(Note: The author, Brian Boley, has a bachelor's degree in Physics and has taught physics labs at the college level. He has also tutored algebra, calculus, physics, chemistry, and biology at both the high school and college level. He has 4 children, all of whom are home-schooled.)
Perhaps the hardest class for most home school parents and professional teachers to teach properly is Mathematics. This is because few among us have truly learned Math -- most of us struggled through the courses. You cannot let most students simply follow a textbook, since this only provides half of the information needed to properly solve Math problems.
Math is very different from English or History -- or even Science -- because Math is the only subject that shares two characteristics:
In other classes such as History, one can start at any point and study just that particular part of History. In English, there are various degrees of correctness and opinions may differ as to what is correct. But not in Math: Math is precise and requires exact execution. Math answers are either right or they are wrong. ("Partial credit" is merely a device used by teachers from the 1970's who were too concerned with the short-term self-esteem of their pupils.)
In this article, we will develop several Concepts, which will enable us to teach Math correctly. As we learn a Concept, we will see how to apply it in the classroom. We will also see the effect that the Concept has on the student.
This is the first Concept: Math answers are either right or they are wrong. There is no opinion -- only proof.
By the time most students get into serious Math -- roughly the 4th grade -- they have been taught that the answers that they are expected to come up with are either correct, partially correct, or wrong. They have also been taught that the question of "correctness" depends upon opinion and that their teacher's opinion of the correctness of an answer is most critical. Thus, you'll see many students look at the eyes of the teacher when answering a question. This is so the student can make an instant determination of correctness based upon the teacher's opinion. If the student can make a strong case, they can improve their grade -- even if the answer is objectively wrong.
This "opinion-based" teaching has developed since the 1970's with the change from strict memorization of facts to the more discussion-based approach to teaching. Whole Language has further emphasized the "opinion-based" view of learning among the students -- "Your opinion is just as good as anyone else's opinion." Self-esteem is preferred to objective answers.
But now we come to Math. 2 x 2 = 4 and it doesn't matter what the students or teachers say or argue. This is the single most important issue that elementary math students (and teachers) must learn!
The Concept seems so obvious, yet is not. The reason it is not is because most of us studied at elementary schools that encouraged fact memorization. The most important things to learn in History about the Civil War when we were in school were the dates and places of battles -- today's History spends the most time in a debate over the causes and outcomes of the war. When we were young, right and wrong were obvious. Today, they are not.
If your student is doing excellent in History, English, Reading, and Science --- yet poor in Math -- this Concept -- the Concept of Right and Wrong -- is what needs explaining.
Explain to the student repeatedly -- the methods of Math determine the answer. What the teacher says does not matter -- the methods determine the answer. While you are going over a problem, explain that the method determines the answer. "What if I said the answer were 8? I'd be wrong, because the Math determines the answer to be 5." Do this repeatedly for months and the student(s) will get the message. This is a particularly important concept to emphasize in the classroom to normal 2nd to 4th graders.
Students that look at your face when you ask, "What is 5 x 5?" are students that believe they need to guess your opinion of the right answer. (Students that get the message will either look away while thinking, or will reply confidently "25!")
Avoid the "partial credit" trap when teaching middle school and high school students. Someday you may drive over a bridge which one of your students designed. Do you expect him to have calculated the loads correctly or should he get "partial credit" for getting a close answer? And all because you taught him that using the right equation was worth 90% of the problem -- and adding 2 + 5 = 8 was only 10% off.
In late 1999, a Mars probe, which cost $125 million, was lost because one team involved used the English unit of pounds for force calculations and the other team used the metric unit of Newtons. They made the "simple mistake" of failing to convert the units. The probe, after traveling about 100 million miles, was 60 miles low when it came time to enter Mars orbit -- and the probe crashed. By "partial credit" standards, the flight team would get a 99% grade. But the probe crashed -- and the taxpayers lost $125 million for nothing. In reality, the team got the answer 100% wrong.
Unlike other areas of study, one can't just pick up a Math book at the college level and begin unless the proper sequence of preliminary courses have been taken. This is because the study of Math is the study of a procedure and of a methodology -- not a set of facts or opinions. A body of facts, such as History or Geography, can be entered at many points. A methodology must be taught from the beginning.
