Lifestyle Education through Discipleship™ musings

Some good advice for teaching beginning arithmetic, from the original Teaching Manual for Ray’s Arithmetic (and other common books at the time).

In the first class, besides those who have never studied Arithmetic, put all who have been poorly trained in the elementary processes. It will be economy of time and labor to do this, even if you have to include the entire school in this division for a time; for, in arithmetic, above all other studies of the common school course, it is of the utmost importance that one step shall be thoroughly understood before the next is attempted. [Good advice even in the home. Lay those foundations solid. -lh] The first two years’ training is of more importance than all the rest the child receives.

Do not attempt to have the children use a book in the primary class. —A book should not be used, because no book contains, and no book can be made to contain, the kind of instruction necessary the first year.

Do not teach the figures in the first lessons, and do not allow the children to do any written work ; but teach orally, illustrating every operation, at first, by means of various objects. —The instruction should be entirely oral, and
should deal altogether at first with concrete numbers. The little child can not grasp abstract ideas. It is true you can teach him to repeat, "2 and 2 are 4;’ "2 from 4 leave 2;" "2 times 2 are 4;’ and "4 divided by 2 equal 2." But, without the proper preliminary work, these words can not possibly convey any clear meaning to his mind. This kind of instruction in a primary class is simply machine drilling on abstract numbers and words which convey no ideas, or at best a mere jumble of ideas to the child’s mind. It is one of the worst, and at the same time one of the most common, faults in the teaching- of arithmetic, and it is one which is very apt to disgust pupils with the subject from the outset.

On the other hand, if the proper method of teaching is pursued, which may properly be called the object method, the children are taught to think; they will be interested at the very beginning, and they will be kept interested by this method until they are successfully carried to the point where the object method is no longer necessary, and their minds are ready to grasp the abstract, through careful preliminary drill on the concrete.

Begin the teaching of arithmetic, then, with objects, — blocks, balls, marbles, sticks, books, kernels of corn, apples, shells, pebbles, etc., etc. The more varied your assortment of objects the better. The numeral frame and other mechanical devices are useful, but should not be used exclusively, or the work will become monotonous and tiresome.

Eclectic Manual of Methods – 1885

I just got a heads up that Ray’s Arithmetic in online on books.google.com.

Here are links of some I’ve found so far. I’ll add more, or update these as I have a chance to go through. Primary, Intellectual, Practical , Test Examples, Key to Higher, Primary Elements of Algebra, Algebra Part2 

For Me and My House ~ At Jesus’ feet,
Lisa @ Me and My House ~ Discipleship for Life!
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Greetings from Me and My House,

Discipleship is one of the key components of Lifestyle Education through Discipleship™, which we explain as “you follow me as I follow Christ.” The Principle Approach describes it as being a “living textbook”. This concept scares the confidence out of many parents. “I don’t know/remember any of that stuff. I don’t have time to learn it. Why can’t I just hand them a self-teaching textbook/workbook?” This is especially true in two main areas, math and language. Many parents are in complete fear that they are going to totally mess their children up by not including everything, or teaching something wrong. This attitude passes on several un-Biblical thoughts toward education, implanting in our children the same humanistic philosophies of education that were implanted into us.

Which leads us to another key component of L.E.D., Renewing Your Mind. You must be transformed in your philosophies (beliefs) about education before you can pass them on to your children. You must be a Learner to be a Teacher; and being a Learner (Disciple) is something we are called to for all of our lives. We are to model Learning to our children. We are not to just learn “spiritual things” as adults, but to continue to learn in all areas of our lives, as God’s Kingdom grows within us. For His Kingdom is His Lordship over EVERY area of our lives.

So, getting to our topic, perhaps you’re saying, “What’s Christian about math? 2 + 2 = 4 for the Christian and pagan alike. It’s just facts; there’s no philosophy to it.” Ah, not so my friend. Math is what God uses to hold our orderly Universe together. In fact, math is what gives our Universe order. Math is the language of Creation. Though atheistic or humanist educators may want you to believe otherwise, there is a deep philosophy of math that only springs from the Almighty Creator God. Math shows forth the attributes of God, and is eternal and infinite, as He is. I’ll let you begin your search is Romans 1:19-20, as I just want to lay down a foundation for you to begin thinking about how you should be teaching math.

