private tutoring for grade school through Algebra
math club classes
Pre-Algebra and Algebra classes
workshops for grown-ups
What is my educational philosophy?
Arithmetic and mathematics are vastly different!
Arithmetic requires strong number skills and a good ability to memorize.
Mathematics requires many skills, not only strong number skills. It also involves strong logical thinking skills, visual-perceptual skills, auditory perceptual skills, verbal skills, and fine motor skills.
Arithmetic:
Adding
Subtracting
Multiplying
Dividing (decimals, fractions, percents)
Managing money
Mathematics:
Algebra
Calculus
Chaos Theory
Combinatorial Game Theory
Complex Systems
Encryption
Engineering
Fractals
Geometry
Logic
Modeling
Number theory
Numerical Analysis
Origami
Physics
Set Theory
Software Design
Statistics
Structural Design
Trigonometry
What do you want your children to be learning?
Why are they spending, on average, eight years of arithmetic and only 2 or three years of mathematics? Mathematics is where the fun is and where the jobs are. We should be spending far more of our time there.
Math education today simply doesn't make sense to most learners. We concentrate on the small details without exposing children to why they need to learn basic arithmetic skills and not the 'big picture' skills of mathematics.
Educational experts and researchers all agree that children will learn best if they are actively engaged in the learning process. Children need to interact with what they are supposed to be learning, not remain passive. Making these experiences fun is the best way of ensuring learning will take place. How do we make it fun?
Quality math games and quality math literature should make up the majority of "math time".
Why Games?
Playing games appeals to all learning styles (auditory, visual, and kinesthetic) because they use all of their senses to participate. They get to interact with others, ask questions in a non-threatening situation and explore new concepts in a fun environment. They even get to act out concepts using physical representations (called math manipulatives). Oh yeah, and they get to work on their calculating skills too.
Many skills and higher order thinking skill levels can be approached through games.
Games offer understanding of concepts such as:
more-or less-than understanding cause-effect relationships
before and after recognizing patterns
categorizing place value concepts
building sequences skip counting
making sets rote counting
reading large numbers recognizing numbers
number pattern recognition fractions and percentages
building memory logical thinking skills
sequencing deductive reasoning
identifying parts of a whole visual and spatial perception
planning and problem solving auditory perception
finding pairs verbal skills
making one-to-one correspondence mental computation
estimating ratios
money skills (such as giving the correct change, rent, salary, loans, budgets, bankruptcy, taxes, interest, inflation, bartering, etc.)
You can introduce new skills using games, drill facts using games, and test them to be sure they remember things using games. Best of all, they will remember math as a fun and interesting time spent with you, and remember math as something they love!
Some game ideas may require materials such as pop sickle sticks, special cards you make out of index cards or an egg carton. In these cases I recommend putting each game in its own small box with a label to be kept with your other board games. You will be surprised how often they get pulled out once made!
Some games will require you to photocopy a game board. In these cases, be sure to glue the game board to tag board or cover the game board with contact paper after coloring it in. Many, many game boards can be kept neat and in order in a three ring binder in this way - just put a pencil pocket with any special supplies you need next to the game boards!
Why Books?
Good math literature captures the imagination of a child - and therefore their attention! They can learn a remarkable amount of number theory and definitions from storybooks carefully chosen by you.
Good literature can introduce to an amazing variety of concepts including those of algebra, geometry, and calculus.
Most of all, math literature is of special (almost critical) importance to 'global' or 'conceptual' thinkers - those right-brained people who need to see the big picture in order to understand the small details.
Many children can know the math concepts but can't make heads or tails out of a page of numbers. If your student understands words better than symbols, math literature may become a valuable lifeline to math. These children can comprehend deep mathematical mysteries if they come in the form of words.
Not all children learn best through this method though. Those children who are very visual learners simply can't 'hear' a book being read to them for very long if they can't follow along with at least pictures. Be sure, if it is not a picture book, to allow these children to build with LEGOs, draw, or similar so that they have a visual context within which to put all those words.
TEACHING ALGORITHMS:
Algorithms are just ways of solving problems. There are very few types of problems that can only be solved one way. Knowing a wide variety of ways of approaching a problem can only strengthen the way a student looks at new problems. Despite this, many curricula today do not teach them and hope students will discover them on their own.
Let's take the example of the multiplication tables. Memorization, in general, only comes easily to a small percentage of students. Those for whom it does not come easily are often convinced by the end of fourth grade that they are bad at math due to the stress on memorization and timed tests. This need not be the case. There are a lot of tips, tricks, and tools to help students learn their multiplication tables that are seldom taught.
Did you know most of us only need to memorize 6 problems? Many do not because you sure don't get that idea from that big table of 100 problems you are shown in third grade!
Why only 6? Here is how I look at it:
1 x any number is itself so no memorization is required
2 x any number is just doubling the number and we learned that when we learned our addition facts.
3 x any number is just doubling the number plus itself and although this might take 2 or 3 seconds to figure out is usually fast enough not to slow you down
4 x any number is simply doubling the double and, again, we already learned that
5 x any number can be solved three ways:
a) skip counting (this takes awhile, but many use it)
b) by picturing a clock face (all numbers on a clock's face are a multiples of 5)
c) using the division method (divide the number you are multiplying in half and
then multiply the answer by 10
ex: 5 x 4 = 4 divided in half or 2.0 and 2.0 x 10 = 20 or
d) 5 x 7 = 7 divided in half or 3.5 and 3.5 x 10 = 35
9 x anything is done by most people using the reverse number patterns.
This leaves only 6 x6, 6 x 7, 6 x 8, 7 x 7, 7 x 8, and 8 x 8 left to memorize.
Luckily for a few, there are even tricks for these! One favorite of mine is as follows:
1) Hold your hands out in front of you with the fingers pointing toward each other. Your thumbs should be closest to your body.
2) Now, number your fingers. Your thumbs are 6s, pointer fingers are 7s, middle fingers are 8s, ring fingers are 9s and pinky fingers are 10s.
3) Now you need a multiplication problem involving 6s through 10s. Let's try 7 x 7 to begin with. Touch your two 7 fingers together (pointer fingers). Each of the fingers touching and toward you are worth 10. In this case those fingers add up to 40.
Now you need to multiply the sum of the fingers not already used on your right hand times the sum of the fingers not already used on your left hand. In this case that is 3 x 3 and equals 9. Add 40 plus 9 for the answer 7 x 7 = 49.
Another example is 6 x 7:
Touch your 6 and 7 fingers together (a thumb and a pointer finger)
Add 10 for each finger touching and toward you (30)
Multiply the sum of the remaining fingers on your right hand times the sum of the remaining fingers,on your left (3 x 4 = 12)
Add the two answers together (30 + 12 = 42)
Answer: 6 x 7 = 42
Please note: the information here is copyrighted by Kathy Wentz and not available for copying whole or in part to email lists, web pages, books or any other public access source without the express permission of the author.
