The teaching of Mathematics - basic skills or concepts?
March 11th 2007 05:08
Signature Tune
The Olympian on line addresses the benefits/disadvantages of studying maths based on two conflicting models.
The article which follows is very long and most won't want so much depth, but some will.
Thrown into the total confusion surrounding modern day education this is a meaningful approach to solving a critical dilemma.
I found it interesting as the dilemma is compounded by the use of computers. How do others feel?
"Readers respond to Olympia's math curriculum
The Olympia School Board is considering two options for a new middle school math curriculum that use different approaches to teach math. The Olympian asked readers to say whether they would prefer a middle school math curriculum focus on basic skills or concepts. A sampling of responses is below.
The Olympian - Click Here
By the time one gets to middle school, basic skills and formulas need to be in place. Really that needs to be in place by end of third grade. So agree my young friends and those with kids in school. Additional concepts and problem solving are rather essential to beginners algebra, geometry and on. That gives meaning to study of algebra and geometry ... how to USE them in life. For those that will not go on in math, the concepts of problem solving and use of some creativity are really good life skills. Those might even keep a few away from involvement with the “legal” system.
Skip Dane, Olympia
Heck, that’s like learning to fly a 747 before a pilot knows how to drive a car.
Dave Gordon, Olympia
Although I’m sure it’s not possible due to cost, I think the best bet would be to offer both options. The students who learn best through the “traditional,” basic skills math should be allowed that option, and those that do better with the conceptual-problem solving math should be given that option.
Back when I was in school, I was a good student who struggled with “basic” math. I could get straight ‘A’s in social studies and English, but when numbers were put in front of me, my mind went blank. Consequently, it took me longer than usual to pass basic algebra. Clearly, I am more of a conceptual thinker, and therefore I chose to go into a field of work as an adult where that type of thinking could benefit me and I didn’t have to use numbers often - I became a mental health case manager.
My husband, on the other hand, is more of a linear thinker. He excels at math and enjoys using numbers. He went into a career where that type of thinking is beneficial - he is a developer at Microsoft.
So, my point is that embracing the type of learning of students rather than forcing them to learn one way or the other can pay off in the long run. Adopting both textbooks and offering students the option that best suits their thinking/learning style is the best idea.
Jennifer A. Flinton, Olympia
They should vote for basic skills. Without the skills concepts and problem solving are almost worthless. You have to understand how to do the math in order to create or adapt concepts that lead to problem solving.
Steve Neuenschwander, Lacey
Why does the answer have to be either, or? I am of the age (66) that we were taught and taught the basics until we understood them completely. Then later on in advanced courses I learned now to apply those skills to concepts and problem solving.
I was able to take those basic skills and advanced concept teaching to become, I thought, was a very successful land surveyor. I don’t think that either one would work very well without the other, but must be done with the understand that each one is important.
I have written and scored many tests during my career and believe that a concept and problem-solving test has the high potential of being written too difficult by being too long for the time frame required to take it. The problems in themselves are not incorrect, but collectively they make it impossible to complete it with a passing grade by a high percentage of the students.
Michael Kinnaman, Olympia
Based on my interpretation of basic skills and formulas, my recommendation would be to stay with the proven method. Every capable person I worked with over the five decades of my working career had the ability to solve any mathematical problem themselves or at least was able to understand a solution as presented.
That skill usually went far beyond the formulas of mathematics, and often reflected comprehensive applications of solutions caused by problems in many work environments. I can say without reservation that nearly every one of those people was educated in the basics. The exceptions were always someone who lacked knowledge of how to work within a formulaic system. Many things have changed with technology but even that is still based on formulas that reflect the truth of “2 2 equals 4.”
Ron Lawson, Lacey
What is wrong with the concept of children actually learning the basics of mathematics? The Olympia School Board should definitely adopt a text on basic skills and formulas. It is invaluable to know these two things. I couldn’t believe my ears when I heard that teachers were happy if a child knew the way to get an answer, but didn’t get the right answer. Is this two plus two equals whatever??
No wonder young cashiers freak out if the register doesn’t tote up how much to charge, or what change to give back. Much of real life actually revolves around math skills ... checkbooks, bank accounts, grocery and other receipts, and much more. How can a child that doesn’t know how to add, subtract, multiply or divide ever be a responsible adult? It is the duty of the now adults to help the children actually learn, not just the concept of learning.
Jeanne Kinnaird, Lacey
I have been a substitute teacher for the Hoquiam School District. I also tutor younger students up to the 8th grade to help them understand math better.
Please let me say that I hope that the new math books focus on basics and formulas? The basics and the formulas are what most of the students that I tutor are missing. These students seem to be so frustrated and fear math simply because they do not have a good grasp on the fundamentals of math. If students miss a step in learning the fundamentals and the formulas then they have a hard time going on to the more complex mathematics that will be needed in high school and college.
