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Build Strong Arithmetic Thinking

by Ruth Beechick

Popular home education writer Ruth Beechnik explores developmentally appropriate ways of teaching math to young children (preschool-second grade). This article offers ideas to help ensure that a child develops a strong conceptual foundation in mathematics. It also offers a "checklist," a sort of scope and sequence, for tracking a child's learning.

Almost everybody knows that abstract thinking comes later after understanding has been built. But, strangely, much arithmetic teaching ignores that fact, especially with young children. To see how this works, try reading the following problem.

ó e ó Ģ ö

Now that was so easy your children could learn it the first day, right? So the next day they can go on to:

ó e ¨ Ģ !

Do we really do that to our children? Yes, in many arithmetic curriculums we do. Letís change a couple of symbols to ones we adults know and make these problems easier to read.

ó + ó = ö

ó + ¨ = !

That helps, but remember that plus and equals signs are still abstract for children. This is a bit better.

ó and ó are ö

ó and ¨ are !

And this is a bit better again.

2 and 2 are 4

2 and 3 are 5

But this is not yet what we call concrete thinking. The digits are still abstract symbols. And the words two, three, and five are abstractions also. What is concrete are two hands, two fingers, two more plates to put on the table.

Concrete Thinking

At preschool ages and in early grades, the best teaching we can do for children is to help them understand numbers-not digits (1,2,3Ö) but actual numbers of things. We can begin to connect the word two with two items, but the digit 2 on paper should wait until understanding has been built up.

Some years ago when Sesame Street was new and popular, most children watched it and our nation dreamed that it would revolutionize early childhood learning. All children would learn their numbers and letters before they even started school. That would make them smarter and would speed up their learning. At that time I asked a first grade teacher if it changed her curriculum any. Now that children already knew numbers, was she able to do more advanced teaching? She answered no. She still had to teach the meanings behind the numbers.

After much criticism from teachers, Sesame Street began to teach more thinking instead of just memorizing symbols. But I donít know how well they do on numbers. The couple of times I have seen it they dwelt on thinking about politically correct values.

Activities for Understanding

So how do you begin arithmetic with your child? First, get rid of any textbooks or workbooks you have for kindergarten and first grade. And second grade, too, if youíre brave. If itís too hard to throw out books you have paid for, put them up on a shelf and in a year or two your child might want to do some pages for fun because he can then understand them.

Everyday activities are far better than workbook activities. Here are a few ideas.

Driving time: "I see a cow." "Look, there are two cows." "Two more traffic lights and we will be there."

Kitchen time: "Take four forks to the table," or "How many forks do we need?" "Please get four slices of bread."

Shopping time: "Do we have five kinds of vegetables and fruits?" (Nutrition teaching correlated with arithmetic.) "Do we have too many items for the fast lane?" "How many carts are ahead of us?"

Work time: "Set the timer at ten minutes and see if we can finish before it rings." "See how many pairs of socks in this pile are yours."

Play time: While playing with blocks, Legos, and such, children will match to make this wall as high as that and other kinds of arithmetic thinking. You rarely need to help this along. Practically all kinds of games provide arithmetic thinking - dominos, Lotto, jacks, board games, pick-up-sticks.

Home-Style Is Best

Once you get the habit, you will find it second nature to prompt your childís observation and thinking about numbers as well as about other mathematical concepts-pound of butter, gallon of water. One family decided in the summer that they were going to homeschool in the fall and the mother immensely enjoyed the next auto trip they took. She talked to the children about what they were seeing along the way instead of just telling them to keep quiet in the back seat. This was simply a change of attitude that she was now the teacher of her children. She didnít have to study or get a certificate to do that.

This home-style teaching works best. Workbook writers sometimes try to begin with concrete thinking. For instance, they may picture three airplanes for the child to count. But then he must immediately write the digit 3. Furthermore, he must add that to the two airplanes in the next row, and at the bottom of the page he gets the whole abstract notation: 3 + 2 = 5. When you work instead with the childís concrete thinking style, you delay the notation until lots of understanding has been built up; you donít get to it at the bottom of each page.

