Vector64 Home Education Math Pages (perpetually under
This page contains general math learning materials and
references to materials and websites that we've used or are using in
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
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.
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
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
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
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.
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
- learner.org (Annenberg Media) has 26 half-hour videos aimed
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
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
which might be an updated version.They also have what appears to be an
Algebra Review at Math
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.
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.
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
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
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
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
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
taken a formal high-school geometry course afterwards.
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
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.
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
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:
Wolfram Integrator which will symbolically integrate
expressions. It doesn't provide an explanation of what it did though.
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.
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.
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
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.
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
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.
Open Courseware Differential Equations, Spring 2006
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.
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.
- 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
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
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
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.
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
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.
Probability and Statistics
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
- De Anza College has an algebra-based stats course with
video lectures at Elementary
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
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.
to Probability, an online textbook from Oilver Knill at
- 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 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.
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
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
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
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.
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.
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.
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
Knuth has a bunch
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,
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