Studying Mathematics: A How-To Guide
Welcome to MTSU and this section of transitional mathematics. You may be a new student at the University or a returning student who wants improved mathematics skills. Either way, you want to maximize your time and achievement. To help you reach peak performance, please read the following guide to studying mathematics. The presentation follows a question and answer interchange between a new student and one of the professors who teach MATH 1000. Former students have asked some of these questions and someone should have asked the rest. You should get a feel for what is expected of you as a college student taking this college credit course.
Student: Excuse me, I’ll be starting at MTSU this semester and I need some help. This is my first time in college. Can you answer some questions for me?
Dr. O: I’ll be glad to answer whatever I can.
1. Getting a Good Start
Student: I’ve been placed in a Transitional Mathematics course. What can I do to get a good start?
Dr. O: The first step is to realize that you are not in high school anymore. College is very different. One difference is that you are ultimately responsible for what you do and do not do. You need to make good decisions about how you spend your time, where and how you study, how you behave in class, and what priorities you set.
Student: Okay, I’ll accept that. Do you have any specific suggestions for the beginning of the semester?
Dr. O: Certainly. You want to register for the correct course. Since you were placed into MATH 1010K, that should not be a problem. Pick a class time when you are working well mentally and physically. You might want to include a break in your schedule so you can review, fill out your notes, and do your homework.
- You will probably have an opportunity to test out of the transitional portion of the course on the first day of class, so get to the room a little early and have your pencil and graphing calculator with you. An answer sheet, scratch paper, and test will be provided. The pretest is for placement purposes and does not count towards your course grade. Give it your best.
- You need to gather some specific information about the course, the semester calendar, and your instructor. Let’s begin with the course.
- You need the correct textbook and whatever ancillary materials your instructor requires. You should get the text at a local bookstore. Be sure that you get the correct textbook for your course.
- Next, you need to know the answers to the questions Who, Where, When, and What of the course and section (example MATH 1010 K10) for which you are registered.
- Find out Who teaches your section of MATH 1010K. For help with this and other problems check with your advisor in the department of your major. If you are undecided, then you need to talk to an advisor in the University College Advising Center, in KOM 102. The advisor will help you find the best available instructor. If your last name begins with A through K, then you will talk to Tanika Mitchell, 615-494-8706. If your last name begins with L through Z, then you will talk to Megan Williams, 615-898-5667.
- Now, find out Where the class meets from your registration materials. Look up the building abbreviations on your campus map and find the building. Physically going and finding the room before classes start meeting is a good idea.
- The When of the class is crucial. You should schedule a time when you are functioning at your best. If you work better in the morning, schedule your math class then. If you don’t get going until later, schedule your class accordingly. Note the days of the week that the class meets. Is it a Monday-Wednesday-Friday (MWF) class, a Tuesday-Thursday (TR) class, or does it follow a different schedule? Know the start and end times of the class. Try to be at the door of the room at least five minutes early so you can get in, find a seat, and get your material ready for class to start. You’ll want to get a good seat, near the front and in the center so you can see the board well and hear your instructor and classmates.
- To find out the What, you need to get the course syllabus. Some instructors distribute a syllabus the
first day of class. Others have their syllabus on-line for you to print. When you
get the syllabus for your class, read it! Find the answers to questions like these:
- What specific materials do I need?
- What is an acceptable type of graphing calculator?
- May I do my work in ink or will I need a pencil?
- How do I contact my instructor?
- What are the assignments? What will I be required to turn in?
- What kind of lab work is required for my section? How will it be graded?
- What is the grading system?
- What kind of tests and how many will I have to take?
- What is the attendance policy?
- Can I make up absences? If so, how?
- Can I make up tests? How?
- Does my instructor offer any extra credit or bonuses?
Student: What information about the semester calendar do I need?
Dr. O: You’ll need to know both some general and some class-specific information. In general you need to know the first and last days that classes meet for the semester, Study Day, the time, date, and place of the final exam, and the dates of holidays and breaks. You can also check the Academic Calendar for additional general information (http://www.mtsu.edu/calendar_academic.php).
The syllabus from your instructor may have specific information for your class such as a schedule of dates for class meeting and topics to be covered, the dates of reviews and tests, and due dates for assignments. Knowing when to expect a test or when an assignment is due will help you keep up-to-date with your work and allow you to schedule extra study time to prepare for tests or presentations.
Student: What other information do I need to know about my instructor?
Dr. O: You will need to know some specifics about your instructor. Find out the following:
- What is my instructor’s name and how do I address her or him?
