We started in the early 1990s with what seem like ancient materials now and materials and methods have changed dramatically since then up through the curriculum. The National Council of Teachers of Mathematics standards released in the late 1980s through the mid-1990s sparked the Math Wars between traditional textbooks and reform textbooks. Traditional textbooks emphasiz procedural mathematics emphasizing cookbook approaches to solving specific problems. Many homeschoolers used a textbook series referred to as Saxon Math (from Saxon Publishers which was in the traditional camp.

The reform approach teaches concepts at the expense of procedural skills and emphasizes conceptual understanding, the ability to communicate mathematics, relationships between concepts and connections between representations. An example of connections between representation would be in teaching fractions, ratios, decimals and percentages as related representations. An example of reform texts would be the Scott Foresman Exploring Mathematics series from the 1990s. We have a set of these for K-8 and used these texts with our children.

The battle over approaches to teaching math also involved an approach known as the New Math which was an approach that begain in the 1960s. This approach posited that teaching set theory, functions, abstraction and topics such as number bases should be be done early so that children could learn theorems easily later on. The New Math has been the object of scorn and ridicule in the United States and I've read that some of the materials and teacher training were pretty bad. We used the textbook series Sets and Numbers written by Patrick Suppes with our children. This series was out of print so we asked Patrick Suppes for permission to copy the texts which he gave us. We interlibrary-loaned the texts and photocopied the textbooks.

Another approach that has gained momentum in the new century is the use of the Singapore Math curriculum. Singapore is a small island off the tip of Malaysia and did very well in the TIMSS (Third International Mathematics and Science Study) tests. The Singapore Math curriculum consists of several very thin student texts, workbooks and teachers books (optional). One of the obvious attributes about the curriculum is that the materials are very inexpensive compared to traditional US textbooks. The curriculum emphasizes practical problem solving, the use of pictures for understanding, the ability to do mental math and a lot of word problems and the homeschooler reviews that I've read on the curriculum have generally been very positive. We've used some of the texts from the Singapore math curriculum and do like them. The textbooks are culturally Singapore but I understand that they have a US-based set of texts now. We used the original texts but the cultural differences were not an issue as our children lived there for a while.

It's pretty obvious that we have a lot of math books in our house for K-8 and that our approach has been to use multiple approaches to teaching math. This approach worked particularly well for our son as he learned K-8 math when he was very young but our daughter has treated math as something that has to be learned instead of something that she wants to learn. I've spent many years doing a selling job on our son for math and I now need to start the same process with our daughter. I think that children need motivation to study mathematics and explaining where mathematics is used in our world is an important process in opening the child's eyes to the possibilities in math.

Over the years, the quantity and quality of materials available to homeschoolers has greatly improved and we now have multiple approaches with which to educate our children in math. When we started out, we mainly used books and my expertise. Later on we used online and college courses and now we're using free videos on the web with our daughter. Some kids do well with self-instruction materials but some also do better with traditional lectures.

In the past, some schools taught the two courses, Trigonometry and College Algebra, to cover the material in Precalculus but the modern approach is to just teach Precalculus.

The topics of high-school algebra are generally taught three times in Algebra I, Algebra II and Precalculus with a little more detail and difficulty each year. My guess as to why it is taught this way is that a lot of things can be going on in the lives of high-school students and that they may miss material or forget material that is covered and that covering the same material three times should result in reasonably good retention. If you have a better explanation, I'd be happy to hear it.

I don't have a good feel for what is currently taught in these subjects as we purchased the textbooks that we use back in the 1990s and I saw no good reason to update them. I know that there was a trend towards integrated textbooks in the 1990s and a major movement to using calculators and computers in teaching these subjects. I'm in the camp where it is okay to use calculators if the student knows what they are doing and if the type of problem requires it. If the student doesn't have proficiency and understanding, then I don't want the calculator or computer doing the work in a magic way for the student.

Other textbooks that have been popular with homeschoolers are the Saxon Math series and the high-school texts from Singapore Math. I haven't seen the two series so I can't really comment other than what I have read about the texts. Parents either love or hate Saxon. I've read good things about Singapore Math and can't recall any negative comments on it. But you'll need to look elsewhere for reviews on these two popular curriculums.

