The "Real" Contest

Have you ever gotten into a "battle" only to find the opponent has changed and redefined all of the terms?

Have you noticed that the publishers often try to change the focus of the disagreements brought by parents. Or they change definitions and throw out new "standards" so that you and I may not even know what the real issues are.

As a lowly intermediate, elementary school math teacher, I often feel that I do not have the qualifications to step up and confront more "learned" people. I think that no one would even give me any time. However, if you won't listen to me, will you at least hear the words of math professors, who know far more than I, and who strongly oppose the teaching of the "fuzzy math"??

A few years ago when I first became aware of the "math wars" that had occurred, and were still occurring, all across the country, I was most impressed by the pleas of professors to rectify and rewrite curricula, professors such as Dr. Wayne Bishop, Dr. James Milgram, Dr. Wu, Dr. Bas Braams, and many more.

Below are a few quotes from Ralph A. Raimi, professor emeritus of mathematics of the University of Rochester, describing the "real contest" in this "math wars" saga and the folly of expecting children to "discover" formulas and procedures on their own.

"But there is a contest, a serious one, and not the one suggested by catch phrases handed out by the publishers of the reform programs. It is not a contest between rote-memorization of meaningless symbols and deep understanding of problem-solving strategies. . . "

"The real contest in Penfield - and hundreds of other school districts across the country - is between mathematics and non-mathematics, between academic content and childish time-wasting, between what children can learn and what the present Penfield curriculum is pretending to have them "develop." A good mathematics program takes advantage of the mathematical discoveries of thousands of years of civilized effort, while Penfield has them counting with sticks, starting history all over again."

And the following from Raimi is scripted so purely and simply, so precisely and beautifully and is certainly worthy of notice. I'll even go so far as to say it is worthy of putting to memory.

"The systems of decimal and fraction notation are marvels of compressed information, intellectual advances that Euclid did not have available. Arithmetic is not trivial mathematics, and it certainly will not be "discovered" by school children. It must be taught and practiced."

The entire article, entitled "Why American Kids Aren't Learning Math," may be found here.

The Ant and the Grasshopper (a modern day story)

Alongside a trail in the open Field of Mathe, a Grasshopper was lazing about, discovering and coloring patterns to his heart's content. A lone Ant passed by, bearing along, with great toil and rigor, a bag of corn he was planning to store, yet singing as he labored. Because he knew the facts, he had accurately analyzed his problem, quickly calculated his needs, while carefully considering the 'rithms' of the seasons.

"Why not stop and discover patterns with me," said the Grasshopper, "instead of toiling and moiling (grasshopper speak for "drilling and killing") your life away? You'll be much happier sitting and chitting about your work than making your repetitious rounds." The Grasshopper continued, "That Trail of Boredom you are on becomes the Path of Monotony just over the hill, you know." (Surely, he thought, this was a factor the Ant had not considered.)

But the Ant was undeterred. And he wasted not a step, still singing as he 'plotted' along, for practice had made his search for the missing factors all the easier.

"By the way, my feelers are 'Connected' to my feelings, you see?" said the Grasshopper, everyday still lazing on the trail. "I'm feeling soooo good about my discoveries. I need to be writing about the work I'm going to be doing after I finish discovering what I'm supposed to learn to do. Care to join me?"

"No, thank you. I'm preparing for what's ahead," said the Ant, as he 'Sing(ingly)Poured' his full bag into his storage bin. And off he went to glean more from even higher p(m)aths. "Quit your Trail lazin' days, Mr. Grasshopper. Get up and get to work."

"Oh," said the Grasshopper, "there's still plenty of time. But for now, this writing about my discoveries is just about filling up my hours. But I'll be 'Blazing this Trail' later."

Sadly, the Grasshopper never got past his Trail lazing days. You see, though over 'Andover' he tried, he found that he had to continually 'spiral back' to the place of his discoveries in the '(o)Penfield'. He never made it off the Lazing Trail nor did he ever reach the top of the Field of Mathe.