Learning Math is similar to learning how to cook. A beginning cook cannot simply walk into a kitchen and bake a cake from scratch. A terminology must be known first: What does "blend" mean? What is a "tablespoon"?
In Math, simple routines and terminology must be learned before advanced routines and terminology can be learned. (I say "advanced" rather than "complex" because higher Math is not complicated -- provided you understand how to do simple Math.) Any student that is not capable of 99% accuracy on basic addition, subtraction, and multiplication is going to have tremendous problems with long division -- simply because multiplication and subtraction are the basis of division. Similarly, any student that has trouble with fractions will have trouble with algebra -- fractions are a vital part of algebraic manipulation.
Hence, it is vital that students understand completely each building block as they learn Math. If you encounter a student that has trouble with division -- throw in a few multiplication and subtraction problems to diagnose potential mis-understandings in those areas. When starting algebra, be sure to thoroughly test your students of their knowledge of fractions and basic area/volume formulas. If a student shows a weakness in these areas, a couple of weeks of review are definitely in order. In some cases, that student may have to spend a couple of months reviewing the missed material.
Many students, taught by the system to learn facts and then forget them to make way for more facts, will forget Math methods after a test. We suggest using the Saxon Publishing Math series as a way to cure this problem. The Saxon books simply do not let a student forget a procedure -- it comes back again and again and again
A simple concept that constantly caused problems with my daughter was her tendency to look at a problem that contained a zero and treat the zero as though it wasn't there. For example, as part of a long division problem of 500/20 = ?, she would look at the zeros and say "how many times does 20 go into zero? -- It doesn't go into zero." She would then go onto the next digit, skipping the place value. Instead, she should have said,” 20 goes into zero exactly zero times" and put a 0 into the place. Clearly, she had not understood that zero was just as important as every other numeral. We had to take a week and do many problems using zero.
Another subtle effect of advanced Math problems is this. Imagine that your student can do single-digit multiplication with 90% accuracy. In a long division problem such as 46535 divided by 15, there will be at least 5 double-digit multiplication problems, or a total of 10 single digit multiplications. If the 90% accuracy holds, then probability says that the student has a 35% chance of getting the correct answer -- assuming that the student makes no mistakes on the many subtraction steps. However, if the student has 99% accuracy in single digit multiplication, the student has a 90% chance of getting the problem correct. This is why it is important to drill and drill the basic functions until they are 99% accurate or better.
Many students that develop Math problems in the middle grades do so because of this effect. The multi-step problems magnify minor inaccuracies into major, test-failing problems.
To correct this problem, give the student a complex division problem. Have the student go slowly through the problem and write down each step separately, spread out across the paper or chalkboard for effect. Now point out the number of steps involved and the fact that every step must be correct for the problem to be correct. And identify which basic building blocks the student is having difficulty solving. Work with the student on those issues.
Student Math difficulties fall into three categories -- Conceptual, Algorithmic, and Process. Let's see what I mean by these categories.
My daughter's problem with zero was a Conceptual problem. She didn't clearly understand the idea of zero. Similarly, students -- particularly those who miss class -- can miss a critical concept that can cause problems for years to come.
Other concepts that are commonly missed include basic geometrical concepts such as area, volume, diameter, etc; negative numbers and their manipulation; the fraction as division; exponents; logs; trig functions; and other items that are covered quickly.
Fixing Conceptual problems is simplest -- explain the concept to the student until they understand it.
Algorithmic problems are problems with the method of solving the problem. For example, a student may simply not understand the step-by-step procedure of solving a division problem. Since we tend to spend the most time on these issues, this is not generally a problem with most students -- it's what we are already focusing on.
The general procedure to solve an algorithmic problem is to first teach the student that an algorithm is a step-by-step procedure, similar to using a recipe to bake a cake. The recipe must be followed, or you will not get a cake. Then, the task is to teach the student the recipe clearly. Feel free to write down step-by-step instructions just as you would for baking a cake. (Warning -- don't try to wing this. Few people can write down an algorithm correctly the first time. You will need to make a couple of revisions, which means that you should do this during class preparation time -- not during teaching time.) Then walk the students through the step-by-step process. Have them take notes and keep the algorithms in a notebook for reference. After they refer to it several times, they will have the procedure memorized.
Process problems are general problems with the manner in which the student solves Math problems -- any Math problems. As such, they are really personal habits that need to be corrected. Process problems cause most of the difficulties Math students have. Yet, we spend the least time fixing these problems, since they are not covered in most textbooks. That is why the next 6 Concepts are devoted to the most common process problems.
