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Seven Components of Math Programs - Zaccaro (THP Winter 06)

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The Seven Components of Successful Programs THP Cover
for Mathematically Gifted Children

Ed Zaccaro,
Dubuque Community Schools, Bellevue, Iowa

What do I do with the third grader who spends his evenings computing baseball statistics? How do I challenge the sixth grader who loves to talk about the big ideas in math and always seems to finish her work before her classmates? Many elementary and middle school teachers find themselves with one or more students in their classroom who are gifted in mathematics. It can be difficult to determine the most effective way to provide programming for these children. Although there is no one perfect program, these guidelines, based on years of experience and study, are designed to help your advanced students experience challenge and enjoyment in math.

1) Challenge and frustration are a part of learning and life. They should both be viewed as a normal part of the learning process.

While most mathematically gifted children enjoy challenging material, some children find the experience of challenge and frustration to be quite stressful because it is a foreign concept to them. Teachers of  mathematically gifted children have the sometimes unpleasant task of helping these students understand that limiting their academics to an intellectual box where there is no struggle or frustration is not healthy and leads to a life that is not as fulfilling or as rewarding.

For example, I share with my students many experiences where adults have made mistakes, including many of my own such as rappelling off a cliff without checking to see if the rope was long enough to hit the bottom. (It wasn't and I was forced to jump 15 feet.) I also have students work on problems like those in the box below where mistakes are frequently made  so they can have the experience of making a mistake and seeing that life goes on.

 No Easy Answers: Problems that Mirror Life

  1. How many square inches are in a square foot?  

  2. What is 8÷1/2?

  3. What is larger, n or 2n? 

  4. If there is a giant piece of chocolate that weighs 250 pounds, how many 4/5 pound pieces can be cut from it?

  5. The unit for weight that is used in the United States is the pound. What   is the metric unit for weight? 

  6. The unit for weight that is used in the United States is the pound. What   is the unit for mass that is used in the United States? 

(Click here to access the answers.)


2) Math is often taught as all scales and no music. Children must have the opportunity to see the exciting and interesting parts of mathematics.

The goal of  many programs for mathematically gifted children is to move students through the curriculum as quickly as possible. This approach can lead to a loss of interest in the subject because it does not nurture a child's passion for mathematics. An alternative approach is to keep gifted children with their same age peers, but give them an opportunity to experience the parts of mathematics that are not only challenging, but also very interesting.

When children first see the wonders of math and science, it is as if they stepped into a room that they didn't know existed.  When they are provided with the opportunity to work with algebra, trigonometry, and physics, children are in awe of what mathematics allows them to do. Examples include

  • Using simple geometry to see how the circumference of the earth was determined for the first time almost 2500 years ago,

  • Using their knowledge of the speed of light to realize that looking at stars is looking back in time a hundred, a thousand, or even a million years,

  • Finding the distance a ship is from shore through the use of trigonometry (Yes, gifted children in elementary school can work with trigonometry), and

  • Finding the height of a tree by measuring its shadow and then using ratios.

Gifted children typically are not given the opportunity to see the wondrous side of mathematics because it is usually taught as all scales and no music. If musicians were not given the opportunity to perform or play music that stirred their hearts, it is unlikely that they would develop a passion for their field. The same holds true for children and mathematics. Children who are talented in mathematics must be exposed to material that lights a fire and nurtures their gift.

3) It is important for children to be shown the fascinating connections between mathematics and the real world.

Because mathematics instruction is often dominated by facts and calculation, children are rarely exposed to important concepts that connect math and science to the real world.

One of these concepts relates to an under-appreciated fact about mathematics and science----Math and science are not like referees and umpires that you can argue with if you don't like what they tell you.  Math and science are coldly and cruelly indifferent to your hopes, dreams and wishes. They  give you an honest and objective look at a situation. Do not ignore their message!!

There are hundreds of stories from history that show this concept has not been followed:

  • The Challenger disaster occurred because the recommendation not to launch made by mathematicians and engineers was overruled by management.

  • The popular singer Aaliya was killed in a plane crash because the pilot and others ignored what mathematics told them. (They knew they were dangerously overloaded, but chose to try to fly anyway.)

  • Racial bias in jury selection was proven by a mathematician. (It was determined mathematically that the probability the jury was fairly picked was approximately 1 in 1,000,000,000,000,000.)

  • About 2500 years ago, mathematicians changed the study of space from one of fantasy and guesswork into a real science.

