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Writing In the Mathematics Classroom Using Heuristics and Algorithms

Abstract: Writing is an integral part of conveying both mathematical concepts and solutions. Often K-12 educators are not provided with an adequate means of teaching writing in the math curriculum without delving into the English curriculum.  By teaching the students to construct heuristics and algorithms (HAL) to derive solutions to mathematical problems, then the student is taught an effective means of problem solving to the extent that they must list the givens, concepts learned, and learn to think sequentially. The construction of algorithms and heuristics stands on its own merits in the area of problem solving and critical thinking. By teaching students not only how to construct algorithms and heuristics, but also a few transitional words and phrases then the HAL can then be used by the student to write an English translation of any given problem and its solution.

Introduction

One of the most difficult concepts for the K-12 student in the mathematics classroom is translating a mathematical solution into words. This may be the result of the level of abstraction needed to make the transition from a numerical solution to the written word. Another potential stumbling block for students when writing solutions to math problems is that students often have a tendency to solve the problem and then immediately jump into writing the solution without ever preparing for the writing process. Finally, writing across the math curriculum has never been incorporated effectively because a method of transcribing mathematical processes has never been clearly defined for either instructors or students.

In English classes, students are taught that the most effective way to write an essay is to first construct an outline. Conversely, math students are told to solve the problem and then explain in English how they solved it. Consequently, students do not have the slightest idea on how to record their abstract thoughts into everyday English. So, students blunder through the process as best they can. There is also the fact that mathematicians are placed in an awkward position when they are asked to teach writing skills for the simple reason that it is not their area of expertise. Further complications arise since the typical mathematics textbook does not present any writing tools directed specifically at the mathematics curriculum. The result is that math educators are forced to improvise when it comes to teaching writing.  Subsequently, there is no uniformity in teaching the writing process in math. The contention is: if math educators had a mathematically based writing model, then a higher quality of writing in the mathematics classroom would be commonplace. To teach a uniform method of writing across the mathematics curriculum we are forced to take an eclectic approach and borrow from the fields of philosophy and computer science. Enter the heuristic and the algorithm.

A heuristic is a generalized method used to solve a problem, and an algorithm is a step-by-step method used to solve a problem. We can categorize any problem that needs to be solved by a heuristic as one that deals with what may eventually become an infinite series of steps when broken down completely. Theoretically, if a problem has an infinite number of steps, then it can never be fully broken down—and so we generalize and in our generalization we create a heuristic. Any problem that can be solved via a finite number of steps and has a numerical solution can be approached as needing an algorithm to generate the solution. Granted, some linear equations have an infinite number of solutions, but the solution of the equation has just a finite number of steps. In teaching both concepts, it is best to generalize the definitions of a heuristic and an algorithm

For arguments sake, we shall say we need a heuristic anytime we solve a problem that is non-numeric. The reasoning is that such problems usually require human intervention, and all human actions/behaviors/thoughts are infinite by nature. Take the example of bending a finger. One is tempted to say this involves only the pulling of the finger toward the palm. In reality, to bend a finger the brain first generates the idea to bend the finger. Once the idea establishes itself in the brain, then signals are sent to all of the muscles involved to either contract or relax at a specific moment in time. But, the brain signal alone contains a series of steps where neural transmitters are secreted and the neurons are turned either on or off. The all-or-nothing quality of neurons themselves involves even more molecular process.  Eventually, we find ourselves at the atomic level, then the quantum level, until, before we know it, we are playing with the very essence of infinity—space-time. But, we still haven’t defined how the initial thought of bending the finger was generated—to do such would guarantee us the Nobel Prize. Anyway, somewhere along the line we have to make a decision of when we can say that we have provided enough information to solve the problem. Hence, we make a generalization and, by definition, we have created a heuristic. An algorithm is any step-by-step solution. Since math education is concerned with deriving rigorously exact and accurate solutions, then the solutions to mathematical problems are, by their very nature, finite. Thus, we can solve any math problem with a finite number of steps and in doing so we establish a need for the algorithm.

It should not be assumed that only algorithms have a place in the K-12 classroom; both the heuristic and algorithm (HAL[1]) can be used extensively in teaching students problem-solving in almost any subject area. But, the beauty of HAL goes beyond the actual problem-solving process. With HAL, not only are educators provided a good problem-solving tool, but they also have a method of teaching writing in the mathematics classroom. Consequently, when a HAL is constructed properly, not only are students providing themselves with a mathematical solution, but they are also constructing an outline from which a powerful English paragraph can be generated. Furthermore, when the student is also taught how to join various transitional expressions with the steps of a HAL, then a nearly flawless paragraph may be constructed with minimal effort. HAL is to writing in math classes what the outline is in English classes.

