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Problem Based Learning And The

Nature Of Mathematics

Brian D. Rude, 2008

 

      The term “modeling” apparently entered the world of mathematics education a few years back. I have been puzzling whether it is something really new, or whether it is just a new term for an old idea - story problems. It seems to be considered something new, perhaps a new perspective. I’m not sure just what is expected of it. Is it supposed to be a revolutionary new way to look at mathematics, and teach mathematics? Or is it, more modestly, just a new emphasis, or new vocabulary, that is supposed to be an improvement on an old way of doing things?

      Modeling does seem to mean applying mathematical ideas to solving practical problems. In algebra we can use an equation to “model” a real life situation. If a person has five coins worth a total of sixty cents, then we can use x to denote the number of quarters, y the number of dimes, and z the number of nickels. Then a “model” for this problem is 25x + 10y + 5z = 60. Or, as another example the equation h = -16t2 + 20t + 80 is a “model” of the height above ground of an object thrown upward at 20 feet per second from the top of a building 80 feet high. All this just seems to be a slight change in traditional vocabulary. A "mathematical model", apparently, just means an equation that represents the problem. Or perhaps the "model" consists of the assignment of variables as well as the equation. Applying a term to the whole idea may be conceptually beneficial. It might be considered that the new vocabulary is more useful than the old vocabulary. I will agree that the term “story problem” does not seem like very sophisticated vocabulary.

      A similar idea to the modeling idea, also apparently somewhat new in the world of education, is “problem based learning”, abbreviated PBL. This seems to be a stronger version of the same idea of emphasizing problems. When we speak of "modeling" we can think of it as a part of a math course, but not necessarily an organizing principle. "Problem based learning", from the examples I have read, seems to include the premise that the course will be organized around problems, perhaps just one big comprehensive problem. Is this sensible?

Math-First Or Problem-First?

      What is the role of problems in teaching and learning math? How should math courses be organized? Should we start with applications, real life problems, as a lead in to a consideration of the mathematical ideas that may come from them? Or should we start with mathematical ideas that arise by extending previous mathematical ideas and then bring in problems that can be solved by these mathematical ideas? The first way emphasizes mathematical modeling, or problems. The second way emphasizes the structure of knowledge of math. The question may be expressed as "problem first" , or "math first"?

      Or perhaps this is a false dichotomy. Perhaps we should use a combination of these two approaches. Or is there some other approach that is separate from either of these?

      I'm not sure if it's a false dichotomy or not, in the final analysis, but I think it is an dichotomy worth considering. I think it is a fruitful topic of discussion in considering how best to teach and learn math.

      One good example of a real life problem leading to math is the Konigsberg Bridge problem. In 1735 Leonhard Euler turned his attention to a puzzle about walking seven bridges in Konigsberg without retracing steps, and gave graph theory its start. I presume historians of math know a lot of other stories of problems that were addressed by mathematicians and resulted in new mathematics. Does this one example tell us that we should always start with problems?

      Euler was a genius. The acts of genius may have some relevance to learning by ordinary people, but certainly cannot be taken as a model. Yes, problems can lead to math. But problems can also just be problems, and problems can bring frustration. It was frustration that led me to frame the question in the first place, primarily the frustration of trying to help students figure out story problems. Even under the best of circumstances story problems can be difficult My perspective is that if the math is not developed adequately then trying to do story problems can degenerate into trying to memorize a recipe. But perhaps I am looking at it wrong. Perhaps to someone holding the "application first" perspective it might appear that the difficulty is in trying to develop the math first. Perhaps story problems would be easier if we started with the problems, instead of starting with the math. How can this question be resolved? Or can it? Where could we look for perspective?

      Consider very young children. Is it possible to teach them math first and then apply it to problems? Perhaps not. First in the mathematical life of the child, I presume, comes counting, and that requires counting objects, and that seems to be starting with an application, counting something, and then moving to math, counting in the abstract. This would seem to support the problem-first idea.

      However I have come across the idea that perhaps may change this view a little. The idea is that very young children go through a stage in which they can count to ten or so, and know to point to successive objects while doing so, but do not have a one-to-one correspondence between their counting numbers and the objects they are pointing to. They may finish counting before they are finished pointing, or vice versa. We might call this the “pretend counting” stage. At some point this stage gives way to the stage in which they understand. Then there is a one-to-one correspondence between the numbers they recite and the objects they point to.

      What is the math, and what is the application, in this situation? I would say that the math is the idea that a specific succession of syllables relates to quantity. The application of this is to use this succession of syllables to a specific set of objects. This certainly does not settle the math-first-or-application-first question. We might say that the math-first idea is supported by the argument that the counting is not real until the idea is understood. And we might say that the application-first idea is supported by the idea that the “pretend counting” stage is a necessary stage. Perhaps no child will get to the real counting stage without first going through the pretend counting stage.

      This counting example reminds me of a similar observation about children, which I believe is true. Children go through a stage in which they like to tell jokes, but their jokes are not funny. This can be explained by the idea that they first understand only the form of joke telling and only later come to actually recognize humor. They first observe that someone tells a short story and other people laugh. But only later do they realize there is more to it than that, that there has to be something funny involved. They get an intuitive, or operational, understanding of the punch line. Then their joke telling is on an adult level.

      Both of these examples, counting and joke telling, involve rather sophisticated thinking, at least in comparison to what came before in the child’s world. And neither example gives any convincing answer to the math-first or problem-first issue.

      I find I come down solidly on the "math first" side of the argument, but not from anything I have said so far in this article. In the rest of this article I will try to explain why this is my conclusion. I believe it is generally better to explain how the new mathematical topic arises out of the old, and then to bring in applications as they are useful to support the understanding of the math. I will try to develop evidence, or at least a line of reasoning to support this claim. I will attempt to use the "nature of math" to support this reasoning. However this is not to say that the “math first” solution, rather than “problem first”, is necessarily clear cut in every particular instance, and I do not wish to be dogmatic about it. Applications should be used in whatever form or amount is most beneficial to learning math.

      In introducing a new topic in mathematics, either in class or in a textbook there is some rationale for starting with a problem. By starting with a problem, it can be argued, we enhance motivation. We establish a context whereby the mathematical ideas we learn will be meaningful. And surely meaningful is good. Thus many chapters in a math book start out with a problem. However I have sometimes found that a problem on the first page of a new chapter somehow does not enhance my motivation. Rather it often seems to squelch it a bit. As a general rule I skip that introductory problem, either as a learner or a teacher, because it doesn’t really seem beneficial. Of course I cannot speak for others. But I do have a few thoughts on why this might be so.

      Beginning a chapter with a problem may increase relevance and motivation, or it may not. I would definitely not argue that we should never start with a problem, but I would argue that in some situations a problem may be more daunting than motivating. Consider this as a problem to introduce a new topic:

      A craft shop produces two products, picture frames and wall decorations. To produce one picture frame requires $4 in materials and 2.1 hours of labor. To produce a wall decoration requires $2.50 in materials and 3.2 hours of labor. The cost of labor is $9.20 per hour. Each picture frame sells for $7.99 and each wall decoration sells for $11.99. Each week the shop has available 100 hours of labor and $220 for materials. How many picture frames and how many wall decorations should they produce each week to maximize their profit?

      This problem is an example of "linear programming", which I have had occasion to teach. This problem is quite manageable at the end of the chapter. The available materials and labor in the problem can be translated into linear inequalities, which can be graphed, and a "feasible area" found. The corner points of the feasible area must then be checked out by a profit function, to be found from the information given, and one of those corner points will give the mximum profit. All of this is quite understandable to college freshmen after a week or so of explanation and work, providing they are reasonably diligent and proficient in algebra. But at the beginning of the chapter I think this problem would be perceived by many students as a “story problem from hell”. It would seem to be just jamming on more and more information that must be dealt with, more and more numbers that presumably must be put into the solution in some way. If a teacher does start the chapter with this problem, I think it should be done very briefly and made clear that it is not supposed to make any sense at the outset. It is presented only to give an idea of the type of problem the chapter deals with. Even then I wonder if it might be counterproductive, because it seems so impossible.

