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# Algebra learning recommendations

This page provides subject-specific learning recommendations for the subject algebra. See all our learning recommendations pages |

NOT TO BE CONFUSED WITHabstract algebra learning recommendations (abstract algebra is a component of upper division undergraduate mathematics) and linear algebra learning recommendations (linear algebra is a component of lower division undergraduate mathematics)

This page includes recommendations for learning basic algebra -- starting with the use of variables to denote unknowns and setting up and solving simple linear equations (typically called "prealgebra" in the US school system) and including all high school algebra.

## Recommendations

### Top recommendations for building algebraic intuition to pave the way for better formal learning of algebra

The process of formally learning algebra can be made smoother and faster if people are already familiar with algebraic rules, even if those rules haven't been formulated in explicitly algebraic language. These rules include rules such as commutativity, the fact that we can add a constant to both sides of an equality and still maintain equality, the fact that any number plus zero is that number, and so on. The key insight of algebra is that these rules hold generally, rather than just in particular instances.

Algebra is about saying that it's not merely true that 3 + 4 = 4 + 3, it's in fact true that (any number) + (any other number) = (the second number) + (the first number). In symbols, that would read for all numbers . However, even if one doesn't use the symbolic representation, one is still doing algebra when one is dealing with generic variables.

Our top recommendation for learning the main rules of algebra is Dragonbox (website dragonboxapp.com). This is a game that teaches the process of solving algebraic equations. The initial levels of the game require very little knowledge of algebra, or even knowledge of the symbols for numbers -- but they introduce the various algebraic rules one by one. Gradually, the pictures are replaced by numbers, and by the end, people are solving the sort of equations one sees in a prealgebra course.

More information on Dragonbox as a potential recommendation is at the link.

### Top recommendations for getting started (with prealgebra)

If this description fits you ... | ... this might be the best recommendation | Cost |
---|---|---|

You want a problem solving-style introduction to the basic techniques, along with supplementary videos and an online practice system. | The Art of Problem Solving book Prealgebra by Richard Rusczyk, David Patrick, Ravi Boppana. The book is part of the Art of Problem Solving (AoPS) book series that has been reviewed very favorably, for example, on Quora. The prealgebra book is accompanied by a series of free videos, which cover all of the topics in the textbook. It is synchronized with the free adaptive online learning system Alcumus. AoPS also offers online classes (paid) that are synchronized with the contents of the books. | ~$50 for the book Videos are free Courses: $285 each for Prealgebra 1 and Prealgebra 2 |

You want a conceptual and visual introduction to algebra that is structured more like a reading experience than a workbook. | Vision in Elementary Mathematics by W. W. Sawyer, ISBN 978-0486425559, Amazon paperback |
~$12 (Kindle), ~$13 (used hardcover, may not last for long), ~$27 (paperback), ~$17 (mass market paperback) |

You are looking for a relatively straightforward introduction to the subject that is very similar to the school curriculum (this may be ideal for students who are not particularly interested in mathematics, or for people learning very far ahead; it can also be used to supplement the above recommendations). | Consider the courses offered by ThinkWell (pre-algebra) and Live Online Math (pre-pre-algebra and pre-algebra). ThinkWell is a little more fast-paced, whereas Live Online Math is somewhat more slow-paced and has more in-video pauses that give people more of an opportunity to digest material. |
$125 (plus extra cost for physical workbooks and CDs of videos) for ThinkWell $100 (plus extra cost for supplementary live tutoring) for Live Online Math. |

### Top recommendations for mastering a range of advanced algebraic techniques (high school algebra and beyond)

If this description fits you ... | ... this might be the best recommendation | Cost estimate (values are approximate) |
---|---|---|

You want a problem solving-style coverage of high school algebra techniques. | Introduction to Algebra by Richard Rusczyk. The book is part of the Art of Problem Solving book series that has been reviewed very favorably, for example, on Quora. It is synchronized with the free adaptive online learning system Alcumus. AoPS also offers online classes (paid) that are synchronized with the contents of the books. | ~$50 for the book Courses: $315 each for Algebra 1 and Algebra 2, $440 for Algebra 3. |

You are looking for an old-fashioned book that covers a wide range of algebraic tricks quickly with as little fluff as possible. | Higher Algebra: a Sequel to Elementary Algebra for Schools by Henry Sinclair Hall and Samuel Ratcliff Knight, Amazon paperback. This is considered a classic resource for advanced algebraic tricks. |
~$20 |

You are looking for a book by mathematicians that covers advanced topics in depth and goes into the logic and motivation behind things. | Algebra by Israel M. Gelfand and Alexander Shen, ISBN 978-0817636777, Amazon paperback. The book has received highly positive reviews, and co-author Israel Gelfand is a highly renowned mathematician. |
~$22 |

You are looking for courses in algebra that closely parallel what's taught in schools | Thinkwell's Algebra 1 and Algebra 2 courses Live Online Math's Algebra course |
$125 for each Thinkwell course (total $250) + $25 for CD set for the course (total $50) ~$100 for the Live Online Math video course (plus more for supplementary tutoring) |

## Comments on accelerated learning of algebra for gifted children

In general, we believe that our recommendations above will continue to hold for gifted children learning algebra at a younger age. Some differences:

- They may need to spend more time on Dragonbox first before they are ready to formally begin learning algebra, to make up for the lack of multi-year exposure to symbol manipulation.
- The imbalance between their verbal and mathematical skills (which could go either way) needs to be kept in mind when selecting appropriate resources.

For more on general factors to keep in mind when adapting resources for use by accelerated students, see recommendation use for accelerated students.

You might also be interested in the section on accelerated learning in algebra learning benefits.