However, the Ant had all he needed. Not only did he successfully reach the top of the Field, he became the King Ant of the Field of Mathe, all due to his rigorous work and efficient use of time. And he ruled over the Grasshopper all his remaining days.

Group Discovery ???

One of the marks of "new math" is discovery learning. But it's more than just discovery learning; it's "group discovery learning". Students are expected to come to knowledge of mathematical facts and procedures through "communal discovery".

You've heard it before. Only what your child discovers himself will he really grasp and remember. He must "own it" to be able to "use it and truly know it".

Now, let me see. If we want everyone to make the discovery himself, why would we put them in a group? That is exactly the opposite way you have every person discovery on his own. The whole group discusses and comes to an action, tried and agreed upon by the group. (Of course, someone in the group had to make the suggestion, to begin with, but never mind that.)

Now, remember through all of this, the purpose is for each student to make the discovery for himself -- so he can "own" that knowledge.

If you're in a group, how can all students come to "discovery" -- of a math procedure -- at the exact same moment? Someone in the group will come to the discovery first. Must he or she remain absolutely quiet? Must he hide his procedure? Remember this is a group discovery.

So, if someone else in the group makes the discovery, I guess no one else ever gets to "own" anything! Be First or Lose Your Shot at It.

By the way, the teacher is very quietly staying to the side, offering no guidance. And once the answer is agreed upon, the teacher is not allowed to check to see that the procedures were used correctly. Later, no drill will be given for students to use for practice. And there is no memorization of facts -- the facts can be 'rediscovered" on the spot, whenever they are needed.

In a traditional classroom, a teacher gives students an opportunity for small discoveries throughout the lesson, but he or she carefully directs the lesson and teaches precise methods, so that all students learn the most efficient procedures. Then the students practice and practice the steps under the supervision of the teacher. Following the guided practice is the independent practice, during which kids do more practice. Teachers are available to guide students and reteach if necessary. It is during this practice that some students "discover" and learn. And as the procedures are practiced over the next days and weeks, still other students may finally make the discovery on their own. But that "discovery" is just as real as the "discovery" made earlier by other students. In the meantime, the student has been able to successfully get the proper answer on the work because the teacher had taught just that.

I disagree that there is no discovery in a traditional classroom. There is. I see it every day.

And I don't think that it is possible for all students in a new math room to "discover" every single fact and procedure without input from the teacher.

Perils of Discovery Learning, Part III: 'Interdisciplinary' Activities

One of the big "buzz words" in mathematic circles today is "interdisciplinary" activities and projects.

If I'm teaching a unit in Science, students might do research in my class or the Library, write papers using skills taught in Writing class, and then generate the paper in the Computer Lab. Or if students are learning about Indian Villages in History, they might work in groups, or individually, to make a village in Art Class.

I often feel that teachers are evaluated (unofficially perhaps) on how much content crosses over into other disciplines, although no requirements have ever been made of me in my private school. It has been "suggested" that I find ways to involve other disciplines, but that's been the extent of it.

I have seen instances when I feel an interdisciplinary project has been very effective and where students are completely immersed. If the other discipline is a favorite of a student, if he loves art, or if she loves to write, he or she will really be engaged. I just don't like the pressure of forcing the project where it doesn't naturally go, where time is lost, all for the sake of "show". We can now brag at how many other disciplines were involved!!!

Dr. Wayne Wickelgren has made studies of interdisciplinary projects and I respect his opinion. In his book, Math Coach, A Parent's Guide to Helping Children Succeed in Math, he contrasts the traditional approach with the interdisciplinary approach:

"The traditional approach, in which classes are arranged by topics, such as arithmetic, fractions, algebra and geometry that build from one level to the next, has been used for decades for good reason. The material within each subject hangs together in logical ways, and is typically broken down into smaller units within which knowledge is even more tightly linked. Teaching students to hang together closely related pieces of knowledge makes sense and produces a deep understanding of a subject."

[That is just beautiful! And it makes such good sense.]

Hangs together in logical ways.