      A picture came to my mind when thinking about these things, a memory from very long ago. I think I was probably about seven or eight years old when this happened, and my brother was a couple of years older. I think he must have just gotten his first bike when this incident occurred. My brother was mechanically talented, and apparently had taken the back wheel off the bike, and took apart the coaster brake. At some point in this process the parts of the coaster brake spilled out of the brake housing onto the floor, a disorganized splash of little steel parts. The image I have in mind is of my mother surveying the chaos. Her approach was not to consider the situation a nice problem with which to introduce a chapter on bicycle mechanics to my brother and me. If she considered this situation to be a potential learning experience, she did not consider it for long. She declared defeat. She scooped up all the parts in a box and took them the next day, along with the bicycle, to a repair shop.

      The “problem” was not motivating to my mother. Rather it was daunting. I think perhaps it could have been an appropriate challenge to my brother. He was mechanically gifted and I think he might have gotten it back together. But that has little relevance to teaching normally talented, or normally untalented, individuals. A problem will not be motivating if it seems impossible

      Of course it might be argued that a problem at the beginning of the chapter is not meant to be solved before proceeding with the chapter. This makes a lot of sense. In fact it may be expected that it will be skipped over at the beginning. It is there simply to give a very brief idea of what the chapter might be applied to.

      But what would advocates of “problem based learning” make of this? I have seen discussions of PBL (problem based learning) in which it is emphasized that the problem comes first. In a common version of PBL the class is divided into groups, and each group is given a problem and they must research to find courses of action that will lead to a solution. It is assumed, apparently, that the research will lead them to the library, not to their textbook (if they even have a textbook). It must also be assumed, I would think, that they will go in search of mathematical principles rather than recipes or shortcuts. This last assumption, I think, is totally unrealistic.

      Or, are the advocates of problem based learning assuming this? Do they see the discovery of mathematical principles as their goal? Do they think in terms of students building mathematical structures of knowledge in their minds? Should they?

      My perspective is very clear. Yes, indeed they should think in terms of building mathematical structures of knowledge in the students' minds. What else might be meant by "learning mathematics"? If we don't move away from problems into mathematical ideas then we have just a utilitarian course.

      It is quite possible for a math course to be a utilitarian problem solving course. I don't know what kind of math would be taught in a vocational heating and air conditioning school, but I can imagine there might be specific types of problems that technicians must deal with, and courses to teach them how. But surely in mainstream math courses, at the elementary, high school, or college level, we would want to provide something more. Rather than a collection of recipes for solving particular types of problems we want to build up structures of mathematical knowledge that have wide applicability and that provide a firm basis for learning more mathematics.

      All this is not to say that problems are not important in teaching and learning mathematics. Indeed, I think it is true to say that for most people, most of the time, most mathematical learning comes in close association with doing problems. I would not advocate dispensing with problems, even if it could be done. My argument is that problems should be used in service to ideas, not the other way around. A math course should not be "problem based", so much as "idea based".

      So the nature and role of problems in the teaching and learning of mathematics needs to be very carefully investigated.

      Can having a problem lead to mathematical principles? I would say yes, it can, but that does not necessarily mean it will. We could take this same question and turn it around. Can learning a mathematical principle lead to doing problems? Again I would say yes, it can. But again that does not necessarily mean it will. Either way, I would argue, requires careful guidance of a teacher.

      Advocates of PBL seem to envision a scenario that I consider unrealistic. Having a problem leads to solving the problem, which leads to learning of mathematical principles, which leads to remembering those principles and being able to apply them to other problems. That is quite a chain of events, and in the more fanciful versions of PBL it all happens with very little help from the teacher. Typically in these scenarios the students of a group, or their chosen representative, "reports to the class" their discoveries, and class discussion ensues.

      But having a problem does not necessarily lead to solving the problem, and solving the problem does not necessarily lead to an understanding of mathematical principles, and understanding of mathematical principles (if we actually get this far), does not necessarily mean that principle will be remembered, or even more important, connected to the larger structure of mathematics so that the principle can be applied to other problems. And there is yet another link in this chain - appreciation. In my old fashioned pedagogy appreciation is a product of competency, and competency comes from the usual methods of teaching and learning - listening carefully, tackling problems, doing homework and having it graded, taking quizzes and tests, etc. Students do all these things by simply following directions from the teacher. It works because teachers, at least with experience, have an idea of what it takes to actually accomplish learning, and can therefore guide the learning process. Competency acquired in this way varies greatly, of course. Students bring widely varying degrees of aptitude, effort and diligence to the task, and teachers bring widely varying degrees of understanding of what is needed for learning. Appreciation is not attained by every student for every subject, but neither is it true that appreciation is rare.

      PBL might accomplish all this. Presumably PBL is meaningful to the students. They become involved, interested, even motivated. Perhaps it can be argued that this chain, from problem to solution to a principle, to retention and appreciation, is at least a logical progression. The students may not necessarily follow this progression, but they might.

      PBL is usually described as a group type of thing. So instead of asking whether or not the student will go from a problem to a solution to a knowledge of math, we would have to ask if the group will make that progression. And how would we know? Advocates of PBL seem to assume that the group has one mind. A fact, or an idea, or a principle that is known and understood by one member of the group is assumed to be known and understood by all members of the group. That is not usually the case. A group of four students has four minds, not one. In the group work that I have had experience with there was no attempt at individual assessment. Much more needs to be said about the complications of trying to do things in groups.

      I have perhaps made the "problem-first" perspective sound unattractive, and indeed that would be my opinion. But what is the alternative? Just what would constitute a "math first" approach?

      In a math-first approach we develop a mathematical concept or topic, and then bring in problems as they are appropriate and beneficial. As an example consider reducing fractions. As I remember in fifth grade, or whenever it was, the idea of fractions was presented as parts. Two thirds means take two out of three equal parts of something. Diagrams were used a lot to present this idea. Then those same diagrams were used to present the idea of reducing fractions. Take a diagram that shows three quarters of a circle and add just a few lines and you have a diagram that shows six-eighths of a circle. From this comes a consideration of two fractions being equal because they represent equal quantities, and next comes a consideration of how to get from one fraction to an equivalent. All this takes careful explanation by the teacher. It also takes doing problems by the students. But the problems are explained in terms of mathematical ideas, of comparing quantities and reasoning about what quantities are equal to what. The problems are simply exercises to do, not “real-life problems”. The problems are in the service of the mathematical ideas, not the other way around.

      At least that is the way I remember learning to reduce fractions. I'll have to admit that memory has never been my strong point, and my memory of learning to do fractions in the fifth grade, or whenever it was, is sketchy at best. And I'll also admit that when explaining "why", teachers often revert to simply explaining "how". But the "how" and the "why" of a mathematical idea are not necessarily mutually exclusive. I can well imagine my fifth grade teacher going back and forth between the "why" and the "how", though I can’t say that I actually remember it. But the idea of going back and forth between the “why” and the “how” is very familiar to me. I do it every day in teaching college freshman math. Students tend to want a recipe for doing the problems. Teachers may not want to oblige them. We want to teach for understanding. We want to explain "why", more than “how”. So we do both. But in my memory, such as it is, fractions in elementary school always made sense. The “why” got through. At least a lot of the "why" got through. Reducing fractions, and adding and subtracting fractions, as I remember, always made perfect sense. I'm not so sure about the "why" of multiplying and dividing fractions. But in general the end result, the structure of knowledge about fractions, was held in place by understanding, not by memorization. I don't think the end result can be held in place by memorization. The structure of knowledge is too big and unwieldy. It has to be understood if it is to be used and retained.

