At this point we should review the questions that were formulated and asked in Step One. We should have answered all of the questions that made up the problem as well as the questions that we used in order to solve the problem. To truly test the answer, it is necessary to go through the entire process, using all previous steps, and to test that your solution works. It is also suggested that you teach your students how to word their answers. In the same way that they translated words into symbols in order to make a plan, they must translate their symbols back to words to answer the question correctly in the solution.
The student should compare the approximate answer with the actual solution to see if the solution is reasonable. One should check to see if one’s solution makes sense. For example, in Problem One, x = 14. One should check by substitution of this value into the original equation’ x + 2x + 42. 14 + 28 = 42. The result makes sense.
Of course, there is more to solving word problems than having a systematic technique. We must also consider one’s retention and sharpness of basic skills, reading habits, verbal skills, and motivation.
The basic skills include any prerequisites necessary to complete a particular unit of work. It is up to you to devise a plan to test whether or not your students have the appropriate skills.
Mathematics can be written in a language that is compact and precise. One cannot be expected to read a passage from a mathematics text like one would read a narrative novel. Concepts are often hidden or assumed. Because of varying sentence structure, literal translation of the material might be confusing. In mathematics one cannot rely on the direct application of basic skills used in reading, such as the use of context clues or structural analysis, to decode and assume a meaning of some vocabulary words. For example, many students read “A number less than 5...” and will translate it as “Five less than a number”. Such words as “constant”, “variable”, “coefficient”, or “exponent” are often translated incorrectly as applied to mathematics. Words like “gross income” and “net income” are also often misinterpreted. Through context clues, one cannot define “gross” or “net”.
Reading habits play an important role in solving any word problem. The student is often not trained properly to read a page from a mathematics text. Important information must be “sifted out” and certain relationships must be noted in order to reach an understanding of the material. We suggest that you have your students read aloud at least once a week to determine any problems relating to reading ability. When one gains in the ability to read and comprehend from a mathematics text, it follows that one will find that mathematics comes easier.
Another problem in the reading is the use of many mathematical symbols that the reader should be familiar with. Many students get “hungup” on the use of symbols. It is up to you to emphasize their usage.
Reading a mathematical passage requires a higher level of conceptual development than most students have achieved. We suggest that a mathematics text be chosen so that the readability level is a grade or two below the actual grade level of the student. There are quite a few reasons for this. Most passages are conceptually packed and have a high content density factor. The reading may require eye movements other than the conventional left to right. The eyes may need to have vertical, regressive, circular, or wordto charttoword movements. We encourage some silent reading in your classroom so that you can watch the student’s eye movements to diagnose some poor reading techniques. If a problem is detected, you should work with the students individually. It is useful to check for these handicaps early in the year so that one would be conscious of any weaknesses.
The rate at which one reads mathematics is usually slower than most other content areas, and one might be required to do multiple readings. This is necessary to grasp the total idea, to note the sequence of order, to relate two or more significant ideas, to find the key question, to determine what operation or process is necessary, or to conceptualize or generalize the passage. The use of symbolic devices such as graphs, diagrams, charts, or other mathematical devices is highly prevalent in the language of mathematics. This language is technical and precise’ common words are used that carry a special meaning. An example of this would be the word “function”.
As was mentioned before, a major stumbling block to the success in solving word problems is one’s weakness in vocabulary. Mathematical vocabulary is not just defining a word, but knowing any symbolic translations that go along with the definition. Using symbols is also using vocabulary.
One of the major objectives in Consumer Related Mathematics is to teach a mathematics course that relates to business. The student should complete the course with a working knowledge of some basic business principles and terms. The mathematics that is taught should be relevant to the student in the event that one elects not to take any further mathematics courses. You might consider this course as a class in survival skills in mathematics. The problems that are taught should be relevant to the students. The subject material of the problems should relate to everyday arithmetic or mathematical skills.
Any business course has within its structure a set of vocabulary words. It is imperative that the student learn these words. A good tactic would be to write problems that contain the business vocabulary within them. The student will remember the vocabulary as well as to translate these words into mathematical symbols. An example of this would be to write a problem relating to a checking account. The problem would have the phrase’ “...reconcile the bank statement”. Here the student should realize that one should check the bank’s balance with one’s own.
We do not recommend just giving your students an isolated list of words to define as an introduction. Vocabulary should be discussed after giving a reading assignment and when you give sample problems to the class. You can give quizzes or tests having the student define certain vocabulary words. An outline for a test should include all important vocabulary. These words should be repeated often and be incorporated into many problems. The student will learn them from constant usage.
As with the Consumer Mathematics Curriculum, the Algebra Curriculum states that the student not only understands any concept taught, but also acquire new skills. The student has many symbols and quite a few vocabulary words to work with. It works well, again, to make up problems which will include practical terms that your students will be using once they have completed their formal training in mathematics.