After you renew your mind, by studying out (4 R-ing, if you will) the Biblical Foundations of Math, the Vocabulary and Principles of Math, and the PIPEline of HisStory in Math, you will have a much deeper understanding AND appreciation for math. Once you begin to see God’s hand in math, you will see wonder, not boredom in it. And you will want to share that wonder with your children. Wow, Discipleship! Far different from the attidude of, “Learn this, you’ll need it someday. Well, no, I don’t remember how to do it, I never use it.” Or perhaps you remember how to do it, but don’t have a clue how to teach it, because you only learned it by rote, on a knowledge level, without understanding or wisdom. Parent, learn it! You aren’t in the grave yet. God still expects you to be learning. There are 2 VERY good reasons for YOU to (re)learn math this way, the same reasons you are going to teach your children math this way: 1)You will come to know God better, 2) you will be better equipped to make Him known to others.

I know most of you are thinking, “But math is hard; I don’t have time;” ad nauseum (whoops, sorry). You’d be surprised at how little time you could learn the foundations to get started in this. Or maybe you’re one of the rare few that is saying, “I know and understand math. I didn’t have this way of instruction and it’s got me where I need to go in the world.” Remember, we are not educating for this world alone. We are educating for Wisdom, not just knowledge.

Let’s take a look at how we can, in Freedom & Simplicity, transform our math Learning and Teaching. A Guide to American Christian Education is probably my foundational resource for developing a Foundational Biblical Worldview of Math. A section is devoted to Arithmetic. In these 20 pages you will probably learn more about math and teaching math than all your years in school taught you; for you will be learning from whole to parts; wisdom, understanding and knowledge. You will see the interconnectedness, the Big Picture; which doesn’t look near as scarey as looking at all those unending disconnected pieces we were taught by. This along with your own Biblical study will take you through many years of teaching beginning math.

Although we also like Ruth Haycock’s Encyclopedia of Bible Truths, and The Noah Plan Mathematics Curriculum Guide, I did not find near as much Foundational and easy to use and understand information in either of those as in the A.C.E. Guide. They are both good for further study, but not necessary at first. Another facinating book, again for later as you have time, not absolutely necessary for beginning, is James Nickel’s Mathematics: Is God Silent?

In teaching you will lay the same Foundation for your children that you learned; teaching them the Biblical Foundations, Source, and Purpose for Math first. Just present what you have learned, on their level. The foundation of HisStory of Math can be presented easily, as the basics of its origin in Asia, development in Europe, and liberty in America; gradually adding more details year by year. You will lay down Foundational Vocabulary & Principles, and again, add to them as you progress. Your beginning child will have the seed that contains the whole, on their own level. And have a Biblical Understanding of Math beyond what you probably had when you began this study.
At Jesus’ feet,
Lisa
P.S. More on Principled Math, like how to actually teach it and what resources to use are in our Lifestyle Education through Discipleship: Freedom & Simplicity in Math book. (In process of being written – in pdf format for electronic purchase.)

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This post continues with our Wisdom’s 7 Pillars resource recommendations, today on the topic of L.E.D. Math.
UPDATE at the end.

LED MATH

Of all the things we teach our children, math is probably the hardest

for parents to implement in a “lifestyle” way, especially beyond the

very young childhood years. In keeping with LED principles, we desire

to keep math instruction as informal as possible for those early years,

and not resort to formal “school book” type programs until the formal

education years (the youth/teen years). We have implemented several

things that have worked well for us. We are working at getting those

things into a “presentable” format for others/you to use. We have also

found & recommend several resources that help us stay true to L.E.D.,

while providing instructional guidance.