      We learn the meaning of fractions with examples, usually diagrams or pictures, which can be considered "real-life problems", but very simple ones. But the examples, diagrams, or pictures, will not stand by themselves. We have to carefully explain them, explain how to interpret them, explain what matters about them and what doesn't, explain how to think about them so that we are thinking about fractions. Then we learn to reduce fractions, which does not lend itself to real life problems. Once the reducing of fractions is understood at an appropriate level of fluency, then, and only then, are we in a position to understand the adding and subtracting of fractions. At this point problems can become more "real-life". At this point we can say "John has 3/4 of a tank of gasoline. If he adds 1/8 of a tank more, then how full would the tank be?"

      Would it make sense to put more real-life problems in the middle of this process? If so, how? How do you make a “real-life problem” of reducing fractions? It can be done, I suppose, but I would ask why. My perspective, as I have mentioned, is that problems should be in the service of ideas, not the other way around. In learning fractions the best problems throughout much of the process are simply exercises that ask one to apply the mathematical ideas. Such problems provide practice. Such practice is needed. Such practice is effective. Such practice leads to satisfaction of accomplishment, which, as I have and will argue, is the basic engine that motivates practically all education.

      Now let's fast forward to another topic, exponents, as another example of the "math first" perspective. Exponents have been important in every college algebra course I have taught. Presumably well prepared students got them in high school, and understood them in high school, but a lot of students didn't. They struggle with them in college algebra. In teaching exponents I explain the mathematics. I don't start with problems. Every textbook I have used explains the mathematics. They don't start with problems. Indeed "real-life" problems never come up with exponents. We start with the idea that an exponent tells you how many times a number, the base, is used as a factor. Then we explain how a few basic rules logically follow from this definitional beginning. We show that x2 times x3 means x times x, multiplied by x times x times x, which can be abbreviated as x5, and so on. Thus the rule of multiplying numbers by adding exponents is explained as a logical consequence of what we mean by exponents. This type of explanation must be given carefully and fully. The process must include doing problems. Explanation must be followed by practice. But again the problems most suitable at this point are simple exercises in which one apples the mathematical principle. The problems are pretty much one step from problem to solution. We learn by doing. We learn to handle exponents by handling exponents. The "doing" in this case is with pencil and paper, and careful thought.

      We learn to handle exponents so we can understand and work with polynomials, and we learn polynomials so we can solve equations with polynomials (quadratics at least). "Real-life problems" do not readily appear in this scenario until this point. Before getting to quadratic equations we do a lot of problems, but they are very short problems that are just exercises to apply the mathematical ideas, not "real-life" problems. When exponents have advanced to quadratic equations we can again bring in real-life problems. We can say "A rectangular garden has a length that is forty feet more than twice its width, and has an area of 540 square feet. Find the dimensions of the garden."

Types Of Problems

      I have argued that problems do not automatically lead to mathematical ideas. However it does seem that some problems lead to mathematical ideas much more readily than others. I think it is useful to classify problems along a range. On one end of this range are problems that lead, or can lead, to mathematical principles. On the other end of the range are problems that are do not naturally lead to mathematical principles, and cannot easily be made to do so. The math must be taught first before these problems can be done. I will call the first type of problems “leading problems”. They lead to mathematical ideas. The second type of problem I will call “trailing problems”. They do not lead to mathematical ideas, at least not easily or intuitively. Rather they use math, and that math must be developed before trying to apply it to the problem. Therefore they must trail the math. I will give examples of each type.

      Arithmetic, as I have argued elsewhere, arises by abstracting from reality. We can start with a problem like this:

      “Mary has three toys. Then her mother gives her two more toys. How many toys does Mary now have in all?“

      This problem leads easily to the idea of addition. It is a leading problem. Or, more accurately, it can be used as a leading problem. Other problems like it can be presented, and it is not a large leap to conclude that three plus two equals five, no matter what objects are used. We develop the idea of addition by abstracting from reality. We use leading problems to do this.

      In upper elementary school problems of paying interest on borrowed money leads to general formulas for interest, so such problems can be called leading problems. In beginning calculus there are many problems that can be used to lead to the idea of the derivative. Galileo's experiments with acceleration down an inclined plane are an example. Problems of rate of change lead to problems of slope on a graph, which in turn lead to the idea of the derivative. And in many other ways we use carefully selected problems because they lead to the mathematical ideas we want to teach.

      But there are also many problems that are not leading problems. They are meaningfull only after the math has been developed. For an example of a trailing problem consider this:

      A floor has an area of 126 square feet. The length of the is five feet longer than the width. Find the dimensions of the floor.

      This, of course, is very much like the garden problem I mentioned a few paragraphs back when discussing exponents leading to polynomials leading to quadratic equations. Could this, or the garden problem, be a leading problem? What would it lead to? Let us assume that students at this point understand area of a rectangle. The way to work the problem is to set up an equation (a model if you will). Let x = the width of the floor. Then x + 5 is the length of the floor. The equation therefore is x(x+5) = 126. Simplify the equation to the form x2 + 5x - 126 = 0. Factor to solve the quadratic. Thus solving this problem requires knowing how to solve quadratic equations by factoring, which requires understanding exponents and polynomials, as previously discussed. Would this be a good problem to present before starting the topic of exponents?

      My perspective is that this problem is very much a trailing problem. To use it as a leading problem might be theoretically possible, but it would require such mental contortions that it would not be at all beneficial. But it works very well as a trailing problem. When students can do quadratic equations then they can be shown how this problem can be set up (modeled) as a quadratic equation and solved.

      The linear programming problem I gave earlier ("A craft shop makes . . .. ) works fine as a trailing problem. A lot of mathematics must be developed before the problem can be solved. This problem does not lead to this mathematics. It uses it.

      Now I would like to turn to another way of classifying problems, and introduce some new vocabulary. I'm not sure that the actual terms I will introduce have a great deal of importance, but I do think it is important to analyze problems and how they are used in the service of learning math.

      In learning the basic meaning of fractions, in fourth grade perhaps, something like leading problems is needed. The fraction 2/5 cannot have much meaning at the beginning as an abstraction. We need something tangible, or something out of real life, to make 2/5 meaningful. We can talk about 2/5 of a pie, or 2/5 of a circle, or even 2/5 of an hour. Then, as the students progress, we can abstract out of these illustrations the idea of the fraction 2/5 as a number

      But is a diagram or a picture of 2/5 of a pie a problem? Just what about it is a problem? There is no question, and therefore no answer. It seems more of an idea, a bit of reality that can lead, with careful guidance, to a mathematical idea, the idea of a fraction. Would PBL advocates insist that fractions must be introduced with a "real life" problem, and reject the pie cutting diagram as insufficiently a problem?

      Ideas like 2/5 of a pie, or three pieces or candy out of four, or one quart out of four that make a gallon, and so on, are not really problems, but they serve as leading problems, leading to the idea of what we mean by a fraction. We might call them "models", but they are certainly not the "models" we considered at the beginning of this article. We could call them illustrations, or examples. Or perhaps "quasi-problems" would be a good term. The examples I have presented would be leading quasi-problems, since they are used to lead to the idea of the meaning of fractions.

      There could also be trailing quasi-problems, which would be examples, models, or illustrations that are best used after a mathematical concept or idea is learned. An example of this would be a diagram such as this.

     

This is a geometric interpretation of the algebraic multiplication of a binomial by itself, (a + b)2 = (a + b)(a + b) = a2 + 2ab + b2. A slight variation of this diagram would illustrate the more general rule that (a + b)(c + d) = ac + ad +bc + bd. A diagram such as these is not really a problem. It can be related to algebra, but the relation is not obvious at first. Problems could be made from it, and that might be beneficial. But as it stands it is not really a problem. We could call it a quasi-problem. It is a trailing problem, assuming the algebra comes first, assuming that knowledge of the exponents and the distributive is used to conclude that (a + b)(a + b) = a2 + 2ab + b2 .

