Mathematics: What Do Our Children Need?
This is the third and last of a series of articles focused on the kind of mathematics our children need. The first article dealt with the reasons to teach mathematics and suggested some non-typical reasons. The second article discussed how our world is changing and the impact such changes will have on our children. This article will examine teaching methods to prepare our children for the 21st century.
In the 1980’s, every industrialized nation in the world was about the business of educational reform – every nation, that is, on different and diverse courses, tenure issues, year round school, middle school vs. junior high school, bilingual education, multilingual education, integrated curriculum and on and on.
Each of these issues by itself has value, but is it where we should focus our attention? American education should be about the business of educating our people; it’s not about the business of teaching; it s not about courses; it’s not about majors; it’s about teaching people to work smarter. Granted, in 1989 the National Council of Teachers of Mathematics (NCTM) published the NCTM Mathematics Standards. These standards were an attempt to define the mathematics curriculum desired for the 21st century.
However, in two separate research studies, it was found that only 15% of the K-3 teachers and 50% of the secondary teachers were even familiar with these standards. I ll bet that less than 10% of the homeschoolers have even heard of the NCTM Standards. I believe the Standards to be a good document and all math educators should know what is stated and aim toward those objectives.
What about those people who call for a “Back to the Basics” movement? My first question is, “What are the basics?” Is reading, writing, and inventing mathematics basic? Is problem solving basic? Is finding square root without a calculator basic? Is computer knowledge basic? What is basic? Is it possible that we need some basics but not the same old basics of 20 years ago.
I believe the truth of the matter is we don t really know exactly what the basics of mathematics will be in the year 2010.
But I believe we know several things it won t be:
- It won’t be long division without a calculator.
It won’t be adding long, large columns of figures without a calculator.
It won’t be memorizing a bunch of math rules.
It won’t be about solving a meaningless set of exercises.
We know that calculators and computers will be basic necessities not only in the workplace but also in our homes. We need to get down to business and look around our world.
Before we proceed further with this discussion, let us examine some myths about mathematics and in particular about manipulatives.
Five Myths about Manipulatives:
Myth #1 It’s nice to use manipulatives if you have the time, money and patience, but they are not essential to achievement.
Just the opposite is true. Research clearly shows that using manipulatives makes lessons stick and boosts achievement. Manipulatives model abstractions and help students build concrete visual images of what’s going on with numbers and shapes.
Myth #2 Manipulatives are appropriate only in the very early grades.
Not true! In fact, learning math requires active participation by learners of all ages.
Myth #3 The teacher s role in the use of manipulatives is minimal. A child s own discovery is what manipulatives are all about.
Simply not true! While discovery is important, it s the teacher who focuses attention on the math concept being explored, who encourages students to think as they work, who helps students make the connection between the visual models and the symbols.
Myth #4 Manipulatives are hard to manage.
Manipulatives do add more activity and noise, require space and more organization, but they are not hard to manage. Simple rules of courtesy, responsibility and cooperation can make it easy.
Myth #5 Picturing manipulatives on a computer and manipulating those images is just as good as hands on.
Absolutely false! Nothing can replace a child’s hands-on experience with manipulatives which model mathematical ideas. The computer is a marvelous device and is wonderful for repetitive rote memorization, simulation problems, etc., but not as a substitute for hands- on. The computer and technology are vital but cannot replace hands-on!
Mathematics can be defined as the science of patterns and relationships. If mathematics is a science, then should it not be taught as a science? I believe it should. So, what basic things does a scientist do? Allow me to suggest four things:
1. Explores or experiments
2. Observes
3. Generalizes or draws conclusions
4. Verifies conclusions (proves)
Yes, your mathematics “classroom” should contain these four components. The NCTM Standards gives five goals for ALL students. Students should be able to:
1. Think and reason mathematically
2. Solve problems
3. Communicate mathematically
4. Have confidence in their mathematical abilities
5. Value mathematics
In reality, no one can TEACH mathematics. Effective teachers are those who can stimulate students to LEARN mathematics themselves. Educational research clearly shows that students learn mathematics well only when THEY construct their own mathematical understanding. That’s where the manipulatives come in. Furthermore, we must stress true problem solving. There is a big difference between giving our children exercises and giving them problems. Let me illustrate:
An example of an exercise:
Kim went to the store and spent $.95 for milk and $1.10 for bread. How much money did she spend?
