Math instruction- best practices and problems (2)

We’ve been looking at the topic — instruction of mathematics, best practices and problems. I thought I would post two articles “Failure of Mathematics Teaching” from the freedom-in-education newsletter and “Teaching Math at Home” by Debbie Mason which address the topic spot on.

Newsletter January 2003
In this issue:
Mathematics
Letters
New: an expanded, printed version of the freedom-in-education newsletter.

Mathematics

Mathematics is the subject in which people experience the most difficulty at
school and is also the subject that causes the most anxiety for home-educating parents. The following article explains some of the reasons |for these difficulties and some solutions.

 

Failure of Mathematics Teaching

 

The recent history of mathematics teaching has been one of declining |standards and of failure. A hundred years ago, the population in general was much more able to perform calculations and to work with numbers than are people nowadays and, in addition, people qualified in mathematics had a clearer understanding of the principles of mathematics than do modern mathematicians. The reason for this decline in standards is complex and involves many interrelated factors:

— Lack of Practical Experience of Mathematics – the best way to learn |mathematics is through practical experience. The modern school system isolates children from most normal, everyday experiences that would involve them using mathematics: shopping, cooking, gardening, crafts, games, etc. ・

— and they therefore miss out on the most powerful teacher of mathematics: practical experience.

* Primary School Mathematics: as with many other subjects, schools succeed in making mathematics something that is essentially enjoyable into a form of torture for young children.

* Metric System ・the metric system is a very clever
invention but it does not make sense to young children and it does not provide a good |introduction to working with numbers. The fact that the school curriculum |often insists on the use of the metric system shows that the people who run schools do not understand either teaching or mathematics.

* Making Maths into a Utilitarian Subject — up until the early eighteen hundreds, mathematics was still seen primarily as a branch of philosophy. However, the practical uses of geometry in building, surveying, |engineering and the military sciences, led to a change of attitude. It was assumed that people could be instructed in the purely practical aspect of |the subject without troubling them with any of the philosophical foundations upon which it is built, thereby robbing the subject of its essence.

* Teaching Mathematics Too Early — every stage of mathematics is |taught too early. When children should be left to learn to count and to play |with numbers, they are taught to add and take away; when they should be learning to add and take away, they are taught to multiply and divide; when they should be learning to multiply and divide, we try to teach them about decimals and fractions; and when they could be beginning to explore |fractions, we are trying to teach them algebra.

* Poor Mathematics Teachers ・this system leads to children nearly |always being taught mathematics by people who do not understand what they are teaching: primary school teachers do not really understand the metric system and secondary school teachers do not understand algebra, analytical |geometry or calculus. It is doubtful whether university teachers have a real grasp of the essential principles of mathematics -a subject which mathematicians down the ages have agreed should be based upon simplicity, logic and beauty.

 

The Difficulties People Have With Mathematics

The consequence of this poor standard of mathematics teaching is that people |in general have lost confidence in their ability: they do not believe that |they are capable of working out even simple sums for themselves and they do not believe that they are capable of understanding the finer points of mathematics.

Naturally, such people do not, at first, consider that they are capable of |teaching mathematics to their own children and are therefore all too ready to submit their children to the same programme of mathematics teaching that |had such poor results in their own case – and so standards continue to decline.

 

People Who are Good at Mathematics

 

Of course there is a small percentage of the population that does not experience any difficulty in learning mathematics. The degree to which this is a good thing depends upon how their mathematical talent is developed. As has already been hinted, above, the modern mathematics syllabus has strayed a long way from its origins. Mathematics was originally used as a tool with which to explore the meaning of life and the nature of the |universe. Ancient philosophers used it to demonstrate the limits of reason and analysis. They showed that no matter how logical an approach was adopted, sooner or later, paradoxes developed within any line of reasoning.

 

This is still known to be the case – but it is not something that is explained to children when they start to study the subject.

 

Instead they are led to believe that they are studying something that is incontrovertibly true.

 

A detailed study of pure mathematics should be delayed until someone is old |enough to see for themselves that it is simply a collection of ideas, none of which represents the absolute truth.

 

The Value of Mathematics
These arguments are not intended to question the importance or value of |mathematics: a really good understanding of numbers is useful in almost |every area of life: it is useful in managing household expenses, in dealing with the tax authorities, the utilities, etc., it is useful when shopping, and in almost every area of work and business.