      Could this situation be reversed? Could geometry be taught first, and used to lead to the distributive law? I'm not sure. It seems to me the more natural way is that counting leads to arithmetic, which leads to algebra, which includes (a + b)(a + b) = a2 + 2ab + b2 . Geometry is developed separately. Algebra and geometry can be merged into analytic geometry, of course. But I do not see any way to easily or naturally think of the diagram as a leading quasi-problem.

      To continue classifying problems, consider 5 + 6 on a set of addition flash cards. That is certainly a problem. It has an answer. But it doesn't seem quite right to call it a "real life problem". It is too simple. A flash card calls for an immediate response from memory, not the assembling of a line of logical reasoning that leads to an answer. I would call this a "look-say" problem, and it operates as a trailing problem. It is used to develop fluency, and a high degree of fluency is needed with the addition table.

      Look-say problems, in my opinion, are not confined to the addition and multiplication tables. I think after fractions and decimals are understood to a reasonably high level it makes sense to treat decimal-fraction equivalents as look-say problems. That means put them on flash cards and drill, which is exactly what I have done in the past.

      At some point early in the study of calculus problems such as this are appropriate:

      Give the derivative, with respect to x, of each expression: x2,     5x3,      3x4     . . . .

      Are these problems of the same nature as the addition facts on flash cards for second graders? Are they, or should they be treated as, look-say problems? Similarly, in learning to deal with exponents it is highly desirable that students get practice with problems such as:

Simplify: 23 = ,     15 = ,      x2x3 = ,     3x-2 =      . . . .

      I think students at times would do well to put a few problems like these on cards and treat them as flash cards, but not for speed drill like with the addition flash cards. The individual answers themselves do not need to be memorized, but the process needs to be developed to a high degree of fluency. These are problems that are one very easy step from problem to solution, but only if the principle, or at least the rule, is understood. Students need to practice on problems such as these, but in order to know the rule, not to know the facts.

      What should we call problems like this? They are not quasi-problems, and they are not look-say problems. I will call them "one-step application" problems. They would normally be trailing problems, because the one step is the application of some principle or rule that has just been learned.

      To continue with the vocabulary of problems, "procedure" is a good name for some multi-step problems, such as completing the square. A procedure is similar to an "algorithm", but I would define an algorithm as a multi-step problem done more mechanically.

      Some problems are proofs. Some problems are derivations. Some problems may best be labeled "story problems", or "real life" problems.

      These labels are not meant to be definitive or exhaustive. They are more descriptive. I think there is a benefit to being descriptive in this way. These terms, and perhaps others, give us a handle to analyze the teaching and learning of math.

      Here is one example of how this vocabulary of problems can be beneficial: How many problems make a suitable homework assignment? Only one? Two or three? Twenty? The answer depends a lot on what type of problems they are. In an advanced math class one proof could be a very heavy assignment. In reducing fractions in the 5th grade a set of twenty problems can be a very light assignment, at least for the students who understand it. In teaching college algebra the number of problems in a typical homework assignment can vary considerably. If the assignment is to derive the quadratic formula, one problem is plenty. If the assignment is to apply rules of exponents, twenty problems is a light assignment.

      And a more fundamental question might be "What is a good problem, and what is a bad problem?" Does the concept of a "good problem" make sense? My perspective, based on the ideas I have developed so far is that a math problem is good or bad only in the context of its purpose. A problem cannot be judged as either a good problem or a bad problem outside of that context. For example consider the following set of problems:

    1) Joey has three pieces of candy and his mother gives him three more. Then how many pieces of candy does Joey have?

    2) How many fives are there in sixteen?

    3) Reduce to lowest terms: 4/8

    4) Mary drives 14 miles at 40 miles an hour on a country road. Then she turns onto an interstate highway and travels 15 miles at 70 miles an hour. Find her average speed for the entire trip.

      Which of these problems are good problems? Which are bad problems? And why?

      Problem number one can be praised as a good problem because it is a "real world problem". But it is a good problem only in a certain context. It is not a good problem for a young child who can recite the numbers from one to ten but doesn't really understand the idea of counting. It is not a good problem for a third grader who should be working on multiplication. It is not a good problem for a fourth grader who should be working on fractions.

      There is only a limited window of usefulness for any problem. For problem 1 this window of usefulness would be probably somewhere in the first grade, when children can count adequately and need to start understanding addition. It might be useful as a review problem at the beginning of the second grade. But thereafter the window of usefulness is closed.

      Consider problem 3. Is it a good problem or a bad problem? I would argue that it is a very good problem in the right context, and a very bad problem everywhere else. The right context is when students are learning to reduce fractions. At this point they need practice. And at this point asking whether an individual problem "reduce 4/8" is good or bad becomes almost meaningless. A more important question at this point is how many such problems are appropriate for a homework assignment? And should tomorrows lesson and homework be the same as today's, or should it progress in some way? And what is the nature of the student's understanding of reducing 4/8 to 1/2? And should the students be given problems such as "reduce 65/104?

      The concept of reducing fractions is important to understand. That implies that some reasoning must be presented, probably involving pictures of fractions of circles. Students must also develop fluency in reducing fractions. That implies that homework must be assigned, discussed in class, and graded, and that quizzes and tests will be used. All of these considerations have little to do with evaluating whether or not a particular problem is a "good" or a "bad" problem.

      My thinking along these lines developed primarily as a music teacher, not as a math teacher. For a number of years I gave private guitar lessons. I always considered a crucial decision in such a situation is what to assign the student to practice over the next week. Sometimes the answer is very easy - the next piece in the book. But if one wants to do better than that, and one certainly should, there can be many things to consider. I never found a guitar method book that I could use indiscriminately. I had many reasons to bring in much supplementary material. There is no end of potential supplementary material available, and I had the means - a computer program - and time to make a nice copy of whatever music I found, edited as I wanted. However, out of a mountain of music that I potentially could draw on, only a small amount was appropriate for a particular student at a particular time. This snippet of music is too easy. That is too hard. This has an F chord in it, and the F chord is notoriously difficult for beginners. That doesn't have any F chords in it, but music with F chords is what is needed at the moment. Just because a piece of music has genuine musical merit is not in itself sufficient reason to make it appropriate in every situation.

      With this expanded vocabulary about problems, perhaps we might consider a few topics and how they would best be introduced or developed.

      Compound interest can lead to the idea of the Euler number e. Real-life problems of compound interest can be leading problems from which we may think mathematically and come up with the concept of e. Perhaps this is an example of the "application first", but it really seems more of a "math first" approach. The number e comes from compound interest only after a very careful algebraic analysis and mathematical consideration of the equation for compound interest. We cannot present a problem or two of compound interest and expect the idea of e to pop into students heads as an easy abstraction from the problems. Rather we have to lead them through the algebra and then bring in the idea of limits in at least an informal way. I think compound interest is the most sensible way to introduce the concept of e, but it is not the only way. We could use a more pure "math first" approach and define the number e abstractly, as a limit, or as a function that is its own derivative. But why would we? Interest, compound or simple, is a part of everyday life for normally educated citizens. It is taught to at least some extent as a part of math at some level (eighth grade in my experience, as I remember). Therefore it is a sensible way to get to the concept of e. For this topic leading with a problem seems very sensible. Whether we decide it is more of a problem first approach or a math first approach is not really important.

      Compound interest provides leading problems for the topic of the number e, and solving exponential equations. But compound interest problems are trailing problems in other contexts. The important thing, I would argue, is that problems should be in the service of building a structure of mathematical knowledge. Problems should always be in the service of ideas, not the other way around.