Now for Dr. Wickelgren's assessment of the interdisciplinary approach:

"Teaching across subject boundaries lacks depth. It may be fun for the students, but it doesn't help the mind organize the knowledge in a logical way, making it harder to remember."

There is also the likelihood that a teacher will overlook an important basic fact or principle that would usually be included in an incremental, structured approach.

I just have to say this one more time:
Hangs together in logical ways.

Perils of Discovery Learning, Part II

Dr. Wickelgren, author of Math Coach, A parent's Guide to Helping Children Succeed in Math, explains that it is not possible for children to "discover" much of what is important for children to learn in elementary grades. How, he asks, can a student deduce, through discovery, the meaning of "3 to the 4th power" or that there are 5,280 feet in a mile. There are many things which must be taught by the teacher and then memorized by the student. Now you may avow that these are things that a student could conceivebly discover, so let Dr. Wickelgren describe how a typical "discovery" session works.

"Using the discovery method, students are given very little of the information necessary to solve problems. Solving problems thus requires gigantic leaps of intuition that virtuallly no students possess, so they flounder. A class typically spends half an hour -- and sometimes as long as two and a half hours -- on a single problem."

Because the "discovery" time is largely unsupervised by the teacher, it proves to be a very inefficient use of students' time.

Dr. Wickelgren tells of a student who "complained to me that his class spent an entire week on one problem before the teacher told the students how to solve it."

And remember that in addition to the inefficiency of this procedure, students are also asked to spend time discussing what to do, and then to spend more time writing about how they arrived at their solution. And what if their solution isn't even correct??!! Remember, the teacher is not involved, is not guiding students back on track when they wander off on a rabbit trail.

As a result of all of the wasted time, very few problems actually are solved by the students. Teachers are giving little or no information, and student proficiency takes a nose-dive.

Contrast this scenerio to the traditional classroom, where children are taught by direct instruction. Students are given most, not all, but most of the information needed for solutions. Smaller amount of insight is needed and thus some students, though not all, solve the problem quickly. Several of these problem-solving activities are given throughout the course of the lesson, affording students additional opportunities to use small bits of insight to solve other problems. (And a teacher steps in to help, giving additional information when needed, if the problem takes too long.) Many students gain confidence because of the successes they make each lesson. And they see the point in what they are doing. The goal is short ranged.

Solutions are found quickly, not hours later, and students do not lose interest because the problem's answer is so long in coming. I'd rather have students engaged for several small problems they have hopes of solving than disengaged for an hour because they see no point in what they are doing.

Perils of Discovery Learning, Part I

I love to watch children learn. When people ask me why I like teaching, my answer is always the same: "I love to see that light bulb go off, to see that elation on a child's face when he 'gets it'". So there is a part of every math lesson I teach where I give students a chance to discover on their own.

But these are not huge, time-consuming tasks which involve groups of students. (I never figured out how you know which child is making the discovery when students are working in groups!) Instead, these are short moments built into every lesson where children begin with a known concept and are led to discover and figure out the new concept.

Later into the lesson, for the students who have failed to make that connection to the new, teacher intervention in a one-on-one basis with that student can pinpoint where the connection failed and the student has another opportunity to discover. I have many memories which I prize, memories of struggling children smiling with great joy. They got it. They figured it out without my actually telling them. I love these "light bulb" moments.

Dr. Wickelgren also acknowledges that students need these moments, but he also describes the perils of great units of time spent on "discovery learning projects".

The troubles of "fuzzy math" failures, which have created an uproar and lead to "math wars" around the country throughout the 90's and since the turn of the century are well documented. Parents have risen up to protest and to request changes in curriculum because of the failures of the programs to teach their children rigorous math.

Dr. Wickelgren describes the failures linked to Standards math (fuzzy math, new math) as follows:

"It dumbs down class content and lowers expectations for all kids. It doesn't adequately tests kids' knowledge. It wastes far too much time on activities that have little to do with math. And despite good intentions, it can actually decrease student participation.

