Living Math  "Living math is math in life, in context, math that has relevance to the learner."  livingmath.net
Math Maturity  The state or quality of being developmentally ready (mature) to learn and understand various math concepts
After all the math research, I believe I've found the two key ingredients to math success. The math you are teaching must be developmentally appropriate and relevant in order for the child to grasp the concepts and succeed.
Jean Piaget's work on children's cognitive development, specifically with quantitative concepts, has gained much attention within the field of education. One contribution of his theory concerns the developmental stages of children's cognition. His work on children's quantitative development has provided mathematical educators with crucial insights into how children learn mathematical concepts and ideas.
In An Easy Start to Arithmetic, Dr. Ruth Beechick writes about three modes of thinking based on Piaget's developmental theory. The first being "The Manipulative Mode". According to Dr. Beechick, children in this mode can work mathematical problems with real objects such as marbles, spoons, people, candy, etc. They must do the arithmetic outside the head, meaning they need to see it, touch it, feel it, rather than think it. The Manipulative Mode usually lasts until about age 6 or 7. The child then transfers to the "Mental Image Mode". This is where the child pictures images of objects in their head. According to Piaget's theory, children use this mode through age 12 or 13 along with manipulatives. In Beechick's final, "Abstract Mode", children are able to think abstractly about mathematical concepts. They don't need to picture 3 M&M's and 5 Fruit Loops in order to add 3+5. They have become so familiar with 3 and 5, that the arithmetic becomes natural.
Whether or not you buy into the full scope of Piaget's Theory of cognitive development, I think there is something to be said about child development and readiness when it comes to academics. If you haven't read The Story of an Experiment by L.P. Benezet, I highly recommend doing so. In the 1920's, Benezet conducted an experiment by omitting formal arithmetic instruction in grades 16. I found the results fascinating.
Paul Ziegler of Systematic Mathematics advocates not starting formal math education before 3rd grade. He states, "Children are simply not ready for formal math instruction before that age. First and second grade math textbooks did not exist before the 1960's." Once starting math in 3rd grade, Ziegler recommends StrayerUpton Practical Arithmetics. These books were originally published in the 1920's by American Book Company and have since been reprinted.
Practical Arithmetics is a three book series covering grades 38. These palmsized hardback books are a real gem. They were written to the student with simple one step incremental instructions. Practical Arithmetics gives problems relating to the life and interest of students. The series also provides carefully planned instruction in problem solving and a series of diagnostic tests so you can gauge if your student is grasping the concepts before moving on. This translates to mastery of concepts. Practical Arithmetics were the books used in Benezet's experiment. I believe the books were originally printed in a six book series, then later combined into three books.
I've created a chart to show the mathematics instruction in grades 18 implemented by Benezet, both formal and informal.
The Story of an
Experiment:
The Teaching of
Arithmetic by L.P. Benezet
Grade

Concepts

Resources

1


Recognize and read numbers up to 100 in natural context

Compare/contrast/estimate using words like: more/less,
many/fewer, higher/lower, taller/shorter, earlier/later, narrower/wider,
smaller/larger, etc. in natural context

Keep count of date upon calendar

Track holidays and birthdays of family and friends

NO Formal Arithmetic Instruction

2


Use of 1^{st} grade comparatives continued

Play games

Begin telling time  hours and half hours

Recognition of page numbers and any numbers naturally
encountered in books (may use index as reference)

Informally learn words like half, double, twice, or
three times

Continue calendar math with days of the week and months
of the year

Discussions regarding penny, nickel, dime, and dollar
as encountered

Incidental explanation of meaning and relation of pint
and quart

NO Formal Arithmetic Instruction

3


Continue teaching significance of numbers naturally
encountered

Play games

Dime = 10 cents; dollar = 10 dimes and 100 pennies;
half dollar = 5 dimes or 50 cents; 4 quarters or 2 halves = 1 dollar

Develop ability to tell time at any particular moment
saying 3:50 rather than 10 minutes to 4 or 2:35 rather than 25 minutes to 3

Develop concept of 60 minutes makes one hour; 7 days
make 1 week; 24 hours make 1 day; 12 months make 1 year; and about 30 days
make 1 month

Encourage child to use numbers (4 digits or less) such
as house number, license numbers, telephone numbers, etc.

