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Mastering the Fundamentals of Mathematics

An award-winning professor teaches you the mechanics behind math. Refresh your skills, keep your mind sharp, and appreciate the beauty of mathematics.
 
 
Rated 5 out of 5 by from Excellent! What a great review of math fundamentals. I am a retiree that wants to brush up on my math skills. I decided to start from the very basics and chose this course first. Professor Sellers is an outstanding teacher; his ability to explain things clearly and easily is remarkable. I really enjoyed this course and I'm looking forward to his next 2 courses in Algebra.
Date published: 2025-09-22
Rated 5 out of 5 by from Great course for review or first-time learning Positives: This is an excellent course. Sellers manages to teach in a way that empathises with his audience without condescending to them, which is a significant achievement, since he is addressing a very wide audience. He doesn't look down at his audience. He recognises that learning these things can appear difficult and daunting --- and *is* difficult and daunting if one is taught by a less-than-exceptional teacher. He has an encouraging manner, a positive attitude without ever being cloying, and a pleasing voice. I would 100% recommend taking this course. === IMPORTANT: === === The rest of this review is addressed to those who have finished the course. === === I recommend NOT reading the rest of this review if you haven't taken the course. === === If you bypass my recommendation and read the rest of this review despite my having just asked you not to, then I insist at least that you do NOT let my review persuade you not to take the course. === === My complaints are far too trifling to justify skipping such an important and well taught course. === === If you are wondering whether to take the course, emphatically please DO take it. (And stop reading this review!) === Negatives: I have five complaints. Three are small, while the last two are of moderate significance. The first of the small complaints is a matter of pedagogical opinion, the second a matter of fact. The first small complaint concerns Sellers's decision to introduce, in Lesson 4, a rule for comparing the sizes of fractions, before the student who had gotten only to that point in the course could really understand why the rule was true. This I fear might lead students to get the sense that they ought to obey rules even when they don't understand them. Now for the second small complaint. Sellers claims that "Many historians" would say that Euclid put "geometry on the map." This is simply not true. Greek geometry began in the 6th century with the work of Thales of Miletus, which included, for example, the so-called "Thales' theorem". And of course Pythagoras was another of Euclid's predecessors. Euclid was working in the late 4th century or early 3rd century. The very title of Euclid's famous book (The Elements) suggests that the book primarily concerns the well known parts of geometry: the elements. They were well known (among specialists) because previous mathematicians had already worked the field. The mathematicians Hippocrates of Chios and Leon each wrote geometrical books titled "Elements": both of these earlier books were overshadowed by Euclid's. In the Elements, Euclid was, for the most part, compiling and rearranging the work of previous mathematicians. Most of the material in the Elements probably derives from previous mathematicians, with some original work by Euclid here and there, especially in the later chapters. Bad to start people off with the idea that Euclid was a genius born from dragon teeth planted in the dirt. He was a genius, but he had a momma and a poppa. My third small complaint is that Sellers says defines scalene triangles as triangles "where each of the 3 angles measures less than 90 degrees." This is simply not what scalene triangles are. They are triangles each of whose angles are different. Now for the first moderately important complaint. Sellers introduces exponentiation by fractional powers, negative powers, and negative fractional powers without first explaining (or even suggesting) that these arise naturally, and aren't the invention of a pedant who wanted to cause you a headache. However, by the time he introduces these things, the student probably will have developed a degree of trust in Sellers sufficient to lead them to assume that there will later be a purpose behind what he is teaching them at that moment. Still, where possible, it is better to avoid the ethic of "Learn first, motivate later" --- which Sellers *usually* manages to do, to his credit. The second moderately important complaint is that Sellers introduces "theoretical probability" in L23 but doesn't, by the end of the lesson, generalize to the *empirical* notion of probability, which is of far greater importance to most people. If A and B are occurrences (events, conditions, etc.), then the empirical probability that A if B (i.e., if B occurs, then A will occur) is given by m/n, where m is the number of occurrences of both A and B, and m is the number of total occurrences of B. This is the concept of probability that people are *usually* using (maybe --- see below), and Sellers' failure to treat of it means that he has failed to introduce one of the crowning concepts of elementary mathematics. Empirical probability is far more interesting, alive, and vivid than theoretical probability. However, Sellers does a great job with what he *does* cover, and I didn't notice any *other* serious missed opportunities. Re: "This is the concept of probability that people are *usually* using." Well, maybe. Charles S. Peirce offers an even more generalized explanation of probability in one of his papers or letters somewhere. But *that* explanation *is* beyond the scope of mathematics fundamentals.