Introducing Students to HAL

All HALs are composed of four parts. The Title, the Initial State, the Goal State, and a series of steps needed to go from the Initial State to the Goal State. The Title is a short description of the HAL problem. The Initial State is where all the given data, and derived data is listed.  The Goal State is where we want to be—all that is required here is that the question be restated as a statement. The actual steps of HAL are the various statements needed to solve the problem, and the last step is always the derived conclusion. The last step can simply be the Goal State restated with the derived solution.

When HALs are initially introduced it is best to do some activities that the entire class participates in. Obviously, the educator should first define a heuristic and an algorithm and have the entire class practice saying the two terms aloud—students actually enjoy listening to themselves stumble over these seemingly impossible pronunciations. Once this has been done the educator should have all students write the following format in their notes:

HAL Title

Initial State: What is given?

1.
2.
3.
.
.
.
N.

After the format is introduced then the teacher should illustrate the differences between an algorithm and a heuristic by using a cooking recipe. Using a heuristic to produce the desired results of a cooking recipe would only require that the HAL format above be used, and then simply copy the steps of the recipe into the steps of the HAL. A cooking recipe is the easiest way to define a heuristic. Using the same recipe it is a simple task to show the students how much more complex it would be to generate a cooking “algorithm.” Such an algorithm would require that every step necessary to fill a cup of flour be listed. For example (explain the below pseudo-code to the students):

1. locate cup
2. extend hand to cup
3. lower hand 12”
4. open fingers
5. lower hand until it rest on cup handle
6. place hand on cup handle
7. grasp cup
8. lift cup six inches
9. locate bowl
10. rotate arm 20” to your right
11. et cetera
.
.
.  now we need to beat the eggs
.
.
. better fill that teaspoon of vanilla

Emphasize that these steps are the steps of an algorithm needed to just lift and pour a cup of flour. It should be explained to students that the above algorithm could also be considered a heuristic since in extending the fingers certain neuronal impulses are sent to the brain, and the brain then sends impulses to the involved muscle groups on whether they should contract or expand. In the above case a simple “algorithm” is used since it would be unwieldy to do otherwise, and students should be made aware of this fact.

Once the students have been introduced and copied the general format then the best thing to do is to lighten things up by having them create a heuristic that requires the seated educator to stand-up, walk to the classroom door, open the classroom door, and exit through the classroom door. Start by writing the HAL title. Students should then generate the Initial State and the Goal State via Socratic dialog. Once this has been done, write a numerical list from one to 25. As each step is successfully executed, then you or a student helper should record the steps. A sample of the Initial State and Goal State of our “Walking to the Door” HAL are listed below.

Initial State: Teacher is seated in the chair with arms on his/her lap.
Goal State: Have teacher exit the door.

This may seem like a simple function at this point, but when certain rules are set it becomes a difficult learning experience. Such a rule is that before any function can be called it must first be defined. Let us now work with the task of standing up from a seated chair.

1.  Lift both arms four inches
2.  Move hands forward until over the knees
3.  Drop hands to knees
4.  Lean torso forward 45 degrees
5.  Place weight on hands
6.  While leaning forward lift the body slowly and straighten out the torso until standing straight

Now we can define the “StandUp” function as being the above six steps. At this point it would be a good time to explain to the class what is happening in the brain and the various muscle groups, not to mention maintaining balance, for performing the simple task of standing up. Have the students stand-up and become aware of what their bodies are doing. Now it is time to take our first step.

The steps required to take a step would be:

1. Lift left upper leg until the femur is parallel to the ground
2. Extend left foot forward 45 degrees
3. Lean forwards 10 degrees
4. Lower foot straight to ground

The same would be done for the right foot. Granted, the above “Step” function could involve more steps. Once the “Step” function is defined then we can create the “LeftStep” and “RightStep” functions to take future steps. This will save us from having to continually rewrite the two “Step” functions. Once the two “Step” functions are defined, then the “Walk” and “Repeat” functions can be defined. For example, Walk=LeftStep AND RightStep. Repeat Walk 20X, will move the person 40 steps in one direction.  With this exercise the educator can play with different ideas and create any function desired, but before any function is created students must know exactly what the function will do.

Once the above exercise has been completed then the students should be assigned some HAL problems for homework and as in class activities. These assignments should require that each HAL have all four parts (Title, IS, GS, and steps), and a minimum of twenty steps. Some homework ideas are writing a HAL on: tying a shoelace, brushing teeth, opening a door, drinking a glass of water, etc. (WARNING: Having students write a get dressed or take a bath HAL may place an educator in a non-enviable position). Almost any human activity can be broken down into a HAL. Another activity is to have the students form teams. Each team is to write a HAL. Once this is done then two teams exchange their HAL and try to replicate (while blind-folded) what the other team has written. Subjective observation indicates that students enjoy writing HALs regardless of their age. To fully utilize HAL it should be practiced throughout the school year.

