Before Math Begins
By Tom and Sherry Bushnell
From the NATHHAN NEWS, Fall 1997
If you have trouble teaching math, do not feel alone!
There is little information for homeschooling parents of special needs children, letting them know that before the skill of counting (beginning math) can be acquired, there comes some even more important learning.
In this article, we hope to share with you some fun ways we use to prepare our disabled children for the skill of math.
First we would like to clarify two important words in this article:
ORDINAL: This term is used to described counting by first, second, third, forth, etc.
CARDINAL: This term is used to described numbers and quantity, not order. 1,2,3,4, etc. is what we are talking about here.
In this article, we will present sections in a logical order of progression.
1. Approximate mental age of math concepts to be acquired.
2. Teaching children to make and identify shapes.
3. Visual perception activities to help number identification.
4. The first steps to understanding cardinal numbers is opposites, sequencing and classification.
5. Activities that help children understand ordinal numbers.
6. Early addition and subtraction: important vocabulary and some early activities.
7. Math activities that come after counting. Estimating Measuring Money and time
8. Excellent hands-on math resources
1. There are the mental ages </u\> usually associated with the following skills. Your child may have splinter skills or be behind, but on the average, this is what to expect
Matching colors ages 2-3
Names colors ages 3-5
Matches shapes ages 2-5
Names shapes ages 3-5
Matching like objects ages 3-5
Opposites ages 4-5
Rote counting 1-25 ages 4-6
Picks out and counts objects ages 4-6
Matches groups of equal numbers ages 5-6
More or less ages 5-6
Matches and finds numbers 1-25 ages 5-6
Before or After numbers ages 5-6
Adds with objects ages 5-7
Adds in head with numbers 1-10 ages 5-7
Counts backwards ages 6-7
Subtracts objects 1-10 ages 6-7
For an older mentally delayed child, here are some steps required to learn early math:
A. Learn the number symbols and their meaning. Count and relate specific quantities to each symbol. This takes practice. This understanding must come from memory and spontaneously before additional math concepts can be acquired.
B. Decode efficiently. Must be able to translate math problems into action. For instance, one apple plus another apple equals two apples. If a learning delayed child must spend too much time concentrating on beginning decoding they are not ready to tackle further addition or subtraction of symbols.
C. To be able to apply memorized facts (1 + 1 = 2, etc.) to produce answers to more complicated questions. Incidentally, we can readily see this when our older learning-delayed child tries to apply multiplication or long division when they have not memorized the times tables.
2. Suggestions on how to teach children to make shapes: Begin with a circle. This seems to be the easiest for children to start with.
Using a chalkboard, put a piece of tape in the middle. Using colored chalk, have you child go around the piece of tape. Make large circles, going around, around, and around without stopping. Next stage is to put a dot on a piece of paper and play racetrack driver around the dot. Use plenty of exaggerated motion and noise!
Next, we make a simple dot-to-dot and they simply connect the dots around, without the cue in the middle.
For a triangle, start with large, dot-to-dot objects. Moving to smaller. Be aware that this takes repetition. It can take days before a child may recognize a circle from a square, from a triangle.
The best way we have found is to simply repeat daily, as part of the
total activity picture for the day. Use a notebook and review the shapes
daily. We use black construction paper cut out in a circle, square or
Trace with finger each time "around and around makes a ... circle, that's right!"
Tracing out the triangle starting at the bottom corner, count 1...2...3.. makes a triangle. A square is similar counting "1...2...3...4...that's right, this is a square!"
Be sure to use a different order so you know they are not just remembering the names in order given but truly recognizing the shapes.
Next we sort shapes. We use cut out, colored construction paper. Different sorting games can be played, especially in conjunction with learning colors.
Present the regular notebook shape and have them find the correct smaller shape from the ones placed beside them on the table.