Date published: 2025-08-17
Rated 5 out of 5 by from Dr. Sellers has convinced me: Math is enjoyable! When I was in school and college, I dreaded math and always felt that there were gaping holes in my knowledge. Now, 30+ years after graduation, I’ve decided to remedy the situation by learning math properly, starting with the basics. To my surprise, I’ve found myself actually eager to do each new lesson in Dr. Sellers’ course; he explains each concept very thoroughly and has a warm, engaging demeanor that makes him so pleasant to listen to. I can’t wait to move on to his algebra courses. Thank you, Great Courses, for choosing such a terrific professor!
Date published: 2024-12-28
Rated 5 out of 5 by from Great Professor! If we are not a student at the age of learning fundamentals of mathematics, for example in school grades below tenth class, we would normally think we know all about fundamentals of mathematics. Of course that would be true about addition, subtraction, multiplication and to some extent division. But beyond that, Professor Sellers has generated such an interest in mathematics, that I have now joined a college course to learn mathematics. That much he inspired me with his classes on decimals, fractions, percentages, square roots, number theory, prime numbers, LCDs and GCFs, divisibility tricks, exponents, negative integers, geometric basics, statistics etc. Probability session was of course difficult for me, but again that is a single introductory class, so no qualms there. I will catch up separately. Now I am ready to start Algebra 1 by the same professor, which will be before I start my College classes in Jan 2025. On a side/different note: I think math is taught differently in the United States compared to how it is taught in India, in some cases. I saw specifically two such areas in this course: 1. finding the LCD has a very simple and straightforward way 2. Similarly finding a square root has a very simple way These two topics as taught in this course is in my opinion are different/harder from the simpler ways I was taught in school in India. But this is not a complaint, just bringing up the contrasts. All in all a great course. I enjoyed the entire course, did all the exercises in the book, and finally I love the real world examples the professor brings up to explain difficult concepts. And of course the pizza, sprinkled all over :)
Date published: 2024-11-19
Rated 5 out of 5 by from Not sure why I took this course, but… Where was this teacher when I was in high school? When I graduated high school I felt I had no real competence in math, in fact I had a bit of a math phobia. I started out college as a history major (not a lot of math there), but became interested in the sciences and switched to a biology major. Obviously, math was to become more important in my life. The clincher for me was my first chemistry course which, of course, required a lot of math, mainly the algebraic manipulation of formulas. I discovered I actually enjoyed it and was good at it. I went on to have a career in science related fields, then later switch to information technology as the money was better. Both fields required a fair amount of math in my day to day work responsibilities. All that to say, is that I do not know why I took this course, perhaps due the lingering math phobia. I was intrigued by the title so I watched the first class and I became hooked. The teacher has such a good style and his explanations are clear and to the point. I also thought it would be good for my retired brain as I am always looking for ways to exercise that organ. I did discovered that I really did remembered more than I thought, and I did pick up a few things either that I had forgotten or never knew. If you need a good remedial course on math fundamentals, look no further.
Date published: 2024-09-24
Rated 5 out of 5 by from An excellent summary of the important fundamentals This program of study provides an excellent summary of the fundamentals necessary to move up the learning ladder and master higher levels of mathematical education. The presenter does a fine job.
Date published: 2024-09-19
Rated 5 out of 5 by from Flawless I'm a mature student doing an MA in International Relations. I've taken a year out to teach myself economics and statistics, but first need to brush up my basic maths. The tutor is flawless, as is the course- he's engaging and obviously passionate about his subject, as well as extremely knowledgeable. The pacing is perfect, the problems are numerous and there are additional question sets in the accompanying book. I'm now looking forward to taking the next maths courses by this professor, algebra 1 and 2. One thing I would like to see is a suggested curriculum pathway for the maths courses. There is a wonderful maths ecosystem on Great Courses, but it's hard to piece the areas together. For example- an Economics Pathway could suggest, 1) Mastering the Fundamentals of Maths 2) Algebra 1 3) ... ? This is where I'm not so sure. Should I study statistics first or the Algebra 2? In any case, this is just a suggest and doesn't detract from the wonderful course and the stellar teacher.
Date published: 2024-07-09
Rated 5 out of 5 by from Great intro course with a master teacher Nothing to add beside what a great course it is and what a skilful teacher.
Date published: 2024-07-03
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Mastering the Fundamentals of Mathematics