- Where is my instructor’s office?
- When are the office hours?
- What is the contact information (phone number and email address)?
- Are office hours drop-in or by appointment?
- Are other times available outside of office hours?
Student: Thanks, Dr. O. I think this will help me a lot. But, if you don’t mind, I have a few more questions.
Dr. O: Not at all. Ask away.
2. Why Take Notes
Student: Why do so many students have so much trouble with math? Why does it have to be so hard?
Dr. O: Mathematics is a subject that requires a great deal of concentrated effort, both in and out of class. The word mathematics comes from the Greek word meaning “something that is learned.” Apparently the Greeks believed that the concepts of mathematics were not intuitive but must be taught. They even have a proverb that says, “Whatever is good to know is difficult to learn.” Students need discipline and dedication to be successful. One reason they have trouble may be that they don’t know how to take good notes in math class. Another reason may be that they don’t study effectively. Learning mathematics cannot be made easy, but it can be made easier when you combine a systematic method of taking useful notes and an effective way to study with self-discipline and a positive, proactive attitude.
Student: Why do I need to take a math class anyway? I’ll never use this stuff in my profession.
Dr. O: As a student now and a professional later you will need to be a problem solver. You will need a method to approach new problems. Solving problems is more of an art than a science, so being able to operate at a high mental level is a necessity. To be an effective problem solving artist, you need to be able to acquire information, process that information into knowledge, and turn your knowledge into a tool to solve problems. Allow me to talk like a professor for a little while. According to what is known as Bloom’s Taxonomy of Knowledge, learning begins with the lower order thinking skills when you receive new information (such as definitions for specific vocabulary). First you acquire information. You develop comprehension when you can explain it in your own words. You’ve completed the lower order skills when you can apply new information to assignments by yourself. Now you have turned information into knowledge.
The higher order thinking skills complete the process of turning knowledge into a useful tool. Analysis begins by examining what worked in the assigned items and considering how you could use that to solve similar problems. Synthesis involves looking for patterns and connections between concepts. Evaluation occurs when you reflect on your knowledge to develop effective ways to use what you have learned to solve other problems that you encounter. You have made a tool for yourself (a pattern of thinking useful for solving problems). Developing these tools provides you with skills and confidence that you can use to solve other types of problems, not merely mathematical items assigned in a classroom.
Bloom’s Taxonomy of Knowledge 1
Lower Order Thinking Skills
Higher Order Thinking Skills
Student: What kinds of information will I need to learn in MATH 1010?
Dr. O: Your instructors will build on the expectation that you already have basic arithmetic skills. Vocabulary is the first type of information you will need to grasp quickly (within a day or two of receiving it!) because learning the language is vital to your success. The language of mathematics consists of terminology, grammar, and syntax. The vocabulary includes the technical definitions of words and symbols. The grammar provides the system of rules that mathematics uses to communicate meaning. The syntax provides the pattern to put those symbols together to form terms, expressions, and equations. Being proficient with the terminology of mathematics helps you to understand and use correctly words and phrases that will have different meanings in a different context. Using the vocabulary of mathematics correctly also helps you think mathematically. One problem students have is that they say the name of the symbol, but don’t know what the symbol means. You’ll also be introduced to concepts, methodologies, and structures. Some will be new and some will be familiar. We will introduce you to calculator skills for the TI-83/ 84 families of graphing calculators that will enable you to get and interpret the data the machine provides.
Student: Wow, that’s a lot! How can I ever grasp all that in the time I have during one semester?
Dr. O: Well, it depends on how efficient you are, how dedicated you are, and how disciplined you are. Basically, it depends on how responsible you are for your own learning and how you use skills and techniques to turn information into knowledge.
3. Learning Vocabulary
Student: How can I develop techniques and a methodical approach to learning vocabulary since it seems to be so important?
Dr. O: Okay. Let me use an example. Here is a definition for term:
Term: the parts of an expression being added or subtracted. The result of combining terms is called the sum.
How would you begin? You might print a copy of all the vocabulary for the unit, or you might copy it down in your notebook. Which one of these ways would be more effective in learning the information? The first thing you need to do to learn new vocabulary is to memorize it (the Knowledge level of Bloom’s Taxonomy). To get new information into your memory requires at least 7 to 13 repetitions. You get at least three repetitions when you copy something down. You get NONE when you print something. If you decide to copy, divide your page into two columns: one for the terminology and one for the definitions. You should begin your daily study session with a trip through the vocabulary. Cover up the definitions column, speak the word or phrase out loud, define it, and then pull down the cover sheet far enough to check your definition. If what you said wasn’t exactly correct, cover up the definition and try again.