NO Formal Arithmetic Instruction

4


Continue informal instruction using numbers and making
comparisons whenever possible

Play games

Develop concept of measurement such as inches, feet,
and yards by first estimating and then actually measuring distances and objects
within surrounding area

Practice reading thermometer and give significance of
32 degrees, 98.6 degrees, and 212 degrees

Introduce terms such as square inch, square foot, and
square yard as units of surface area

Use toy or real money to practice making change in
denominations of 5’s only (this should all be done mentally)

Toward end of year, children should have done a great
deal of work with estimating areas, distances, etc. and checking by
subsequent measuring

Time should be taught involving seconds, minutes, and
days

Relation of pounds and ounces should also be taught

NO Formal Arithmetic Instruction

5
1^{st} semester


Count by 5’s, 10’s, 2’s, 4’s, and 3’s in this order,
starting mentally first

Use toy or real money to practice making change amounts
up to 1 dollar including pennies

Continue informal work in estimating distance, area,
time, weights, measure of capacity, and the like

Play games

Compare the value of fractions and discover for
themselves that 1/3 is smaller than ½ and greater than ¼ ; that the larger
the denominator the smaller the fraction  illustrated concretely or by
pictures

NO Formal Arithmetic Instruction
Toward
end of 1^{st} semester, students were given Practical Problems in
Mental Arithmetic, grade IV to stimulate quick thinking and get children away
from using the fingers to do the work of the head.

5
2^{nd} semester


Count by 6’s, 7’s, 8’s, and 9’s mentally

Discuss relative value of the fractions ½, ¼, 1/5, and
1/10 using concrete examples from prior money practice

NO Formal Arithmetic Instruction
Continue
Practical Problems in Mental Arithmetic, grade IV to stimulate quick thinking
and get children away from using the fingers to do the work of the head.

6
1^{st} semester


Teach processes of addition, subtraction, multiplication,
and division avoiding purely mechanical drill and problems involving long
numbers that cause confusion

Insist on accuracy at the expense of speed or covering
ground

Processes should be mental whenever possible

Before starting a problem, ask student to estimate or
guess what the answer will be and have them check it upon completion

Teach fractions and mixed numbers being careful not to
confuse students with involved or complicated problems

2025 minutes per day – Begin formal arithmetic
StrayerUpton
Arithmetic, Book III, first 108 pages

6
2^{nd} semester


Learn multiplication tables and tables of denominate
numbers

Continue work from 1^{st} semester

25 minutes per day
StrayerUpton,
Book III, beginning with page 109182
StrayerUpton,
Book IV, first 50 pages

7
1^{st} semester


Review tables of denominate numbers, including US money

1760 yards in a mile

880 yards in a half mile

440 yards in a quarter mile

Great deal of mental arithmetic

25 minutes per day
Strayer
Upton, Book IV, beginning at page 51

7
2^{nd} semester


Remember that ability to reason the problem correctly
is far more important than errorless manipulation of the four fundamental
processes

25 minutes per day
StrayerUpton,
Book V, first 100 pages, omitting 1010, 28, 7177

8
1^{st} semester


Continue practice of estimating before answering
problems

Tables of denominate numbers are kept fresh

Continue estimating lengths, heights, and areas of
familiar objects – then check actual answers

30 minutes per day
StrayerUpton,
Book V, beginning with page 101, omitting pages 127134
StrayerUpton,
Book VI, first 32 pages

8
2^{nd} semester


Summary of everything learned in arithmetic, but above
all, the ability to approximate and estimate in advance the probable answer
is most important

Much of the work is completed mentally

30 minutes per day
StrayerUpton,
Book VI, beginning at page 35 and omitting pages 36, 4648, 5759, 8082,
9293, 104, 158188, 194, 203204, 206208

I am not advocating doing nothing. As you can see from the chart in Benezet's experiment, the children learned about a variety of math concepts. However, they learned math in the context of every day life, making it relevant. They were not given formal instruction from a text book. The students were taught using living math, playing games, counting real money, referencing page numbers in the books they were reading, etc. It was amazing to see how advanced the students in the experiment were by the upper grades when they were able to learn at their own pace in the lower grades.
I have also created a chart to show the scope & sequence recommended by Dr. Ruth Beechick for grades 13. In Beechick's book she also gives ideas, activities, and games for teaching mathematical concepts at the child's pace without formal curriculum.
An
Easy Start to Arithmetic by Ruth Beechick
Grade
13 Math Scope & Sequence Chart
Topic

