Math1.doc Dale R. Reed Seattle, Washington USA firstname.lastname@example.org 3-26-2000
Suggestions for helping your children enjoy mathematics
(and be better thinkers)
This is a collection of math suggestions from mostly homeschoolers. Few of the "I"s and "me"s in this document are me(Dale). The document is growing as a result of my posting it to various discussion lists requesting people to reply with what they do about math in their families. Also I added some(remember I have not checked out all these suggestions so buyer beware) ideas from the SeattleTimes/Real_Life/Parenting column. Please treat this document as a menu to choose from in your attempts inspire your children to enjoy using mathematics.
This is a list for all of you to use in whatever manner you wish. Each and everyone of you can rename, edit, cut and paste, add you own ideas, include it in your newsletters in whatever you want. It has been written to benefit you and your children.
The purpose of these suggestions is to help your children develop their mathematical skills and gain an appreciation for the fact that Mother Nature and all her wonders can be better understood and enjoyed with mathematical analysis. Also mathematics is a universal language that different generations and cultures can use to communicate with each other.
It is my opinion that what separates us from animals and robots is the creative use of our brains. I think that we must consciously develop our brains ability to think new thoughts by exercising are brains in our work and play. I think that strong flexible brains are made by thinking exercises. For the same reason that every(well most) day I lift weights, do twenty chinups and juggle balls, rings, and clubs to keep my upper body strong and flexible and I walk briskly for about an hour to keep my lungs and legs in shape.
So let us get to work.
Five days a week, about one half hour a day, the student should work through a mathematics text book. He or she should maintain a notebook of carefully/neatly worked problems done with a quality mechanical pencil, pink-pearl eraser and a straight edge. Each homeschooled youngster should keep this notebook with his grammar, history, and other notebooks that make up his portfolio of studies and accomplishments.
Prior to purchasing a book be sure to evaluate your child’s capabilities. For example Saxon provides placement tests that you can download at http://www.Saxonpub.com. They have several different levels from K-2 up to Algebra 2. Even if you don't want to use Saxon, the middle grades test do a great job of showing up weak areas such as fractions or decimals.
Before buying a math book borrow(from another homeschooler or the library) a copy and use it at home for a couple weeks. Also be aware that all(including Saxon) materials, change over the years. You might like an older version but not the newer one or visa versa. Be aware that many(especially Saxon) textbooks are very expensive but built tough so you can often buy used at local bookstores and/or online. Many homeschooling Mom’s, sometimes with their children’s help, have collections of used books for sale.
I started homeschooling my daughter in 1st grade and continued on with Modern Curriculum Press workbooks for math because that is what she was used to and Saxon for the lower grades was far beyond my budget! They were ultra simple and easy to use for both of us. She is in 3rd grade now and enjoys math. She seems to be able to move more quickly that the MCP books so we gave an ACSI workbook a try. So far so good.......
My kids and I have been using Open Court for 3 years now, and I am impressed with it. We used Saxon prior to that (early grades), and I did value the attention given to repetition with math facts and use of manipulatives. My oldest wanted more "real world" problems, so we decided to use Open Court, which - according to the owner of Math "N" Stuff in Seattle is not out of print, just being printed in limited amounts, since SRA/McGraw Hill bought it. The SRA/McGraw Hill program is almost the same as Open Court, just a bit more glossy and high grade paper, hence more expensive. We really like the "Thinking Stories" that are part of the program. I was fortunate enough to get 4th and 8th grade when I needed them... and we loved them! My youngest is doing the 8th grade level now (at age 11) and enjoying it. I liked using the 8th grade one with my son... made it so he didn't have to do Saxon Algebra 1/2... went straight to Algebra I. And THAT is one of the main reasons he can do most of the problems in his head. Open Court is KING when it comes to encouraging mental math... they even teach shortcuts for things like 43x47... by the time you are done, you KNOW how to complete a square in factoring polynomials, and it makes sense.
First of all, Open Court math books are not
workbooks/worksheets. They are textbooks that you don't write in. You just
write the answers down on a piece of paper, and some daily lessons are ALL
discussion or experimentation. These books involve the kids in discussion like
you'd expect from a literature course or something... they're something else!