Another way to study vocabulary is to make vocabulary study cards. Write the word or phrase on one side of the card. Include the course and unit or chapter information so you can keep all the vocabulary for a chapter together. On the other side copy the definition. Now you can test yourself two ways: by reading the word and speaking the definition and by reading the definition and speaking the correct word. As you begin to comprehend the terminology you can add examples and expanded definitions from your class notes and textbook. Pick a card size that will allow you to carry the cards so that you can study vocabulary whenever you have a chance, such as waiting in a line or between classes.
Did you notice that the definition of term contained the definition for sum and requires you to know what is meant by expression? As you progress through the course, your definition of term should increase, as well as your understanding. Does this give you some ideas?
Student: Yes, can you show me how to make a card for this definition of term?
Dr. O: I’ll be glad to.
Sample Vocabulary Card
Unit 1. Vocab: MATH 1010 K 12
The parts of an expression being added or subtracted. The result of combining terms is called the sum.
Ex. 4 or 3x are single terms.
In the expression 2x + 3 both 2x and 3 are terms
See: expression, sum
Student: Yes, thanks. It looks like something that seems so simple at first can be really complicated.
Dr. O: Not necessarily complicated, but complex, meaning that it has more than one part. That is why you must start with merely memorizing and work every day to increase your ability to think with the new information until it becomes knowledge and then a tool. Very little in mathematics stands alone. Math resembles a freeway interchange more than it does a one-way street. The concepts of math connect in more than one way, so look for connections to something you already know and patterns and similarities.
4. Taking Notes
Student: What methods do you have for learning the mathematical concepts that I’ll need to work problems?
Dr. O: First, take readable, organized notes. You’ll need to take notes from at least three sources: your textbook, your on-line math lab work, and your in-class work. Shall we start with the textbook?
Student: Please. Why would I need to take notes from the book when it’s all right there in front of me?
Dr. O: Your question suggests the answer when you think about what “it’s all right there in front of me” means. You have more information in the text than you need for review. You should have a goal of reading the textbook assignments all the way through only once. After that, you’ll refer to your notes, so take good ones.
For each assignment preview the section of the text. Begin by reading the title and specific concepts covered at the beginning of the section definitions. Next, read the headings and subheadings, look for words and phrases in bold print, illustrations, charts, graphs, marginal items such as “Study Tips,” “Cautions,” and “Student Notes.” Write the Chapter and Section number, section title, and test number (assignment for test 1) at the beginning of your notes. Now you are ready to begin taking notes in your notebook.
Use the same organization as in the text and don’t forget to record the page numbers. You want to record the most important information in the section, not copy the whole thing, so be selective. Use a two-column method. In the left hand column, about 2 inches wide, write main ideas such as topic name, anything in bold print, key concepts, and so on. In the right hand column write a brief explanation. If you have trouble understanding what you wrote at a later time, the page number allows you to find the exact place in the text quickly so you can refresh your memory and expand you explanation. As an alternative, you can make notes in the text itself by writing key words in the margin and highlighting or underlining the gist of the idea for definitions and concepts. To study from both note-taking methods, cover up the right hand column and use the left hand column to fashion questions. Answer the questions then uncover enough of the right hand column to check. Strive for 100% accuracy.
Examples are a different matter, however. When most folks try to read an example of how to work a math item, they take way too many shortcuts, such as reading the name of the symbol rather than what the symbol stands for, or they merely glance at the example without trying to grasp what is displayed. When you take notes from an example, ask yourself the following questions.
- What are the directions for the item?
- What is the goal for the item?
- How does the item begin?
- What are the steps involved and why are they taken in this order?
- How can I tell when I’ve finished the item?
- What can I do to self-check my work?
- What vocabulary is involved?
- What calculations are necessary?
- What patterns emerge that will apply to similar items?
- What do I need to gain from this item that I can build on for future items?
As you study the examples, write down the answers to these questions that apply to the example in a side-by-side format. I suggest that you keep a list of these questions handy and add to them as you go along. It will be a good idea to review your list before each study session, especially for the first month or so until they become second nature to you.
If you are having trouble understanding the text, try to find an alternative text that covers the same material perhaps first looking in the library.
Student: That sounds like a whole lot of work. Why do you think it’s a good idea?