Process problems cause the most Math difficulties and lead to the most problems and furthermore, process problems are the cause of long-term Math difficulties. What are some sample process problems?
Sloppiness -- If your student doesn't write neatly, he will consistently line up columns incorrectly, resulting in mistaken addition and subtraction. This problem is particularly a problem with bright students who "get it" in the early grades. But as the problems get more and more complex, their sloppiness will tend to prevent them from getting correct answers.
Taking Shortcuts -- Students who can solve simple problems easily will often have trouble with complex problems because they are used to taking shortcuts rather using a robust methodology.
No Checking Methods -- Students who get good grades rarely check their answers. Once again, this comes back later to haunt them.
All of these Process problems are related -- they are most commonly found in students who succeed in Math for years -- and then start having problems. Process problems can be identified when a student starts making "dumb mistakes" or "simple mistakes" such as not carrying a number properly, or adding 8+7=17. This commonly occurs in the middle school grades, but can sometimes not surface until high school. A few students will experience the problems first in long division.
The root cause of all Process problems is a lack of self-discipline. A student with a spotless room rarely will have Process problems for long, while a student slob will have a difficult time correcting Process problems -- yet can benefit from it in many ways. Students without Process problems will often be described as Methodical. Students with Process problems will often be described as Brilliant but having difficulties. We will explore the most common Process problems, their effects, and cures in the next few Concepts.
There are 3 reasons why the military emphasizes cleanliness and neatness:
· It saves time since things don't get lost.
· It leads to self-discipline and planning.
· It leads to greater self-confidence and pride.
These three benefits also apply in the area of Math.
In Math, Neatness makes a big difference once 4-digit addition is introduced. It is vital that the student keep the columns in alignment in order to get the right answer consistently. A sloppy student will accidentally misalign columns of numbers and add them incorrectly.
There are several points where neatness must be upheld in Math:
· Columns must be neatly aligned vertically -- particularly in long division.
· Decimal points must be aligned vertically in arithmetic problems.
· Numerals and variable letters must be precisely and neatly written.
· If unclear, the numeral "7" should be written with a dash across the midpoint to clearly distinguish from "1".
· In the same manner, "Z" should be written with a slash to distinguish from "2".
· When solving equations, each step should be written on a separate line.
· "=" signs should be lined up vertically.
· After the introduction of variables, the "x" sign should be replaced by the "*" for multiplication.
· After the introduction of fractions, the fraction bar should replace the simple division sign.
· Solve equations 1 step at a time. Avoid trying to do multiple steps at once because you will get confused.
· Only 1 equal sign should exist in any equation.
After equation work has begun in Pre-algebra, be sure that students always start with plenty of room. Don't let them zigzag across the page. Instead, insist upon a vertical column of precisely aligned equations.
To get the point across to the students, one day without previous notice, pass out a handwritten assignment page, which is SLOPPY! Violate as many of the neatness points above as you can, and have the students work on the page as a typical homework assignment. Be sure to pass out the assignment late in class so that the students will have to sweat over the assignment at home and be unable to ask you questions easily. Make it really hard to tell your 1 and 7 apart and the difference between your 2 and Z. Misalign your columns on purpose and make it hard to tell whether you meant x as a variable or x as a multiplication sign.
The next day, be prepared for comments. Encourage a gripe session about the difficulty of reading the problems. After the gripes are going strong, explain to the students that it was to illustrate a point. (If you have time, you might even grade the homework in class -- be merciless in your grading!) Then begin your lecture on neatness.
For weeks afterward, grade papers on neatness. Get your students in the habit of being precise. That precision will begin to show in their work, also, as neatness-related Process mistakes gradually disappear.
One of the common problems that develop both in the early middle school years and again in the high school years is the Process problem where the student approaches each problem as a totally different problem. This leads to confusion and a tendency to get the process right once, but then to drift off course. The solution is DITS-WET - "Do It The Same Way Each Time".
Encourage your student to look at categories of problems. Each type of problem has one method to solve that problem. Emphasize that once a student finds a way to solve a category of problem, he or she can use that method to solve ALL problems that are in the category. Then, use that method over and over and over again. If the student understands and uses this Concept, then they will concentrate on the higher-level question of "which category is this problem a member of?"