I view this as a mere exercise. Let’s look at a problem using the same data.
Kim went to the store and spent $.95 for milk and $1.10 for bread. The clerk gave her change for $3.00 If she received change in only nickels, dimes and quarters, how many coins could she receive? Explain.
This, my friends, is a good problem!
An exercise is a question which can be answered without much thought. The operation is usually obvious and this type of question is typically done for drill and practice.
A problem is a question which seeks an answer which is not obvious nor is the procedure
A good problem will contain a number of the following components:
- 1. Answer is not obvious.
2. There is more than one way to solve it.
3. There is more than one answer.
4. There is extraneous data.
5. It challenges the student.
6. It requires some thought.
7. It may not contain sufficient data.
8. It is of interest to the student.
9. It is within the student’s ability to solve.
10. It spawns other questions.
Another good problem:
Examine and explain your thinking. What is the relationship between perimeter and area? That is, if the area remains the same, will the perimeter remain the same or change?
Here’s a good algebra problem.
Use only these numbers to fill in the blanks. You cannot use any numbers except those in set S.
S = {0,1,2,3,4,6}
x = _______ x + y = _______
y = _______ 5y = _______
x2 = _______ y – x = _______
xy = _______ y2 = _______
Answers: x = 4, y = 6, or: x = 0, y = 6
The new learning holds that math can be learned more efficiently in groups: hands-on is better than hands tied behind the back. And counting on fingers is evidence that a child is “trying to figure it out” – a thought process to be encouraged rather than punished. It further establishes that reading and writing are as essential to learning math as they are to the study of any other subject. Blocks, games, puzzles, balance beams, fraction pies, calculators, and computers – these “manipulatives” and new tools of math have made its study the most enjoyable part of the day for children in schools that are engaged in this new learning.
Yet, whereas it is true that every child can learn math, it is a fact that under the present math program few of them do. Small wonder. We teach a math program consisting of nine years of drill in arithmetic, followed by algebra taught as a foreign language in the old way – by memorizing word lists and grammar.
For most students, the final bow is inflicted with a blunt instrument: plane geometry which is too abstract and difficult for many survivors of the death march from arithmetic to algebra. This course has a reputation so bad that less than half of U.S. students even attempt it, and of those who do, most do not learn it. This must and need not continue. First, the calculator and the computer have made skills in rapid paper-and-pencil computation, flash cards, and “rote and drill” obsolete. Second, the ability to analyze real- life problem situations and to express them in mathematical terms has become far more purposeful than the ability to apply the right formula to obtain the correct answer to a textbook math problem.
For these reasons, problem posing, problem solving, and collaborative learning are at the heart of the new learning in math.
Each day, nearly all children are subjected to instruction in the old-style math, and few of these children are learning it.”
-From Everybody Counts
Theon to Hypatia once said, “Reserve your right to think, for even to think wrongly is better than not to think at all.”
B.F. Skinner said, “Education is what survives when what has been learned has been forgotten.”
Oscar Wilde said, “Education is an admirable thing, but it is well to remember from time to time that nothing that is worth knowing can be taught.”
Folks, we must change our educational system. We must change the way most of us homeschool. We must change before it’s too late. We can no longer teach our children the way we were taught. You can’t just give a child a workbook or a textbook or a video and walk away thinking everything will be okay.
My father once said that if we always do what we’ve always done, we ll always get what we’ve always got…And that’s no longer good enough.
There are four basic skills though that every child will need in the 21st century:
- 1. The ability to think.
2. The ability to use logic and reasonableness skills.
3. The ability to effectively and efficiently communicate.
4. The ability to get along with others in a group.
Mathematics can help children accomplish some of these objectives but remember, “Not everything that counts can be counted and not everything that can be counted counts.”
Make your mathematics “classroom”:
- * Discovery orientated
* Problem solving based
* Logic centered
* Reasonableness assured
* Thought provoking