We have noticed through our own teaching that many secondary students may not have the mathematical insight necessary to solve seemingly simple word problems. We feel that a broader set of experiences involving the more difficult concepts may help with the vocabulary, a dominate weakness. The use of color in the diagrams you present, using the overhead projector to illustrate as many concepts as possible, and such activities as trips to local businesses, using a weekly subscription to a mathematics journal in the classroom, and using charts and illustrations will bring more reality to the concepts being used. (Problem Four)’
Tell your students that they are given $ 1,500.00. Using the newspaper, have the students furnish a 12 feet by 16 feet living room. They must choose the furniture through ads, cut out the ads, and keep an accurate record of the money spent (including a 7 1/2% sales tax).
It should be noted that many students who may not possess the cognitive skills for solving problems in a structured manner do, in fact, possess a strong intuition for solving problems. We, as teachers, should not discourage this intuitive approach. We believe that there are two different types of thinkers’ the structured and the intuitive. An intuitive thinker will constantly attack the problem in a seemingly haphazard fashion with the emphasis on redesigning the original problem. For example, given the equation’ x + 8 = 14, the intuitive thinker asks, “What number plus 8 gives me 14?” The structured thinker may use the additive inverse of 8 and add to both sides of the equal sign.
As is true in any classroom situation, the student will learn more if motivated. For instance, a good motivational question for our classes would be to find the thickness of a piece of paper, and to ask for suggestions as to the different methods to solve the problem. One suggested method would be to measure 500 sheets of paper and then divide by 500. You can also talk about the possible error involved. With this type of question, the students may need some coaching or steering into the proper direction. It usually is to the benefit of the student if a method is discovered on his/her own.
Teachers have unlimited sources of motivational techniques. We will include some possibilities in our unit, but in no way are you limited.
Study guides and teaching guides are very helpful to the student. Here you outline the material and stress the vocabulary that is essential in that section or chapter. Before something is to be read by the students, you should point out the important ideas, symbols, and vocabulary. This gives the student some purpose to read the material, and helps him/her make note of the points that should be stressed.
Pulling the vocabulary apart and looking at the prefix, suffix, and root is helpful in analyzing many words. For example, the suffix “nominal” means term. One’s knowledge of the prefixes “mono”, “bi”, “tri”, and “poly” will aide in the introduction of the words “monomial”, “binomial”, “trinomial”, and “polynomial”, in Algebra.
Giving assignments can be a drawback if not done properly. When giving assignments, it is a good practice to include a study guide, vocabulary aide, and/or background experiences necessary for any symbol interpretation so that the student has a goal to work toward. I t is senseless to have someone do an assignment incorrectly. One would probably have to unlearn and relearn the material again’ this is much more difficult and very frustrating.
Skills and abilities do not grow in isolation from different content areas however different subjects do develop their own language and symbols. If a person trains oneself in the symbols of mathematics, it usually makes it easier to see the logic involved in the representations used in different areas.
The broader an education one attains, the more effectively one can integrate the skills needed to relate to one’s environment. We believe that our classes should be interdisciplinary and stress certain social requirements. It is true that our students will need skills, knowledge, and sensitivity to come to feel comfortable in their world. We suggest that you visit other classrooms in other subject areas. This way you will not only see different teaching techniques, but get a good idea of what particular topics your students are discussing. You should integrate material obtained in the other disciplines into your own word problems.
Most children are taught to solve word problems from memory. The problems are written in such a way that they are nearly identical to the other problems that one has seen. There is very little thinking or comprehension in this method. There is nothing wrong with either you or the student rewording a problem into a form in which one is familiar. In fact we encourage it.
We find, especially in the lower levels, that the ability to analyze is not yet developed. Much time should be spent on this, and the students should practice with many problems of the same type. It is our suggestion that the wording in the problems be changed. This is something you do not find in many mathematics texts.
Using quotations either orally or posted around the classroom are always useful in getting your students to be creative, and to realize that mathematics is indeed integrated among all aspects of daily living. A good reference will be found in the
Mathematics Teacher
, January 1976, pages 4044. These are a collection of quotations compiled by Barbara Curcic of Bloom Township High School in Chicago Heights, Illinois.
Here is a list of some other motivational techniques we have been using in our classrooms’ discuss the historical mathematical events about the ideas you are introducing, have the students write their own word problems and use creativity in their illustrations, have the students find word problems in their private reading, write problems that are individualized to the students (use their names and special things about them)’ have the students solve puzzles which contain words relating to mathematics, and help them to use calculators correctly in computing the basic computations.
Some motivational techniques that we suggest are trips to various businesses, teamteaching, or the use of the computer.
Examples Mapped Through the S.M.S. Method
Let us look at this example in a Consumer Mathematics Course and use the S.M.S. Method to solve it: (Problem Five)
Amy wants to buy a new winter coat. She found the one that she wants to buy in O’Maller’s Store. The original price was $ 89.95. She decided that she could not afford it. The following week the same store was having a 30% off sale. How much money will she save on the coat from the original selling price and what would be the sales price? If there is a 7 1/2% sales tax, what would be the total bill?