 

The value of pure mathematics –geometry, algebra, etc. may not be so |immediately apparent but historically it has always been regarded as one of |the branches of learning that does most to develop the intellect and sharpen |the wits.

 

Good Mathematics Teaching

 

Here is a list of suggested dos and don’ts for parents who want to help |their children to enjoy mathematics.

 

• Don’t expect someone else to teach your child mathematics.
• Do take responsibility yourself for helping your child to understand mathematics.
• Don’t buy or follow a maths course.
• Don’t worry if your child does not appear to be making progress n the subject.
• Do involve your child in as many practical activities as possible: shopping, cooking, crafts, gardening etc.
• Do play lots of games with your child: cards, dice, board games etc. Games develop all the skills that are most useful in the understanding of mathematics.
• Do practise basic arithmetic as much as possible: adding up, taking away, multiplication and division – mental arithmetic and sums on paper.
• Don’t expect your child to understand millimetres, centimetres, millilitres, cubic centimetres, metres, litres, kilometres, kilograms, etc. even if you understand them yourself.
• Do use metric measurements when they are appropriate.
• Do use imperial measures when you can: feet, inches, yards, miles, pounds, ounces, pints, gallons, etc. (They involve the use of fractions such as |halves, quarters, etc. and provide a good introduction to the principles of |arithmetic.)
• Do give your child a watch with a clock face (when they are old enough to |have a watch) and help them to use it to tell the time.
• Don’t push your child.
• Don’t let your child become upset by being made to do mathematics.
• Do let your child use a calculator when they want to.
• If you want your child to progress in mathematics, do study mathematics yourself. Surprisingly perhaps, these dos and don’ts are applicable for very young children — children enjoy counting things even before they are able to speak — right up to teenagers and beyond. Everyone wants to be good at working things out and to be able to understand weights and measures, but |very few teenagers really have a spontaneous desire to know about algebra or geometry: traditionally such things have been studied by older people.
• Instead of imposing a maths curriculum on someone, that may put them off the |subject and which certainly will not inspire them to really understand it, it is far better to leave the subject to one side until someone either has a |practical need for it in something that they are doing or else becomes |interested in it as a intellectual exercise. Either of these things can happen at any time during one’s life, and when they do, mathematics reveals itself to be a source of fascination and pleasure.

 

The Royal Road to Geometry

 

Alexander the Great is reputed to have asked Menaechmus to teach him geometry concisely but he replied “King, through the country there are royal roads and roads for common citizens, but in geometry there is one road for all.”

Unfortunately, this important lesson has not been understood by modern mathematics teachers who try to pick out the useful and important bits of the subject without explaining their context. The result is that almost everyone is left confused to a greater or lesser extent by their experiences of mathematics at school and university.

 

For an introduction to geometry:

http://www.freedom-in-education.co.uk/euclid.htm

The following article was written for The Elijah Company’s eNewsletter by our
good friend, Debbie Mason of Charlotte, North Carolina. Debbie is a longtime
board member for the state organization, North Carolinians for Home Education
and is the mother of four children ranging in age (as of 2002) from 14 to 22 .

——————————————————————————–

Teaching Math at Home
by Debbie Mason

As a homeschool mom with a math education background, I am seen as the answer
person for homeschool math questions, such as:

a.. What is the best curriculum?
b.. When do I need to start teaching math?
c.. Do I need to drill the times tables?
d.. How much math do I need to teach?
Because I love math, it is my favorite subject to teach. My passion has caused
me to develop some opinions about how to teach it. I do not claim to be a math
education expert. My opinions come from twenty-one years of homeschooling four
children, now ages fourteen to twenty-two, my interactions with other
homeschooling families, and my personal study.

One of the reasons that I chose to homeschool was that it provided the freedom
to tailor the education of my children around their unique interests, abilities
and the family’s priorities. I didn’t want to do it the way the school did it; I
wanted to do it differently and better. When homeschooling parents adopt the
institutional approach to education, they miss out on so many of the beauties of
homeschooling. If you are going to homeschool, take advantage of its advantages.
Three of the goals that I had for my homeschool were that my children would love
to learn, know how to learn and be allowed to learn at their own pace. I saw
many problems that were caused by children being pushed to do something before
they were developmentally ready. We often see children pushed in reading, but it
also happens with math.