Next, older children and some younger ones, can relate shapes to objects
in the room. For instance, when you hold up a circle, they can find a
paper plate, the end of a spool of thread, the rim of a bowl. For a
3. If our children do not have the ability to recognize spatial relationships (circle, square, triangle, rectangle, etc., they will not go very far in math. So, if you see trouble in identifying and making early shapes (barring fine motor skill problems), work on visual perception first. An easy way we like to work on visual perception is a kind of dot-to-dot activity. Here is how we do it:
Materials you will need:
* 6 sheets of Index card stock (8" by 11 1/2" sheets)
* A black marker than can produce dark clear 1/2" dots
* Washable marker, black
* Access to a laminating machine
Using a piece of card stock, map out lightly with a ruler a grid so that you can make 5 dots on a page, equally spaced.
Now make a second one.
Next, take a third sheet of card stock and make a simple grid of 9 dots.
Now make a second one.
Finally, take the last two sheets and make a simple grid with 25 dots [illustration]
After these are made and dark 1/2" dots are made at the appropriate intersections, have them laminated at your local stationery store.
To begin with, make simple line to line shapes with your erasable marker and have your child copy them on his card of dots. Start with easy to more complicated: Lines that come close are good enough. They need not be absolutely straight as drawn with a ruler. Just so you can see that they can perceive which lines should interconnect, if any.
4. Sorting according to common themes is a prerequisite to understanding cardinal numbers. Here are some grouping words that are important to have our children remember: All, some, only, none, anyone, everyone, and nobody.
Picking up toys is an excellent way to practice grouping and sorting. All the legos go in this box, the farm animals go in this container, and the Lincoln logs all go in this tub. Sorting silverware, cups, plates and bowls is another of our favorite ways to practice classification.
An activity to practice matching and classification is to color different colored squares on cardboard to create a tree, flower or animal shape. We have the children crumple wads of corresponding tissue paper, beads, buttons, or fabric and glue them onto the dots corresponding to the squares of the same color. This makes a mosaic of sorts.
There are hundreds of sorting tasks in a household. No need to buy expensive sorting beads with card, etc. Having our children naturally learn to sort by working together is all part of a happy, peaceful home.
Next, we can practice counting. Songs and rhymes when we start teaching numbers may help with remembering the names of numbers, but these names must mean something in order for math concepts to be understood.
The first skill required in the sequencing process is learning opposites. Here is a good listing of comparatives that are important because they have to do with ranking:
Bigger Biggest Smaller Smallest, Taller Tallest Shorter Shortest, Higher Highest Lower Lowest, Thicker Thickest Thinner Thinnest, Longer Longest Shorter Shortest, Wider Widest Narrower Narrowest, Deeper Deepest Shallower Shallowest, First Last Second Next Before and After
5. Sorting objects into a specific order (smallest to biggest, etc.) is paramount to understanding ordinal numbers.
Hands-on activities are best when teaching sequencing concepts, i.e.
Who arrived first? Who arrived first, second, third, in the room? Who gets to wash dishes tonight, tomorrow? Which apple is biggest ... smallest? Rank vegetables by size. Rank bowls by size. Food such as meat can be thicker than bread. Crackers can be thicker than a sandwich.
Going up stairs (who is on the first step, second step, etc.) is a fun way to work sequencing in during the day. For older children who struggle with sequencing, give them a routine to perform in a certain order: First, run, Second, jump across the bench, Third, roll to the wall, Fourth, run back to me.
Many appliances and their operations teach sequencing: First plug in, then push this button, then set this knob, etc.
Vocational training for older mentally delayed children is simply practicing sequencing. For instance, learning to fold an item, wrap item, pack in box, close box top and set the closed box in the tub is a complicated but familiar sequencing useful for job training.
Important note: Avoid comparing people to each other, although details may be blatant. The Bible teaches that we should not compare ourselves one to another.
6. Addition and Subtraction
How quickly our little ones learn that 2 cookies are better than one, and if someone takes one of their cookies and leaves them with one, they are not better off!
Here are some important vocabulary words our children need to know before entering into addition and subtraction: Add, take away, decrease, increase, limited and unlimited.