Before we look at the S.M.S. Method, please observe the way the above problem is written. Most mathematics texts shorten the problem and make the wording more concise. Few people will think of a problem in that shortened concise manner. To the students, the above problem is more realistic because it tells a story of events. Also notice that the problem asks several questions. Here the teacher will find which student will read the problem and answer all of the questions. Too many students fail to answer all of the questions.
As a prerequisite, the student must have solved several problems dealing with every question asked in the problem above. This might be a typical questioning technique used on a worksheet’
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I. Know your problem:
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A. Read the problem carefully.
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B. What is the unknown(s)? answers:
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1. How much will she save from the original selling price?
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2. What is the sales price?
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3. How much is the sales tax?
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4. What is the total price including the sales tax?
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C. Are there any key words and what do they mean? answers:
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1. Original price
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2. 30% off sale
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3. Save
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4. Sales price
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5. 7-1/2% sales tax
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6. Total bill
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D. Can you compare your problem to other problems similar to it? answer: Yes, there are three questions; there are 3 separate problems.
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E. What data is given: answer:
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1. The coat costs $ 89.95.
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2. It is on sale at 3% off.
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3. The sales tax is 7 1/2%
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F. Which form will the solution(s) be in? answer: Each solution will be in dollars and cents.
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G. Can you draw a diagram or graph? answer’
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(figure available in print form)
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II. Make a Plan’
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A. Write any equations or expressions answer:
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$ 89~95
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Original price
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X 30%
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What you save from the
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original selling price
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what you save
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sales price
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sales price
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sales price
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x 7-1/2%
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+ sales tax
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sales tax
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total bill
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B. Approximate a solution answer:
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You will save a little less than $ 30.00. The sales price will be a little more than $ 60.00. The sales tax will be about $ 4.20. The total bill will be about $ 64.00.
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III. Carry Out Your Plan answer:
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(figure available in print form)
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IV. Test Your Solution’
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A. Did you answer all of the questions? answer’ yes
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B. Check your work; go through all of the steps.
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C. Did you write your solutions so that anyone will know what solution goes with what question, and did you write the solution(s) in the proper form? answer:
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Yes.
Some other possible problems that deal with everyday situations and can be used with the S.M.S. Method are listed below’
Problem 6:
You will be having a Disco Party next Friday evening. You have invited 52 other people, including the chaperone. All of the recipes that you plan to follow are written to serve eight people. By how much do you need to increase each ingredient in the recipe to feed everyone, including yourself? Do not hesitate to have “leftovers” because your Aunt just bought you “SealAMeal”. What is the minimum amount that you should cook?
In problem (6) you are being creative in the introduction of the problem which adds to the motivation; your students will want to read the word problem story. Once again, you have not only asked more than one question, but in the answer you have left many possibilities for your students to work out. The use of proportion is illustrated, and when you include yourself you will have 53 people, not just 52.
Problem 7:
You live approximately one mile from the school. If you walk three miles per hour and you must be in homeroom at 8:30 a.m., what is the latest time that you can leave your house?
You are reinforcing the skills of fractions and units of measurement in hours and minutes. They will feel more comfortable with this problem because it is very real to them. It is also possible that the solution to this problem will be a constant reminder to them to be on time to homeroom in the morning.
Problem 8:
You have received the following grades during the second marking period at R. D. Tee High School’ 85 in mathematics, 95 in Spanish, 70 in history, 75 in gym, 80 in science, and 80 in art. In order to make the school “honor roll” your average must be at least 85, and you can not receive a grade lower than 70 in any course you take. Did you make the “honor roll”?
The skills necessary in the concept of averaging are included in problem (8). Once again we need to emphasize that the above problem includes the basic skills of addition and division with the introduction of the concept of averaging.
Problem 9:
Your mother has agreed to pay you $ 2.00 an hour for you to baby sit with your little sister, Tachema, during the weekends this summer while your parents go away on their boat. You have offered your brother 10% of your weekend salary if he and his wife will watch Tachema on Saturday night from 9 p.m. until Sunday morning at about 3:00 a.m. for you to go out. Your parents usually leave at about 7:00 p.m. on Friday evenings and return at about 10:00 p.m. on Sundays. How much money will you receive for an entire weekend? How much will you have to give your brother and sisterinlaw? How much will you have left as your earnings?
By using lengthy and complex stories, you are reinforcing good reading habits, giving an appropriate practice with basic skills, and introducing new concepts.
Problem 10:
You have borrowed a book from the New Haven Public Library which is now overdue. They charge five cents a day for each day the book is overdue during the first month, and then they charge eight cents a day after the first month. If your book was due on April 11, and it is now May 29th, how much money will you have to pay when you return the book today?
In an Algebra Book you will usually find the problems which involve a mixture of prices and ask for a total cost of a combination. The above problem introduces the same concept differently.