Math and the Young Child
In her book titled An Easy Start In Arithmetic, Ruth Beechick says there are
three modes in which children think about math: manipulative, mental and
abstract. These modes also correspond to the developmental stages of a child.
First, young children learn through the manipulative stage. They need to touch,
feel and move. When you, as an adult, see the problem 2+3=5, you think in the
abstract mode. You understand the concept of two and three. You do not have to
see and touch two blocks and three blocks. You don’t even have to picture two
blocks and three blocks in your head. Preschoolers cannot do this; they are in
the manipulative stage. Later, during elementary school, they develop the
ability to do math in the mental mode. They can picture the number and the
addition process, but they are still not able to understand the abstract concept
of a number. This ability to understand the abstract concepts of math develops
around age twelve.

It’s best for a homeschooling parent to keep these developmental stages in mind
while teaching math. During the early years, math concepts need to be taught
with things that the child can touch, feel and manipulate. This need usually
corresponds nicely with the real life of the child. Children need a lot of
real-world, concrete experiences before they can internalize the meaning of
numbers, arithmetic operations, geometric shapes, proportion and all the other
terms, ideas, processes and relationships that are a part of mathematics.

One of the best things a homeschool parent can do is to get a good elementary
math book and read it themselves. Learn the terms and concepts, and then apply
these terms and concepts to your child’s everyday life. Many homeschool parents
hate math and do not feel very competent to teach it. If this is true of you,
you need to do some homework. The more you understand the concepts yourself, the
better off your children will be. Now I’m not talking about algebra; I’m talking
about early elementary math.

Children come into contact with math everyday. When children play with building
blocks, puzzles, toy cars, when they have a need for counting, patterning,
comparing, estimating, etc., they are building a repertoire of concrete
experience. Helping mom in the kitchen or dad in the workshop offers many
opportunities for real-life math. Gardening, playing a musical instrument,
grocery shopping, setting the table, and playing board games are all examples of
activities that provide children with context and a frame of reference for
future math learning.

It is so important that your children have these experiences before they start a
formal math program. I actually don’t start a formal program until third grade.
Until that time, I do real- life math, read math books, do math activities and
play math games while occasionally throwing in a few math lessons. A book that
is fun to use during the elementary ages is Family Math. Also, the library is
full of picture books with a mathematical theme. Once a child has the
developmental maturity of a third grader he is able to cover K through second
grade math rather quickly and can then move on to the third grade math. You
could probably wait a little longer, if you’re patient and have the nerve to
stand up against peer pressure. A student who has had a life rich in
mathematical experiences will be better able to understand the math exercises
that he is now asked to do. During this stage, he is able to do mental math. He
doesn’t always need the manipulatives because he is able to picture them in his
head. Manipulatives are sometimes helpful, especially when learning a new or
difficult concept or process. However, be careful that you don’t expect him to
be able to do the abstract thinking that is required for many math processes. He
is not ready for that yet.

Flexibility
Flexibility is one of the greatest advantages of homeschooling. Your child can
move as quickly or as slowly as necessary. If your child is getting the material
quickly and seems bored or frustrated with doing all of the problems, then cut
out some of the problems and move on. You have an advantage over the classroom
teacher of knowing your student very well. You will be able to determine whether
he is getting it or not. If he is having difficulty, then slow down, or take a
break and come back a few weeks later. If you decide it is time to take a break
from progressing in math, don’t just stop doing math altogether. I know you have
heard the expression, “If you don’t use it, you lose it.” This is especially
true of math. Keep doing review even during the breaks from introducing new
material.

My oldest daughter, who is now a college graduate with degrees in math and
music, was always good at math. I could tell from early on that she was good in
this area. When she was going into the third grade, I started her in a sixth
grade math book and spent two years in the book. Because this book started with
so much review we were able to start in this book without having to do very much
outside review. This worked, in a way. She did well academically, but she hated
math. This bad attitude was a major red flag for me. One of my goals was for my
children to have a good attitude about learning. I had to do something to fix
this problem, especially since I knew she was gifted in this area. So, we took
the next year off from progressing in math. I did consistent review for this
year. However, the review did not take much time, so it really was a break. When
she came back to learning math during the sixth grade, she had a much better
attitude. (I need to add here that it was MathCounts, a math problem solving and
competition club for junior high students that turned her into a math lover. To
find out more about MathCounts go to http://mathcounts.org.