At a time when the WASL is so stressed in our school systems for a school board not to be looking at reinforcing the basics seems strange. The teaching of a system that stresses the basics which includes the formulas helps a student can feel confident not only to succeed but to explore and feel comfortable doing so. This in turn allows a student to address concepts and problem solving in a confident manner. But to bring in a system that stresses concepts and problem solving rather than basics and formulas can at times produce students that are timid accidentally and these same students loose any motivation of going on in school. This it seems to me is counter productive and is against all that a school should stand for.
Progress sometimes is being able to look back at what you have done or where you have been and being able to say that was something that worked.
James Rutz, Hoquiam
If you’re in a position in which you have to choose between a textbook and curriculum based on either basic skills or problem-solving, then you’re considering the wrong text books.
It is ridiculous to think that a Math curriculum doesn’t have the foundation in basic skills which works towards applying problem-solving in real-world scenarios.
I’ve worked with students that have collected their own information based on questions from within their class about issues around their community. The data they generate becomes an ongoing inquiry that succeeds through each successive term.
These real data have meaning beyond formulas and concepts, because the students have ownership to the information. They create paper graphs and employ various math techniques to solve their questions. Students as young as fourth-graders go on to use Excel to create comparative and reporting graphs from information/data that they’ve generated. Make it real. Relying solely on book learning has considerable limitations.
Dan Hannafious, Belfair
I do not live in the Olympia District and am not a mathematics teaching effort. Having said that, I am completely convinced that students need to learn basic skills and then move into concepts. Lack of basic skills is, in my view, the reason kids are struggling with math and the math WASL. It appears we don’t really know how to teach math and should back up a few decades and deal with basics as used to be the case.
Dick Nichols, Tumwater
Without basic skills first, the concepts and problem-solving will not happen!! Middle School is exactly that - “Middle School” and should focus on honing those skills for young minds - basic skills and formulas. Nothing new about mathematics ... all the same and math can be easy to grasp if taught in sequence. Start skipping around and you’ll confuse the pupil. KISS - Keep It Simple Stupid!!!
Ricky L. Johansen, Jr., Lacey
As is so often the case, this is a false dichotomy. The answer is, both. I was born in 1937, and I can still multiply and add faster in my head than on a calculator, at least for many problems. I am no mathematical genius, I simply remembered and practiced what I learned in elementary school.
However, I also studied statistics in graduate school, and realized that, while my math skills were helpful, I needed far more knowledge of math theory, algebra, probability, calculus, problem solving, and number systems. Finally, I think that students need to not only learn to use a computer, but something about the mathematics that it is based on.
Now, I know that we can’t teach all of this in Middle School, but we can teach some of it. We can introduce the basic concepts, probabilities, and algebra at least.
Earl M. Wajdyk, Olympia
Like Yogi Berra said, “This is deja vu all over again.” In the early ‘70s, I had freshman university students that came out of ‘60s schools in Washington which used the “concepts” and “problem-solving” approaches to education. They could not do simple math, spell or construct a grammatical sentence. They were left in the dust by students from traditional schools. One student told me he expected his future “secretary” would correct all his mistakes.
The questions that need to be asked are: What are the desired end results of the education process? Would you rather have a surgeon (or apply the professional service of your choice) who has the basic skills and knowledge, or one who “understands the concepts” and uses a “problem solving approach” to surgery?” Or at least, “Where will the “secretary” come from?
Patrick Coolen, Olympia
I think we should get back to basics and forget all the fluff. The 3 R’s have worked for generations, but in this computer age we aren’t allowing students to grasp the basic stuff. In addition to the math concepts, we should focus on basic science and technical skills. It doesn’t do the students any favors if they use calculators and computers to arrive at an answer since they won’t know how they arrived at it. Let’s get back our school days, learn how to make change, make pie charts, and allow students to play “tag” on the playground. School shouldn’t be fun and it shouldn’t be easy.
Teachers that only work for a paycheck should be weeded out and ones that teach as if it was a calling should be rewarded. Our kids deserve to be kids and learn how to interact with each other in productive ways. The “old” math will work just fine if it is taught in a manner that allows the students to learn at their own pace. There could be remedial classes for those that “just don’t get it”, but basic math and reading skills are all that is needed to move to the next level. It might not be fun and it might not be easy, but neither is the real world.
Hank Toombs, Olympia
Without any hesitation whatsoever, I will say this loudly: teach our children how to add, subtract, multiply and divide. All of the formulas dumped on children all across this nation have not taught them how to do simple arithmetic. When these concepts are clearly and completely mastered by young minds, they can better comprehend more advanced math. Then and only then can we hope to get our students out of the quagmire of mediocrity.