      What about imaginary numbers? Should an introduction to imaginary numbers precede the topic of solving quadratic equations, or follow it? Until recently the answer to this was quite obvious to me. It would make no sense to introduce imaginary numbers until you have gone into quadratic equations. Quadratic equations are leading problems for imaginary numbers. However the college algebra text book I am now using puts a section of a chapter on imaginary numbers ahead of the section on quadratic equations. This makes quadratic equations into trailing problems. I tried doing it that way. It works. Or at least it can be made to work. If imaginary numbers precede quadratic equations then the number must be defined purely abstractly as the square root of minus one. This is a math first approach. We start by asking about the square root of minus one, observe that neither a positive number nor a negative number will produce a negative number when squared, and then simply define i as the square root of minus one, and investigate the consequences of this definition. Then after some practice problems (one step application problems) we can bring in quadratic equations that we previously decided simply had no solution.

      Either approach to imaginary numbers, I believe, provides a nice setting to say a few words about the concreteness of numbers in general. This is a good time to point out to students that all numbers are "imaginary" in a sense. The term "imaginary" arose from the idea that there is no square root of -1. We are just pretending. Imaginary numbers, in one sense, don’t really exist. However that can be said to be equally true of negative numbers. I can hold three eggs in my hand, but I can't hold minus three eggs in my hand. But then I can go ahead and point out that holding three eggs in my hand is not the same as holding the number three in my hand. The number three, the simple, positive, counting number three, is just as “imaginary”, or just as abstract, as any other number.

      Is this discussion a good use of class time? I'm not sure. It's at best a five or ten minute expenditure of class time, so I think it's worthwhile. It is not a problem based explanation. It is very much an idea based explanation. I tend to think it's an idea that merits a little class time.

      I continue to hold the opinion that it's better to go from quadratic equations into imaginary numbers, and not the other way around. However a more important point here is that either way problems are not as important as ideas. Problems should be used in the service of ideas.

      Consider the idea of equation solving at the beginning of elementary algebra. Should the topic be introduced with real-life problems, or purely mathematically, or some combination of math and problems, or perhaps by quasi-problems? I do not have a great deal of experience with this, so I am not sure. It would seem that the pure PBL approach would not be ideal. Starting with a real life problem would seem to invite solutions by arithmetic, and solutions by trial and error, but the whole idea of equation solving is to move beyond arithmetic and trial and error. The goal is to build mathematical structures of knowledge, not to just solve problems by any means available. I would think a math-first approach would be best. Equations would be presented as a new mathematical idea, and explained as a new mathematical idea. It is an idea that can arise from many practical situations, and it is an idea that has use in the real world, but it is the mathematical idea that we are most interested in. A math-first approach emphasizes that the concept of equations should be carefully considered and developed. Quasi-problems, comparing and equation to a see-saw, in which changing one side necessitates a change in the other side, can be useful. Another quasi-problem, relating an equation to a statement that can be true or false, depending on what value is chosen for x, can also be valuable.

      This approach to equations emphasizes the “why” of equations, but students will want recipes. They will want the “what” and the “how” more than the “why”. They can be given recipes. Indeed when an assignment is given they should be told exactly how to do the problems, and the problems at this beginning stage would be mostly one-step application problems. But, just as with the “how” and “why” of reducing fractions, the “how” of solving equations is not harmful providing the “why” is explained adequately. The teacher may go back and forth between the “how” and the “why” a number of times until the idea of solving equations is understood and learned to an appropriate level of fluency.

The Nature Of Mathematics

      Next I wish to address some ideas about math and learning math that I think should be of concern to math teachers, and that can be better understood with the perspective of all of the ideas developed so far in this article. I put "the nature of mathematics" in the title of this article. But we have to subdivide the issue. We have to know math in order to use it, and we have to learn math in order to know it. So instead of just asking "what is the nature of math?", I think we should also ask, "What is the nature of knowing math?", and "What is the nature of using math?", and then we can go one step further. We can ask, "What is the nature of learning math?". Considerable reflection has convinced me that these are best treated as separate questions, because they have separate answers. I will not pretend to answer these questions fully, but I think they are worth considering. Most importantly, I think, is that a consideration of these questions can be of great benefit to the mathematician in understanding all the trouble and frustration non-mathematicians have in learning and using math. In other words, teachers need to know this.

      The nature of mathematics is that it is a logical structure. Every element in the structure is either an axiom (or something like an axiom), or it is a logical consequence of axioms. Everything, except the very beginning, arises logically from what came before. The ways that the elements of a structure of math are expressed can vary considerably. The axioms sometimes are explicitly stated and recognized, but often the axioms are more intuitive than explicit. In the main trunk of the tree of mathematics, the way it is usually perceived by most people at least, the beginning point is counting. Counting is not explicitly defined in most peoples' minds, but we expect five years olds to be able to do it. It is defined in the sense that we can easily agree on what is correct counting, and what is not. Once counting is in place in one's mind then addition can start to make sense. Once addition makes sense then subtraction can make sense. Then multiplication and division can make sense. Then fractions can make sense, and signed numbers, and decimals, and on and on and on. Algebra arises out of arithmetic. Calculus arises out of algebra.

      But to say that everything in math is a logical consequence of what came before is to say something about the nature of mathematics, not about the nature of knowing mathematics, or of using mathematics, or of learning mathematics. Math is not useful just because it exists. It is useful because we know it, and know how to use it. But we make mistakes. A consideration of how mistakes happen can shed light on the nature of using and knowing math.

      To use math means we must use logic, and using logic is one source of mistakes. We tend to think of logic as quick and easy. Sometimes it is, indeed often it is. But applying logic is also often slow and halting. In many situations a logical conclusion is quick and easy only in hindsight, only when we already know it, not when figuring it out. When one is figuring out math, logic can be exceedingly slow and laborious.

      The difficulty in understanding proofs illustrates that logic is often hard. One step in a proof leads to another, or several steps together lead to a conclusion. But no matter how easy and logical it may seem to the teacher, students may not see the logical connection. So we explain. We give more examples. We explain again. Success often comes, but success is seldom perfect. Many students understand the logic, some very easily. But others do not. It is not realistic to expect that every student in a class will understand every step of logic in a proof. Logic is not always quick and easy. Sometimes it is very difficult.

      Problems of any type, not just proofs, can be difficult for students. Problems use deductive logic, just like proofs, but unlike proofs, the logic is usually not formal. And students often want a recipe for doing problems. Telling students "how" often satisfies them for the moment, and often enables them to proceed with homework problems. But I think it is also true that students are very much aware that it is important to "understand" the problems, even when they do not fully know just what it means to understand. Perhaps it is accurate to say that they are more aware when they don't understand than when they do. They often want a recipe so they can work the problems, but they also want it to make sense. They will be frustrated, even resentful at times, if it does not make sense. Students know that logic is important, even though it is often difficult.

      I think it is true that logic is most understandable when it is intuitive, when it fits situations in everyday life. Indeed logic is a ubiquitous part of everyday life. When a person says “I have to go to town this afternoon”, rather than, "I think maybe I'll go to town today", we may be pretty confident that that conclusion is a logical result of facts and circumstances, but the chain of logic that leads to that conclusion may be rather obscure. In response to the question, “Why do you have to go to town this afternoon?” one might get a statement such as “Well, it’s Thursday isn’t it?” That hardly completes a coherent chain of logic. However the complete thinking might start something like this: “I have to go to town at least twice a week because I really like tomatoes, fresh tomatoes, and the grocery store gets produce in on Mondays and Thrusdays, but Monday is not good because school gets out at 3:30 and I have to pick up Alice, and that hardly gives me time to shop before I need to be home to start supper, whereas on Thursday I don’t have to pick up Alice, because . . . . . .” and on and on and on.

      If we can handle such convoluted chains of logic in everyday life without difficulty then why can't we do it with abstract mathematics?