I tell you this story to show an example of the creativity you can have in the
homeschool environment. You are not locked into the traditional program of
conquering a math book a year and staying on grade level. Also, don’t feel that
you have to use the same curriculum every year. You can alternate or switch
books if you don’t like the one you are using. Different programs work better
for different kids. I also recommend that you do math along with your student
when at all possible, especially if this is a first child. It will help you
review the math concepts, and is also good for your student to work closely with
you. Remember that your attitude about a subject is more important than your
knowledge about it. I remember pretending to be fascinated with the discovery of
bugs and snakes when my children were little. If I can do this, you can pretend
to like math. You don’t have to pretend to be good at it, though. It is good for
your child to see you learning with him.

I am often asked about memorizing the math facts, such as the times table. I
take a middle of the road approach on this topic. While I do think it is
important that children know the math facts, I do not agree with stopping
everything else until they are memorized. I usually took some time to work on
memorizing the facts, and then I moved on before they were perfected. I found
the more math the student did, the more these facts became a part of his
knowledge. So I used a combination of some work on memorizing the facts and a
lot of work on using them.

Problem Solving Skills
I usually teach problem solving skills while my children are doing their final
years of elementary math and along with algebra. Many people tend to ignore
these skills because they are not covered well in most textbooks, and covering
them would take extra time and resources.

One reason I take two years to cover algebra 1 is to have the student work on
problem solving. In real life, math problems do not come with a label. You have
to figure out what kind of problem it is and then how to solve that problem.
Textbooks usually have exercises after a lesson is taught. The student exercises
the skills that he has just learned by doing the exercises. Problem solving, on
the other hand, teaches the student to look at a problem and determine what kind
of problem it is and how it should be solved. This skill is much more helpful in
real-life than the ability to work exercises at the end of a lesson. The
textbook exercises provide skills and knowledge that are necessary for solving
problems. The MathCounts competition is very effective in teaching problem
solving skills. These skills will help your student in all areas of math, and it
is great preparation for the SAT. This competition for seventh and eighth grade
is especially good for children who are good at math. It is a challenging
program, however, and can be discouraging for those who are weak in math.

Another math problem solving program is the Math Olympiad program. It is a
problem solving competition for grades four through eight. The book, Math
Olympiad Contest Problems for Elementary and Middle Schools, by George Lenchner,
is a great supplement to your math curriculum. This book is also a good choice
for junior high students who find MathCounts too difficult. I use the Math
Olympiad book (it is an easy book to use) during the last couple of years of
elementary math as a supplement, and the students love it. I remember one
student, who had hated math before she started doing Math Olympiad problems,
stating, “Math is so cool!” after just two days of doing these problems. I never
participated in the Math Olympiad competitions, but they are available.

Beyond Elementary Math — Is Higher-Level Math Important?
Once elementary math is conquered, it is time to move on to algebra. I usually
take two years to do algebra I. It is important that the student have a good
understanding of algebra because it is the foundation of the rest of math. Don’t
rush it, and don’t move on until each concept is grasped. I absolutely love
Elementary Algebra by Harold Jacobs. All four of my children have used this book
successfully. The more I use it, the more I like it. I also like Jacobs’
Geometry.

Many people have difficulty in seeing the practical application of teaching the
higher levels of math. They don’t see that they use algebra in their daily
lives, so they wonder why they need to learn it, or teach it. I am convinced
that higher levels of math teach us to think more clearly and logically. How
important is this to your daily life? The process of thinking that is taught in
algebra and geometry teaches us to process information in a logical way. Other
than my love for math and the connection with thinking skills, I do have other
reasons for having my children learn higher math. I want to prepare my children
to be able to do whatever they are called to do. Even if they are not called to
do anything mathematical, they may be called to do something that requires
college, and college requires three to four years of high school math. Until my
children reach high school, I homeschool them the way I want. We study what we
want, how we want, and when we want. When they get to high school, I make some
compromises to prepare them for college. I make sure that they have two years of
foreign language and three years of science, whether I feel that they need it
for their life or not.