INFORMAL

As a part of Lifestyle Education through Discipleship principles, we

believe most academic instruction, including math, should be fairly

informal, up to about the age of 12 or 13. The math concepts can

mostly be taught orally and informally through household objects and

dialog, with a little paperwork instruction included as needed. Our

children’s absolute favorite, and highly effective, math instruction

for the early years, we called “Money Math”. Very simply teaching them

to add and subtract through the use of money. They were “paid” for

doing things, and they “paid” for receiving things; beginning with

pennies, and progressing through nickels, dimes, quarters, half

dollars, and bills. They had to keep a running total, adding what they

were to receive, and subtracting what they were to pay. Money is

definitely the easiest and most logical way to teach our base ten

system, decimal placement, and negative numbers. We offer a booklet

called Freedom & Simplicity in Math (for the Childhood years), that

includes our “Money Math” ideas, as well as some charts and tools for

those early years. Calendars, clocks, and measuring devices round out

the *needs* of the early years.

We also love to incorporate Cusineaire rods (and counting bears) in our

hands-on, build it, see it math instruction. Yes, you can do the same

thing with popsicle sticks and dried beans, but the rods (and bears)

are so colorful, sturdy, uniform and fun. They aren’t just a “school”

tool; they are a fun “toy”, for building houses, complete with a family

to live in them. Our toddlers love the Jumbo rods and bears, too.

Our favorite resource by others, for these early years, which covers

all the topics typically taught in K through third grades, is “An Easy

Start in Arithmetic”, part of the 3 R’s series by Ruth Beechick. (We

recommend the whole series.) Mrs. Beechick includes ideas for teaching

each topic informally as you go through your day in managing your

household with your child alongside. Some written work is also

incorporated, as you design it (very simple, no daily worksheets to

write or such). A few simple chart ideas are also included. To

continue on with these ideas and foundation, Mrs. Beechick has also

written “How to Teach Your Child Successfully”. In it she gives ideas

& guidelines for older children (4-8 grade), in all foundational

subjects. We believe that “An Easy Start in Arithmetic” is all you

would *need* to teach your child math in those early years. Add on the

“Freedom & Simplicity Math”, Cuisenaire rods, and counting bears for

some practical, hands-on application ideas. “How to Teach …” will

guide you through the concepts needed to continue the rest of the later

childhood years, but many will probably want to add a little more from

other resources at this stage.

CONCEPTUAL

For those that desire to add a little more to the subject including

more written “paperwork”, we still recommend keeping the instruction

conceptual, concrete and informal, enjoyable, and “un-school bookish”

as possible. (We don’t feel these resources are necessary, but some of you may.)

The “Miquon” math series of “workbooks” utilizing Cuisenaire rods, by

Key Curriculum Press, provides this type of resource. It translates “real life” math,

that we’ve been doing concretely (with real objects through “An Easy

Start ,”) to paper. If you choose to utilize the “Miquon” books, we

still recommend using “An Easy Start in Arithmetic” as your foundation,

and adding the “paperwork” as an occasional exercise, as enjoyed by the

child, not an “everyday have-to”. There is no reason to begin this

“paperwork” at the same time you start your informal instruction with

“An Easy Start”. It can be begun even a few years later.

DRILL

Along the way, as your child learns the “math facts”, for

addition/subtraction, and in later years for multiplication/division,

you may want to work on these facts becoming “automatic” for him to

remember. This can be done by several methods. The most notorious is

by plain old fashioned flashcards, but many other ways are much more

fun and just as effect, if not more so for various types of learners.

“Math-It” turns the drill into a game, for both sets of facts using

“Add It” and “Timz It”. (Also available are “Pre Math-It”, covering

learning the addition facts through dominos, and “Advanced Math-It”,

covering percents/ decimals/ fractions.) We have devised our own type

of drill similar to “Math-It”, included in our “Freedom & Simplicity

Math”. Our children also all enjoyed learning sing-song jingles of the

facts from audio tapes, such as “Skip Counting”. The hands-on drill

tools called “Wrap-ups” are great for your kinesthetic kids, but we

found that some kids memorized where the string went more than the

actual facts. An alternative to the “Math-It” games, (I mention

because they are extremely popular with home ed parents) are the

“Calculadders” drill sheets. These are timed worksheets, and by far

the least fun and “informal” of the resources mentioned here.

Occasionally these can be used with older children for brush up, but we

don’t use them with our younger “informal stage” children. Freedom &

Simplicity Math (or “Math-It/Advanced Math-It”) and the “Skip Counting”

tapes are our recommendations.