Trailer

Addition and Subtraction

01: Addition and Subtraction

This introductory lecture starts with Professor Sellers' overview of the general topics and themes you'll encounter throughout the course. Then, plunge into an engaging review of the addition and subtraction of whole numbers, complete with several helpful tips designed to help you approach these types of problems with more confidence....

33 min
Multiplication

02: Multiplication

Continue your quick review of basic mathematical operations, this time with a focus on the multiplication of whole numbers. In addition to uncovering the relationship between addition and multiplication, you'll get plenty of opportunities to strengthen your ability to multiply two 2-digit numbers, two 3-digit numbers, and more....

31 min
Long Division

03: Long Division

Turn now to the opposite of multiplication: division. Learn how to properly set up a long division problem, how to check your answers to make sure they're correct, how to handle zeroes when they appear in a problem, and what to do when a long division problem ends with a remainder....

31 min
Introduction to Fractions

04: Introduction to Fractions

Mathematics is also filled with “parts” of whole numbers, or fractions. In the first of several lectures on fractions, define key terms and focus on powerful techniques for determining if fractions are equivalent, finding out which of two fractions is larger, and reducing fractions to their lowest terms....

32 min
Adding and Subtracting Fractions

05: Adding and Subtracting Fractions

Fractions with the same denominator. Fractions with different denominators. Mixed numbers. Here, learn ways to add and subtract them all (and sometimes even in the same problem) and get tips for reducing your answers to their lowest terms. Math with fractions, you'll discover, doesn't have to be intimidating-it can even be fun!...

32 min
Multiplying Fractions

06: Multiplying Fractions

Continue having fun with fractions, this time by mastering how to multiply them and reduce your answer to its lowest term. Professor Sellers shows you how to approach and solve multiplication problems involving fractions (with both similar and different denominators), fractions and whole numbers, and fractions and mixed numbers....

30 min
Dividing Fractions

07: Dividing Fractions

Professor Sellers walks you step-by-step through the process for speedily solving division problems involving fractions in this lecture filled with helpful practice problems. You'll also learn how to better handle calculations involving different notations, fractions, and whole numbers, and even word problems involving the division of fractions....

32 min
Adding and Subtracting Decimals

08: Adding and Subtracting Decimals

What's 29.42 + 84.67? Or 643 + 82.987? What about 25.7 - 10.483? Problems like these are the focus of this helpful lecture on adding and subtracting decimals. One tip for making these sorts of calculations easier: making sure your decimal points are all lined up vertically....

30 min
Multiplying and Dividing Decimals

09: Multiplying and Dividing Decimals

Investigate the best ways to multiply and divide decimal numbers. You'll get insights into when and when not to ignore the decimal point in your calculations, how to check your answer to ensure that your result has the correct number of decimal places, and how to express remainders in decimals....

32 min
Fractions, Decimals, and Percents

10: Fractions, Decimals, and Percents

Take a closer look at converting between percents, decimals, and fractions-an area of basic mathematics that many people have a hard time with. After learning the techniques in this lecture and using them on numerous practice problems, you'll be surprised at how easy this type of conversion is to master....

32 min
Percent Problems

11: Percent Problems

Use the skills you developed in the last lecture to better approach and solve different kinds of percentage problems you'd most likely encounter in your everyday life. Among these everyday scenarios: calculating the tip at a restaurant and determining how much money you're saving on a store's discount....

31 min
Ratios and Proportions

12: Ratios and Proportions

How do ratios and proportions work? How can you figure out if a particular problem is merely just a ratio or proportion problem in disguise? What are some pitfalls to watch out for? And how can a better understanding of these subjects help save you money? Find out here....

30 min
Exponents and Order of Operations

13: Exponents and Order of Operations

Explore a fifth fundamental mathematical operation: exponentiation. First, take a step-by-step look at the order of operations for handling longer calculations that involve multiple tasks-complete with invaluable tips to help you handle them with ease. Then, see where exponentiation fits in this larger process....

31 min
Negative and Positive Integers

14: Negative and Positive Integers

Improve your confidence in dealing with negative numbers. You'll learn to use the number line to help visualize these numbers; discover how to rewrite subtraction problems involving negative numbers as addition problems to make them easier; examine the rules involved in multiplying and dividing with them; and much more....

32 min
Introduction to Square Roots

15: Introduction to Square Roots

In this lecture, finally make sense of square roots. Professor Sellers offers examples to help you sidestep issues many students express frustration with, shows you how to simplify radical expressions involving addition and subtraction, and reveals how to find the approximate value of a square root without using a calculator....

31 min
Negative and Fractional Powers

16: Negative and Fractional Powers

What happens when you have to raise numbers to a fraction of a power? How about when you have to deal with negative exponents? Or negative fractional exponents? No need to worry -Professor Sellers guides you through this tricky mathematical territory, arming you with invaluable techniques for approaching these scenarios....

31 min
Graphing in the Coordinate Plane

17: Graphing in the Coordinate Plane

Grab some graph paper and learn how to graph objects in the coordinate (or xy) plane. You'll find out how to plot points, how to determine which quadrant they go in, how to sketch the graph of a line, how to determine a line's slope, and more....