4) Children who are gifted in mathematics must learn to appreciate their gift.

Can you imagine what it feels like for an athlete or musician to have hundreds of parents and classmates cheering for him or her? Add to that the newspaper articles, trophies, medals, and other awards. This kind of reinforcement pushes athletes and musicians to excel. It is unlikely that this kind of motivating environment will ever become routine for those students who excel in math and science. Because there are precious few opportunities for gifted children to be formally recognized and honored, it is important that teachers make students feel that their gifts are something to be treasured.

I have found the use of Einstein Awards for extraordinary problem solving to be a very effective tool to motivate children. When students solve a very difficult problem (called Einstein problems), they receive an award with a picture of Einstein presented in class.

The response to these awards has been dramatic. One parent called and said that she had to limit her son to two hours of math each night because he was constantly trying to solve Einstein problems. Another parent called and said his 5th grade daughter was so excited by  her first Einstein award that she told him that it was the best thing that ever happened to her!

When children see that an area in which they excel is valued by those around them, their interest and passion for the subject can increase dramatically.

5) Parents and educators must understand that a child's interests and passions do not necessarily correspond with their areas of giftedness.

The experience I had with my oldest child led me to a clear understanding of the importance of allowing and encouraging children to follow their passions, which may or may not be their area of giftedness.

Luke was a very talented violin player and also gifted in mathematics, but Luke's two areas of passion were soccer and voice. Two very worthy areas of passion, but unfortunately they were areas where Luke not only did not possess giftedness, but had clear weaknesses.

Luke played soccer for years and eventually because of his perseverance and passion for the game, made and played on his high school team. He also had an ambition to make the All-State choir. To that end he practiced for years without success. The noises coming from his room were not always pleasant, but he continued to rehearse several hours each day until he finally made All-State in his senior year of high school.

Today Luke is not involved in either mathematics or violin. He is the vocal director and teacher at a high school in Wisconsin and also the varsity soccer coach and, from what I can see, very happy and content with his life's direction.

 6) Mathematically gifted children must be given material that truly challenges them and appropriately challenges them.

Bright math students usually pick up concepts so quickly that they are left with very little to do intellectually while the rest of the class masters the new material.  In addition, the consequences of not challenging elementary children can be serious because children who are bored tend to develop thinking skills and work habits that are less than ideal.  

One solution to this dilemma is to differentiate instruction. Look at the problems below. Notice how the complexity and difficulty increase as the levels increase. When children are presented with various degrees of difficulty, they are able to attempt problems that are just within their cognitive grasp. This not only enables them to grow intellectually, but also helps nurture a passion for mathematics.

Level 1:  Eight gallons were poured into a gas tank that was 1/4 full. Now the tank is 3/4 full. How many gallons does a full tank hold?

Level 2:  Sound takes 5 seconds to go one mile. Clark is standing near a rock wall and when he shouts, it takes 20 seconds for the echo to reach his ears. How far away is the rock wall?

Level 3:  A sprinkler that waters in a circular pattern shoots water to a distance of 15 feet. If the sprinkler is set in the middle of a 30 foot by 30 foot yard, how many square feet of the lawn does the sprinkler miss?

Einstein:  Sara, Claudia, Karen and Kath are sisters who inherited money from an uncle. Claudia received 1/5 of the money while Karen received 1/2 of the money. Kath received 1/4 and Sara was given the rest. If Sara received $1750, how much money did all four sisters inherit?

7) Highly able children must have the opportunity to work with children with similar abilities.

The importance of having the opportunity to work with children of similar abilities cannot be overstated because the value of this kind of interaction is not limited to the intellectual growth that it can foster.  The social and emotional development that can occur as a result of healthy disagreement, discussion, and debate can have a profound impact on  mathematically gifted children. An additional benefit is a reduction in the social isolation that these children sometimes experience.

In summary, meeting the needs of mathematically gifted children can be difficult.  As teachers try to develop mathematics programs that provide appropriate challenges and also teach basics skills, decisions need to be made concerning acceleration, enrichment and differentiation. As these decisions are made, it is imperative that teachers also keep in mind that they must help students take intellectual risks; learn to think deeply and with insight; see the magic and wonders of mathematics and help students understand and appreciate mathematics and its place in the world.

Ed is a retired teacher and the author of several books that provide curriculum for mathematically gifted children. He can be reached at

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