The fourth grade book, for example, starts out by having kids (and parents!) estimate how many apples are in a bin. You get to make successive guesses as they give you more and more information, and you learn a lot in the process. There are pages where part of the problem is blotted out by "accidental" ink spills, and the kids figure out what they can still determine, given the information left. There are problems for which "Cannot determine from available information" is a valid answer. It really IS "real math"... from real life... with just enough silliness thrown in to make you smile while you're doing it.
McGraw Hill has several different math series. So far our favorite is the Explorations and Applications.
Shane's been working in Harold R. Jacobs, "Mathematics: A Human Endeavor" for a year and loves it. It doesn't dumb him down and he's loving Math. It's challenging, but he keeps at it. Jacobs also published Algebra and Geometry books that are worth considering.
A math book I have used with both kids and loved is Family Math by Jean Stenmark, Virginia Thompson, and Ruth Cossey. Lots of easy to make and play games for all levels (mostly elementary, but some middle school). My kids at 7 had a hard time translating math to the written page, so we just didn't do it. If you think about it, it is a hard concept, taking a concrete concept such as 7 apples minus 2 pears and reducing it to symbols on a written page.
So, for younger kids, the more concrete and hands on, the easier math is to grasp. We played dominoes and added totals (good for adding into the 100s), counted and graphed stuffed animals and toys, used large size number lines and hopped from number to number by 2s, 3s, 5s, etc.
For fractions, I bought a set of "fraction bar games", which are plastic bars divided into halves, thirds, etc. There are suggestions for games to play to familiarize players with the concepts of equivalencies, adding fractions, naming fractions, etc.
We also did a lot of oral word problems early on, even with multiplication and division (although they didn't know they were doing that). To pass time in car rides, or other boring situations they often beg me to give them word problems. Only they called them "Betsey problems" because they often featured a character named Betsey.
Other fun books are Math for Smarty Pants and other books by Marilyn Burns, though those are more for 9 and up (in my experience).
I started coming up with a mental list of situations where
Saxon works, and where Saxon doesn't work.
First, Saxon uses a repetitive review approach. You keep doing a few of the same kinds of problems nearly all the way through the book. As such, it is designed to help kids prevent mathematical facts from falling out of their heads. Some kids need this (and those kids know they do); some kids don't. Some kids, in fact, have minds that NEVER forget a math fact. I was one of those kids, and Saxon would have driven me batty if I had to use it to learn from. (Now that I already know algebra, I do the problems alongside my son for recreational purposes, but I'd have hated to learn from it.) Face it, some kids like to tackle a new concept, nail it down, and move on... using it as necessary, but never having to LEARN it again. Saxon isn't good for those kids.
Also, there are other math books that use the repetitive learning approach. Rod and Staff (which, unfortunately, only goes through 8th grade, but is a good pre-algebra book at that level) not only reviews continuously, but notes, with each review problem, what lesson the concept was originally taught. In addition, it has WONDERFULLY comprehensible explanations. When my son was in 4th grade and struggling with adding fractions with different denominators, I turned to the 8th grade Rod and Staff book. There were 2 pages of explanation, which my son instantly understood. We'd spent hours and hours and pages and pages trying to get this through his head (and I'm a good explainer). So I was impressed.
University of Chicago's math program puts out another continuous review higher math series. Their book is preferred over Saxon by many public schools. It is more colorful, more engaging, and it doesn't neglect showing kids a REASON for learning all this stuff.
Which brings me to another drawback/asset (depending on the kid) of Saxon's books. Saxon is intensely effective on training kids to follow steps to a solution. They memorize the steps. They do not always get the concept behind the steps, unless they are naturally geared that way. My son very conceptual. He does Saxon math problems in his head and skips the steps. (We race to a solution, and he sometimes wins.) But he HATES writing down the steps, passionately. We still use Saxon, but we adapt it.
If you have a kid who whines, "But why do I have to LEARN this? What use is it?" and you are attempting to use Saxon, be forewarned that you will have to supplement the text to answer those questions. University of Chicago's stuff answers them while it teaches.