Another ridiculous teaching in the books is to learn key words found in story problems. How many "in all" is a key telling the child to add. How many are "left" tells him to subtract. Why do we analyze the wording of story problems? To help us do story problems on tests. This is "teaching to the tests," which, of course, is not teaching for real life.

The first stage of concrete thinking is having physical blocks or toy airplanes to manipulate. The next stage is to picture the items in the mind. Workbook pictures may be halfway between those two stages. But learning that certain words on a page translate into plus or minus is artificial in the extreme. We get ourselves into this situation by making workbooks for such young children and by pushing down abstractions lower and lower in the grades. Learning key words is not going to help when the child comes to geometry and other higher math where he must be efficient at abstract thinking. The way to prepare for advanced abstract thinking is to build understanding at the lower levels of arithmetic. You build it in concrete ways. We worry in our society that this is slow, but in the long run it is faster, as well as stronger, learning. And it has the additional advantage of not burning the child out on workbooks by third grade.

Track Your Childís Learning

You can keep a checklist to record your childís progress in arithmetic. From time to time you will observe that the child has achieved one or another of the skills on the list which accompanies this article. Check or date the item on the full list, or if you prefer, add each new observation to a growing list you keep in the childís portfolio. Feel free to add items you observe that are not on this list. If your child achieves the listed items, consider that a good normal level to reach by third grade.

Remember, these skills are to be learned with real objects, not with problems on pages of a workbook. Not with problems that look like this:

ó e ó Ģ ö

Arithmetic Checklist

Preschool to Grade Three

  • Counts to 10___, to 100___ to 200___. More than just chanting, the child must be able to count out 15 sheets of paper, etc.
  • Uses and understands ordinal numbers up to 10th___. Examples: we are third in line; itís the fifth house on the left. On calendars may go up to 30th___.
  • Recognizes groups such as dots on dominos or dice: up to five___, to 10___.
  • Adds two groups up to sums of 6___, to 12___.
  • "Takes away" a group from 6 and tells what is left___, from 10___, from 12___. (Donít do the other kind of subtraction yet-the kind which compares two items and asks how much larger or smaller one is.)
  • Counts by twos to 10___ or higher___. Use pairs to practice this-eggs in a carton, socks in pairs.
  • Counts by fives and tens to 30___, to 100___. Use an abacus, piles of checkers, nickels and dimes.
  • Experiences the use of
    • nickels___, dimes___, quarters___,
    • ruler (inches only)___, feet (no fractions)___,
    • time in hours___, half hours___,
    • whole measuring cups and spoons___.
  • Understands a fractional part of apple, candy bar, etc.: one-half___, one-fourth___, one-third___. (Only numerators of 1.)
  • Uses words of quantity, size and shape:
    • tall, taller, tallest___, large, larger, largest___, less, more___, top, bottom___, circle___, square___, hour___, minute___, other___.
    • Advanced concepts: add___, plus___, equals___, subtract___, minus___, left, right___, dozen___, quart___, pint___, pound___, other___.
  • Solves real-life problems___. Examples: What coins do we need to buy this stamp? Do we have all the books to return to the library? Advanced (Use after most of the above are checked.)
  • Reads digits from 1 to 10___, to 30___, to 100___, writes digits to 10___, to 30___, to 100___.
  • Writes addition and subtraction problems in horizontal sentence form___, and vertical form___.
  • Understands two "times" a group___, three times___.
  • Can divide a larger group into twos___, into threes___, fives___, tens___. (Use real objects only, not written problems.)

Copyright, 2004. The Old Schoolhouse Magazine. Reprinted with permission.

About the Author:

Dr. Ruth Beechick is a well-known and respected teacher, professor, and curriculum developer. Her popular and affordable early education books for reading, arithmetic, and language are available through as a set (including all three books and a wall chart) for under $10.

Growing Together Family Learning Newsletter, Vol. 1, No. 1, page 9

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