1^{st}
Grade

2^{nd}
Grade

3^{rd}
Grade

Numbers

Count
cardinal numbers to 100 with understanding of what numbers mean
Count
ordinal numbers to tenth

Read,
write, & count up to at least 200
Count
ordinal numbers to tenth

Read,
write, count, and use numbers up to at least 1000

Place
Value

Understand
tens and ones place in numbers up to 39

Review
ones and tens place
Teach
hundreds place noticing use of zeros


Groups

Recognize
groups such as dots on dominos
Count
by 2, 5, & 10 with manipulatives

Count
by 2 to 10 or higher
Count
by 5 & 10 to at least 30 or 100 if possible with manipulatives

Become
proficient in counting by 5 & 10 up to 30 or higher
Count
by 2 up to 20
Try
counting by 3 & 4

Measurement

Clock,
calendar, ruler, measuring cup & other measuring devices, money to
include pennies, nickels, and dimes

Time
in hours and half hours
Length
in inches and feet

Pennies, nickels, and dimes; possibly quarters & half dollars
Understand
pound, pint, quart, & dozen

Work
with all coins & dollar bills writing amounts, using decimal points &
dollar signs
Build
understanding of hours, half hours, and quarter hours
Days,
weeks, months, and years
Use
measures of inches, feet, and yards converting one to the other
Use
pints, quarts, gallons, ounces, and pounds in realistic situations
Read
scales and thermometers

Vocabulary

Use
words having to do with size, quantity, & shape

Use
words such as subtract, minus, plus, add, equals, and comparisons like long,
longer, longest, left/right, top/bottom, etc.


Addition

Add
two groups with sums of six or less, not including zero

Add
2 numbers up to 12
Add
3 numbers
May
add 2 digit numbers without carrying if child understands place value

Master
all addition facts up to 9’s
Learn
carrying, borrowing, checking, and bridging

Subtraction

Take
away a group from six or any lesser number and tell what is left

Take
away a group from 12 or any lesser number and tell you what’s left
May
subtract 2 digit numbers without borrowing if child understands place value

Master
all subtraction facts up 9’s
Learn
carrying, borrowing, checking, and bridging

Multiplication

Begin
to understand meaning of combining similar groups two times or three times

Understand
that multiplication is a way to add equal groups
Learn
2’s, 3’s, 10’, 5’s, & 1’s multiplication facts
May
introduce multiplying 2 digit numbers that don’t require carrying


Division

Begin
to understand dividing a large group into smaller groups of 2’s, 3’s 5’s, or
10’s with manipulatives

Understand
that division is a way to subtract equal groups.


Fractions

recognize
1/2, ¼, and sometimes 1/3 by cutting apples or candy bars

Understand
fractional parts of 1/2, ¼, 1/3, 1/5
Learn
to write a few fractions which have 1 as a numerator
Some
children may understand introduction of other fractional parts with real
objects


Notation

Read
and write digits 110

Learn
to write addition & subtraction
problems in both horizontal and vertical form using and knowing signs


Problem
Solving

Much
experience is usually given in problem solving using manipulatives

Continue
expanding experience in solving realistic problems solving mentally, or with
hundred chart, or number line, some may be solved by writing figures on paper
in their proper positions