      One answer to this question is that word, “abstract”. We get a lot of math by abstracting from reality, and it appears that the abstract is simply harder for the mind to deal with than the concrete. Our brains have evolved to deal with the concrete. Indeed one might wonder how we are able to deal with the abstract at all. There must be survival value in it.

      Another difference between the everyday logic we use and the mathematical logic we ask students to use is time, and our awareness of time. The person who explains why she has to go to town on Thursday may have spent literally hours thinking about it before coming to the conclusion that she has to go to town this afternoon. However that thinking typically takes place in the interstices of everyday life. She may have mulled over her plans while brushing her teeth, and while driving to work, and while working, and while playing with her children, and while fixing dinner, even while talking with friends. A problem such as this, without heavy emotional weight, can easily enter and leave one’s mind as time and circumstances permit with little notice on our part. Perhaps it took a day or so of this kind of thinking before it is obvious to her that she has to go to town this afternoon, and not some other time. But of course she doesn't count the minutes she spent on thinking about it. But when doing math, which usually means doing problems for homework or for study, which means one is taking a course and wants a decent grade, one is more aware of the time.

      And yet a third difference between everyday logic and mathematical logic is motivation. In the going-to-town example, the person does want a satisfactory answer to when to go to town, for personal and practical reasons. A satisfactory answer prevents the frustration of wasting time and effort. The motivation to use logic in mathematics is usually more artificial. A person may be generally positive about school and learning, but still want the math to come without being too taxing or taking too long.

      We may also imagine what happens when a person, by their natural interest, gives thought to mathematical topics in the interstices of everyday living, without begrudging the time and mental effort it takes to figure out things, indeed being unconscious of it. Such people do exist, of course, and some of them become mathematicians. Indeed I think it is not uncommon for mathematical thought, of one degree of abstraction or another, to occur in normal people. However I would argue that it is not realistic to think that such thinking is anywhere near adequate for the normal person to achieve the mathematical knowledge that we expect every citizen to acquire.

      A second reason we make mistakes in logic is that the elements to be put together logically may not be present in one's mind. This is an important idea, that leads to some thoughts on learning math. Consider this mundane example, which I have used before:

1. Three people are needed if we are going to go the the movie.

2. John wants to go.

3. Judy will go only if Joe doesn't, but Joe won't go because he doesn't like bad weather and the weather is bad.

4. Therefore the trip is on.

      Does this make sense logically? Or is something missing? The premise requires three people, but only two are established in these four steps as wanting to go. Therefore logic does not lead to the conclusion. There is a missing element. (And a consideration of how long it takes a person to realize an element is missing says something about logic not being always quick and easy.)

      Now compare the above situation to the fuller situation as shown in this diagram.

      Logic can not be appled to a situation with missing elements. Can it happen that in learning mathematics a student can sometimes not apply logic because some element is missing? Of course. It happens all the time. We could ask why an element is missing. (Or we could turn the question around and ask why an element should be present. That can be a productive thing to do.) In the diagram above all elements are present that are needed to come to the conclusion. But, and this is very important, logic does not work in your mind because the elements are all present in the diagram. The logic works in your mind because the necessary elements are present in your mind. In figuring out this example the necessary elements can easily enter your mind just by reading the statements in the diagram. But this is not instantaneous. It takes a bit of time to read each statement, decode it in your mind, and then read other statements, and only then apply the logic. The logic may seem quick and easy, but it is not instantaneous. And it will not happen until the necessary elements are lifted off the paper and fixed in your mind, if only for the few seconds for your mind to work on it and come to the logical conclusion.

      In this example the elements tend to be separable, discrete and concrete, and that makes it easier . Each part in the diagram makes some sense it itself, or in connection to only one or two other elements, and relates to everyday life. It can be much more difficult to apply logic when the elements are more abstract, or ill defined, or unfamiliar. An example of something not very concrete might be like this: "This relation is not a function, so even though it is true that 7 in the domain is paired with alpha in the range, we can not conclude that observing alpha means that we must also observe 7." Did that make sense? If you know a lot of mathematics, sure. But the elements involved in this example, "relation", "function", "domain", "range", "7", and "alpha", are not part of everyday life like "movie", "spite", and Tuesday. If we think of "movie" and "Tuesday" as being as concrete as bricks, we might say that that "relation", and "function" are about as concrete as drifting clouds in the sky, morphing from one form to another in spite of our best efforts to pin them down and understand them. At least they are to students struggling with these concepts. A concept that is so amorphous is, in a sense, missing, even though the students may use the words and have some idea of what the concepts are.

      Here is an exercise that I think can be revealing. Study the above diagram just enough to be sure that it makes sense, to be sure that the conclusion does indeed come from the premises. Then cover up the diagram and reproduce the logic on a piece of paper from memory. Likely you will not be able to do it. You may remember the conclusion, that the trip is on. And you may remember that a premise is that three people are needed. And you might even remember who those three people are. But will you remember all the details that lead up to each person's decision to go. If you do, you must have a pretty good memory. But the important point here is that either by memory or by looking at the diagram, the logic will not work until all the elements are simultaneously present in your mind.

      But, and I think this is important, once you do get all the elements in mind, and it makes sense, you turn your attention away from it, and assume you understand it. Yes, you do understand it, in one sense. Your sense of satisfaction for understanding it is not totally misplaced. But it is a mistake to think that this sense of understanding means you remember all the details. All the details of a logical conclusion must be present in mind for it to make sense. But once that sense is made, details begin to drop away. Once we put all the pieces together for a logical conclusion we have established something very important, that the pieces do indeed fit together. In many situations that is enough. In this example we know for sure that the trip is on. That is the important information that needs to be established. Once established it doesn't matter if the details drop away. But many times in learning math we do not want the details to drop away. Many details must be retained, until at least after the test. And more importantly, many details must be retained if the structure of mathematical knowledge is to be built on, to be expanded. So memory is important in knowing and using math, both short term and long term.

      So what must be held in memory when trying to figure out a bit of math? Consider a problem in adding fractions. You must remember that only fractions with like denominators can be added. That must be immediately available from memory. It cannot be something that is to be figured out anew each time. It can happen, of course, that in a specific situation a person may have need to add fractions, make several mistaken attempts resulting in the wrong answer, reflect on the problem, and then realize that he was trying to add different denominators, try again a time or two, and eventually get to the right answer. But if every simple problem of adding fractions must entail an inefficient and time consuming process of trial and error and reconstruction of what should be obvious, one has a very poor foundation on which to build more math.

      Also in this example memory must supply the addition facts. Again it is possible that a person might forget the sum of 6 and 9, but count on fingers and then arrive at the right answer. But again this is inefficient.

      And, in this example, reducing fractions or raising them to higher terms must be an automatic and efficient process. Again we can envision situations in which a person eventually gets to a right answer. But in the bigger picture adding fractions must be made effective and efficient if one is to build more mathematics. A lot of things have to be on an immediate access level in memory in order to use math. It is not sensible to say that a person could do a problem involving the addition of fractions if that person doesn't know his addition facts, doesn't know his multiplication facts, doesn't understand changing fractions to higher terms or reducing fractions, and doesn't appreciate the need for a common denominator.

      In this example of adding fractions the particulars of the problem can be held in place by short term memory. But the general facts about adding fractions, common denomimators, addition and multiplication facts, and so on must be immdiately accessible, and that can only happen with long term memory.

      So knowing math means having knowledge. Some critics may disparage knowledge as "mere facts", but that is unfortunate. It is true enough to say that facts are not enough, but it is also absolutely true to say that facts are indispensible.

      I have said that using math (or doing mathematical thinking) mostly means putting together chains of logic. However there is another perspective of using math that I think is also very valuable. Math, in many situations, is a way of coping with problems that you can’t "wrap your mind around". Math allows you to do those problems anyway. You translate the problem into symbols that represent mathematical elements, then you follow known and proven rules about manipulating those symbols until you get your answer.

