Writing In the Mathematics Classroom Using Heuristics and Algorithms
February 2003Ó
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.
Please note: This document was created with MS Word 2007. Consequently, the HTML is Word HTML and is not W3C compliant.
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.
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:
Initial
State: What is given?
Goal
State: What is asked?
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.
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. Let us 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 = ½ bh
4. Area of triangle = ½ (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:
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 =
½ bh, it follows that the area of the triangle =
½ (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 = 170in2. Ergo, 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.
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.
Bibliography
Seech,
Z., Writing Philosophy Papers, Wadsworth Publishing Company, Belmont,
California, 1993.
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