Now, I don’t completely cave in to the traditionalists. I do a lot of things in
a homeschooly way. I integrate many of the subjects, emphasize math and science
for those students with that bent, and emphasize history and government for the
students with that bent. My children with a musical gift are active in music
activities while the future politician will be in debate. So each student’s high
school experience will be different.

Earlier I mentioned that many people don’t think that algebra and geometry are
needed for every day life. I don’t believe that this is true. Because I am
comfortable with this knowledge of math, I do occasionally find myself using
this information to solve real-life problems. Also, if your children end up
being homeschool parents, it will be very helpful for them to know these
subjects. To sum up the reasons why I think higher level math is important:

a.. It teaches logical thinking.
b.. It prepares a student for his potential calling.
c.. It prepares a student for college.
d.. It teaches math skills that may be needed for real-life.
e.. It prepares future parents to pass along important math skills to the next
generation.
Being a math major with a love of math, I place a high priority on learning
math. However, I realize that all homeschoolers don’t have the same priority.
Each homeschooling family brings its own set of priorities to the homeschooling
situation. God knows this when he places children in families. He knew that I
would place a priority on math and not on foreign language, for example. So, I
will take all of my children at least through pre-calculus. However, it is hard
to say what level of math is right for every student. Each student is different
and has a different calling on his life. We as parents need to study each child,
praying for wisdom in planning each one’s education. As we pray and plan, God
will lead us to make the right decisions for each child. He knows the plans He
has for each of them.

Recommended Resources available from The Elijah Company

(All Special Prices reflect special discounts available through December 31,
2002)

a.. Ruth Beechick’s 3-R’s is a packet of three little booklets one of which is
titled “An Easy Start in Arithmetic.” These 3 little booklets (plus wall chart)
are sold as a packager. Dr. Beechick shares simple, practical, non-stressful
ways to introduce reading, writing and arithmetic to preschool through 3rd
graders. Special Price: $8.95
a.. Family Math: The following is an excerpt from a review in the Scientific
American Magazine.
Family Math is what it claims, a guide for parents and children to work
together, not on passive tasks of rote learning but in active game play, solving
problems, experimenting and even discovering. The tools are many but familiar:
cups and playing cards and beans, paper and pencil and scissors, for some a
watch with a second hand, and a $5 calculator. What to do is neatly shown in the
100 or more math activities that are laid out in this cheerfully illustrated,
informal book, High aims include the growth of confidence no less than skill and
a glimpse at how careers open in this world with math. This is a curriculum for
informal math education, so badly needed. Special Price: $17.95

b.. Mathematical Reasoning: Mathematical Reasoning, Book 1 (grades 1-3) and
Book 2 (grade 4-8). I’ve never met a child who did not thoroughly enjoy these
highly visual books. These are about the only “workbook” math programs that are
truly FUN as they present just enough of each of the different math concepts to
teach a child without burning him out. Unless you are a very busy Mom, we don’t
think the Teacher’s Manual is necessary for Book 1, but do recommend it for Book
2. At the end of Book 1, some multiplication is introduced; so, if your child
has not yet learned his tables, set the book aside until he does. Each book
contains age-appropriate sections covering Number & Numeration, Geometry,
Operations, Measurement, Relations, and Tables & Graphs.
There are two ways to use these books: 1) to simply present the pages to the
student to be worked (used by many parents as a math program); or, 2) use the
Teacher’s Manual along with the Student Text to go beyond simply solving math
problems to developing the “reasoning” behind the solutions. Special Price:
$18.95

a.. Mathematical Reasoning Teacher’s Manual Book 1 & 2: These texts go beyond
just giving the answers to the problems in the student texts. The TM’s take the
child deeper into the “thinking skills” stage of math problem solving. If you do
not have time to work with your child, you will probably not need the TM for
Book 1, but you’ll need it for Book 2. Special Price: $10.95