In addition, I found this website and this website that give parents steps to help your child develop an understanding of math concepts.
Another math program I've been researching based on Piaget's Theory is Math on the Level. Their website states...
Math on the Level is a flexible, maturationbased, homeschool math curriculum with a very effective review system. It's unique approach is designed to turn your home environment into a rich, reallife learning experience and make math education meaningful and fun for your whole family.
 Teach when it is easy for your child to learn
 Use everyday activities to make math real
 Review for long term retention without busywork
The creators of Math on the Level realize not every child is ready to learn the same concepts at the same age. "Therefore, rather than prescribing a required order in which to teach concepts, Math on the Level includes every concept you need to teach your child but lets you choose the order and pace at which to teach them. The four Teaching Guides show multiple ways to teach every concept but do not assign an age or a gradelevel to any concept. You can teach in whatever order works best for each child and at a pace that adapts to each child's maturation." Math on the Level is a set of books to guide the teacher/parent in teaching math concepts at each child's level/pace. You can review as much or as little as needed.
I'm seriously considering Math on the Level (MOTL). This sounds exactly like what Ruben needs. Ruben is gifted in many areas, though he struggles in others. I've learned from past experience with him, it's all about timing. When he's ready to learn it comes so easy, but when he's not, there's great frustration and it's a battle. MOTL would allow me to teach concepts as he's ready.
MOTL would also give me some tools to reinforce concepts with RileyAnn. She's a little freaked out by the idea of not having a workbook and given the fact that she'll be starting 4th grade in the fall, as well as her personality, we will continue with some type of math book for her. I'm certain the program we choose for her will be sequential mastery and not spiral. However, I'm undecided if I should continue with MathUSee or switch to something else. I just don't feel like MUS has enough practice in prior concepts. For example, once RileyAnn finished Alpha and Beta, Gamma offers no review or application of subtraction. So she's not seeing how it applies to real life, making it meaningless. Hence, she's forgotten how to approach multiple digit subtraction. There's a compatibility tab on the MOTL website that shows how you can use the program with other methods of teaching. If I decide to go with MOTL, there may be options for using it to supplement MUS, making it more meaningful to aid understanding and retention.
Some options I have on the shelf are...
StrayerUpton Practical Arithmetics First Book  Traditional, Sequential, Mastery
Life of Fred  Reform, Spiral
Misc. 4th grade textbooks  Traditional, Sequential
RightStart Mathematics  Reform, Sequential
Saxon  Traditional, Spiral
Some options I plan to explore at an upcoming homeschool convention are MCP Mathematics  Traditional, Sequential, Mastery and Math Mammoth  Traditional, Sequential, Mastery.
Though I have not made a final decision on which program to teach our kids math in the future, I have learned a great deal in my research. You can view former posts in this Math Mania series here...Post 1, Post 2.
I've come to the conclusion that math doesn't have to be dry and boring. Children are born with a natural sense of curiosity and I need to nurture this curiosity by making math relevant in our daily lives. I've also come to understand that it's not so much about the curriculum and enforcing some contrived scope & sequence, rather it's more about presentation and capitalizing on those teachable moments. I'm suddenly viewing teaching math more like the approach I take to teaching history. Choose a great "living book" or text and use it as a spine or guide to teach various concepts when the child is developmentally ready. This notion fits very well into the Charlotte Mason method of spreading a feast before the child and letting them take what they are ready for and leaving the rest.
I will followup some time after convention and let you know what we decide to use for math in the upcoming school year. I also plan to complete a post in the next couple of weeks on math and the Common Core Standards since several homeschool math publishers are updating their curricula to align with these standards.
Are you preparing for the upcoming school year? What math programs are you looking at or currently using successfully? Please leave your comments below.
We're using MUS and it is working well for the 6th, 2nd, and 1st graders. Next year they will keep going with MUS. My upcoming K son is a math whiz doing most of what his next two older siblings math covers in his head. We're avoiding a curriculum with him and focusing on Living Math until he's older so worksheets don't suck the joy he has out of math.
ReplyDeleteYou should also check out these
ReplyDeletehttp://www.triviumpursuit.com/blog/2014/02/12/delayingformalmathhistoryparti/
http://web.archive.org/web/20080201171823/http://www.mountainlaurelsudbury.org/Rithmetic.asp
http://:www.angelfire.com/nm/shalom/teach/history.htm
Oh yes forgot to mention
ReplyDeleteTake into account that Benezet actually wanted to start formal instruction in 7th vs 6th. What's interesting is Aristole's timeline. No formal math instruction until 15. And as recently (well somewhat) before 1850 it could be 1518 when a student stared formal math instruction. The elite schools rarely taught. It was thought to be a poor man's subject as you mostly found math being taught in schools for the poor.
Sorry for Al the posts forgot to post
ReplyDeletehttp://www.triviumpursuit.com/blog/2014/02/12/delayingformalmathhistorypart2/
http://www.triviumpursuit.com/articles/research_on_teaching_math.php