Dr. O: The work you do outside class will make taking tests and understanding new concepts and procedures easier and more profitable for you. The harder you work in the beginning, the more success you will have in the end. The more energy you put into effective study, the less energy you’ll need to exert to be successful on tests and the final exam.
So, when you study from your textbook, you’ll want to shift through “it’s all right there” to get to and record the most important information. Then when you review you can concentrate on what you need to turn the information into knowledge.
Review before each class meeting to refresh your memory, set up expectations, and prepare yourself to ask and answer questions about the material. New material will be easier to connect to previous material when you have the information in the front of your mind, rather than buried underneath something else.
Student: That makes sense, I guess. You said something about taking notes from an on-line math lab?
Dr. O: Yes. One component of this course is the on-line math lab (called MyMathLab). You’ll notice that you are assigned homework and tests on-line that supplement and reinforce what we do in class. When you work on-line be very careful to follow the directions for entering your responses. The computer works within very narrow limits on what is acceptable. For example, you must be careful when you write a slash as a fraction bar. The on-line lab might not interpret the slash as you intended. You should use the fraction tool button from the symbol palette to enter a fraction, especially one that contains variables. Make notes on what is and what is not allowed as you work through the on-line homework before you take on-line tests.
When you work through the homework portion, take notes as you would from the text examples. Work the on-line homework and test items on paper and be sure to keep a hard copy in your notes for review. Record the number of the item, the chapter and section it comes from, and anything about it that gave you trouble. When you take the on-line test, note which items were used and where they came from. You can bring these hand-written problems to class and ask questions over them if you have them. You can bring these hand-written problems to class and ask questions about them.
Student: Okay. What about in-class notes?
Dr. O: In class we’ll cover vocabulary, examples, concepts, patterns, using problem solving strategies and approaches to exercises, recognizing goals, procedures, reaching goals, checking, key ideas, and avoiding common errors. Your instructor will be asking, explaining, writing, and demonstrating.
Of course, you need to be in class to benefit from your instructor’s insights.
Student: Is that all? How will I know when to take notes and when not to take notes?
Dr. O: A rule of thumb for any class you take, “What goes on the board goes in your notes.”
When your instructor writes it, repeats it, or asks it, it must be important. Note it! Do not rely solely on your memory. You will need the cues that a good set of notes gives you. Start the notes for each class meeting with a heading that includes the unit or chapter, the topic, and the date. Use the two-column method that we talked about for taking notes from your textbook. When you record examples from the board, identify the source if it is an exercise from the text (15/2.3 would mean exercise 15 from chapter 2, section 3). Write down the directions for the item, each step, the explanation for the step, and the check. Be sure you know how to tell when you’ve successfully completed a problem.
When calculator skills are explained record each step. Write down the buttons used, the exact order to follow, and what is supposed to happen, and how the results should be displayed. Copy any example down as exactly as you can so that you can duplicate what happened when you are by yourself. Practice until you can perform the task correctly without hesitation.
You’ll need to study your notes as soon as you can on the same day you take them. Perhaps you’ll need to rewrite some of them or fill in gaps. You want to make clear connections with what has come before. Be sure that your notes answer these questions:
- How does today’s information fit what we learned earlier?
- What is similar?
- What is different?
- How does the new build on the old?
- What calculations do I need to be able to make?
- What special calculator skills were introduced?
- How much practice will I need so that I can be proficient (this means fast and accurate).
- What will the results look like?
- What is the goal?
- How do I start, proceed, finish and check?
Good notes will tell you all of this once you have organized the information. After all, organizing information is the first step to . . .
Student: . . . turning it into knowledge. I’m beginning to catch on.
Dr. O: Yes, you are. Good for you.
5. Developing Proficient Study Habits
Student: Now that I’ve written my notes, how do I use them to develop proficient math study habits?
Dr. O: Develop consistent and effective study habits. Study every day. Schedule a definite time to study as soon after math class as you can. Choose a place to study that is free from most distractions, is well lit, and not too comfortable (you don’t want to fall asleep). Consider using the math lab at least some of the time for a study location. You will need to schedule at least 8 to 10 hours per week, probably more, just for math. Study in sessions of 45 to 65 minutes at a time, take a short break, and continue until you have finished all of the assigned work. You don’t want to fall behind. Review your notes, textbook, assignment list, and vocabulary before you begin your homework.
When you study your notes, you want to do more than read them. If you use the vocabulary card method, add examples and expanded definitions from your notes. To help you remember rules and concepts, you might use a mnemonic device, such as a word, phrase, or sentence. An example is the sentence “Please Excuse My Dear Aunt Sally” to remember the order of operations.