Addition is taught first. Simple addition may be all that some of our
children ever acquire. We need to be grateful for the place are children
are at mentally and not feel like a failure because we can't seem to
We also highly suggest encouraging the use of fingers in counting. A good resource for this is The complete Book of Finger Math: Simple, Accurate, and Scientific by Edwin M. Lieberthal. This book is #368 in the NATHHAN Lending Library.
Addition can be taught simply, without complicated programs or work sheets. In fact, if your child is kicking out page after page of addition worksheets, yet cannot perform a task requiring adding like items, perhaps centering on making hands-on problems is a good idea.
Please bring me 5 forks, 7 spoons and 3 knives. Now, add to that 2 forks. How many forks do we have now? Take away 2 spoons. How many spoons do we have now? Add 4 knives. How many knives do we have now? How many pieces of silverware does each person at the table have now?
7. Incorporating math into every day activities becomes a way of life for most families with pre-school children. Luckily, with homeschooling, it doesn't stop when our children reach first and second grade, because most homeschooling families keep on applying math to every day life. In fact, as our older children start to show an interest in a particular trade, we can tailor make math programs that give them extra practice in areas of calculating that they will really need.
The same is true for our special needs children. Beginning with math, like colors, opposites and shapes are important and adding and subtracting are more useful for some children than others in their daily life. Putting math into perspective for our children begins with observation from us parents.
Does your child spend time counting objects that are meaningful in their lives?
Does your child seem frustrated when he or she doesn't bring enough silverware over for the table?
Is your learning disable son or daughter needing practice in measuring to go much further in learning skills for running a business or household?
Perhaps your child could care less about integers, algebra, and finding the area of a quadrilateral.
Major on important, interest holding math, when working with children who have a hard enough time remembering to read or write. Stick to life math. Make it meaningful.
A good book for meaningful math is: Math for Every Kid: Easy Activities that Make Learning Math Fun by Janice Van Cleaves.
Here are some very practical math skills that our children may be able to understand and perform BEFORE even mastering addition and subtraction.
How many do you think we'll need for...
This is a very useful skill. It enables our children to grab just a few or quite a few, depending on what is asked of them. This is especially helpful for our children who are learning disabled and saves considerable time over laboring over every item to be counted out when asked to go get "something" for someone. This skill just plain takes practice. Start with visual perception skills. Is your child able to copy simple line drawings?
Nest move to comparing which containers will hold more...cereal, water, rocks, sand.
After this, move to fun activities like guessing how many spoons each person will need for dinner.
Start with simple counting of persons or items needing something.
Move on to how many of what ever item is needed that each person will need.
Next practice grabbing handfuls that look close to 3, 5 or 10.
Have them set each handful in a bag and count the handful as they go.
Once again, the skill of opposites is general a prerequisite for
understanding measuring items. Biggest, smallest, higher, lower, these
types of ranking are important.
Many measuring skills require counting. However, there are a few ways we
can get around this. For instance, we can instruct our child to measure
a "foot" (about their foot with shoes on hopefully). We can measure
their hands or two hands and say, "Set these down, using both your hands
as a space." We use this concept when teaching printing to our children.
When measuring the temperature outside, use a large, outdoor hanging thermometer, color-coded. When the hand is in the blue part of the thermometer, we tell our children to put on a coat. When it is in the orange part, we tell them to put on a sweatshirt. If the hand is in the read part, they do not have to wear a coat or sweater. This makes telling approximate outdoor temperature easy. (Eliminates arguing too!)
When starting fractions (we won't tell them this is what we are doing) words like whole and part are pretty easy to figure out. Point out a whole pizza. Minus one piece, minus another and so on can become a fun game at dinner. When half is gone, point this out. All gone is important too.
For our children who can count to 25, a useful measuring skill is to teach them to measure an item by another item. A ruler need not be present every time we measure something. Show them how to take a pencil, bottle cap or whatever is handy and count how many pencils or bottle caps long the item needing measuring is. This way they can duplicate the measured item without a ruler at hand.
