Succeeding In Homeschooling of Mathematics
by Hal Schneider
Most homeschooling parents will agree with the proposition that, of all subjects, mathematics has proven to be the most difficult. Most parents of non-homeschooled students would also chime in: 'Right On !'. Unlike many other subjects, a very common lament of Math is: My child simply doesn't get it!
Most parents, if they are honest about it, will also admit: I never really got it either! Which is, of course, a contributing factor to the problem.
Yes, Math is different because of the various concepts that need to be truly understood (vs. being exposed to and remembering facts of other subjects such as History, Geography, and English). Unless a student is properly taught the 'fundamentals' of Math early in the process, success becomes 'iffy'. To me, 'fundamentals' is much more than remembering facts such as the multiplication tables or the learning the concepts of basic arithmetic (e.g. adding and subtracting).
Starting with 4th-5th grade or so, it is critical that relatively simple, but critical concepts be properly introduced. For example, give them a problem like: 3 cats plus 2 dogs equals???? Naturally, many students would instantly yell out '5 animals'! Isn't that correct ? In Math terms, NO !!... because one of the 'fundamental' Math concepts is: (1) Only the values of like 'things' may be added to or subtracted from each other.
Since a cat is not like a dog, they may not be added to one another! (End of statement.)What is also important in this problem is that we moved our student forward into the 'real world' where numbers typically have 'things' associated with them -- a critical aspect of the practical usage of Math.
Other critical 'fundamentals' (the criticality of which may not be obvious) include:
(2) The result of multipling or dividing a 'thing' by 1 equals the original 'thing'
5 times 1 = 5 and 3 dogs times 1 = 3 dogs
5 divided by 1 = 5 and 3 dogs divided by 1 = 3 dogs
This concept also can be considered in a related, but in a different and more useful sense:
(3) The value '1' can be written as any 'thing' divided by itself
1 = 4 divided by 4 1 = 3 dogs divided by 3 dogs
Another simple concept is:
(4) One may substitute one form of a 'thing' with another 'thing' of equal value
1 dime = 2 nickels 1 dollar = 4 quarters 1 half = 2 fourths
These four Math concepts are building blocks to a very easy understanding of many more complicated concepts such as fractions, word problems, etc! Really? Yes, indeed!
Suppose we consider the denominator of a fraction to be a 'thing' (e.g. ' halves', 'fourths') in the manner as we would for 'dogs', 'cats', 'miles', etc.).
Thus, the fraction 2/4 can be thought of as '2 fourths'. Therefore,
2 fourths + 3 fourths = 5 fourths ( 2/4 + 3/4 = 5/4)
So now we introduce the problem:
1/4 + 1/2 (1 fourth + 1 half) = ?
Like our 'dogs' and 'cats' problem above, there is no immediate solution because the problem involves addition of two unlike 'things' (a 'half' is not like a 'fourth'). However, if we were to use fundamental (4) and substitute an equal value for '1 half' that uses 'fourths', we would then be able to do the addition. How do we do that?
Using fundamentals (2) and (3), the value of '1' can be expressed as anything divided by itself.
In this case, changing a 'half' thing to a 'fourth' thing involves breaking the half into '2' pieces. So, we choose the '2/2' form of '1' and multiply: 1/2 times 2/2 = 2/4 (1 half times 2 halves = 2 fourths)
We now substitute '2 fourths' for '1 half' (equal values in different forms) to give us two like things so that we can add: 1/4 + 1/2 = 1/4 + 2/4 = 3/4 (1 half + 1 fourth = 2 fourths + 1 fourth = 3 fourths)
Note how much time it took us to arrive at an answer that many students would arrive quite easily and quickly. My experience is that the typical student's trait to 'rush to the answer' is our worst enemy!! The 'Why I am doing something.' is one critical key to success.
Other important 'fundamental' concepts are:
(5) Any number followed by a word equals the number times the word
2 dogs = 2 times dog
(6) The values of like or unlike things may be multiplied or divided by each other
2 times 2 dogs = 4 dogs (a number 'thing' combined with a dog 'thing')
60 miles divided by 2 hours = 30 miles divided by hour ('miles' and 'hours')
3 men work 15 hours = 45 man-hours of work ('men' and 'hours')
Does you child do fairly well in Math with the exception of math word-problems? If so, join the crowd! Here's a tip: Many commonly-used, non-mathematical words or phrases have precise mathematical meanings. Simplest example: 'and' = add. Can you come up with such common, non-mathematical words that mean 'multiply' or that mean 'divided by'?
Many such 'key words or phrases' are used in math word problems. Shouldn't the student be taught their Math meaning when s/he comes across these key words? Are you homeschoolers doing it?
Another favorite technique of mine for word problems: Absolutely prohibit 'premature' arithmetic operations until everything that needs to done is defined (hopefully, in an equation, but at least in short, written steps). Why? If you ever look closely at most students' work sheets (aka scribbles) when solving a word problem, you will quickly see how disorganized it is. Should I add, subtract, multiply or divide which of the multiple given values . . . why?. . . what's the next step(s)? Performing any premature arithmetic can readily disrupt the logical thought process needed to reach the end result. Attempting to write an equation using given facts is just the opposite . . . it can only be developed using logical thought. Also, always include the 'thing' associated with a given value when writing.
Although there are many more 'fundamentals', they are beyond the scope of this article, you may visit my website for a free test (mostly practical math word-problems) which involve the use of the above and many other 'fundamentals'. All the problems include detailed solutions.
In summary, based upon standardized international testing, it is no secret that Math education in the United States needs lots of improvement; homeschooling Math is even more difficult! Unless we first teach these 'fundamental' concepts in a 'proper' manner (i.e. using enough one-on-one detailed explanations in a manner similar to above), use of good, but complex integrated 'Math Teaching Programs' for homeschooling may be for naught. Hal Schneider, e-mail : email@example.com (URL): home.earthlink.net/~mathaid/mathbook.htm
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