Saxon also neglects practical math. Its problems touch richly on scientific applications, but not on day-to-day mathematical uses. Your child will not learn to balance a checkbook, read a meter, complete an income tax form, calculate interest on a loan, or maintain a revolving credit account wisely. These things are easy to teach in a homeschool setting, but remember you will have to do it; the text won't fit it in neatly between percentage calculations and factoring polynomials.
Saxon teaches geometry, but barely touches on proofs. I ran across a description somewhere of classical education's main reason for teaching Euclidean geometry: It was a LOGIC course. So... if you want THAT kind of geometry for your kids, you'll need to use another geometry text. And if you are like me, wanting your kids to have exposure to constructing proofs, but not be buried in them, you may find that the Bob Jones geometry book is a happy compromise, with clear explanations and just the right dose of fun. Be warned, however, that BJU does NOT do repetitive review. They expect you to get it the first time around and then be able to use it.
All this said, there are definite instances where I'd recommend Saxon. For kids who don't mind tedium, are geared learning steps before concepts, and appreciate John Saxon's dry humor and don't miss the "why I have to learn this" explanations, Saxon sets mathematical technique FIRMLY in the student's head. For kids who can learn from anything, anywhere, Saxon works well, too, and it's admittedly readily available and easy to resell.
We use A Beka for almost all of our work. We are now starting our 5th year of use and I have found that for my eldest it is just what she has needed. The first 3 years were very colorful and had a wonderful way of explaining the concepts. It has the concept of building math one step upon each previous step with just a touch of review in each lesson. We have now reached a point that she is able to teach herself. I just set the lesson schedule and she knows what she needs to do. My youngest is flying through her math and seems to have a grasp of what it is all about. One of the things I have liked is how they work their way through basic math and gently work a child to beginning algebra with a introduction to the first skills needed in 3rd grade. I have looked at Saxon and it appears to be similar and we should be able to easily switch if we decided to at any point now. I would highly recommend the A Beka for the first 3 years at least. From there it appears that the two are fairly equal.
We use a fair amount of A Beka, but have always avoided their math. I can't say we tried it and didn't like it. We never tried it because the pages looked too busy and they seemed to jump from one concept to another without enough practice in between for the younger grades. However, I have not looked at the upper grades and look forward to hearing the other responses.
2. The three most important math skills are: estimation, measuring and logic/problem solving. Before children can begin to really learn these, or any other "math skills" they first need a strong concrete understanding of math. By this I mean real world experience, not abstractions. Before you can estimate the distance from here to that tree you must have a confident sense of how long a foot actually is. This requires ample time to play and fiddle and figure out how things work.
I(Dale this time) am presently working through “Made Easy” by Silvanus P. Thompson and Martin Gardner, 1998. This first complete revision in over 75 years of the million-copy bestseller—including more than 20 new problems. Those of you that have subscribed to Scientific American know of Martin Gardner’s many popular scientific and mathematical publications over the last 50 years. I bet most 12+ year olds will enjoy learning from this interesting book.
3. Families can play mathematical games together.
One way parents can be creative is to make good use of games. Games are not only educational, they also each children many social skills. children learn how to listen, to follow directions, to agree on rules, to take turns, to plan ahead and to act cooperatively. They also help to bring families together for fun, laugher and communication.
Games: "Tic Tac Twice" (a strategy game) and "True Math" from Aristoplay (800 634-7738) (Their games "Quickword" and "True Science" are great, too) Also, Muggins Math games (they even have algebra games) available from most major home school suppliers.
Other games and activities are Tripoly (a card game involving poker and Michigan Rummy), Cribbage, Pool (great for geometry and physics), Darts (adding and multiplying on your feet, literally). While playing Concentration with my 4-year-old I told him to say the number on the card to himself before turning it over, and that would help him to remember it better the next time around. He followed my advice and greatly improved his game. He also began to carry over this technique to other situations he encountered and his memory greatly improved. Also try Hearts, Cootie, Go Fish, Crazy Eights, Spit and of course Jigsaw puzzles(either done by one person or with everyone working on it) are especially nice for long rainy weekends. There are competitive games like Monopoly and GO and cooperative games like Community, Our Town and Harvest Time for younger children. There are also games families can play that can start a lively family discussion. Such games are Scruples for Kids and the Self Esteem Game. War game is a good mental math game. Both of our boys enjoyed just doing simple problems like; what's half of X, what's 4 times X, etc. When they got stumped, usually they found the answer if we could relate it to money. I have picked up some easy logic material from Critical Thinking Books and Software. But most of their stuff is for much older kids, at least it stumps me quite often!