TOPICAL

Shortly before your child advances to formal math instruction you will

want to make sure they have not only the basic facts “drilled”, but

also all their basic arithmetic down pat. This “transitional” stage is

usually between the ages of 10 and 12. During this time they will

begin doing regular “workpage” “math problems”, covering all concepts

of addition, subtraction, multiplication, division, fractions, decimals

(money), and percents, as well as measurements, time, place value,

negative numbers, and Roman Numerals concepts. Much of this will be

review, but also solidifying and pulling it all together. The “Key to

” series by Key Curriculum Press has booklets covering Fractions,

Decimals, Percents, and Measurements. Also, you could begin utilizing

the program recommended below, by covering each concept (not workpage)

in the beginning levels and using some of the word problems and/or

tests from the “extra practice” booklets/pages as indicators that the

child has good comprehension.

INCREMENTAL

Formal instruction, including formal math instruction, begins for our

children around 12 or 13 years of age. We believe the best math

program, that includes all elementary and high school math concepts,

taught in a line upon line, incremental, conceptual (understanding of

the principles, not just rote memorization) and concrete way (with

manipulatives) is “Math U See”, by Steve Demme. So, if this program is

complete, conceptual, and concrete why do we bother recommending the

other resources and not just start with “Math U See” and use it all the

way through? We feel “Math U See” is an excellent program, & love the

way it integrates conceptual teaching & concrete manipulatives for

applying math to “real life”. However, and for us and Lifestyle

Education this is a big however, we do not feel such a formal

structured program should be implemented for young children. We do

feel it is the best program for our older students, in those formal

instruction years.

All the resources recommended here have their own strengths for their

own stages of learning. We have picked the best in each category (as

we see it, from the wide variety we have used and thoroughly reviewed,)

for working within the principles of Lifestyle Education through

Discipleship. In review, those are: “Freedom & Simplicity in Math” and

“An Easy Start in Arithmetic” for early Childhood stage (and

optionally, “Miquon” math w/ Cuisenaire rods); for the later Chilhood

stage, “You Can Teach Your Child Successfully”, “Freedom & Simplicity

Math” (or”Math-It”), “Skip Count” tapes, “Key to …” series, (and

optionally, “Wrap-ups”); and for the Youth/teen stage, the

complete series of “Math U See” (which goes up through Trigonometry, if

you so desire).

One key component of Lifestyle Education is through “Notebooking”. Our

“Freedom & Simplicity in Math” includes guidelines for Notebooking

Math. As with all the other topics/subjects we Notebook, the goal is

to produce the child’s own Book of what they’ve learned and now know,

their own “teaching” guideline, – their own “text”, if you must. This

provides the “written work” the child does and provides him his own

personal reference book for looking up concepts and procedures for

figuring out problems. As far as we know this is the only resource of

this kind. It will, Lord willing, be available for purchase later this

Spring. Please pray for its timely completion as we put the finishing

touches on this “published version” of our L.E.D. Math methods for

Freedom & Simplicity™ in YOUR homeschool.

UPDATE: We have not changed our opinions on the above mentioned resources. We still like them and all the “pros” of them still stand. However, there is one more addition to our list, and also another preferred resource.

I would add Making Math Meaningful by David Quine of Cornerstone Curriculum, as another option alongside of Math U See. It too teaches math in a real life setting, using manipulatives, and teaches reasoning. There are some lesson “perspectives” I prefer in it, but some presentations, (like Steve’s video examples, and the uniformity of manipulatives) that I like better in MUS.

My preferred resource now though is Ray’s Arithmetic. It is a completely non-consumable program (a big plus for big families), and teaches completely through the principles of math, real life application, and reasoning skills. It is a series from the 1800’s, and therefore a bit more advanced than today’s teaching, but the books are not “grade-leveled” so that doesn’t really matter. It does not come with manipulatives, but it is expected that you will use real objects to present the lessons. It explains how you reason through to the solution of the problems. It also expects early lessons to be done orally and mentally with manipulatives, not paper and pencil, so it can be used with our informal early teaching.

At Jesus’ feet,

Lisa

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