Practice any calculator skills that you are required to use. Review those previously learned and any new ones recently introduced. Know how to use your calculator to confirm the results of your own work.
For the items that were worked out in class, write the expressions, equations, or functions on a separate sheet of paper, then try to work them out correctly without looking at your notes. This will tell you how well you are grasping the concepts used in the example. Use the side-by-side method, working on the left side and making notes on the right side, something like this:
Evaluate the expression 3x – 2 for x = -2. Evaluate means find the value of the expression by replacing the x with (-2). Then follow the order of operations to compute the value.
3(-2) – 2 Multiply before adding.
-6 – 2 Apply the definition of subtraction.
-6 + (-2) When combining two numbers with the same sign, add the values and keep the sign.
-8 The value of the expression is –8 when x = -2.
Once you understand this example, you can follow this pattern to work similar items from your homework below it. When working on your assignment, note similarities and differences between the example and the assigned items. Should you become confused on an item, go back, look at the example, and try again. If you still have problems, you can go to the math lab or ask your instructor during office hours. The on-line homework has tutorials available, and you can also go to the University Studies Tutoring lab, KOM 124, for help.
6. Test Taking Strategies
Student: I have trouble taking math tests. How can you help me?
Dr. O: Math tests provide a means for the instructor and the student to check progress. Your ability to perform well on the tests will be a direct result of how well and effectively you have prepared yourself. Keep up with your assignments. You do not want to cram for a test. Keeping up is more important in math than in nearly any other subject because catching up is extremely difficult. Practice working the assigned homework items correctly will lead to easier tests and higher scores. Most students do not practice enough to truly understand the concepts, the symbols, the vocabulary, and the syntax of mathematics. When you have done an item correctly once, then you need to try it again without looking at your previous attempt. Review each and every day what you have been taught. Since math is cumulative (like a snowball rolling down a freshly powdered slope), review is somewhat built-in. You especially need to concentrate on items that were troublesome for you, as well as vocabulary and basic concepts. Try writing out a test using items from your notes and textbook, especially ones the instructor worked out in class. If available, use a test review to select items. Make out vocabulary quizzes from your cards, both matching and opened ended items.
On test day, get to class early. Be sure that you have all the materials you will need, such as pencils, scratch paper, and graphing calculator. Use your vocabulary cards and notes for a quick review. As the test starts you might want to do a memory dump on your scratch paper first. Write down formulas, rules, and key concepts that you want to remember. Preview the test before you begin working out the problems. You might see something that reminds you to add to your memory dump. You don’t necessarily want to start with the first question, but with an easy one to build confidence and jog your memory. Know how much time you have to finish the test and use all of it. Be sure to pace yourself so that you don’t get stuck on a question. Check your work as you go. If the test contains multiple-choice questions, do not change an answer unless you are certain that the earlier one was incorrect. Try not to leave any questions blank, especially if your instructor gives partial credit.
When tests are returned, keep a list of the problems that you missed, why your answer was incorrect, and how to correct what you did wrong. Practice these so that you never make the same mistake again on a test. Mistakes happen because of carelessness, using processes incorrectly, and incompletely understanding concepts. All of these difficulties will be reduced when you prepare properly. Make and correct mistakes before tests so that you avoid them during tests. You do this by working lots of homework problems. If time is given during class, ask for clarification of anything about the test questions and scoring that confuses you, otherwise schedule an appointment with your instructor.
7. Getting Help
Student: Where can I get help outside of class?
Dr. O: You can go to the math lab in the Stark Agribusiness and Agriscience Center (KOM 124). The Math Lab phone number is 898-2465. Tutors will help you with your homework assignments. Computers in the lab allow you to work on-line where you can get on-line tutorials. The tutors will help if you have problems using the computer. Take all the material you will need with you. Remember: be responsible for yourself and your learning. You might want to schedule study time in the lab as well so that help is readily available when you need it.
Also, know your professor’s office hours and visit during them. You can discuss problem areas and get help with specific questions. Don’t go to the office and say. “I need help with everything.” Have a specific question or an item that you want help with for a start. If you are off campus during office hours, you can call. Be sure to tell your instructor your name, course and section number (MATH 1010 K10), and why you are calling. If you need to leave a message for your instructor on voice mail, speak slowly and clearly, identify yourself, the course and section, and briefly state your message.