- learner.org (Annenberg Media) has 26 half-hour videos aimed at high-school, college classrooms and adult-learners. The series emphasizes real-world examples and explains concepts that many students have trouble with. Algebra In Simplest Terms. Registration is required and it uses Windows Media Player.
- The University of Idaho has video lectures on Algebra I in Real Player format. I think that these lectures are aimed at a college or adult audience but it may be usable by younger children. The course number is Math 106. Another Algebra course they have videos for is Math 108 which might be an updated version.They also have what appears to be an Algebra Review at Math 143 which looks like it moves more quickly.
- Tallahassee Comunity College has Algebra I tutorials at MAT 0024 Video Tutorials 7th Edition . I have not reviewed these yet.

- Tallahassee Comunity College has Intermediate Algebra tutorials MAT 1033 Intermediate Algebra . These are pretty good for instruction,

- The University of Idaho has a set of Math 143 College Algebra Videos that are well done. They also have video lectures for Math 144 Trigonometry
- Math TV has trig videos in RealPlayer format at Math TV Trigonometry Videos .
- St Petersburg College MAC 1105 College Algebra (registration required)
- Tallahassee Comunity College has College Algebra tutorials at MAC 1105 College Algebra, Trigonometry tutorials at MAC 2114 Trigonometry and Precalculus tutorials at MAC 2140 Precalculus . The College Algebra and Trigonometry tutorials are pretty good. The Precalculus tutorials are more explanations of examples in the book.
- NC State MA107 Precalculus

Perhaps a class or miniclass on proofs before the geometry course would make teaching a traditional geometry course a little easier on the student.

But that may be moot point anyways as I've heard that the proofs part of the traditional geometry book has been removed. Geometry may be integrated today as well - I don't know as it's not an approach that I care for.

We used Jacob's Geometry and and it's good. You can see the reviews at Amazon.com Geometry by Harold Jacobs. I recommend the Teachers Edition to save parents time in correcting the exercises.

I had a look around the house for something useful to gradually introduce proof to teenagers and I found two books: The Nuts and Bolts of Proofs by Cupillari and Introduction to Logic by Copi. Unfortunately, they're both aimed at undergraduates. Cupillari is terse while Copi is deals with argumentation and debate ahead of deduction. Copi is an excellent book but unsuitable as a gentle introduction to proof ahead of geometry.

I think that the material covered in Article on Mathematical Logic for the Schools by Patrick Suppes provides some helpful background for reasoning and proof. This book was written in the 1960s and you won't find it on the web. It also uses language from that era so students may feel that they are in a time warp from the language. You may be able to find it on interlibrary loan and photocopy it if you can get permission to do so from Suppes. I have a copy somewhere at home and the permission email from Suppes that I received in the early 1990s. I think that an introduction to proofs book aimed at middle-school students would make for a great writing project.

A member on the home-ed mailing list suggested Introduction to Geometry by Richard Rusczyk. I had a look at this text as a coworker's daughter used it and liked it. The book seems to be aimed at middle-school students from the size of the type and language. It looks like a very readable text but it does not appear to me to be a complete geometry textbook in the sense that Jacobs' text is. The website does state that the book can serve as a complete geometry course but I think that I'd disagree with this. My coworker's daughter worked through this text in a month and has taken a formal high-school geometry course afterwards.

- The Foundations of Geometry by Hilbert
- Foundations of geometry for university students and high-school students by Ruslan Sharipov.

I've read that some colleges and universities may be denying college credit for good AP exam results and I wouldn't have a problem with this. I think that it is possible to cover three semesters of calculus in a semester and a half but it would take the average student a lot of time and effort which could have adverse impacts on other courses that the student is taking. The homeschooler may be able to cover the material with self-study or using an online AP Calculus program such as the one offered by PA Homeschoolers. Another option is to take three semesters at a nearby college or university.

