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Primary and secondary teachers could buy large-print
math workbooks for grades 1 to 6 (Primary School Mathematics) in duplicate
to post their pages on walls under the title tutor training program
for students to see and review what they should know from present or
earlier studies. Students in difficulty can be asked to explore the posted
material to regain confidence and fill gaps in their knowledge. The JumpMath
books described next are not big-print.
Elementary Mathematics: To learn to
read, write and spell, students need to master the alphabet - learn it and not
forget it. Anything less would lead to difficulties or fear in or of reading,
writing and spelling. Likewise, to learn high school and college
mathematics, and to avoid fears and difficulties,, algebra, geometry, trig and
even calculus, students need to master the following efficiently and fully to
the point of automation, the how with and if necessary without comprehension
of why: addition and times tables, decimal methods for arithmetic; angle,
length and time telling or measurement; fraction skills and sense
besides calculator usage skills. Alone or with help, parents, teachers and
older students, those taking charge of their own education, need to
check mastery, develop the missing ones, or verify the missing
ones are being develop in school.
Toronto JumpMath
Work Books (Grades 3 to 8) for home and school
The jumpMath home
and school mathematics program asserts the following:
One feature
distinguishes our workbooks from regular math textbooks, however: in the
JUMP workbooks, teachers are consistently shown how to help students who
are having trouble moving forward by breaking mathematical concepts and
operations into the most basic elements of understanding and perception
in its Teacher Manual -
Fractions, page 2. The jumpmath
publication page offers a downloadable fraction unit and describes
workbooks for home and school (grades 3 to 8). Copies of the workbooks for
grades 3 and 4 bought for inspection are well-done. The workbooks may
cover more than necessary.
-
The jumpmath program
publication page offers a downloadable fraction unit and describes
workbooks for home and school (grades 3 to 8) - Schools should see the
bulk order prices. Parents should consider chaperoning their children,
grades 3 to 8, through the home version of the workbooks.
The jumpMath program
(created by a mathematician) appears to cover the middle years of primary
and high school instruction well. So I recommend its consideration besides
any other program parents and teachers choose for students in grades 3 to
8 in school or for remedial instruction. See what is what is
best.
Older Site Material Questions and activities in
the webpages
[ Level I ] [ Level II ] [ Level III ] [ Level IV ] [ Level V ]
will allow you to judge the mathematics and logic
skills of your charges in primary school and high school instruction.
A simpler route may be to acquire the jumpmath workbooks for home
schooling (I have seen those for grades 3 and 4) and chaperone your
offspring through them - time and patience required.
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Secondary School Mathematics
The page Where is it going
gives or suggest reasons for high school that seem to have been
forgotten.
Teens and pre-teens need to obtain fraction sense and an efficient command of
operations with fractions and whole number without a calculator. If that
is not done, all further mathematics instruction is sabotage or
suboptimal. The teacher who finds his or her students do not have the
prerequisite for the current course needs to review and consolidate those
prerequisites - not doing so may turn instruction into a formality. The
new site area Solving
Linear Equations with Stick Diagrams and Fractions introduces
algebra while illustrating and reinforcing fraction sense and skills. As parent,
you could take your son or daughter through it carefully.
The ability to add, subtract, multiply and divide fractions whose
denominators and numerators range from 1 to 100 say is sufficient provided
students also have a good command of what is a fraction. That be said,
fraction sense and skill is needed to understand and do algebra beyond being
given a formula and numbers to use in it. The algebraic way of writing and
reasoning needed the further or proper mastery of analytic geometry,
trigonometry and calculus relies on fraction sense and skills. While some
students and teachers unfamilar with the next level in mathematics may object to
or not know the foregoing, the efficient mastery of fraction sense, what they
are, and fraction skills is a must for going further in high school
mathematics. That mastery should be consolidated between ages 10 to 13
say. Anything less slows or stops learning. All topics before students
talk about solving equations may be used to emphasize fractions and allied
concepts: ratios and proportions. The aim of site areas in fractions,
algebra, geometry and so on is to provide students or their teachers and
tutors (parents included) clear directions for understanding and explaining
mathematics and its logic.
Site books (online in full so you do not
have to acquire them) may help some students 14 plus to adult learn
mathematics and logic, and parents guide their children.
The advice offered here is approximately correct, for some
circumstances not all. Pick and choose that which applies to yours.