      I will illustrate this idea with a language example, as math is close to language in many ways. We are able to decode language, and decoding language involves logic. Consider this example of decoding language:

      “Do you deny that it is not true that Mr. Jones said that it is not true that you were in the tavern at 9:00 PM when the crime occurred, and that Mr. Smith said that he was not there, though he, and indeed others, said he was, or that he was but he said that he wasn’t, or that it doesn’t matter, and that you know the difference? Your answer please, yes or no. The court is waiting.”

      If I were a witness in a court and were asked this question I would have no idea how to answer it. I cannot begin to wrap my brain around that. But consider all the small parts of this as data to be processed. When asked if you deny something, your data processing brain considers that something with the alternatives of either truth or falseness. Did you steal the cookie? You can admit guilt or deny it. The brain must hold the question in mind, and come to a conclusion, but in normal circumstances that is not hard. You can wrap your mind around it very easily. You won't get much sympathy by claiming you don't understand the question.

      Consider the part about “Mr. Jones said”. Your brain can bring in the words of Mr. Jones and reach a conclusion. He either said such and such or he did not, according to your memory.

      Consider the part about “not true”. Your data processing brain can handle that. The word “though” can be handled by your brain. The idea of either-or can be handled by your brain. The phrase “when the crime” occurred can be handled by your brain. And your brain is capable of handling subordinate clauses, prarallel clauses, and other complexities of language. You can wrap your mind around all the small parts of this example.

      But can your data processing brain handle the whole paragraph as presented? Mine cannot. It’s not even close.

      Can your data processing brain handle this?

      If the number which is the power to which a second number must be raised in order to get a third number is added to the number which is the power to which the second number must be raised in order to get a fourth number, then the sum is the number which is the power to which the second number must be raised in order to equal the product of the third and the fourth number.

      I would find that very hard to decode if I didn't know what it said. What is says is one of the basic rules of working with logarithms, that the log of c plus the log of d equals the log of cd. We learn this when we learn about exponents. In symbols it is easy to work with. We don't have to wrap our brain around that clumsy paragraph. But we do have to know the rules of working with logarithms. A lot of using mathematics, to repeat once again, consists of translating a problem to symbols, using known rules to manipulate those symbols, and then translating the result into the answer to the problem. We take a complicated problem translate it into mathematical concepts, follow the rules we have learned to manipulate those concepts, and the whole process is relatively easy. We do not apply logic, so much as we apply rules. Logic may be slow and cumbersome, but if we are fluent in the rules we may avoid much of the logic. We know the logic will work because we have laboriously worked out the rules and we know they work.

      A fifth grader may read a written problem that requires adding fractions. He adds the fractions in a mechanical way, because he has put in enough time and effort to learn it. Having learned it thoroughly he can do it quickly and efficiently. A tenth grader may read an algebra written problem, set up variables and an equation, and solve the equation in a mechanical way. She can do this because she has put in the time and effort to learn the algebra. That's the easy way to do math, and the effective way.

      A lot of mathematical ideas are developed by the reverse of this process, by setting up symbols, finding the logical consequences of our definitions of these symbols, deriving rules of manipulating these symbols, and then exploring the logical consequences of all these ideas. For example in learning about exponents we start with the idea that an exponent is simply a convenient abbreviation. The exponent tells you how many times you use the base as a factor. But from this beginning of convenience we derive a body of knowledge, a structure of knowledge, about exponents. Similarly we can explain counting to a young child as a matter of notation. "5" is just the number of fingers on one hand, and "2" is . . . . . . this many. (You'll have to visualize.) From this very concrete and linquistic beginning we can explore the logical consequences of what we observe and the result is arithmetic.

      A young child may have no trouble "wrapping his mind" around the idea that two fingers and three fingers make five fingers. But when we ask that same child, a few years later, to analyze and solve a written problem that involves the adding of fractions, we want him to "wrap his mind" around only the outline of the problem, the big picture. When that big picture of the problem is reduced to numbers and operation symbols the actual computation should be done by following procedures that have been learned to a high level of automaticity.

      For example the problem might be this: "The distance from the cabin to the lake is 4 3/8 miles. How much farther does Katrina have to go when she has hiked 1 7/8 miles?" The child working with this problem must use logic of some sort to translate the problem to 4 3/8 - 1 7/8. But once that translation is complete the logic should be complete. Then the child should simply "turn the crank" to get the answer. Hopefully he or she will again apply some logic at the end to determine if the answer obtained seems reasonable and appropriate to the problem.

      This is not to say that I am advocating doing problems by memorizing recipes. What I am saying is that knowing mathematics means having a mass of ideas, laws, rules, and recipes on a level of immediate random access. That is the nature of knowing mathematics. This immediate random access is not a result of applying logic. It is a matter of simply drawing from memory. If you need the product of six and eight, you should just pluck it out of memory. You should not figure it out. If you need to solve a simple equation you should be able to just do it, not spend time first wondering about strategies that might or might not work for solving equations.

Memory And Undersanding

      We do not solve problems by memorizing recipies, I have said, but one might argue that the previous paragraph says that memorizing recipes is a part of solving problems. I would agree. But it is very important that memorizing recipes is not the primary strategy in solving problems. The primary strategy in solving problems in math is to relate the structure of the problem to mathematical ideas that we know about. Then we can use what we know about the mathematical ideas in order to know about the problem. Solving written problems in algebra usually means setting up an equation that represents the facts of the problem. Once this is done we can use what we know about solving equations in a mechanical way. We are using logic to solve the equation, but the logic of equations solving has been previously worked out and put into memory. Now we simply use that knowledge. We manipulate symbols in the ways that we have already determined to work. So in a very real sense we do use memorized recipes.

      But just a few paragraphs back I said that what I learned in elementary school about working with fractions is held in place in my brain by understanding, not by memorization. Now I am saying that math must be held in place in our brains by memory, not by understanding. Is this contradictory? No, it is not contradictory. The answer to this dilemma is simple. Both are needed. Both are absolutely needed. One may certainly be tempted to ask, "But which is more important?", but I think that is not a productive question. They are both important. They are both absolutely essential to learning and using math. Understanding is absolutely needed to remember and use math. We have to build chains of logic. That's what math is. But memorization is also absolutely needed to remember and use math. You can't expect to reinvent every bit of math you need every time you need it. Both understanding and memorization are necessary conditions for knowing and using math. Neither is a sufficient condition for knowing and using math. This is a very important point that I will emphasize throughout the rest of this article.

      Of course not everything we use in math must be memorized. Some elements in a chain of logic must be on a "reference level of access", which means you look them up as needed. If you need the specific gravity of a 20% saline solution to do a problem, you look it up in a chemistry reference book. But knowing that you need a common denominator to add fractions can not be on a reference level. If you have to look that up, then you can't add fractions, not with the proficiency needed to build more math. Many things in math must be known on a immediate level.

      And many things in math must be on a "ciphering level", meaning that we must figure them out when we need them. The product of six and eight must be on a "look-say level". There shouldn't be any figuring at all to get the answer of forty-eight. The product of sixty-three times twenty-seven can be on a ciphering level. We can figure it out if and when we need it. So mathematical knowledge, as is true of any knowledge, can be partly on a look-say level, partly on a ciphering level, and partly on a reference level. Just what part of math should be on which level of access is a matter of opinion and trade offs. But it is totally wrong to think that no knowledge should be on an immediate look-say level. There is a great deal of knowledge that must be on an immediate look-say level. That is what it means to know mathematics, as opposed to being mathematically inclined or mathematicall talented.

