Mathematics Ages to 5 to 14

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.

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.

Four Cautions

The advice offered here is approximately correct, for some circumstances not all. Pick and choose that which applies to yours. 

1. 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.
   
2. 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.
   
3. 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.
   
4. 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.
   

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. 

 1         41      100
 5         34      456
+3        +23     +342
 9         98      898

The above examples indicated three counts may added or assembled via a decimal place value COLUMN method for addition.  The extension of the foregoing to include carries is left to reader. 

The above remarks provide a primary school base for the arithmetic how-TOs above. 

Rote learning is alive and well in mathematics, or should be.  The foregoing identification of the counting principles in primary school mathematics, and arithmetic how-TOs provide a thought-based development for decimal methods for addition, comparison and subtraction.  Here the comparison by conversion method is a prequel to the thought-based art of subtraction. That being said, students should see how repeated addition leads to multiplication - a thought-based introduction of multiplication. But decimal methods for multiplication and long division should be mastered by rote in a manner that leads to repeatable, reproducible and reproducible results. The thought-based development would be too long for primary school students.  That being said senior high school mathematics in the development of multiplication methods for polynomials may easily digress to include a thought-based mastery of decimal methods for multiplication (easy) and for long division as well. That being said, primary school students be taught how to check the results of long division via the calculation:

dividend = quotient*divisor + remainder.

To learn more, see the 

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