Find one or more classmates who will be your math buddies. Form a study group with your buddies. Set a regular meeting time so you can discuss problems, check your notes so you don’t miss anything, and give quizzes to each other. One of the best reasons for going to college is to start networking, gaining friends and future colleagues.
8. Preparing for the Final Exam
Student: When do I need to start studying for the final?
Dr. O: Start on the first day of class. Note the date and time of the final exam and mark it on your calendar. At some point your instructor will want you to print the final exam review and answer key. As you work through the examples on the review, refer to similar items from you notes. Be certain that you can work the review items correctly. Remember: MATH 1010K is cumulative, meaning it covers everything you were assigned during the semester. Review previous items at least once a week as long as you work them correctly, more often for those that cause you to struggle. Select similar items from the text and on-line homework and tests for additional practice. On many items, you can change the constants to give yourself a similar item. Check your work yourself and with the tutors in the math lab. Devise practice exams from the information you have and practice taking them. The final is a timed test similar to the pretest, so you need to practice working with a clock until you are as comfortable as you can get. By the time the final rolls around, it should be the easiest test of the semester because you will be familiar with the concepts and the format of the items and how the result for an item should look. Every time you study, every test you take, every on-line lab assignment you complete, every meeting with your study group helps prepare you for the final exam.
Student: This sounds like a lot of work. I’m not sure I can keep up.
Dr. O: You can when you are disciplined. You can count on spending at least one to two hours of study time a day on math. And remember that you start small with a few concepts then build on that foundation. The more you apply yourself, the easier your routine will become to follow and the more you will accomplish. You are ultimately responsible for yourself and your learning, as Dr. Seuss pointed out.
“You can get help from teachers, but you are going to have to learn a lot by yourself, sitting alone in a room.” Dr. Seuss2
9. Getting Organized
Student: Do you have any tips on organizing material to help me understand?
Dr. O: Keep all your math materials in one place. Study in the same place at the same time each day. Avoid interruptions and distractions by turning off your phone, I-pod, and TV.
Making lists of important concepts is a method of organizing. You might use the initial letters of the items in your list to form a mnemonic, as we talked about when we discussed developing study habits.
Two additional ways to organize information are charts and mind maps. Following are an example of each method.
You can summarize information in a simple chart similar to this:
|When the slope is||The graph is||The Equation (could be)||The Item is||The y-Intercept is|
Example: m = 2
y = mx + b
y = 2x + 3
Example: m = -2
y = mx + b
y = 2x + 3
Example: m = 0/1
y = b
y = 3
Example: m = 1/0
x = a
x = 4
NOT a function
NOT a function
None (unless the vertical line is the y-axis)
Simple charts allow you to place related data in a compact form that you can use for study. You may also notice similarities and differences in the data that will help you understand and use it correctly.
An option for organizing information is a mind-map. You can see the big picture by diagramming a topic and the relationships between its concepts. Mind maps are developed from the middle out. The main idea is written in the middle of the page and related concepts radiate from it. The lines and arrows show the connections between the parts. You can build-up a mind map from your notes and use it to review.
Here is an example of a completed one for polynomials4:
Dr. O: A great deal of information can be expressed in a small space using a mind map. Do you have any other questions?
Student: Well, what about word problems? I can work it out once I know what to do, but how can I figure out the equation?
Dr. O: You need to know and use Polya’s Method5 for problem solving.
Step 1: Understand the Problem
Step 2: Devise a Plan
Step 3: Carry Out the Plan
Step 4: Check
Step 1: To understand what you are required to do in the problem, ask yourself some questions:
- What is my goal (the main questions being asked)?
- What information am I told in the item?
- What constraints or conditions are imposed by the problem?
- What will the results look like?
- How can I state the goal in mathematical terms and symbols?
- What additional information do I need, such as a formula, a definition, or a concept?
- Do I need to make a sketch?
- If I don’t know the additional information, where do I find it?
- What do I need to determine first to answer the question that is asked?
- How can I organize the information to reach my goal, that is, to answer the question?
Step 2: To devise a plan, you start by writing the English version of the answers to some of the questions from step 1, then translating them into mathematical expressions and equations. Once you have an equation that will satisfy the goal, you are ready for the next step.
Step 3: To carry out your plan, solve the equation or equations you devised to give you your final result. Make sure that you follow the order of operations and the correct processes needed.
Step 4: To check your result, make sure that you answer all the questions asked in the problem. Your result needs to satisfy the constraints established by the problem and be reasonable.
As with any practical method, practice makes perfect when you are applying it correctly. Let me demonstrate.