If taking calculus at a college or university, I strongly suggest looking carefully at the course offered. I have seen colleges that offer a calculus for science and engineering majors and also offer a calculus sequence for non science/engineering majors and I think that these courses should be avoided unless that's what you really want. Some colleges use a Calculus A, B, C, D sequence as equivalent to the normal Calculus I, II, III sequence. I've seen other colleges use the designation Calculus I, II, III and also the designation Calculus I for Science, Calculus II for Science, etc. I think that the best thing to do is to look at the number of credits for the course. It should be four credits and if it isn't, you may need to examine the course description to see what you're getting.

One other option is the honors approach. The honors course in high-school seems to have been on the decline for quite some time as taking an honors calculus course would probably mean that you wouldn't perform as well on the AP Calculus exam. There is a lot in the way of applications in AP Calculus and taking time for theory takes away time from spending time efficiently to get the best possible AP exam score.

Many universities offer an Honors Calclulus course and these are typically heavily theory-oriented. Common textbooks used in these courses are Calculus by Michael Spivak and Calculus by Apostol. These books have a long lineage compared to books in common use today. For reference, I took Honors Multivariable as a freshman and we used Calculus by Salas and Hille from the 1970s. Apostol is a tough, tough book which teaches theory and applications though the book is dated. I prefer Spivak as it provides a nice presentation to a difficult approach.

For standard college coverage, we have Osterbee and Stewart. I think that Stewart is by far superior to Osterbee in topics covered and ease of understanding. I think that it's not that bad an idea to have a few calculus books around, especially if your children are going the self-study route. One other point about Stewart and Osterbee is that you can buy Student Solutions Manuals for them which can make the texts easier for self-study. One other thing that I found about Stewart is that many of the problems in the book have solutions on the internet. Just type in a few words from the problem in Google to see if your particular problem has a solution publically available.

And one honorable mention textbook on our shelf is Calculus by Sherwood and Taylor, 1946. A book with applications and proof that is far smaller and lighter than your typical modern $150 textbook. This book cost $3.75 new according to the inside cover.

One site that may be useful if you have Stewart is Stony Brook Mathematics Calculus Web Pages which has syllabus and problem information for many calculus variants. We've found the syllabus for MAT 131 Calculus I Fall 2005, MAT 132 Calculus II Fall 2005 and MAT 205 Calculus III Spring 2005 useful. For those considering a theory approach using Apostol, there's MIT's OpenCourseWare 18.014 Calculus with Theory I, Fall 2002 which provides lecture notes, recitations and assignments and the course layouts at Stony Brook Honors Calculus I and Stony Brook Honors Calculus II.

Useful Calculus Links:

- The Wolfram Integrator which will symbolically integrate expressions. It doesn't provide an explanation of what it did though.
- Using the TI-83 Graphing Calculator The TI-83+ Graphing Calculator can be required for math and science courses in high school and college. Personally I hat the concept of buying a brain-dead and expensive piece of technology that doesn't have modern technology because schools are afraid of students cheating. I'd rather use a laptop with Mathematica or GNUPlot or an HP calculator or a quality PDA. At any rate, this site has a lot of useful stuff about using the TI-83.
- Calculus by Gilbert Strang online textbook in many pieces from MIT Open Courseware. Also includes a student study guide and a teachers edition which has answers.

