Beginning Algebra Pdf

[Barb Magario]

Traditionallythe study of algebra is separated into a two parts, elementary algebra and intermediate algebra. This textbook is the first part, written in a clear and concise manner, making no assumption of prior algebra experience. It carefully guides students from the basics to the more advanced techniques required to be successful in the next course.

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Beginning Algebra Made Useful addresses the needs of learners to make sense of algebra by quantifying and generalizing everyday occurrences such as commuting to work, buying gas or pizza, and determining the better deal. It requires learners to actively engage with algebraic concepts through physical and thought experiments in ways that help them connect ideas, representations, and contexts, and solve problems that arise in their daily lives. The text helps learners grow their brains and develop growth mindsets as they learn algebra conceptually. Problem sets continue the process, extending work begun in each lesson, applying new understandings to new contexts, and considering ideas that arise more fully in upcoming lessons. Longer assignments that can be used as group projects are included in the text. Group work is encouraged throughout the text; suggestions for orchestrating group work are included.

The text is open access and free for download by students and instructors in .pdf format. In the electronic format, graphics are in full color and there are live html links to resources, software, and applets.

Beginning Algebra, Mathematics, Conceptual Understanding, Real Life, Contextual, Experiential, Growth Mindset, Growing your Brain, Group Work, Remedial Mathematics, Middle School, High School, College University

In addition to the permissions granted by the CC-BY-NC-SA license, the author grants permission for bookstores and/or print shops at K-12, colleges, and universities to reproduce and sell print copies of this book. Other commercial uses of the book are not permitted.

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Molly ONeill is from Daytona State College, where she has taught for 22 years in the School of Mathematics. She has taught a variety of courses from developmental mathematics to calculus. Before she came to Florida, Molly taught as an adjunct instructor at the University of Michigan-Dearborn, Eastern Michigan University, Wayne State University, and Oakland Community College. Molly earned a bachelor of science in mathematics and a master of arts and teaching from Western Michigan University in Kalamazoo, Michigan. Besides this textbook, she has authored several course supplements for college algebra, trigonometry, and precalculus and has reviewed texts for developmental mathematics.

I differ from many of my colleagues in that math was not always easy for me. But in seventh grade I had a teacher who taught me that if I follow the rules of mathematics, even I could solve math problems. Once I understood this, I enjoyed math to the point of choosing it for my career. I now have the greatest job because I get to do math every day and I have the opportunity to influence my students just as I was influenced. Authoring these texts has given me another avenue to reach even more students.

Recently, I was reminded in Melvyn Nathason's first year graduate algebra course of a debate I've been having both within myself and externally for some time. For better or worse, the course most students first use and learn extensive category theory and arrow chasing is in an advanced algebra course, either an honors undergraduate abstract algebra course or a first-year graduate algebra course.

(Ok, that's not entirely true, you can first learn about it also in topology. But it's really in algebra where it has the biggest impact. Topology can be done entirely without it wherareas algebra without it beyond the basics becomes rather cumbersome. Also, homological methods become pretty much impossible.)

I've never really been comfortable with category theory. It's always seemed to me that giving up elements and dealing with objects that are knowable only up to isomorphism was a huge leap of faith that modern mathematics should be beyond. But I've tried to be a good mathematican and learn it for my own good. The fact I'm deeply interested in algebra makes this more of a priority.

My question is whether or not category theory really should be introduced from jump in a serious algebra course. Professor Nathanson remarked in lecture that he recently saw his old friend Hyman Bass, and they discussed the teaching of algebra with and without category theory. Both had learned algebra in thier student days from van der Waerden (which incidently, is the main reference for the course and still his favorite algebra book despite being hopelessly outdated). Melvyn gave a categorical construction of the Fundamental Isomorphism Theorum of Abelian Groups after Bass gave a classical statement of the result. Bass said, "It's the same result expressed in 2 different languages. It really doesn't matter if we use the high-tech approach or not." Would algebracists of later generations agree with Professor Bass?

A number of my fellow graduate students think set theory should be abandoned altogether and thrown in the same bin with Newtonian infinitesimals (nonstandard constructions not withstanding) and think all students should learn category theory before learning anything else. Personally, I think category theory would be utterly mysterious to students without a considerable stock of examples to draw from. Categories and universal properties are vast generalizations of huge numbers of not only concrete examples,but certain theorums as well. As such, I believe it's much better learned after gaining a considerable fascility with mathematics-after at the very least, undergraduate courses in topology and algebra.

Paolo Aluffi's wonderful book Algebra:Chapter 0, is usually used by the opposition as a counterexample, as it uses category theory heavily from the beginning. However, I point out that Aluffi himself clearly states this is intended as a course for advanced students and he strongly advises some background in algebra first. I like the book immensely, but I agree.

There's a big difference between teaching category theory and merely paying attention to the things that category theory clarifies (like the difference between direct products and direct sums). In my opinion, the latter should be done early (and late, and at all other times); there's no reason for intentional sloppiness. On the other hand, teaching category theory is better done after the students have been exposed to some of the relevant examples.

Many years ago, I taught a course on category theory, and in my opinion it was a failure. Many of the students had not previously seen the examples I wanted to use. One of the beauties of category theory is that it unifies many different-looking concepts; for example, left adjoints of forgetful functors include free groups, universal enveloping algebras, Stone-Cech compactifications, abelianizations of groups, and many more. But the beauty is hard to convey when, in addition to explaining the notion of adjoint, one must also explain each (or at least several) of these special cases. So I think category theory should be taught at the stage where students have already seen enough special cases of its concepts to appreciate their unification. Without the examples, category theory can look terribly unmotivated and unintuitive.

I wasn't going to weigh in on this as I think that this is very definitely "subjective and argumentative" (particularly the later), and when before I spoke up in favour of category theory in undergraduate education, it sparked a few comments and I was reminded of why I like the fact that discussion is suppressed in MO. But given that one side of the argument is already here, and the other is not so well represented, I'm going to answer.

Let me start by declaring: "I am not a category theorist". I am a differential topologist. Foundational questions leave me cold, size issues just don't bother me. I'll accept any axiomatic framework if someone wants me to (I'm a fully-paid-up member of the "Axiom of Choice" party). To enter Greg's culinary world for a moment, such things are bit like Norwegian cheese. I can see that to the right person, it's delicious. But I'm not that person.

To continue the analogy, category theory isn't an ingredient that can be added for Extra Flavour, but which not everyone likes. Category theory is like cooking with freshly harvested, organic ingredients as opposed to dull, insipid, shrink-wrapped stuff from the vast conglomerate supermarket. Just making one ingredient organic doesn't have much effect on the flavour of the whole dish, but changing the whole lot does.

But to the matter in hand: undergraduates and category theory. I believe that category theory is an excellent way to understand and express mathematical concepts. I find in my own work that, time and time again, when I express my ideas using categorical language then it makes them clearer both to me and to others. Believing this, as I do, why on earth would I want to deprive my students of the same benefits?

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