"But the most important downfall of the approach is that it often results in only cursory knowledge of the nuts and bolts of math -- including basic aritimetic facts . . . and how to solve a variety of problems. This severely weakens the math curriculum because basic mathematical knowledge and problem-solving skill are the key ingredients of math proficiency. Mastering basic facts early is critical becase they form the basis for a huge amount of mathematics that follows. A child who doesn't know those facts by heart -- and how to use them in problems -- is at a serious disadvantage, even if he or she understands the concepts of addition, subtraction, multiplication, and division."

And this stunning statement from Dr. Wickelgren:

"The primary reason for the downfall: excessive reliance on student discovery of facts and principles instead of explicitly teaching them."

And another:

"Discovery sounds good on paper. In practice, it is time-consuming, inefficient, and results in little learning."

Discovery Learning; More from Math Coach

Dr. Wayne Wickelgren, author of Math Coach; a Parent's Guide to Helping Children Succeed in Math, has much to say about "discovery learning" and the "new math", often called "fuzzy math" (and which he calls "Standards math").

Here are some of his insights:

"The most pervasive theme echoed throughout the Standards is their emphasis on student exploration and discovery. Instead of presenting information to the class, Standards math teachers ask their students to discover mathematical concepts while solving math problems. "Typically, students break up into small groups of four or so to solve a problem. The teacher circulates among the groups to observe the discussions but otherwise does not interfere with their learning by providing too much information."

"This framework is intended to let each child's natural creativity in math blossom, enabling children to discover important concepts and problem solving methods on their own. . . Teachers encourage students in the groups to do a lot of talking and writing about their thinking process that led to a solution . . ."

"To facilitate discussion, the groups are often working on math problems that are somewhat different from traditional computational or story problems. . . grounded in real-world situations. They are open-ended and contain many parts and many possible answers. . ."

"One example: A middle school math teacher demonstrates a pendulum made from a string and a weight and asks students to construct a pendulum, investigate how it functions, and formulate questions that arise."

"Traditional problems by contrast, have a single correct answer and focus on closely related mathematical ideas and facts."

Dr. Wickelgren explains that hands-on, real-world activities dominates "Standards" math classrooms. Students are never encouraged to memorize addition, subtraction, multiplication, and division facts, but rather to manipulate objects to find the answers, rather than using pencil and paper, repeating the activity each time a set of facts are presented.

"Memorization is regarded as dull for kids and also ineffective as a learning method, as it seems at odds with really understanding the material."

All teachers know the value of manipulating objects to teach the four basic operations, but for a child to have to repeatedly group objects to solve 8 X 6 is standard operating procedure in a "fuzzy math" classroom.

Dr. Wickelgren also describes the organization of the textbooks:

Rather than dividing a year by topics, such as two-digit multiplication, fractions, long division, and decimals, the year is organized into group projects which link to other subjects to create interdisciplinary studies. Rather look at how the textbook is organized:

"A year's worth of mathematics might look like this: A Wagon Train's Journey West, A Genetic Study of Fruit Fly Reproduction, Managing a Supermarket, A Month in the Life of a Real Estate Broker, A Voyage to Mars."

The "fuzzy math" crowd design these activities because they believe that prior to adolescence, children's minds can handle such "concrete" mathematical concepts, mentioned above, "but are not mature enough to handle abstract numbers and operations."

And so your child, being too immature to memorize facts, may be asked to design and create a pretty portfolio with attractive bindings to hold their writings about how they feel about math. All of this is during math class!

Dr. Wickelgren acknowledges such positive features of Standards math as "emphasis on understanding, solving more challenging problems, enriching math curriculum with more probability. . . early study of coordinate geometry". But he also warns of the perils of the "discovery learning", which will need to be discussed in a later post.

[Dr. Wickelgren also acknowledges that the Standards have been modified, but warns that the changes are slow to make their way down to the classroom, especially since there are so many who refuse to see the error in the "new math" approach. Curriculum writers take it very personally. Their "works" are "their babies". ]