- The Illinois NetMath distance education program at the University of Illinois at Urbana Champaign uses the curriculum from Math Everywhere, Inc that runs on Mathematica. The courseware modules cost $45 which is very inexpensive compared to college textbooks sold in the United States. Mathematica has a number of purchase options for students though I've found their policies on transferring licenses a bit hard to work with. The courses are visual and hands-on and students receive help from online mentors under guidance from faculty. There is a lot of Mathematica courseware at http://calcand.math.uiuc.edu/courseware/. I don't know if that's what they use for the courses but the materials there could be used for teaching mathematics starting at the precalculus level. Costs run about $278 per credit hour at the time of this writing which is average for state universities (my experience). I'm an old-fashioned guy and prefer the textbook-lecture or just textbook-alone style but this course offers college credit and the ability to start and end at any time. We're pretty happy with our son's results and the rigor of this program.
- Distance Calculus at Suffolk University is more or less the same program that NetMath offers but the mentors are Professors instead of students. Costs are about double those of NetMath but it appears to me that they have much less paperwork to get in and only the final exam needs to be proctored. I had a conversation with the Professor running the program and asked about theory and basically theory that you'd find in honors courses is an endangered species. The courses are difficult and require work but it looks like more application and less theory.
- EPGY at Stanford University provides math education from K12 through University using online software. We're using their Symbolic Logic course right now which is based on a course developed and used from the 1970s. Testing or other demonstration of ability is required to get into their program. They apparently have an online high-school now according to Wikipedia.
- MIT Open Courseware Differential Equations, Spring 2006 an engaging professor and a video camera operator and a wireless mike make the presentation pretty good. I only looked at the first video. This isn't a course for credit but the materials that you'd need, including video lectures, for self-study.

- University of Houston Video Calculus Calculus video topics. They also have PowerPoint lectures for AB and BC Calculus at PowerPoint Lectures. Our daughter likes the pace (pretty slow) but not the presenters accent (southern). I have no idea why she doesn't like southern accents.
- Lamar University has a Calculus Course in the form of many Windows Media files. I had a look at the first review file and it looked like someone was just writing on a piece of virtual paper.
- The Calculus Lifesaver videos from Princeton are meant to be review sessions for Calculus I and II. I had a look at the first video and the guy looks to be a good presenter, knowledgeable and enthusiastic. I will ask our daughter to take a look at these too. They move fairly quickly and he assumes that you know high-school math well.
- NC State Calculus for Scientists and Engineers I and II, and Calculus III using Stewart.
- Tallahassee Community College has Calculus tutorials at MAC 2311, 2312, 2313 Calculus with Analytic Geometry I, II, III. These are pretty good.
- RPI's Prof. Schmidt works out solutions to problems in Boyce and DiPrima though it is for the 7th edition of the book. We have the eighth edition which is somewhat similar. There are some nice non-multimedia resources at Math 2400, sections 5-8 INTRODUCTION TO DIFFERENTIAL EQUATIONS.

Discrete Structures or Discrete Mathematics is usually a requirement for Computer Science degrees and it may be the first course where the student encounters working with proofs. I've seen versions of the course that are relatively easy and versions that are quite difficult. The most common textbook that I've seen used is Discrete Mathematics and Its Applications by Kenneth Rosen and I do like this book (we have three different editions of it on our home library) though using it for self-study would be rough.

I prefer the book Discrete Structures, Logic, and Computability, Second Edition, by James Hein for self-study. It also cover more material though some of it is at a superficial level. I think that having both of these books around is a good way to go for self-study or studying it using a university course.

The books that I used for Discrete Mathematics were Discrete Mathematics with Computer Science Applications by Skvarcius and Robinson at the undergraduate level and Mathematical Structures for Computer Science by Gerstang at the graduate level. I would generally recommend the newer texts over the older texts.

Our son took Discrete Structures I at the University of Massachusetts in Lowell using Rosen and then took Discrete Structures II but that course had a modern algebra approach which a lot of the students had trouble with. It was Mathematica-based and very visual and that approach may have made it harder to understand the material. He ultimately withdrew from the course because he had too heavy a workload and because of health reasons.

A year later, though, I had a look in my old discrete math book by Skvarcius and found the topics covered in our son's second course. I recall that that materal was pretty easy when I took Discrete Math and had our son read the chapter. He found the approach quite a bit easier to understand. Modern Discrete Math books seem to skip coverage on modern algebra or cover it very, very lightly. So perhaps there is some benefit to using a little material from an older book now and then. The Skvarciue book is out of print according to Amazon but I imagine that there are lots of used copies floating around.

He is currently taking Discrete Structures II using another textbook.

A good book that covers Discrete Structures is Concrete Mathematics by Knuth, Patashnik and Graham. This is a fairly difficult text to go through but is great as a reference resource or if a student wants to read about a lot of interesting mathematics. It's also a great resource for those taking Discrete Structures courses in college.

