It isn't enough just to teach learners addition and multiplication facts and computational algorithms: We need to teach arithmetic in such a way that they think it is cool to solve word problems because they understand how to do it and are challenged by it. Before I teach the addition and multiplication facts or the paper and pencil algorithms, I teach an operation in the context of word problems using pennies, rulers, tables, and calculators in order to teach how arithmetic works.

Many kids hate math and especially word problems because they "don't get it." We don't like what we don't understand. If word problems aren't an intriguing challenge for a learner, we haven't done our job properly.

I have a number of purposes in mind for this program: 1) to supplement school programs for children with exercises that will help them understand arithmetic, 2) to help teach arithmetic independent of a school program, 3) to provide a minimal program for adults who don't have the time or inclination to learn paper and pencil math, 4) to provide a review for parents and tutors, and 5) to point to additional resources for those who want to dig deeper.

I have found that children often guess at which operation is needed, and they usually don't check their work. They haven't been taught that there are three essential steps in solving math problems:

1. THINK: Analyze the problem. This is the first and most crucial step. What is wanted, what is given that is relevant, and what is at least one way to solve the problem? What operations are appropriate? How does one translate what needs to be done into the language of math?
2. COMPUTE: Perform the appropriate operations. This is the mechanical part of problem solving.
3. CHECK: The only thing worse than not being able to solve a problem is to solve it incorrectly. Learners should get in the habit of doing their computing neatly and checking their work thoroughly. This includes 1) rereading the problem to assure that you understood it and that you analyzed it properly, 2) checking the numbers used and checking your computations, and 3) asking yourself if the answer is reasonable.

There are two basic groups of arithmetic operations. The first is combining amounts as in counting, addition, and multiplication. The second is separating amounts as in counting backwards, subtraction, and division. Understanding this helps in thinking about word problems. I start by teaching the learner to count pennies, fingers, and beans. Counting is the simplest way to combine amounts; you keep adding the same amount. Counting backwards is the easiest way to separate amounts; you keep subtracting the same amount.

In order to achieve mastery you may need to provide more word problems at an appropriate level than I have included. You may find them in the child's school math books. If the learner needs more practice on skills, go to AAA Math. Also consider using Math for Your 1st and 2nd Grader, cited in Math Resources. It has word problems and lots of number problems to hone the skills.

Students must learn the addition and multiplication facts and the computational procedures, but they also must learn to use a calculator. When was the last time you completed your tax return without a calculator? The calculator enables the student to solve with ease more interesting and realistic problems. When a child does homework, he should abide by whatever limits the school places on the use of calculators.

There has been a long-running debate about how to teach elementary math that is similar to the reading instruction debate. One side favors the traditional skills approach, which is the tell'm, show'm, guide'm, drill'm, test'm, and review approach.

The National Council of Teachers of Mathematics issued math teaching recommendations in 1980, 1989, and 2000, which stressed the idea of letting children discover how to perform math operations. Parents often dubbed this approach "fuzzy math." A later recommendation issued on September 12, 2006, specifies the topics that should be covered in each grade. The national press interpreted this report as a return to basics. Hardly! See Topic: Curriculum focal points released. To see the NTCTM recommended topics for each grade, go to Number and Operations Standard.

It is hoped that parents and tutors will find the scripted mastery of skills approach emphasizing problem solving relatively easy to learn and to use in teaching. If a child's school teaches "discovery math," the parents should, first, try to understand it, and, second, try not to be hostile to it in the child's presence but proceed to supplement it with direct instruction. During the summer you have a freer hand on how and what to teach.

If you suffer from severe math anxiety, here is a chance for you to start over, and review and master arithmetic in a step-by-step way.

In teaching math skills remember these points:

• Preview each lesson so that you will be prepared.
  
• Act as if arithmetic is fun, like solving puzzles.
  
• Move along at a speed that is comfortable to the learner.
  
• Have a relaxing break every 15 minutes or so.
  
• Keep reviewing a skill until it is mastered.
  
• Celebrate achievements, even small ones. Be the learner's cheerleader.
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Your feedback will be welcome. Please send an e-mail message to me, Bob Parvin: bandcparvinXhotmail.com (Substitute @ for X. I'm trying to hide my address from spammers.)

ACTIVITY 1: ARITHMETIC SKILLS

Start this program by checking on the learner's knowledge of each skill and remember that there is more than one way to solve problems and make computations. One way to check skills is to pick problems from the various areas in this program and see if he can solve them. Try to determine whether the learner understands them or is just doing them by rote.

Schools don't have much time to drill the addition and multiplication facts. This is something that parents can do a lot better in one-on-one tutoring. Computer programs and games can help, but some old-fashioned drill may be necessary.

This program is scripted so that you read aloud the explanations to or with the learner, depending upon his reading level. Special instructions to you, which are not to be read aloud, are bracketed and italicized.

Make as much use as you can of the child's school textbook. Use the scripts in this program to supplement the textbook where they may be useful. In tutoring adults you may not need a textbook.

Here are the teaching scripts for you to use for the various skills:

Script for Counting
Script for Addition and Subtraction I
Script for Addition Facts
Script for Multiplication and Division I
Script for Multiplication Facts
Script for Number System
Script for Addition and Subtraction II
Script for Multiplication and Division II
Script for Fractions and Ratios
Script for Measurement Math I
Script for Measurement Math II
Script for Mental Arithmetic and Estimating

Other Skill Building Resources

For online resources to use in teaching the addition and multiplication facts and for the paper and pencil operations go to AAA Math.

