Vector64 Home Education Math Pages (perpetually under construction)

This page contains general math learning materials and references to materials and websites that we've used or are using in our Home Schooling program. 

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.

Secondary School Math

The subjects traditionally covered in secondary-school math are Algebra I, Algebra II, Geometry and Precalculus. Many high-schools also offer Calculus  but I'll discuss Calculus separately.

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.

Algebra I

The textbooks that we have used and are still using are traditional. We've used Harold Jacobs' Elementary Algebra and I still love the text for the many enjoyable puzzles, jokes, comic strips, fun facts and math-related artwork. It's dated of course and the current events items may feel foreign to teens today but it has a feel to it that I like for teaching. Of course it won't have calculator and computer exercises, the fancy pictures and the surprising heft of modern textbooks but that's the price that you pay for using old and proven materials. I'm not going to do a review of the text as there are may good reviews already out there. I like the review at PA Homeschoolers: Review of math textbooks by Harold Jacobs. Parents might want to consider the Teacher's Guide which is really more useful as an answers book. You can find a table of contents at WH Freeman Elementary Algebra by Harold Jacobs.

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.

Multimedia Resources

Algebra II/Intermediate Algebra

Finding Jacob's Elementary Algebra made for a simple decision on Algebra I but it made it comparatively harder to find an Algebra II textbook. I had a look at Stanford's EPGY program and found that they used textbooks by Lial and Miller and picked Intermediate Algebra by Lial, Miller and Hornsby for this subject in the Sixth Edition. It is a reasonable text though the edition that we have is fairly small by today's standards and was devoid of the use of technology (calculators and computers) other than an early section on how to use a scientific calculator. It appears that Intermediate Algebra is separate from Algebra II though we combined the two.

Multimedia Resources

Precalculus

We use Precalculus by Lial, Hornsby and Schneider and it appears to be the first edition from 1997. This book does weave in graphing calculator exercises and exploration and adds and is moderately bigger than the Intermediate Algebra book. The current book by Lial seems to be pretty popular and seems to have good reviews at Amazon. I believe that Precalculus is a combination of the older College Algebra and Trigonometry.

Multimedia Resources

Geometry

Geometry used to be the class with an introduction to proofs for high-school students. My personal feeling is that this approach has its advantages and disadvantages. The advantage is that the student has had algebra and can understand moving from expression to expression and provide the reasons why they can do so so that they should be ready for proofs. The disadvantage is that the student is essentially learning two subjects, Geometry and Proof at the same time and that may be a bit overwheling for some students.

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.

Multimedia Resources 

Online Texts


Calculus

The calculus scene is a little complicated today given the amount of clout that the Advanced Placement Calculus Exam has obtained over time. The premise is that the Advanced Placement Calculus Curriculum and achieving a high score on it is equivalent to two or three semesters of college calculus. A complaint about AP Calculus is that having a test results in high schools teaching to the test at the expense of everything else. Indeed, there is a lot of material to cover for the BC version which covers three semesters.

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.

Self-Study

For self-study, you can simply go straight through a textbook reading the lessons and doing the exercises. But actual college calculus courses seldom, if ever, work this way. Generally a selection of problems from the text or the professor are assigned. So you could actually just get a syllabus with readings and problem assignments off the web to approximate a college course. Of course you don't have the lectures available but many texts can be used reasonably well to learn calculus.

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:

Online Courses

In the fall of 2006, our son wanted to take a Calculus III course but finding a classroom version was difficult as Calculus III is usually taught in the fall. We could of course, have gone to the University of Massachusetts at Lowell but the cost would have been over $4,000 and it would have met four days a week which would have meant a lot of driving for me. So we looked at online courses and found three interesting options.

Multimedia Resources

Discrete Math/Structures

Speaking very simply, Discrete Mathematics is the mathematics of countable sets. For a much more complete and accurate description, please see the Wikipedia entry for Discrete mathematics.

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

Multimedia Resources

Abstract Algebra

Online textbooks

Complex Variables

Online textbooks

Linear Algebra

From Wikipedia:

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis. Linear algebra also has a concrete representation in analytic geometry and it is generalized in operator theory. It has extensive applications in the natural sciences and the social sciences, since nonlinear models can often be approximated by linear ones.

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

Number Theory

Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. (from Wikipedia Number Theory entry  This subject can be taught at a number of levels including elementary and secondary school. The folks at Art of Problem Solving have the book Introduction to Number Theory by Mathew Crawford which is aimed at secondary school students. Graham, Knuth and Patashnik have a very nice section in Concrete Mathematics, Second Edition, Chapter 4. Some Discrete Structures books do have sections on number theory though Concrete Mathematics is the best that I've seen at the undergraduate level.

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.

Resources


Probability and Statistics

Resources

Miscellaneous Resources

The remainder of this document will list various resources that we have used in the past and how they may be useful to others. If you have materials to suggest, drop me a line for consideration. For now, you can refer to My old math page for my resource lists from the past.

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

Videos

Some students like videos and some hate them and the current state of the art of free, online videos on the web aren't great but they may be useful for students that want to learn materials with a presenter. I will try to add videos here when I run across them.

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.
Updated on May 26, 2008.
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