Regular math competitions will provide a motivational theme for the club. Students in 4th and 5th grades will be encouraged to participate in the Mathematical Olympiads for Elementary and Middle Schools competition. All mathletes will be encouraged (required if they hope to be promoted to the next belt level) to participate in the Heatherwood Math Olympics (HMO). Two HMO tests will be administered during during the year and the last meeting of the year will be an award ceremony rewarding the mathletes for their good work with belt promotions and mathletic medals to continue the athletic analogy.

5. Solution Procedure

Remember the old joke, hitting a drive in golf can be broken into a set of discrete steps. The first thing you have to do is address the ball. "Hello, ball!" It's the same thing in solving math problems. It's important for the mathletes to understand the general process of solving a math problem, any math problem, although a salutatory greeting will not be part of that process.

An effective math problem solving procedure generally can be broken into four steps:

• Understanding the problem.
• Developing a strategy.
• Executing the solution.
• Checking the solution.

Good problem solvers follow this approach implicitly and work through the steps very quickly. This will be a recurrent theme in the Heatherwood Mathletes meetings, and in helping your child solve problems at home, it would be good to keep these stages in mind.

a. Understanding. Before writing anything down, the student needs to read the problem carefully and think about it. As a coach, in working with the mathletes, you should allow them to ask questions as long as the questions concern the problem itself rather than how to solve the problem. To help the mathlete at this step, you might ask: What is the question you're being asked? Does the problem give you enough information? Or, what will the answer look like? In this final vein, the student should always ask this for him or herself: Will the result be a number? If so, what does the number represent? Will it have "units" -- like miles/hour or square-meters?

Particular difficulties may arise from the language of the problem. For example, the students may be asked to find the number of prime numbers less than 50. Here it may be clear that the student does not know what prime numbers are. This presents the opportunity to talk about prime numbers and to take the problem up again after this excursion. More subtle language problems may also appear that would be harder to diagnose. For example, the problem may refer to a workweek and mean five days rather than a full week. Some problems, like this, may really not be posed very well. In Bill Cosby's version of Noah talking to God about building the ark, God exhorted Noah to build an ark with a volume specified in cubits. Cosby had Noah ask the right question: "Right! What's a cubit?" As coaches, remember that problem-posers are fallible and part of the understanding of the problem may involve figuring out what the problem-poser really meant. Although this is not, strictly speaking, a mathematical issue, it's good experience in learning to deal with one's superiors, like teachers, parents, and employers.

It is often useful in this step for the students to summarize the information in the problem by writing something down. For example, consider a problem that states that Aaron has 6 apples, Bertha has 5 apples, and Carl has twice as many apples as Aaron and Bertha together. Algebraically fluent students would immediately write down something like: A = 6, B = 5, C = 2(A+B) = 2(11) = 22. In this case, a good choice of notation and an efficient summary lead to a rapid solution. Students, particularly the younger ones, may have trouble defining the notation needed for this kind of problem. Many will be able to solve it in their heads anyway, but harder problems won't yield to that solution procedure. We will need to work with the mathletes to develop effective means of summarizing the problem and developing notation that is clear to them and leads to solution. Examples shown during the regular meetings and in consultation with parent-coaches at home are needed to help the mathletes overcome this hurdle. This will take time.

b. Strategy. After the problem's understood, the mathlete needs to develop a plan of action for solving the problem. This is the part of the procedure that we're probably most interested in, as it is where the creativity and a large measure of the fun comes in. There are many different types of problems with a wide variety of different approaches or strategies. Any one problem will usually yield to more than one strategy. Some strategies will be more efficient than others will, but the most efficient strategies may not be practical for some of the mathletes. For example, a strategy may be to convert the problem into an equation or set of simultaneous equations, and the mathlete may not yet have experience either in translating the problem into these terms or solving the resulting equations once translated. So, other strategies will need to be employed in this case.

Dr. Lerchner lists some of the most commonly used strategies:

• drawing a picture or a diagram,
• finding a pattern,
• making an organized list,
• making a table,
• solving a simpler problem,
• trial and error,
• acting out the problem,
• working backwards, 
• using deduction.

Sometimes children may come to one of these strategies on their own, but in general we should not expect the mathletes to use strategies that are unfamiliar to them. Problem solving is a learned discipline, and it is learned by exposure to a wide variety of problems over an extended period of time. We view the exposure to many problems and a variety of problem solving strategies as one of the fundamental purposes of Heatherwood Mathletes. We will try to devote a part of every meeting to discussing a problem and the multiple strategies that can be used to solve the problem.

c. Execution. The problem solver must then implement the strategy to arive at a "solution" to the problem. If a student is following a strategy that appears to be fruitless, the coach will need to encourage the student to find another strategy.

Many of the problems that the mathletes will encounter will involve reducing the problem to an arithmetic computation. Most of the effort that the mathletes expend in their school work involves developing these arithmetic skills: addition, subtraction, multiplication, division, fractions, decimals, etc, etc. This is true enough that many of the students may enter the Mathletes equating arithmetic with mathematics. Nothing could be further from the truth. Mathematics is a creative discipline, as is problem solving in general, but at the heart of every problem (or mathematical proof) is the execution step that needs to be effected accurately -- and in the context of a competition like the Math Olympiad -- rapidly, as well. We hope that if the mathletes develop a love of problem solving, they will also develop a greater appreciation for the importance of working on rapid and accurate arithmetic skills. This is why Heathewood Mathletes devote so much attention to arithmetic skill development and it is important for the parent-coaches to stress this point with their mathletes.