a.. Math-It!: This is the Davis’ favorite math drill facts program and is
especially valuable for the child who is a kinesthetic learner or who likes math
tricks (figure the square root of 5625 in your head in the next 15 seconds!).
There are four levels of “play”: Add-It (additions facts), Double-It (doubling
facts), Half-It (halfing facts necessary to multiply large numbers by 5), and
Times-It (multiplication facts). Each level comes with its own 8×10 cardstock
answer board and a set of 1-1/2″ x 1-1/2″ problem cards. Each problem card has a
corresponding answer block on the answer board; and, when all the problem cards
are placed on the answer board, the board is full. When the answer board can be
filled within 60 seconds, the child is considered to have “mastery” of that
level.
We recommend using Math-It! for any age child to make sure the child has total
mastery of all his drill facts. Included with Math-It! are an instructional
tape, an illustrated instructional book (which takes math facts way beyond the
times-tables and introduces lots of “math tricks”), and an 81 page comprehensive
Guide to Learning Math Concepts for children through age 14. This is truly an
amazing and wonderful program! Special Price: $44.00

b.. Math Olympiad Contest Problems for Elementary & Middle Schools: This is
truly a book to teach thinking skills. For example: Problem One asks, “If today
is Friday, what day will it be 100 days from now?” What makes this book unique
is its layout: Section One teaches the process of problem solving. Section Two
covers the problems themselves. Section Three has the answers. If the child
really doesn’t know what to do with one of the problems, Section Four offers
simple hints to the student. Section Five is a step-by-step Solution Manual for
the student who is completely stuck. This is followed by several Appendixes
dealing with everything from Basic Concepts for Young Mathletes to Working With
Exponents. Not just fun, but useful. Special Price: $21.00

a.. Math Word Problems Levels A, B, or C and Math Word Problems Set of Three:
A, B, and C. We are enthusiastic promoters of some of the Critical Thinking
titles (they publish Mathematical Reasoning, described above). These word
problem books are the original expensive one-volume text divided into more
useable books for specific grades. The publisher recommends the following order:
Level A (whole numbers & fractions) is for grades 4-6; Level B (decimals &
percents) is for grades 5-8; and Level C (whole numbers & percents) is for
grades 5-10. Choose the title that best fits your child’s needs or purchase all
three at a savings. Math Word Problems Levels A, B, or C Special Price: $11.00
each. Math Word Problems Set of Three: A, B, and C Special Price: $23.95
a.. Jacob’s Elementary Algebra Kit and Jacob’s Geometry Kit: A few years ago,
we were introduced to Jacob’s Math texts by the author of this article and,
since then, have sold more Jacob’s Math than we have sold Saxon. Most students
have testified that when compared to Saxon Math, Jacob’s Math is far easier to
understand and is much more interesting since it contains problems that are more
“real.” The “Kits” include student book, parent’s/teacher’s manual, and test
booklet (tests for each chapter, mid-term and final exam). The Teacher’s manual
is written to be used in a classroom with overheads which (overheads) are not
available; and are even (in our opinion) unnecessary, due to the
self-explanatory nature of the student text. The main value of the TM is that it
contains answers to some of the problem sets. Jacob’s Elementary Algebra
contains most of what is contained in Saxon Algebra I and Saxon Algebra II.
(Saxon does not have an independent Geometry text, but includes a small amount
of Geometry in its level II text). Jacob’s Geometry is a complete geometry text.
This is important because the College Entrance Exams often contain more geometry
than they do algebra. Jacob’s Elementary Algebra Kit Special Price: $79.95
Jacob’s Geometry Kit: Special Price: $84.95
a.. Algebra and Trigonometry: Functions and Applications Kit by Paul Foerster:
Harold Jacob did not write an advanced algebra/trig text; but, when asked which
one he would recommend, he said this text by Foerster is the closest to what he
would have written. We sell the Student Text and Solution Manual together. A
teacher’s manual exists, but it is nothing more than the Student Text with the
material from the Solution Manual in the margins of each page. We consider this
distracting to the student and, therefore, do not offer it.
Foerster presents his problems is such a way that students learn that there is
more than one way to solve most problems. Sections include Functions &
Relations; Linear Functions; Systems of Linear Equations & Inequalities;
Quadratic Functions & Complex Numbers; Exponential & Logarithmic Functions;
Rational Algebraic Functions; Irrational Algebraic Functions; Quadratic
Relations & Systems; Higher-Degree Functions & Complex Numbers; Sequences &
Series; Probability, Data Analysis, and Functions of a Random Variable;
Trigonometric & Circular Functions; Properties of Trigonometric & Circular
Functions; and Triangle Problems. There are also several appendixes and a
Glossary.Special Price: $125.00