To find out about software programs, go to SuperKids. For a review of a program to teach the facts go to Nothin' but the Facts, a SuperKids review.
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ACTIVITY 2: SUPERMARKET MATH

The supermarket is a good place to practice arithmetic, but we don't actually need to go to the store. We can simulate transactions using a supermarket flyer, which is like having a new workbook every week. We can also use catalogs of interest to the learner to simulate transactions.

In the simulation you may play the role of the customer, and the learner plays the salesperson.

You may use real coins (3 quarters, 2 dimes, 1 nickel, and 4 pennies) and play bills (4 $100s, 1 $50, 2 $20s, 2 $10s, 1 $5, and 4 $1s.) Early on you will need to teach the learner how to count out the change by starting with the amount of the purchase and counting change up to the amount tendered.

With each problem the learner should remember the three steps in problem solving: analysis, computation, and checking. Most of the computations should be made with a calculator because the focus is on problem solving and learning to use a calculator.

Checking when using a calculator means, first, checking one's thinking. Were the right operations performed? As each item is entered, the operator should check the display to see that it was entered correctly. When the operations are finished, he should ask himself, "Is my answer reasonable?" One might make a rough estimate to help make the test of reasonableness.

Here are examples of types of simulated transactions that you might use based on a supermarket flyer after having taught the skills:

1. Buy two nontaxable items, e.g., $2.45 and $1.76, and ask for the total charge. Have him express the number question and enter it in the calculator. He enters 2.45 and checks the display to see that he entered it correctly, presses +, enters 1.76 and checks, presses =, and reads 4.21 or $4.21.
2. Pay for the above items with a bill to give him practice in making change. He should hold the bill to one side until you confirm the change. Teach him how to count change up from the purchase price to the amount tendered. He should count it out aloud to you: "Your purchase comes to $4.21," and then he counts out the change starting with pennies. "This makes 22, 23, 24, and 4.25, and (adding quarters) 4.50, 4.75, and $5.00. Thank you."
3. There are two ways to do addition on a calculator. One is the usual way, and the second is to combine amounts in the memory bank. Have him add two amounts such as $1.45 and $3.27 in the usual manner. Then guide him step by step making the addition in the memory. First, he presses MC to clear memory. Then he enters 1.45, checks display, and presses M+ to put the amount in memory. Next, he enters 3.27, checks display, and presses M+ to add the amount to the first amount. To read the sum, he presses MR. He would subtract the same way by pressing M- instead of M+. To clear memory, he presses MC. If MR and MC are on the same key, he presses it once to read and again to clear.
4. Buy two or more items sold by the pound such as 3 pounds of carrots and 2 pounds of bananas. First, he will press MC to clear memory. If the carrots are $0.69 per pound, he will enter .69 x 3 = (2.07) and press M+ to add the product to memory. If the bananas are $0.57 per pound, he will enter 2 x .57 = (1.14) and press M+. Then he will press MR to read the sum of the two products in memory.
5. Make a purchase including a taxable item. All he needs to know about percent at this point is that the sales tax rate is a percent. An 8% tax means that for every 100 cents spent of taxable items a sales tax of 8 cents is added to the price. Buy, for example, a bar of soap for $0.75 and add your local tax. For example, if your tax rate is 8%, he will enter .75, press +, enter 8, press %, read 0.81. This has the same result as 0.75 x 1.08 = 0.81 or 0.75 x 0.08 to compute the tax which is then added to 0.75. The learner should be familiar with each approach after he has studied decimals and percent.
6. There are three types of division problems. Start with this type. Suppose that kiwis are $0.35 apiece. You buy all you can get for $2. The question is how many sets of 0.35 can one separate from 2. He enters 2, ÷, .35, = (5.7142857). This means he can separate 5 and about 71/100 sets of $0.35 from $2. Since they sell kiwi in round numbers, change is given amounting to 0.7142857 x 0.35 = 0.2499 = $0.25.
7. Buy something to practice a second type of division problem and to practice rounding. Suppose corn is three ears for a dollar. Buy one ear. He can picture $1 representing three equal sets of cents, each set being the price of one ear. So he divides 1 into three equal sets by division. He enters 1, ÷, 3, = (0.333333), which means each set is 33+ cents. Some supermarkets round down. The most common rounding rule is that if the digit to be dropped is less than 5, round down (0.33). If it is 5 or more, round up. So 0.666666 is rounded up to 0.67.
8. The third type of division problem is a comparison problem. A 6 oz item sells for $1.50. The 18 oz item sells for $4.00. How many times larger is the 18 oz item? How many times more does it cost?

9. Buy a "X pounds for $Y" item. Suppose oranges are 5 pounds for $2. Buy 2 pounds. Here is a two-step problem. The first step is to find the price per pound by dividing $2 into 5 sets: 2 ÷ 5 = (0.4) or $0.40 per pound. Second step is to multiply the price per pound in the display by the number of pounds. So he will press x, 2, = (0.8) or $0.80. The neat thing about an electronic calculator is that you can use the answer from one operation in a second operation without reentering it. This is called "chaining."
10. Ask which is the best buy between two brands. For example, is it brand A cereal, which is 24 ounces for $2.98, or brand B, which is 2 pounds for $3.29?

You can think of many other simulated transactions that will give the learner more practice where he needs it.

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Copyright (c) 2001 and 2006 Robert G. Parvin. Parents, tutors, and teachers may download, copy, rearrange, and revise this material for their own teaching purposes. This web site is made available free of charge "as is," with no warranties whatsoever. If you are dissatisfied with this web site, or any portion thereof, your exclusive remedy shall be to cease using the web site.