The HAL Writing Function

Once students have mastered creating HALs then they are ready for using the HAL Writing Function. The first thing that students should be introduced to are transitional expressions. With HAL we will primarily use two types of transitions (Seech, 1993), the premise and the conclusion. The premise is defined as any statement in which evidence has been offered. Conversely, the conclusion is any statement that is supported by evidence. With a HAL the Initial State is the premise, and the Goal State represents the conclusion. Each step or a series of steps within the HAL also contain a premise and conclusion. Before using the HAL function, students should be taught a transitional quantifier (Given . . ., There exists . . . etc.), a conditional statement (If . . . then . . ., Since . . . then . . ., Given that . . . then . . . , due to the fact . . . then, etc.), and a term to be used for the conclusion (Ergo, Therefore, etc.). Almost any book on writing can provide the reader with more transitions (Troyka, 1990).

Since one purpose of HAL is to make writing easier, then students should not be overburdened with transitions. The emphasis should be placed on the construction and use of HAL. An interesting observation is that the more exotic and foreign sounding the word, then the more students seem to enjoy using it--once such word is ergo. Also, any educator who works with this method will be pleasantly surprised by their students’ use of the words. And, after time, educators should be prepared to hear arguments where the words: conversely, nevertheless, eventually, furthermore, etc., are thrown their way when students are denied a trip to the lavatory.

Quite simply, it is best to define the HAL Writing Function as a function that inserts after a transition a fragmented sentence. Each step of HAL, including IS and GS requires the use of only periods and commas, and possibly the conjunction and, and the disjunction or which should be used in the non-inclusive sense. When each sentence fragment is inserted between the transitions, then the result is a complete sentence. To use the HAL Writing Function effectively requires a minimal amount of memorization by the student, and the key to writing effectively using this tool is in creating a sound algorithm or heuristic (this students can master easily). This is best explained by way of example.

Suppose we are asked to solve the problem below.

Problem: There is a figure where a triangle sits on top of a square. The width of the square is 10”, and the height of the triangle is 14”. The base of the triangle is equal to the width of the square. Find the total surface area of the figure.

The first thing to do is to give the problem a name...so let's name it:

Finding Figure Surface Area

Next we need to construct the IS. In generating the IS all of the given data, as well as the missing dimensions, of both the square and triangle are listed. In this instance the IS should also describe how the missing dimensions were derived. Again, the purpose of this step is to list all of the necessary data needed to generate a solution. Our IS will be (transitions in boldface):

Initial State: A square with a width of 10” and a triangle sitting on the square with a height of 14”. Since a square has a width of 10”, then its length is 10”. Since the base of the triangle is equal to the width of the square, then its base is 10”.

Instead of writing the entire question down, which we can do, let us merely paraphrase the question so that our GS is:

Goal State: Find the surface area of the figure.

Now it is time to actually construct each step needed to go from IS to the GS. It should be noted that we have listed all equations that will be needed to solve the problem and are assuming that our students are familiar with these equations. Our algorithm is (note the insertion of conjunctions in bold and the omission of all punctuation except commas):

1. Square l = 10”, w = 10” and area of a square = lw
2. Area of square = (10”)(10”) = 100in2
3. Triangle b = 10”, h = 14” and area of a triangle = ˝ bh
4. Area of triangle = ˝ (10”)(14”) = 70in2
5. Area of figure = area of square + area of triangle
6. Area of figure = 100in2 + 70in2 =170in2
7. Ergo, area of figure = 170in2

Above is a typical algorithm that students would generate after minimal training. The algorithm above only requires the memorization of one conjunction by the students—and, and the transition ergo. Again, with this HAL we are assuming that students know the formulas for finding the area of rectangles and triangles, consequently the student had to derive only one equation and that was the area of the figure. Students should fully understand the construction of HALs, and as an educator you will want to work several of these examples in front of the class using Socratic dialog.

The above algorithm is of average length. It could be lengthened if we were to break Step 1 and Step 3 into two additional steps. For example, Step 1 could instead be divided into: 1) Square: l = 10”, and w = 10”, and 2) Area of square = lw. Accordingly, steps within an algorithm can be joined together. It is not the length of the algorithm that is important, but rather that all of the data be incorporated within the algorithm. Also, all solution steps should be listed sequentially so that if anyone were asked to read the algorithm then they could replicate the solving of the problem.

It is now time to call the HAL Writing Function. HAL will use the following pseudo code to covert the HAL into English:

Insert Optional Title

There exists <Insert IS>. Since <Insert Step 1> then <Insert Step 2>. Given that <Insert Step 3> then <Insert Step 4>. If <Insert Step 5> then <Insert Step 6>. Ergo, <Insert Step 7>.