Krypto is produced by The Making People Happy Games Co., PO Box 1125, Fairfield, CT 06432. To play, each player is dealt five cards with numbers on them. A target card is turned over and then you try to add, subtract, multiply or divide the five cards in such a way that you equal the target card. Julie says, "I got a deck for Christmas from a friend who homeschools. It says for all ages but I think that is a bit optimistic. My seven year old and I tried to play it today. I played his hands, too, but it seemed like it would be a very fun game once we get going with it."
Idea: We have a "Do It In Your Head" policy in our house (most of the time). This morning, my youngest son (12) and I played "Where In the World" (also by Aristoplay). Everyone adds up his own score at the end of the game. Each card is worth 1 to 60 or so points. We each add our cards up in our heads then switch cards to double check one another. He added his 20 cards faster than I added my 16 cards. He also beat me fair and square in the game by identifying 20 European countries on a blank map, including two I had never even heard of.
My kids and I play a lot of "21" or "Blackjack" ... which is great for quick, mental counting. We play with pennies, nickels and dimes and quarters.
Simple Games are: Cards(War, FanTan, Hearts), Dominos, Cards(Rummy) and Triminoes. Make up new rules for dominoes. There are many different ways of playing dominoes that you and your children can invent. It is very important that parents play the games with their children and multiple children can keep score to correct each other. Harder problems can be worked on a calculator but easier ones can be worked in the parents and children's head with older children using advanced scoring levels while younger children can help them and/or confirm some intermediate calculations.
Don't forget the old standbys of Popular Board Games for simple math concepts are: Monopoly(Parker Brothers) and LIFE(Milton Bradley), checkers(regular and Chinese), chess, and backgammon.
Other favorites of one family are: Bowling (ignore the computer and use a score sheet). Tangoes (Chinese tangrams), and Coupons (not really a game, but a neat activity and a way to comparison shop)
"Adult" Games for intermediate math concepts are: Rail Baron and The Stock Market Game from Avalon Hill, Executive Decision, Stocks and Bonds, Stock Market Specialist and Win, Place or Show-Horse Racing Game all from 3M Bookshelf Games.
Gambling Games such as one that uses a Roulette Wheel. Once they understand the big advantage the house has your children will be amazed that anyone gambles in Reno or Las Vegas.
Keno (Our state lottery has Keno everywhere and the State House advantage is obscene. Danny likes to pick four numbers and follow their progress while we eat lunch. We always "lose," so I figure this is a pretty good lesson to learn now, while the money is pretend).
Play calculating games with very young children but use: poker chips, playing cards, dominoes, cuisenaire rods, wood blocks, measuring cups, clocks, calendars, calculators, tape measures, puzzles and tangrams. We keep all of our toys in bins on shelving. Just cleaning the playroom requires sorting and classification. We have a marvelous dollhouse that is made from modular rooms that can be stacked in a variety of configurations. Sometimes we use the poker chips to count, but mostly we stack them in interesting patterns. Since they lock together they will stand in a tower. This makes them infinitely preferable to counting chips. For little kids, games like Candy Land and Cooties work to develop the skills they need--counting and sorting.
4. Parents must indicate their enthusiasm for applying math in their everyday living.
Determine how much cheaper(in percentages) it is to buy food in larger quantities, how much more an automobile costs if you buy it on time, etc.
5. Parents and their children would enjoy learning how to program the computer in LOGO(I have a Logo1.doc File that I(Dale this time) will send you if you whistle) and of course Visual Basic, HTML, and C++ for the older youngsters that are interested in learning a valuable “real” programming language. With Logo your children can create 2d and 3d knots, beautiful fractals(trees and ferns for example), even chaos as they learn. I am beginning to learn how to program with the L3 code from http://www.magicsquare.com. It is very similar to Logo but is easier to program and runs much faster. But L3 has the disadvantage that there is no active user community to exchange ideas and programs with. There is an almost entirely graphical programming language described at
http://www.toontalk.com that some children find interesting.