Suppose you were assigned the following application (a.k.a.: word problem):
The measure of the second angle of a triangle is three times the measure of the first and the third is nine degrees less than five times the first. Find the measures of the angles.
Step 1: Understand the Problem
Goal: Find the measures of the angles.
Information given: a triangle, which means three angles. I’m told about the measures of angles 2 and 3. What about the measure of angle 1? It is the basis for finding the measures of both angles 2 and 3.
What do I need to know about the measures of the angles of a triangle? The sum of the measures of the three angles is 180°.
Organizing: Because I need to know the measure of angle 1 to find the measures of angles 2 & 3, I represent the measure of angle 1 by the variable a.
Let the measure of angle 1 = a.
Now I must meet the constraints imposed by the problem to write the representations of the measures of angles 2 & 3.
The constraint for angle 2: The measure of the second angle of a triangle is three times the measure of the first, so I must state the measure of angle 2 as 3 times a.
Let the measure of angle 2 = 3a.
The constraint for angle 3: and the third is nine degrees less than five times the first. The key words require me to write the measure of angle 3 as 5a - 9.
Let the measure of angle 3 = 5a - 9.
Sketching and labeling a drawing of a triangle may help me see the situation more clearly.
Step 2: Devise a Plan
I need to write an equation that will let me answer the question of the problem. Since the sum of the measures of the 3 angles in a triangle =180°, I can write:
(a) + (3a) + (5a - 9) = 180°
Step 3: Carry Out the Plan
Begin by combining like terms in the left side expression (to the left of the equals sign).
|9a – 9 = 180||To get the term 9a by itself, I need to add 9 to both sides.|
|9a – 9 + 9 = 180 + 9||Simplifying produces|
|9a = 189||My goal is to isolate a, so I’ll reduce the factor of 9 to a 1 by dividing both sides of the equation by 9.|
|9a/9 = 189/9||Simplifying both sides gives|
|a = 21°||The measure of angle one is 21°. Have I reached the goal? No, I still need to find the measures of angles 2 & 3.|
|Angle 2 = 3a||Substitute the value of angle one for a and evaluate.|
|Angle 2 = 3(21) = 63°||Now find angle three.|
|Angle 3 = 5a- 9||Substitute the value of angle one for a and evaluate.|
|Angle 3 = 5(21) - 9 = 105- 9 = 96°|
|Angle 1 = 21°, Angle 2 = 63°, and Angle 3 = 96°|
Step 4: Check
My angles need to meet the constraints of the problem and have a sum =180°.
First, review the problem and note the constraints.
The measure of the second angle of a triangle is three times the measure of the first and the third is nine degrees less than five times the first. Find the measures of the angles.
Triangle means three angles. No explicit constraint is placed on the first angle except that it is the basis for the other two. Let the measure of angle 1 = a. Since the measure of the second angle of a triangle is three times the measure of the first, I let the measure of angle 2 = 3a. The measure of the third is nine degrees less than five times the first, so I let angle 3 = 5a- 9°.
I need to answer these three equations correctly and have the sum of the measures equal 180° to solve the problem. I calculated the measure of angle 1 as 20°.
Now I write my equations and check to see if my solutions meet the constraints.
Angle 1 = a = 21°
Angle 2 = 3a = 3(21) = 63°
Angle 3 = 5a - 9° = 5(21) - 9 = 105- 9 = 96°
The values I computed for measures of the three angles meet the constraints of the problem. To finish my check, I add the three measures together. If the sum is 180°, then I have the correct measures.
21° + 63° + 96° = 180°, so I have my solutions.
The textbook also introduces a five-step approach that is a variation of Polya’s in Chapter 2.
11. Combatting Math Anxiety
Student: Math can seem so overwhelming. I’m concerned that I won’t have the time to do everything. I sometimes get anxious during tests. What can I do?
Dr. O: Many people struggle with math anxiety. Here are some tips that might prove helpful:
- First, believe in yourself. You can and will be successful. Maintain a positive attitude so that you are proactive rather than reactive in your approach to mathematics.
- Remember we talked about math buddies? Find some classmates and form a study group. Encourage each other and share the study load.
- Get comfortable asking for help, not only from the members of your group but also from tutors and your instructor. We are here to help. An ancient Chinese proverb says, “One who asks a question is a fool for five minutes. One who does not ask a question remains a fool forever.” The responsibility is yours, so ask.
- Before tests, find out as much as you possibly can about the test. What type of problems? Is vocabulary included? Is there a study guide? Does my instructor review before the test?