- Learning takes time and effort. Your child or teen should know that or be
told, especially if you have succeeded in protecting them from worry. Do not
assume he or she knows that learning takes time and effort. Marks in schools
may be too generous to the extent that students do not receive this
message from teachers. I am a skeptic.
- Your charge also needs to be told that notes and work for doing problems,
written on paper, needs to be written precisely. Ideas or work written
incorrectly will be a source of error. For instance, methods for arithmetic
(addition, multiplication, subtraction, and long divisions) rely on numbers
being written in the proper place or column. Imprecision in location or
alignment of numbers is a common source of error due to a change in meaning
or interpretation between writing and reading. Likewise, the algebraic
way of writing and reasoning requires a proper and precise command of
notation, otherwise what is meant or intended at time of writing will not be
misinterpreted a moment or period later at the time of reading or further
reasoning.
- Too many high school and college students not interested in mathematics
believe arithmetic should be left to decimal computations with
electronic calculators. They forget how to do arithmetic by hand met
if not learnt in earlier years. But in algebra and beyond, operations
with fractions appear and they need to be done exactly - the decimal
approximations provided by electronic calculators cannot be used for the
exact derivation of formulas. At the high school and college level,
please check whether or not your son or daughter can add, multiply and
simplify fractions efficiently. You may be surprised, sorry.
- Online help in reading, writing or mathematics has one limitation at the
moment. The written work of students needs to be seen and corrected
repeatedly for errors in presentation and notation. In mathematics
especially, a student may master an idea (almost) without being to write
arithmetic or algebraic calculations precisely and exactly on paper. The
errors here in notation are dangerous. A student may write one thing
while meaning another, and then later read what is written inexactly.
Imprecision in reading what is written is accompanied by an imprecision in
writing mathematical thoughts or calculations on paper. Imprecision in one
implies imprecision in the other. The written work of students needs
to be read and marked carefully, so that imprecision in writing (notation)
is corrected. Here if there are few errors all should be identified, but if
there are many, the most important ones should be identified and some left
uncorrected in order not to discourage a student too much.
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Primary School
Mathematics Revisited
(Added October 8, 2008)
Early schooling and parenting may and should provide a good operational
command of decimal arithmetic from working with one thousandths of a unit
to working with millions of them. It should also provide an operational
command of fractions. Whole numbers N and general fractions p/q like five
and three quarters should be regarded as multipliers. They say how
many wholes and unit fractions are present.
In common language a unit fraction (1/q)
appears when we speak of a half (1/2), a third (1/3), a quarter (1/4), a
fifth (1/5) and so on. The three quarters identifies, as it
should, 3/4 with three times a quarter. In general the
fraction p over q or p q-ths should be identified a p times a q-th.
To be more precise, it saying three quarters in words or 3/4 in fraction
shorthand notation, the three (3) is the multiplier. It counts the
number of quarters 1/4. When you have a unit that can divided into four
parts of equal value (4 identical parts would suffice), each of those
parts equals a quarter of the unit or 1/4 unit or
unit
4
Here the denominator 4 may be regarded as a divisor,
while the unit fraction 1/4 can be regarded as a multiplier. Following
that, the fraction 3/4 alone or as in 3/4 units may be regarded as a
multiplier.
Irrational numbers too, for example pi, act
as multipliers too. The circumference or perimeter of a circle is pi times
the diameter d = 2r or p = pd = 2pr.
Counting with Decimals. Early schooling provides a gradual
knowledge of the alphabet, say a to z. It takes time for learners to
recognize and then write the letters, and then to form
words. Likewise, time is required for learner to meet and master
whole numbers. The latter mastery may proceed in sequences:
- 1, 2 and 3
- 1 to 10
- 11 to 20
- 21 to 100
- 100 to 1000
- 1000 to 10000
and so on. Student learn the principle of counting one at a time,
and how to count by re-arranging objects into sets of 0 to 9 ones, 0 to 9
tens, 0 to 9 hundreds, and 0 to 9 thousands, and so on.. That leads to the
development and mastery of the decimal number notation: For
example, 546 indicates 5 hundreds, 4 tens and 6 ones. Decimal
notation points to counting and accounting by grouping of objects, real or
imagined, into sets of sets of 0 to 9 ones, 0 to 9 tens, 0 to 9 hundreds,
and 0 to 9 thousands, and so on. Decimal notation provides a
counting method. Students learn to add one, ten, hundred and thousand in
in and with decimal notation.