For computer programs there is Sammy's Science House and Thinkin' Things I, both by Edmark, and Tesselmania by Mecc. Math Blaster offers Basic Drill and Practice in an Arcade Game, School Mom is a Tutorial, and Windows Arithmetic offers Examples, Problems and Solutions.
6. We began getting ready for our homeschool SAT test by drawing a small poster, Math Champion, for Sara & Jon's work areas. They took markers and made these posters as the beginning of our full-on math studies. We all throw in the phrase, Math Champions during our studies to know what we are working towards. I give them silly math breaks: go find something blue, they bring it back and I ask them to add the corners in the object or some such math question. Or, pretend your sister doesn't know how to multiply, teach her! Or, quick run out to the garden pick a bouquet with 2 flowers of each color & tell me how many you have, what's the equation? Just some of the silly things I have to do to keep math fun. The more I do this, the longer the stretches of actual study time.
7. The parents must keep up to date on what tests the children will be expected to do well in the future and make sure the parents and the children can do the problems. There are not only the Certificate of Mastery tests mentioned below but I predict more and more testing for all of us in the future not only to obtain government licenses to do certain jobs but also testing at work. Or to put it another way I expect the end of compulsory education as we shift toward mandatory(at least if you want the job) knowledge and skills.
For instance the State of Washington(and most other states) has its Commission for Student Learning has been publishing samples of the soon_to_be_mandatory Fourth, Seventh, and Tenth Grade Tests. Sometimes there are articles in the paper with sample problems. Of course you can always visit your local government school and find out what math books they are using. Maybe even talk to a teacher for she may be supplementing the material.
8. When taking trips together, over the dinner table… whenever there was a chance I recommend that you set an example of a person who is continually calculating/approximating numbers in your head.
9. Another fascinating and potentially useful mathematical brain teaser for young and old is the study and tying of knots. There is lots of info in cyberspace, for instance http://www.cs.ubc.ca/nest/imager/contributions/scharein/KnotPlot.html
but you can also get started by simply buying some rope, finding a Boy Scout Manual and start tying. My(Dale) two sons and I enjoyed tying knots when the boys were smaller. Especially the knots that tie a fish hook on the end of a fishing line. And tying and untying knots is good for exercising arthritic hands. So it is something grandchildren and grandparents can enjoy doing together.
Exhibit an interest an interest in things mathematical. There are reference books such as Encyclopedias(World Book), The World Book of Math Power, Schaum's Outline series http://www.mcgraw-hill.co.uk/schaum/index.html, and of course The Dover Book Company. For instance Dover publishes the two very interesting books for older students and their parents by Morris Kline, "Mathematics and the Physical World," and "Mathematics for the Nonmathematician." Some older children will appreciate Douglas R. Hofstader’s ideas as published in his many books including “Matamagical Themas: Questing for the Essence of Mind and Pattern,” 1985.
Check out the materials(they will send you catalogs chock full of interesting learning materials) at
And there are hundreds of interesting mathematical sites in cyberspace. Some web sites that I am aware of are
Unbelievably beautiful fractals and interactive exercises such as those at
I am always finding new ones that amaze and offer something to think about. Katy is continually being interrupted from her work by Pooh(that's me) calling her to "look at this Piglet!" as we drink in the beauty of our mathematical world being displayed on the 17" color monitor.
to see that results of the latest research on Mother Natures beautiful secrets.
to learn about some of the things I studied in Antarctica 40 years ago.
To follow the adventures of explorers who use mathematics to find the way back home surf to http://www.goals.com/.
http://www.howstuffworks.com/ has clear explanations complete with excellent graphics about the world we live in.
These daily practices and exposures should wet your future "astronauts" appetite for further exploration of Mother Nature's fascinating world. Dale
PS If you are looking for curriculums, including math, please consider the offerings at http://www.robinsoncurriculum.com/
There is an interesting, mostly negative, review of this curriculum at http://eho.org/features/robrev.htm.