One thing that I've found is that programs in Discrete Structures vary widely in what they cover. In my undergraduate course, we covered a little Abstract Algebra which was also covered the first time our son took Discrete Structures II. It seems that there are many courses around that don't include this material. I think that predicate calculus is also lightly covered in discrete structures courses; and sometimes not covered at all. My feeling is that the more math, the better, and that university students should supplement what they learn in their courses by self-study or available video lectures.

Online Textbooks

- Graph Theory with Applications by Bondy and Murty, an introductory text.
- The University of Hamburg hosts the graduate text Graph Theory, Third Edition by Reinhard Diestel.

- MIT Open Courseware has readings from their course at 6.042J Mathematics for Computer Science, Fall 2005. This course provides lecture slides, course notes, problems and solutions.
- UC Berkeley has an audio course at Math 55 Discrete Mathematics (in progress as of October 2007). I don't know how useful an audio course is on this topic as you need to see things on a board but this course might have some use for students interested in the topic.
- ArsDigita University has videos on Discrete Mathematics in Real Player format. The course texts are Discrete Mathematics and its Applications by Rosen and Concrete Mathematics byt Graham, Knuth and Patashnik. The course site is Discrete Mathematics . The instructor is dynamic and seems to be able to get his material across. I reviewed parts of the first lecture and it looks like a rigorous course. Highly recommended.
- Texas A&M has the course Math Problems II and the
course website is at Math 646
- Mathematical Problems II with lectures and
other course materials. The course deals with a lot of interesting
problems but the topics correspond with many of the topics in Discrete
Structures and might be interesting for the student of Discrete
Structures.

- Elements of Abstract and Linear Algebra by Edwin H. Connell at the University of Miami.
- Robert B. Ash at the University of Illinois has Abstract Algebra: The Basic Graduate Year, aimed at graduate students.

- Robert B. Ash at the University of Illinois has Complex Variables.

I took an undergraduate course in Linear Algebra many years ago and just remember working with vectors and matrices. I've had a look at the NetMath Linear Algebra course from UIUC and the material in that course is more difficult and covers a number of topics that weren't covered in the course that I took. Notably image compression. Our son likes the online course at UIUC and I think that it's pretty good (coming from one without a strong background in the topic) as an introductory course if you can deal with the issues of online courses.

Additional resources

- MIT Open Courseware 18.06 Linear Algebra, Spring 2005 with video lectures. A textbook is used in the course. The courseware includes readings, assignments, exams and video lectures. The video lectures are jerky and the presenter could be a little more polished but I think that the videos could be used to supplement self-study with the textbook and other course materials.
- Linear Algebra by Jim Hefferon provides a free, online Linear Algebra textbook that can be downloaded. Answers to all exercises are provided in a separate download.
- A First Course in Linear Algebra by Robert Beezer, University of Puget Sound.
- Introduction to Matrix Algebra by Autar Kaw at the University of South Florida. It appears to be an algebra-based text.
- Advanced Linear Algebra lecture notes, by Keith Matthews

Number theory can be a lot of fun for younger children as patterns can be explored by curious children without turning them off with formal proofs. Secondary school children may be able to handle formal proofs and maybe even the material from Concrete Mathematics.

- For some quickie undergraduate material, see MIT Open Courseware Computational Math lecture notes on Number Theory, and Theory of Numbers, Spring 2003 which provides handwritten notes of the lectures.
- Elementary Number Theory online textbook from W. Edwin Clark at the University of South Florida.
- Theory of Numbers by Leo Moser. An online textbook for a second course on number theory.
- Algebra and Number Theory lecture notes from A. Baker, University of Glasgow. No-nonsense presentation that looks reasonably easy to read.
- Elementary Number Theory by Peter Hackman. It looks useful as a supplement to a Discrete Structures course. It's terse as a book and flowery as lecture notes.
- Elementary Number Theory Course at Harvard Fall 2001 by William A. Stein. A lot of nice notes useful for Discrete Structures including RSA encryption.