- Practice. Practice. Practice. Do all your homework. Review every day, both the vocabulary and examples, before trying to work your homework exercises. Make and take practice tests to develop proficiency and confidence.
- Learn some relaxation and breathing exercises to keep the oxygen flowing to your brain.
- Be dedicated and conscientious in your studying so that you build confidence in yourself and your skills. As your confidence grows, maintaining your positive attitude will be easier.
References and Acknowledgments
Student: We’ve talked about so many things. Can you summarize them for me?
Dr. O: I’ll try.
Getting a Good Start: Register for the correct class at a time you work best. Make good decisions and set your priorities so that they help you reach your college goals. You need specific information about the course, the semester calendar, and your instructor. Remember to answer the questions Who, Where, When, and What about your class.
Why Take Math: You need to be able acquire information, process it into knowledge, and turn your knowledge into a problem solving tool. Learning mathematics requires you to use higher order thinking skills. We talked about Bloom’s Taxonomy and math to help you see how you transform information into a productive tool.
Learning Vocabulary: Speaking the language of mathematics is critical to understanding. The first step is acquiring vocabulary. Writing your vocabulary provides extra repetitions. A two column method or vocabulary cards give you an effective way to study.
Taking Notes: Take good notes from at least three sources, textbook, on-line homework, and in-class. Use a two-column format for concepts and a side-by-side format for examples. Be brief but not so that your notes confuse you. Review immediately after class to fill-in gaps. Study by self-testing.
Developing Proficient Study Habits: Develop consistent and effective study habits. Study every day at a definite time in a suitable place, perhaps using the math lab. Schedule at least 8 to 10 hours per week in sessions of 45 to 65 minutes at a time until you finish all assigned work. Review your notes, textbook, assignment list, and vocabulary before beginning homework. Use mnemonic devices to help you remember concepts and lists. Practice required calculator skills. Use the side-by-side method for problems (steps on the left side and notes on the right side),
Test Taking Strategies: Performing well on tests is a direct result of effective preparation. Keep up with assignments. Correctly work the assigned homework items several times each. Review each and every day. Concentrate on troublesome items, vocabulary, and basic concepts. Writing out a test using items from your notes and textbook. Use any available test review. Make out vocabulary quizzes.
On test day, get to class early. Have all your materials, such as pencils, scratch paper, and graphing calculator. Use vocabulary cards and notes for a quick review. Do a memory dump. Preview the test before you begin working. Start with an easy question to build confidence and jog your memory. Know how much time you have and use all of it. Pace yourself and don’t get stuck. Check your work as you go. Try to answer all of the questions, especially if your instructor gives partial credit.
When tests are returned, keep a list of the problems missed, why your answer was incorrect, and how to correct what you did wrong. Practice so that you do not make the same mistakes again. Ask for clarification of anything about the test that confuses you, or schedule an appointment with your instructor.
Getting Help: Use the math lab in SAG 202. Tutors will help you with your homework and with MyMathLab assignments. Take all the material you will need with you.
Know your professor’s office hours and visit during them to get help with specific questions. If you call, be sure to tell your instructor your name, course and section number (MATH 1000 K12), and why you are calling. If you leave a voice mail message, speak slowly and clearly, identify yourself, the course and section, and briefly state your message.
Form a study group with one or more classmates.
Preparing for the Final Exam: Write the date and time of the final exam on your calendar. Work through the examples on the final exam review. MATH 1000 is cumulative. Review previous items regularly. Select similar items for additional practice. Devise and take practice exams. The final is a timed test, so practice working against the clock. Be familiar with the concepts and the format of the items and how the result for an item should look.
Getting Organized: Keep all your materials in one place. Study in the same place at the same time each day. Avoid interruptions and distractions.
Two ways to summarize information are simple charts and mind maps.
A Methodical Approach to Problem Solving: Use the four steps of Polya’s Method to solve word problems.
Combating Math Anxiety: Use the seven tips provided to help you and the members of your study group build confidence and combat math anxiety.
Student: You’ve given me plenty of tips. They should help me get the best grade I can this semester. Is there a way I can test on my math study skills and check my progress toward being an efficient learner?
Dr. O: Yes there is. Here is a diagnostic inventory of math skills that you can use to review what we’ve discussed.
Student: Thank you for your time and good advice. I feel better about taking my math course now.
Dr. O: You are most welcome.
Dr. David A. Otts
University Studies Department
Middle Tennessee State University