DITS-WET is particularly good for students that have difficulty with multi-step problems (such as long division or basic algebra), and with word problems. Imitate a machine or a computer as you do a sample problem (such as an 8-digit addition problem). Emphasize DITS-WET each cycle of the problem.
Another manifestation of this problem is my daughter's previously mentioned difficulty with zero. In 3rd or 4th grade, many students transition from a memorization-based arithmetic (multiplication tables) to an algorithm-based approach. They have a bit of difficulty understanding that the method doesn't depend upon the numbers involved. "0" and "1" appear to be exceptions to the methods to many of the students. So they are still caught in this middle ground where they are trying to learn a method, yet they know that the numbers make a difference to the outcome, but don't understand that following the method will handle the difference in numbers.
We used the DITS-WET Concept with our daughter to emphasize that it did not matter WHICH numbers were in the problem, the METHOD was the same. We emphasized the difference between Method and Data, explaining them as concepts. Then we drilled over and over again that you don't treat "0" different from the other digits. DITS-WET.
Later, DITS-WET was invoked with basic geometric questions of area of a circle and volume of a cube. I remember using the procedure in college to emphasize the step-by-step procedure for solving a word problem. (Details below).
Many students of Math fall into a common trap -- A simple problem is presented as an introduction to a class of problems. The brighter students quickly realize that the problem can be solved by mental shortcuts and don't really study the method rigorously. A few weeks or months later, a much more complicated instance of the problem appears and the shortcuts don't work because the complexity is too great.
In effect, avoiding shortcuts is a variation on DITS-WET. It is important for students to learn the methods that always work. Teaching shortcuts is something that should only be done with students who have demonstrated a mastery of the complete method. Worst is a method of teaching which I've seen occasionally, where a simple method is used to solve a simple problem. But the method doesn't work with more complex problems. In effect, the student must learn several methods to solve problems of increasing complexity, when one moderately complex method would always solve any of the problems. In the long term, it is to the student's benefit to have one thoroughly understood method that always works, than be faced with the added complexity of having several methods for different situations.
To break a student of using shortcuts, toss in complex problems in a sea of simple problems. Or better yet, quickly progress the student to the level of solving the complex problems regularly. With more mature students, the benefits of learning DITS-WET can be taught and the shortcuts kept in their proper place.
Many students never check their answers. It's particularly important as multi-step problems appear. And reversing the process can check almost any problem.
The best way to teach checking is to require it. Require a student to solve the problem and put the reverse check beside the problem.
Many students don't work with units for several years after they are introduced. Units tend to be added at the end of the problem as an afterthought rather than pulled through the method with the numbers. The result is that many mistakes, which could be caught by paying attention to units, are not caught.
For example, if a student divides 2 hours by 150 miles to get miles per hour, he will catch this mistake (Oops, that's hours per mile -- obviously wrong) if he uses his units, but may not catch it if he simply adds "mph" at the end.
Require units be pulled through the entire problem and count the problem wrong if they are not.
No student can develop the step-by-step concentration that Math requires in an environment surrounded by rock music and television. That is because modern music and TV are designed to grab and hold the listener or viewer's attention.
Mathematicians almost without exception have noted that the best environment for doing difficult Math problems is either a quiet place (Newton invented calculus sitting outside under the trees.) or refer to baroque music such as that of Bach, Vivaldi, Telemann, or Handel. This is because the baroque music is very orderly and is without attention-grabbing climaxes. Certain quiet "New Age" and Jazz music has the same quality.
To encourage your students to study with this music, consider using Bach as a background to seatwork. Exposure will gradually convince students of the merits of the music -- Bach's "Jesu, Joy of Man's Desire" became a top 5 hit in the 1970's after a Boston radio station gave it heavy air play as part of an experiment.
As problems grow more complex, luckily they begin to have more and more relevance to the world. I encourage each of my students to look at their answers and give it a reality check -- "If I drove to Chicago from New York at 60 miles per hour, does it make sense that I would get there in 2 hours?"
The best way to begin to encourage this is first to explain the technique. The next day, tease the students when they have obviously impossible answers. Do this a couple of times and emphasize the reality check Concept.
Word Problems cause many difficulties for many students. Yet, there is a straightforward Process that allows all Word Problems to be methodically solved. Most Math students are not taught this process and are not held to the process. Thus, they have considerable problems with this class of Math problem. The following method works beginning with Pre-Algebra.