The actual HAL created paragraph is (transitional words/statements in boldface):

There exists a square with a width of 10” and a triangle sitting on the square with a height of 14”. Since a square has a width of 10”, then its length is 10”. Since the base of the triangle is equal to the width of the square, then its base is 10”. Since the base of the triangle is equal to the width of the square, then its base is 10”. Since square l = 10”, w = 10” and area of a square = lw, then the area of square = (10”)(10”) = 100in2. Given that triangle b = 10”, h = 14” and area of a triangle = ˝ bh, it follows that the area of the triangle = ˝ (10”)(14”) = 70in2. If the area of the figure = area of the square + area of the triangle, then the area of the figure = 100in2 + 70in2 = 170in2Ergo, the area of the figure = 170in2.

In creating the above paragraph from an existing HAL the student would, at the most rudimentary level, only need to memorize three parts for HAL paragraph construction—they are: state what is given via the transitions Given: or There exists. (All good problem-solving solutions begin with listing the givens (Polya,).) Though we used three conditional statements (Since . . .then, Given that . . .it follows, and If . . .then) for illustrative purposes, a student would only need to use one when first starting to write a HAL paragraph. Finally, in this case and in all future solutions, the student only needs to memorize one transitional conclusion—Ergo, in our case. The last thing to do is to place our periods. The rule of thumb for period placement is to place a period at the end of each line that precedes a new transition. Observation has shown that students do not have any problems in memorizing a few transitional words and period placement. They seem to delight in using structured logic since it is a unique way of communicating (there may also be a sub-conscious appreciation of its beauty). I intentionally used ergo, since this is the term of choice among my students. After exposing a set of students to this technique it always seems to evolve into a friendly game of competition to see who can create the most sophisticated sounding paragraph. This observation seems to hold truer for high-risk students and those hailing from lower socio-economic backgrounds.

The beauty of the method is that even for complex problems the HAL paragraph will not be much wordier than the algorithm. Since the HAL generated paragraph will only be slightly larger than the actual heuristic or algorithm. Accordingly, if students are taught a minimal number of transitional words/phrases then the probability of creating similar HAL generated paragraphs is very high. Some may suggest that this then becomes a contrived manner in which students express themselves mathematically. This may be true, but it is also true that the students are not only solving the problems correctly, but are also expressing themselves according to the educators’ expectations. It could also be argued that students now have a means in which to express themselves mathematically, versus frustration and no expression at all. Finally, since HAL is used to help students initially in expressing their problem-solving endeavors, then it is hoped that with mathematical-cognitive maturity that students will eventually supplement the HAL method of communication.

Conclusion

Writing is an integral part of conveying both mathematical concepts and solutions. Often K-12 educators are not provided with an adequate means of teaching writing in the math curriculum without delving into the English curriculum.  By teaching the students to construct heuristics and algorithms (HAL) to derive solutions to mathematical problems, then the student is learning an effective means of problem solving to the extent that they must list the givens, concepts learned, and learn to think sequentially. The construction of algorithms and heuristics stands on its own merits in the area of problem solving and critical thinking. By teaching students not only how to construct algorithms and heuristics, but also a few transitional words and phrases then the HAL can then be used by the student to write an English translation of any given problem and its solution. Obviously, HAL is a form of pseudo code.

Students are taught to use HAL as an outline to writing and, obviously, a tool for solving problems. A HAL consist of four parts: the Title, the Initial State where all of the givens and derived data are listed, the Goal State where they restate the problem, and the numerically steps where they list all steps needed to go from the Initial State to the Goal State.  When using the HAL as a writing function students only need to memorize one conjunction (and), a quantifier such as given where they will list all of the initial conditions (IS), a transitional statement such as If . . . then, and a conclusive transition word such as Therefore.  Once the student has learned to construct a HAL, and learned their transitions then it becomes a simple matter of garnishing the HAL with transitions to construct an acceptable English paragraph. Obviously, the algorithm is a form of pseudo code and the students are taught how to restate this code into properly constructed English sentences.

The time commitment to teach the HAL method of writing is minimal. Educators can teach it in about six contact hours, and it takes students about twice that long to become proficient users of HAL. Students should be taught the technique at the beginning of the school year/quarter/semester and be required to practice a minimum of one HAL problem a week. Subjective observation suggests that students enjoy participating in HAL activities, and find it significantly easier to express their mathematical solutions in written form. HAL is not only effective in math courses, but is a successful means of teaching writing in any course that uses problem-solving and requires critical thinking (science, debate, logic, computer science, etc.). Finally, HAL gives the math educator a tool that helps students express their mathematical solutions verbally, motivates the students, and is a process that both students and educators will find enjoyable.

[1]HAL was first introduced to the public via Arthur C. Clarks’ book 2001: A Space Odyssey. The term HAL originated in the labs of IBM, and was an acronym for “heuristic algorithm.”

Bibliography

Seech, Z., Writing Philosophy Papers, Wadsworth Publishing Company, Belmont, California, 1993.