Money and Time:
Recognizing coin values take practice. Some of our children will never understand the value of money. (Some non-delayed adults don't either.) We can help them, however, learn to keep their money organized. This makes counting out a total at the check stand less complicated for the teller. It is much safer than allowing our children to hand the checker a wad of bills or a pocket full of change.
Use a sturdy wallet with compartments. One with zippers is preferable, putting pennies in one, nickels in another and quarters in another. Bills can be grouped together according to pictures.
Next step is teaching our children how many pennies are in a nickel and dime. Then comes how many nickels in a dime. "How many coins in a quarter" becomes more difficult. If our children can figure this out, they probably will be able to put together change at a later date with practice.
Also, we teach our children social politeness in regards to money, by not letting them flaunt their money or count it in public around other children who may want it.
Value is another hard subject to teach. How many of us have experienced one of our children handing us a few coins and having them point to a nice new toy for us to purchase for them. Life has been the best teacher for this important subject. Counting helps too!
Finding grocery items in the ad section of the newspaper (color pictures are preferred of course) and having our children practice counting out the correct amount helps our older ones before the real crunch time at the check out stand comes and pressure is on.
Telling time has become much easier with digital clocks and watches. Our children need only learn cardinal numbers before they can tell us the time. However, for complete understanding of time, the old fashioned clock with hands that move and are placed on numbers which mean something to our children is best. The first step to understanding the concept that time has limitations is the buzzer. For instance, "In 10 minutes we are going to have devotions; let's pick up the toys now," or "We are going to leave in 5 minutes," or "Sit beside me here in this chair for 3 minutes to settle down."
Next, having our children set the buzzer themselves to "race the clock" because a fun way to make time go all too quickly!
Counting by fives is the next step. We always make counting by fives the important passage to wearing a watch. It is a great incentive. Counting by fives around the clock starting at 12 is the next logical step.
Four our children who will never count by fives, helping them learn to watch a clock closely and "When the big hand is on the 10, you will need to put awy the toys," helps.
One to one correspondence
Once again, some of our children may be a long way off from understanding any number above one. However, some may be able to understand one-to-one correspondence rather than counting.
For instance, One knife, one fork, one spoon for each person at the table.
Draw contours on place mat of a cup, plate, silverware, and set a place mat for each person around the table. Each jar needs a lid. Each plate needs an apple, a sandwich, and a piece of cheese.
One button hole for each button. Each egg carton holds twelve eggs, fill it up and close it. Make a fruit or vegetable salad using two of each fruit or vegetable, etc.
8. Some of the resources that have been a help to NATHHAN families and come highly recommended are below:
Math Their Way - order through bookstore. A complete math program for kindergarten through second grade. A slower moving program for pre-school level. Much time is devoted to readiness concepts. Lessons are presented through hands-on activities that are really attainable at home. It is very parent friendly and unifix cubes are recommended. This is not a workbook but reproducible activity sheets are included.
We also suggest: Making Math Meaningful - Levels K-7 It moves slower than a lot of math programs and is meant to be used on a one-to-one basis with each child. Available through Lifetime Books and Gifts, 1-800-377-0390.
Mignon Math has been also used by NATHHAN families. Available through Lifetime Books and Gifts, 1-800-377-0390.
Math-U-See is also excellent for children moving toward understanding addition and subtraction.
References for this article:
Helping Children Overcome Learning Difficulties, 3rd edition By Jerome Rosner
Practical Activities for Teaching Concepts By Svein Nymoen, Lena Richter and Sissel Hofgaard Swenson
The Teaching Research Curriculum for Moderately and Severely Handicapped - Self-Help and Cognitive Prepared by The Staff of the Teaching Research Infant and Children Center
Instant Math for Beginning Skills & Concepts Hands-on Manipulative Activities By Sam Ed. Brown, Ph. D.