Make a drawing or chart of the information contained in the problem.
Step 2 -- Write down as a vertical list a set of equations expressing known facts. For example:
Total Area = Area1 + Area2 + Area3
Height = 5 cm.
Area = Length x Width
(Advanced students can use variables to represent words i.e. A=LW).
Length1 = 10 cm
Length2 = Length1
Width1 = 5 cm
Be sure to list ALL the known facts -- either values or relationships -- as equations.
Step 3 -- Write down all the unknown variables under the know facts.
Length3 = ?
W = ?
Important Note: Keep it neat and organized!
Step 4 -- Solve any equations that you can and substitute values into other equations to solve them.
Gradually put numbers where variables or words used to be until you have numbers for everything.
A good workman needs tools. And he needs the right tools for the job. In any profession, better tools are invented as time progresses -- tools that allow difficult tasks to be handled quickly and easily. For example, today’s modern carpenter begins work with a hammer, nails, and a handsaw. Yet he has available an electric skill saw, various specialty forms of handsaw, an electric table saw, a band saw, and even more advanced tools. Yet Jesus was a carpenter and had no electricity in His shop (other than in His words!). He was able to accomplish his carpentry tasks using very basic tools -- it just took longer and took more effort to cut a board than it would have if He had owned an electric table saw.
Is the table saw more difficult to use than the handsaw? Even though carpenters begin with the handsaw, the handsaw probably requires more skill and practice to use correctly. So the handsaw (the simple tool) is more difficult to use than the table saw (the advanced tool).
In the same way, mathematicians started with addition and subtraction. Multiplication was simply a tool that sped up addition. Algebra is a wonderful tool for solving many different types of problems and trigonometry saves considerable time and effort in calculating heights and elevations for surveyors. Calculus was fully developed to make solving the motion of the planets much easier, but is invaluable to the civil engineer and economist. And complex mathematics considerably simplifies the development of complex electronic circuits.
There is a tendency among Math students to say, "______ (division, algebra, geometry, trig, calculus, etc) is supposed to be really hard." (Invariably the level of math in question is a couple of years ahead of the student.) This frightens the student and adds to their Math anxiety.
Point out the Toolbox Concept. Point out that the various tools actually make solving problems easier. For example, using a handy pattern of tiles (linoleum or ceiling tiles), have the students determine the area of a large area. If there are several students, have half the students determine the area with nothing but counting. Let the other students use multiplication. See which ones finish first and are more accurate. (If there is one student, have him perform the counting task first, timing his speed. Next, have him multiply for the result, once again timing him.) Which method is easier and quicker? Now explain that this is what the higher Math tools do -- they make problem solving easier. Help the students understand that they are filling a Toolbox with Tools.
As you can see, the keys to success in Math are not Brilliance and Intuition -- rather, a step-by-step approach and mentality is required. The Brilliance and Intuition in Math is required for new Math discoveries -- Newton with Calculus, Godel with Incompleteness, and Einstein with Tensor Calculus. But the task of Math up until at least the level of Differential Equations is the task of the good workman.
Math success is driven by Persistence. Ideally, solving a complex algebra problem should have the same feel as painting a house -- gradually the outlines of the beautiful ending appear as the step-by-step transformation occurs. Mathematics has a Concept of Elegance, which is defined as a step-by-step solution to a problem which is clear, easy to follow, as short as possible, and comes quickly to the correct answer. As you paint a house, you cover the walls step by step. When you solve a Math problem, you go step by step.
In the same way that a brilliant artist is not the type of person who you'd want to paint a house (imagine the odd colors!), a brilliant student can have difficulties with complex Math problems -- not because he can't develop an approach to the problem, but because he won't stick to the mundane Processes required to solve the problem step-by-step. The brilliant student needs to be reminded to stay on course and continue step-by-step neatly and precisely until the problem is completed.
As a workman, the Math student needs to keep his toolbox organized. Know what tools are available and what types of problems they solve. This isn't a function of brilliance -- only of organization. Make a list if necessary -- formulas for area, volume, perimeter; methods of algebraic transformation; definitions of trig functions; a card catalog of algorithms; a listing of integration techniques, etc.
Copyright Ó 1999 - Brian L. Boley. All Rights Reserved.