Although rapid and accurate computation is an important part in solving the problem, it is important to understand this step in the broader context of the four-step process. Coaches should probably be most concerned and give greatest attention to the first two steps of the procedure (understanding and strategy). That said, the mathletes will receive the greatest satisfaction when they complete the problem and actually get it right!

d. Check. Children typically think that once they get an answer the problem is solved and their work is done. This is probably a symptom that they don't understand good problem solving procedures yet, and also that they really don't care that much about the answer. If they were doing a calculation to determine if they had enough money to buy a new bike, outfit, or video game that they desperately wanted, they'd probably check to see if their calculation was right. If it's a school problem, then maybe checking the answer is not seen to be all that important. Once it's understood that checking the result is an important part of problem solving, an interest in the checking step can be seen as a barometer of the mathlete's interest in the problem. We, therefore, must seek to encourage their interest in the problem in general to ensure that they check their answer thoroughly.

But, what does checking an answer consist of and how can the parent-coach help the mathlete perform the check for him or herself. (1) The first question is if the answer is reasonable. If you're balancing your checkbook and find that your balance is greater than your yearly salary, you've probably made a mistake (or should seek better investment advice). Alternately, if you get a negative balance, you may want to recheck your result (or transfer funds). Often a student can see immediately that the answer was wrong simply by asking if it seems right. Dr. Lerchner recommends that the student be asked to state the answer to the problem in a complete sentence. In doing so, the student may see that their answer was absurd. (2) A second approach that the student can be asked to consider, is to summarize the question and their strategy and ask if, in retrospect, it was actually an effective strategy. On reflection, the student may realize that their strategy did not answer the question that was asked, but may have answered another question. (3) Finally, the student should always be asked to check if the arithmetic was carried out accurately, or the table made right, or the diagram drawn right, etc. If the strategy produces an answer in several parts, each part would have to be checked. They should be encouraged to consider doing the arithmetic differently, if possible. So, for example, if they added something, try again by chunking into a different set of summands. Students also tend to add when asked to subtract or multiply sometimes, and should always address whether they have carried out the operation that they have intended.

Good problem solvers do these steps automatically, but as coaches we will have to work with the mathletes to help them follow these steps until they become automatic.

6. Being an Effective Coach

Being a good mathletic coach isn't easy, and we all are learning on the job. One of the key issues to learn will be how much help we should give to the mathletes. Too much help will leave them with little to do, and too little help may lead to frustration. If we hit the mark right, the students will experience the challenge of the problem and the pleasure of discovering its solution. The letter ff the law is to ask only questions and let the mathletes work independently. But, both Belt Coaches and Parent-Coaches will have considerable leeway. Sometimes it will just be best for your child to see you lead through a step in the solution, other times you can best help by asking leading questions. Let your knowledge of what your child needs be your guide.

Dr. Lerchner warns that many students mistakenly believe that successful problem solvers are those who are able to determine a strategy and carry it out almost immediately. He tells us that children need to understand that it is all right to experience difficulty in the process and it will be common that they will need to test several strategies before finding the appropriate one for them.

He also gives some advice about how to deal with wrong answers. He suggests avoiding expressions like "No", "Wrong", or "No good". I like "Go directly to jail, do not pass go" or "You are sentenced to the fires of hell for all eternity". Actually, he suggests praising what's right, clearly identifying what's correct and using expressions like "Close" or "Try again" for the parts that need work. It's really just a matter of common sense. If the child clearly does not understand the problem, has the wrong strategy, or has made a simple computational error, you should help to rectify the situation explicitly by asking what the problem means, suggesting another strategy, or circling the part of the computation that they should recheck.

7. Expected Mathlete Decorum

It is unfortunate that evenings, when the parents are most able to participate, are the time of day when many of the children will not be at their best intellectually. Some of the children will be running on empty, others will be sleepy, others will want to run around, others will see the meetings as a good time to interact with their friends. But, the purpose of the meetings will be to prepare the mathletes for creative problem solving. Hence, our challenge. Nevertheless, under any circumstance, the mathletes will need to stay focused on math during the meetings. Disruptive behavior by even one student may affect all of the mathletes. We will need to stress the 3-Cs to the mathletes with respect to our expectations of their behavior during meetings. These are:

• Concentration -- Mathletes need to concentrate on the work they are doing,
• Courtesy -- Mathletes must be courteous to other mathletes and coaches, particularly by concentrating on their work and behaving in a non-disruptive way, and
• Cooperation -- Mathletes need to cooperate with the coaches to be courteous to other mathletes and concentrate on their work.

Children who have trouble abiding by the 3-Cs will be asked to give up their belt or under worse circumstances miss a few meetings until they are able to convince their parent-coaches that their behavior will not be disruptive to other mathletes. Continuing problems will result in students being asked to rejoin Heatherwood Mathletes in the subsequent school year.