- Stat 2 Introduction to Statistics (Fall 2006) Course lecture videos from Berkeley. The instructor seems to operate in a start and stop mode and seems a little awkward. This is an algebra-based course.
- De Anza College has an algebra-based stats course with video lectures at Elementary Statistics.
- Applied Probability is a short program at ArsDigita University with lecture notes, videos and problem sets. I haven't reviewed this yet.
- Dartmouth has produced a nice textbook titled Introduction to Probability which is the most rigorous probability/statistics online textbook that I've seen. There are quite a few out there with no theory and this book stands out for being brave enough to include integral signs in the text. Mathematica modules for learning with the text are available at the University of Wisconsin-Parkside. Wolfram makes their Mathematica Player available for free now and maybe students can use the modules without having to buy or rent Mathematica.
- Robert B Ash at the University of Illinois has lecture notes on statistics at Lectures on Statistics with problems and solutions.
- Introduction to Probability, an online textbook from Oilver Knill at Harvard.
- Stanford University has Lecture Notes for their course Introduction to probabilistic systems analysis at Lecture Notes for EE178 and handouts including problem sets and solutions and exam sets and solutions at Handouts.
- St Petersburg College STA 2023 Elementary Statistics

The Role of Axiomatics and Problem Solving in Mathematics, CBMS 1966 A rather interesting symposium on teaching high-school mathematics using axiomatic approaches. This stuff is dinosaur territory these days as the math teaching world is running and screaming in the opposite direction.

We used Houghton-Mifflin Video DVDs for precalculus for our daughter. They are designed to accompany Houghton-Mifflin textbooks, perhaps if the student misses a class or two. We borrowed the DVDs from a university library and our daughter indicated that they are well-done. She had covered most of the materials when she started going over them for review. There are 12 DVDs for Precalculus. It appears to be a college series and I did see calculus DVDs. This might make for a useful option if you have access to these DVDs in an inexpensive form. I wasn't able to locate the DVDs for purchase so I don't know what they cost.

The National Library of Virtual Manipulatives has a number of visual examinations into patterns in the areas of numbers, operations, algebra, geometry, measurement, data analysis and probability. These are done in Java and span from K-12. They look like interesting math games.

There is a very nice page at NYU with pointers to free online textbooks and other materials. The material tends towards undergraduate math which might be interesting to some homeschool students.

The Centre for Innovation in Mathematics Teaching has an amazing set of teaching materials including textbooks, lesson plans, slides, tests and practice books at their site, all available for download or interactive practice. Levels run from K-early college.

NC State University has videos for their course Mathematics of Finance which look pretty interesting and may be a nice break for science and engineering students.

Colorado University at Denver has Windows Media videos on a range of math topics including Linear Algebra and Differential Equations.

- Lecture Notes of William Chen at MacQuarie University. Topics are Elementary Mathematics, First Year Calculus, Discrete Mathematics, Linear Algebra, Miscellaneous Topics, Multivariable and Vector Analysis, Complex Analysis, etc. The Discrete Mathematics notes look very good.

The Mathematical Sciences Research Institute has a bunch of videos at their Streaming Video page and at their VMath Video Archive that cover many areas in mathematics, computational mathematics and other areas of interest to mathematicians. It seems to be aimed at the graduate level and above but some homeschoolers and undergraduates might enjoy the lectures. The videos require RealPlayer and handouts generally accompany the video presentations. The nice thing about these presentations is that the presenter assumes that the audience is comfortable with mathematics and can relax somewhat in presenting the material. I think that these presentations may be interesting to homeschooling children to see people that are very comfortable talking about mathematics.

Knuth has a bunch of videos that might be interesting to the homeschoolers with a heavy interest in mathematics and related fields. These are categorized by Musings, Problem Solving, Mathematical Writing and Other.

MIT Open Courseware has a huge number of cours materials and some of these courses have videos.

- Mathematical Methods for Engineers I, Fall 2005 Linear Algebra review, differential equations of equilibrium, calculus of variations.
- Mathematical Methods for Engineers II, Spring 2006 Numberical methods, initial-value problems, network flows, optimization.

Updated on May 26, 2008.

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