are the only one who knows you are confused until you say something. Even if you’re not comfortable asking a question right then, at least think of questions and write them down so that you can get those questions answered soon after.

### Annotate your notes

While you are taking notes during class, it is important to take detailed notes. These notes should include more than just what your professor is writing down; they should include bits of your inner dialogue as you were sitting through class. If something seemed really crucial, put a star next to it. If one of the worked examples was unclear, put a question mark next to it and a brief note of why you were confused. If the professor reworded their explanation of a concept and you had an ah-ha moment, write those new phrases down. Without these additional entries, looking at your notes will be just like looking at a textbook. Your notes should be in your own words rather than an identical copy of what your professor chose to write on the board. If you write your notes in a way that makes sense to you, you won’t have to spend additional time deciphering them and your annotations will act as a guide once you leave class.

## After Class

### Confidence in the new material

Now that you have attended class and actively participated, asked questions, and taken annotated notes, what do you do with all this new information? For your brain to hold onto this new information tightly enough to apply later on, now is a good time to review all the notes you took, follow through with getting your questions answered, and get your murky spots solidified. You can try to answer your own questions, but you should reach out for help if you can’t figure it out yourself. Professors often hold office hours for just these types of questions, or you could organize a study session with some of your peers. Either option will keep you from falling behind. Your goal is to be super confident in everything you’ve just learned. In fact, helping other students with their questions is a great way to develop confidence in what you’ve just learned. If you can answer other students’ questions, you have a strong grasp of the material.

### Chase a deeper understanding

To truly understand a concept and be able to apply it as knowledge in the future, you must understand the why. Why does it behave that way? Why do we use it for that? Why is it important? This way, when you see a new problem that looks scary, you can figure out how to tackle it step by step based on your understanding of why you can and why you do certain actions in math. Deeply understanding a concept allows you to fully put it into practice and develop new ideas on top of it having strong roots.

### Carry out practice, practice, practice

Although parents will often say they don’t have a favorite child, most math instructors will say that they have a favorite study skill. And I am willing to bet that ninety-nine percent of math instructors would agree that THE most important study skill is practice.

As we’ve discussed, math is an active learning subject. You cannot sit passively in a lecture, read the textbook, and then go in and ace an exam. If you have not practiced the application of the content regularly through practice problems (hint hint, that’s why teachers give you homework!), the assessment will be a struggle. You might be able to recall some of what you saw or read, but it will be very hard to apply it.

In math, you are lucky to have a subject that typically provides many worked examples with correct answers for you to look over. However, simply looking at these examples will do you no good. Remember, *active* learning. Write down a few example problems and try to do them yourself without the guidance of the book unless you get stuck or end up with the wrong answer. Then use that detailed solution to guide your learning and understanding of how to solve future problems of that type. In fact, go find more of that problem type and just keep trying until you are getting to correct solutions without looking at the worked steps in the book.

Another critical part of practicing math problems is showing all your work. Write down all details of your work instead of just the answer. If you are trying to solve a multistep problem (which is pretty much all college-level math) in your head, you are much more likely to make mistakes. Solving in your head will also make it impossible to go back and figure out which step, or part of the problem, you are introducing an error to. Learning from your own mistakes is one of the most powerful ways to learn, but you won’t be able to without first being able to see and search for the mistake.

If you practice enough, you will master a concept backwards and forwards and be able to tackle all different versions and types of questions.

### Connect ideas

The best way to keep these study skills rolling into your next math class is to have a big picture of how all the math concepts you’ve been learning fit together. Can you make a web of math ideas and how they are all connected? Remember the example at the beginning of the section about the relationship between addition and multiplication. If you had created a concept map as you learned addition and then multiplication, you would better understand how they fit together, and how knowing one skill will help you conquer the other skill. This map allows you to visualize and identify connections between concepts, so that every new concept you learn won’t feel like you’re starting to build your understanding from scratch.

# Getting the Most from Your Math Notes

### Learning Outcomes

- Compare effective note-taking strategies for quantitative courses against those for other courses

In grade school, 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 fifteen to twenty 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.

### The Value of Preparation

How much value do you think Greta is gaining from her math lectures in each of these scenarios? Do you think fifteen to twenty minutes really makes a difference? If you haven’t before, try spending fifteen to twenty 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 a 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? Three-ring binder, folder, spiral notebook? Taking notes with a laptop 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?

### Organizing Your Math 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 options:

- 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 a **homework journal **that records your online homework.

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.
- Compare your class notes to the assigned reading—maybe you can reorganize your thoughts, or answer one of your own questions.
- 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?

# Reading Quantitative Texts

### Learning Outcomes

- Identify strategies for reading quantitative texts

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:

- 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 Notations 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 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 Support at FSW

Students looking for help with math assignments at FSW should reach out to the Academic Support Center. There is a center on each campus where students can walk in for immediate support. Support is also available online through the math center’s virtual services and through tutor.com. Students should access tutor.com through the link in their Canvas course to ensure they receive the benefits available to FSW students.

### glossary

**homework journal:** a week-to-week written record of your online math homework, which will help you keep track of the material as you progress through it

a week-to-week written record of your online math homework, which will help you keep track of the material as you progress through it