Many grown up people cannot make accurate change. They cannot keep their check books straight. They cannot fathom what it means to do a budget. These are basic math skills.
What has happened to our teaching methods? Granted I was raised in a different time and in a different state but achieving the next grade level at school absolutely required learning what was being taught in the current year. Every child learned. There were no exceptions. Sometimes special help was required to understand a concept but it was given with a firm hand and the confidence building knowledge that the student finally, “Got it.”
Nothing but nothing builds confidence like truly succeeding. Success is always measured in small portions. Not a single person ever born knew everything the minute they walked into school. Teaching takes patience, a kind heart, and the desire to impart truth and knowledge to the next generation of scholars.
Anne V. Norskog, Lacey
The question I think is too simplistic and implies that middle school math curriculum is a ‘stand-alone’ process that has no relationship to the preceding base of elementary curriculum.
WASL issues set aside; I would think it imperative that any teaching curriculum be based on a progressive scale. This is equally important regardless of the subject matter at hand.
What and how students are taught math in grades K-5 drives the foundation for what needs to be taught next. If you work on the presumption that elementary school children have successfully been taught the basic math functions and have absorbed the rote memory processes of addition, subtraction, multiplication and division [all of which is really just addition and subtraction], AND that it has been recognized from the start that math in and of itself is a conceptual and problem solving tool [how many apples does Johnny still have if he starts with 5 and gives 2 to his mother], then continuing and expanding the math curriculum so that more complicated concepts and problem solving tools can be applied and used should be a natural progression at the middle school level that will continue in grades 9-12. If however, the elementary school process currently is just a matter of rote memorizing of 1 1=2, throwing the switch at grade 6 and expecting students to quite literally think about concepts is most likely a step backwards rather than allowing them to follow a natural and logical progression. With math this is especially critical since quite literally it is a ‘concept’ by nature, and only serves a purpose as a tool for problem solving.
Math is essentially a foreign language using a totally different alphabet, and each symbol has both a defined meaning and a concept behind the combination of that alphabet. Besides teaching just the alphabet, the process of what it means conceptually needs to start at exactly the same time as the language is being taught. When you convert 1 1=2 [the formula] into words you get ‘one plus one equals two’. It is a concept that has to be taught to have a reference in the world of words.
If by the time students have gone through the elementary school curriculum and do not have the foundation of basic math along with the foundation of how to use it as a tool at the level consistent with that curriculum, the question is not what should they be taught at the middle school level, but why is the elementary curriculum not successfully providing the base from which the students can move forward from?
Education regardless of the subject matter at hand, is a gestalt not a lets do this from time ‘A’ to ‘B’, then move to a new totally foreign process without having built a foundation.
Gary Kaufman, Olympia"
A satisfactory foundation is what seems to be required and is so often missing.
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Comment by Ash
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I have always loved Mathematics and have done some with my other studies at University and reasonbly recently. I know a lot of people have trouble with maths and they are generally talking about the simple stuff.
I was a very high speed performer with algebra and calculus and trigonometry and co-ordinate geometry, had less interest in Arithmetic.
None of my kids had any trouble either, I guess your skipping a grade would not have helped.
For some types of maths it's practice, which I never needed, but some folks just never get beyond simple arithmetic and nowadays a lot of school teachers are deficient which never happened to me.
The education system nowadays is a little crazy, they teach a lot about various things, but mostly nothing much that the kids couldn't teach themselves.
I've never had the advantage of computer studies, just picked it up on the run, but I have done some computer software design amongst other things.
It is the use of them that causes the biggest problems as I found out when University required their repetitious use and I had to learn on the run, which wasn't much fun and many of the teaching staff if you could call them that, simply demonstrate to those who are already "off the ground"
Can't see my doing any more but often one is required to use certain skills for certain tasks. I prefer to actually know what I am doing.
But, it is specifics I need, not a lot of hogwash that I am never going to use. But I'd really like to do some serious computer studies but it doesn't fit with my circumstances, one day I may, just for fun.
katyzzz
Comment by Anonymous
But I agree that the perceived dichotomy is false. If only one thing is to be taught, it should be fundamental skills. But when students learn concepts as well as techniques, they learn how to think, how to apply their techniques to varying situations, and earn a sense of satisfaction: we all like to understand "why." Teachers can and should teach both.
By the way, I would not say that addition, subtraction, multiplication, and division are just addition and subtraction. The real numbers make up an ordered field, which is a set with two distinct operations: addition and multiplication, which satisfy all the properties the kids learn about in primary and secondary school. If anything, say that the four operations are addition and multiplication (though the two are linked by distribution).