- 17+1 = 18
- 4567+10 = 4577
- 3456 +100 = 3456
The foregoing increases the count of ones, tens or hundreds by
one. But continued counting leads to conversions of 9+1 ones,
tens, hundreds and so on into one ten, one hundred or one thousand.
There-in lies the first appearance of conversion in decimal
addition.
First Steps in Addition. The question why we take 3+5 to
be 8 stems from the physical addition of a set of 3 objects to set of 5
objects
o o o + o o o o o = o o o o
o o o o
3
5
8
Thus a non-overlapping count of 3 with a count of 5 gives 8. The
objects may be dots, animals, coins, and so on. Primary school students
may spend years seeing how the addition of single numbers in context
implies the addition table for all pairs of digits 0 to 9. That
answers the question why 1+1 = 2 in human languages. Hands-on
experience with objects or their pictorial representations (dots etc) lead
students from the physical situation
3 units + 5 units = 8 units
to the arithmetic leap and assumption
3 + 5 = 8
Thus 3 +5 = 8, an addition of multipliers, stems from experience. Its
empirical. It appears to be repeatable, reproducible and thus verifiable
result. The next result is
3 units + 5 units = (5+3) units
The latter done backwards (with the aid of the
following arguments for the counting being independent of order) gives
(5+3) units = 3 units + 5 units
That illustrates and implies the distributive law for
multipliers, or multiplication of units by multipliers.
Counting is independent of grouping: Beyond the foregoing,
counting with decimal grouping in units, tens, hundreds and thousands
appear to give a unique result. That if a set of 3 48 units is
counted one way and then another, then both counts should lead to the same
number units and tens and hundreds. Albeit, a group of 348 units may be
divided into 3 groups of a hundred, 4 groups of ten and 1 group of 8 unit
in many different ways
The count of how many different ways would be an exercise for
the high school or college lesson on combinatorics - the art of
identifying different combinations and counting them.
In counting small sets in terms of the numbers
- 1, 2 and 3
- 1 to 10
- 11 to 20
- 21 to 100
the principle that the count is independent of the order of counting is
met and applied in practice, and then extrapolated. but not vocalized.
An pattern that is assumed is called an axiom in mathematics. The
pattern may stem from experience. Now that is stated, we do not have to
refer the experience.
The principle
Axiom (Assumption): The decimal count of a set of units is
independent of how the count is conducted.
needs to explicitly adopted for the logical development of arithmetic
with decimals. As an illustration, counting the dots
o o o o o o o o
from left to right, vice-versa, or an random order (there are 8!
possibilities - why) leads to the number 8. Further remember, decimal
notation points to counting and accounting by grouping of objects, real or
imagined, into sets of sets of 0 to 9 units, 0 to 9 tens, 0 to 9 hundreds,
and 0 to 9 thousands, and so on. Decimal notation provides a
counting method independent of how units are arranged into decimal groups
of 0 to 9 ones, 0 to 9 tens and so on.
The above axiom as is or strengthened (point to ponder) will be
employed below.
Addition with Decimals - place value methods: Here we assume
students know how to add all pairs of numbers 0 to 9. That we assume
master of the 10-addition table. Addition of decimal counts is based
on the counting via sorting into groups of 0 to 9 units, 0 to 9 tens, 0 to
9 hundreds, and 0 to 9 thousands, and so on.
5
34 456
+3 +23
+342
8 57
798
So (A) 5 + 3 ones gives 8 ones; (B) 4 one + 3 ones gives 7 ones, and 3
+ 2 tens gives 5 tens; and (C) 6 ones + 2 ones gives 8 ones, 5 tens
plus 4 tens gives 9 tens; and 4 hundreds plus 3 hundreds gives 7
hundreds. The foregoing place value addition is based on counting in
groups of 0 to 9 units, 0 to 9 tens, 0 to 9 hundreds, and 0 to 9
thousands, and so on. But in (C), there one set thought of as being
grouped into 4 hundreds, 5 tens and 6 units, combined with another set
grouped into 3 hundreds, 4 tens and 2 units, yields (4+3) hundreds, (5+4)
tens and (6+2) ones. |