Monday, March 11, 2013

Math Mania - Part 3

Living Math - "Living math is math in life, in context, math that has relevance to the learner." - 

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 1-6.  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 Strayer-Upton 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 3-8.  These palm-sized 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 1-8 implemented by Benezet, both formal and informal.

The Story of an Experiment:
The Teaching of Arithmetic by L.P. Benezet

-       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
-       Use of 1st 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
-       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
-       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

1st 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 1st 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.

2nd 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.


1st 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
20-25 minutes per day – Begin formal arithmetic

Strayer-Upton Arithmetic, Book III, first 108 pages

2nd semester
-       Learn multiplication tables and tables of denominate numbers
-       Continue work from 1st semester
25 minutes per day

Strayer-Upton, Book III, beginning with page 109-182

Strayer-Upton, Book IV, first 50 pages

1st 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

2nd 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

Strayer-Upton, Book V, first 100 pages, omitting 10-10, 28, 71-77

1st 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

Strayer-Upton, Book V, beginning with page 101, omitting pages 127-134

Strayer-Upton, Book VI, first 32 pages

2nd 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

Strayer-Upton, Book VI, beginning at page 35 and omitting pages 36, 46-48, 57-59, 80-82, 92-93, 104, 158-188, 194, 203-204, 206-208

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 1-3.  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 1-3 Math Scope & Sequence Chart
1st Grade
2nd Grade
3rd Grade
-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
-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
-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
-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.

-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
-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

-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

-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. 

-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
-Read and write digits 1-10
-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, maturation-based, homeschool math curriculum with a very effective review system. It's unique approach is designed to turn your home environment into a rich, real-life 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
One curriculum purchase covers your math needs for the whole family from pre-K through pre-algebra.

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 grade-level 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 Math-U-See 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...

Strayer-Upton 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 1Post 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 follow-up 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. 


  1. 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.

  2. You should also check out these

  3. Oh yes forgot to mention
    Take 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 15-18 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.

  4. Sorry for Al the posts forgot to post