30 min
Geometry-Triangles and Quadrilaterals

18: Geometry-Triangles and Quadrilaterals

Continue exploring the visual side of mathematics with this look at the basics of two-dimensional geometry. Among the topics you'll focus on here are the various types of triangles (including scalene and obtuse triangles) and quadrilaterals (such as rectangles and squares), as well as methods for measuring angles, area, and perimeter....

31 min
Geometry-Polygons and Circles

19: Geometry-Polygons and Circles

Gain a greater appreciation for the interaction between arithmetic and geometry. First, learn how to recognize and approach large polygons, including hexagons and decagons. Then, explore the various concepts behind circles (such as radius, diameter, and the always intriguing pi), as well as methods for calculating their circumference, area, and perimeter....

32 min
Number Theory-Prime Numbers and Divisors

20: Number Theory-Prime Numbers and Divisors

Shift gears and demystify number theory, which takes as its focus the study of the properties of whole numbers. Concepts that Professor Sellers discusses and teaches you how to engage with in this insightful lecture include divisors, prime numbers, prime factorizations, greatest common divisors, and factor trees....

31 min
Number Theory-Divisibility Tricks

21: Number Theory-Divisibility Tricks

In this second lecture on the world of number theory, take a closer look at the relationships between even and odd numbers, as well as the rules of divisibility for particular numbers. By the end, you'll be surprised that something as intimidating as number theory could be made so accessible....

32 min
Introduction to Statistics

22: Introduction to Statistics

Get a solid introduction to statistics, one of the most useful areas of mathematics. Here, you'll focus on the four basic “measurements” statisticians use when gleaning meaning from data: mean, media, mode, and range. Also, see these concepts at work in everyday scenarios in which statistics plays a key role....

32 min
Introduction to Probability

23: Introduction to Probability

Learn more about probability, a cousin of statistics and another mathematical field that helps us make sense of the seemingly unexplainable nature of the world. You'll consider basic questions and concepts from probability, drawing on the knowledge and skills of the fundamentals of mathematics you acquired in earlier lectures....

32 min
Introduction to Algebra

24: Introduction to Algebra

Professor Sellers reviews the importance of math in daily life and previews the next logical step in your studies: Algebra I (which involves variables). Whether you're planning to take more Great Courses in mathematics or simply looking to sharpen your mind, you'll be sent off with new levels of confidence....

31 min

Overview Course No. 1014

Multiplying large numbers. Simplifying fractions. Handling square roots. These and other skills are the veritable foundation on which all of mathematics rests. Whether you're a high-school student, an adult preparing for a new career, or someone who just wants to keep his or her mind active and sharp, these 24 lectures offer a well-rounded approach to the basics of mathematics. They zero in on topics that everyone needs to know: adding, subtracting, multiplying, and dividing whole numbers, fractions, negative numbers, and decimals; converting between fractions, decimals, and percentages; solving real-world problems involving ratios and proportions; and working with whole-number exponents and square roots. With each topic, Professor Sellers shows you how to approach, understand, and solve problems of varying complexity. And, later in the lectures, he offers brief introductions to more advanced areas of math and prealgebra, including two-dimensional geometry, elementary number theory, and basic probability and statistics. To help you solve problems with greater ease, these lectures come packed with invaluable tips, tricks, techniques, and shortcuts you'll find yourself returning to again and again.

About

James A. Sellers
James A. Sellers

If you are shaky on basic math facts, algebra will be harder for you than it needs to be. Spend every day reviewing flashcards of math facts, and you will be surprised at how much better at math you are!

INSTITUTION

University of Minnesota Duluth

Dr. James A. Sellers is Professor of Mathematics at the University of Minnesota Duluth. He earned his B.S. in Mathematics from The University of Texas at San Antonio and his Ph.D. in Mathematics from Penn State. Professor Sellers has received the Teresa Cohen Mathematics Service Award from the Penn State Department of Mathematics and the Mathematical Association of America Allegheny Mountain Section Mentoring Award. More than 60 of Professor Sellers's research articles on partitions and related topics have been published in a wide variety of peer-reviewed journals. In 2008, he was a visiting scholar at the Isaac Newton Institute at the University of Cambridge. Professor Sellers has enjoyed many interactions at the high school and middle school levels. He has served as an instructor of middle-school students in the TexPREP program in San Antonio, Texas. He has also worked with Saxon Publishers on revisions to a number of its high-school textbooks. As a home educator and father of five, he has spoken to various home education organizations about mathematics curricula and teaching issues.

By This Professor

Mastering the Fundamentals of Mathematics
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Algebra II
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Algebra I
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