      But to say that both understanding and memory are necessary to know math is not to say that they work in the same way. Memory and understanding work in quite different ways. Perhaps most importantly, in the routine use of math, logic is only a back up system. It is a very important back up system, but still it is only a back up system. Logic is used only to check, to confirm, even to correct the knowledge of mathematics itself. Sometimes the logic is so easy that it is brought to mind when a fact is brought to mind. When I remember that two and three are five I may visualize two objects and three objects and see in my mind that indeed the make five together. But when I remember that six times nine is fifty-four, it is not so easy to visualize. I simply bring it from memory. But I could use logic to check it if I needed to. Similarly when I multiply x2 times x3 to get x5, the logic may be so easy that it is indistinguishable from remembering the simple fact that you add exponents to multiply numbers. However when I solve an exponential equation by taking the log of each side and then proceeding, I do not have the logic behind all these steps at the top of my mind. But once again I can reconstruct that logic, if need be. When I explain what I am doing to another person I can call forth any part of that logic as needed. Knowing mathematics means remembering the ideas, concepts, rules, procedures, and so on, with logic as a back up system when needed. Using mathematics means using knowledge and logic toward a goal.

      As a parallel, consider this. How will I find my way home after work? Will I use logic, or a memorized recipe? I will use a memorized recipe in the sense that I know exactly how to get home. I do it everyday. But I have logic, and a mass of related knowledge, as a back up system. If road work disrupts my usual route I will use that logic and related knowledge and still get home. If nothing disrupts my usual route I will use the usual memorized recipe. It would be foolish to do otherwise.

      The term, "memorized recipe" can have connotations that can be misleading. It can seem to imply that nothing else is known other than the steps of the memorized recipe, and therefore it is inflexible. That obviously can happen. Sometimes a mememorized recipe does exist in isolation. My route home could be isolated memorization if I have just moved to a new city. I may remember to turn at a specific corner, turn again on Pine Street, turn again on Hiway K, and then I'll see my house. At this stage of learning one wrong turn could make me totally lost. As another example, one who knows very little about computers can be taught recipes in isolation to do a lot of tasks, and remain very vulnerable to total breakdown when something goes wrong with a recipe. Indeed that is probably the case in many work situations. Employees are taught what keys to punch on a computer to do a specific part of their job, and know nothing more about it.

      But in everyday life memorized recipes usually do not remain isolated. In most situations they become embeded in thick matrices of related knowledge. If I move to a new city I will find my way home by an isolated memorized recipe for a little while, but not forever. I will gain a lot of related information of the geography of the city. In a new job I might memorize isolated recipes on the computer for a while, but eventually I will gain knowledge that gives me more perspective Chains of information become nets of information. A chain is susceptible to total failue when one link is broken. A net is not.

      All knowledge that we use, or that we learn to any great extent, becomes embedded in a thick matrix of related knowledge, and this certainly includes academic subjects. A structure of knowledge can be subdivided into "framework structure", "supporting structure", "auxillary structure", and "overhead". Framework structure would be the essential elements of a subject, the facts and the connections that cannot be absent if the information is to exist at all. Supporting structure would consist of facts and connections that are important, but not essential. The auxillary structure would be less important, but still desirable to know or have in mind. "Overhead" is just a catch-all term for information, or requirements, that we would be glad to get rid of if we could. It would normally be framework structure that is used in logical chains, but supporting structure and auxillary structure can be valuable as a glue to keep the framework structure in place. And supporting structure and auxillary structure make the knowledge into a net more than a chain. The more supporting structure and auxillary structure we have in a structure of knowledge, the more firmly the framework structure is held in place, and the more amenable to checking and repair it is. Of course it takes learning and memory to fix this supporting and auxillary structure in mind, so trade offs must be made.

      At this point I have talked about the nature of math - it's a logical system - and the nature of knowing math - you've got to have a lot of facts, ideas, meanings, and connections in mind - and the nature of using math - apply logic to knowledge to come to a logical conclusion. So next we must consider learning math. What is the nature of learning math?

      I would say that the nature of learning math is that we build logical structures, usually under careful guidance of a teacher, and then commit to memory the elements of those structures that have to reside in memory in order for those structures to be useful. In learning to add fractions we use logic to understand the need for a common denominator, and then commit to memory the simple fact that we have to have a common denominator. The "logic" in this process is usually not very formal. We don't have formal axioms when teaching fifth graders to add fractions. But if the logic is informal, it is still absolutely essential. Everything we do, in adding fractions, or anything else, must make sense in terms of what has come before.

      So learning math has a component of logic, and it has a component of memory. But, once again, they are not separate. The logic is not always easy to follow, but remembering elements of a logical chain can aid in understanding the logic. And ideas that fit together logically are relatively easy to commit to memory. So memory aids logic and logic aids memory. For both logic and memory, practice is needed.

      Knowing math and using math are not always distinct. Thoughtful reflection on how to do a new problem is a matter of using mathematics, not a matter of knowing mathematics. But as a result of engaging in thoughtful reflection on a new problem one adds to one's mathematical knowledge. When one problem is figured out it enters into the mathematics that we know, and makes the next problem, if it is related, much easier and quicker. We can only use the mathematics we know, but using math increases the math that we kow.

      To illustrate a bit more about how these ideas fit together in learning math, consider this example. A child, perhaps fifth grade, is learning to add fractions. The child is doing homework at home, and is getting help from his father. The problem is 1 5/8 + 1/2. The child is a pretty good student. He listened at school when the teacher explained this type of problem. He and most of his classmates did several on the board, and started on the homework during the last ten minutes of the math class, before moving on to history or science or whatever. But now at home he does the problem like this: 1 5/8 + 1/2 = 1 5/8 + 1/8 = 1 6/8 = 1 3/4. He thinks about it, feels it's not quite right, and asks his dad about it. His dad quickly sees the problem and points it out. The child has to think a few minutes, nods in satisfaction, and then does the problem right. The nature of mathematics itself did not change in this process. But the nature of knowing mathematics did change, or at least the mathematical knowledge in the student's mind changed. It changed from general idea of changing the half to an eighth, to the more complete idea of changing the half into four eighths and then adding. The learning involved both logic and memory. He remembered that a common denominator is important, but didn't quite have it right. He got it right by using logic, with the guidance of his father. As a result of his effort he has not totally "mastered" the addition of fractions, but he has made progress.

      The nature of learning mathematics, then, is that we use logic to assemble ideas into meaningful structures of knowledge, but then use memory to hold them there. In another context I have used the terms "structure building" and "brain packing". Structure building simply means using logic, some sort of logic, to assemble ideas into meaningful structures. Ideas that make sense have at least a chance of being remembered. But making sense by itself will not guarantee that they will be remembered. Brain packing simply means doing whatever is necessary - practice, drill, review, or whatever it takes - to commit the ideas to memory.

      The nature of math is logic, but the nature of knowing math is information The nature of using math is is to bring logic and information together in ways that fit the problem at hand. The nature of learning math is to use math to develop more math, and then somehow commit to memory the information that is necessary to use that newly gained math.

      So we might go one step further and ask what is the nature of teaching math. The nature of teaching math is to guide the process. That means to explain, to provide problems, to guide the process of applying logic and previous knowledge to the problem, to guide the process of identifying information that must be remembered to continue the process. Of course there is a lot more to be said about this, but that seems to be the essential nature of teaching math.

      I want to finish with an idea that I think is worth some consideration. Starting with a problem is like having a need for a product before going to a store. That sounds reasonable. Why should we go to a store unless we need something? However that is not all there is to it. There is an advantage of knowing something about the store, of getting a general inventory of what is available, in advance of needing any specific thing from it.

      In buying things one may be prescriptive, or one may be opportunistic. When one is prescriptive one specifies what is needed, and then tries to get it. When one is opportunistic one first finds out what is available, and then decides which of many available products would be worth getting and which would not. These two perspectives are not always distinct. The borderline between them may be vague at times. And both perspectives have their place, and have their own advantages and disadvantages. But I think these two perspectives can be very useful

      Advertising is a form of public education. This may not seem sensible if we think in terms of advertising as an attempt to induce us to spend our money foolishly. That is an aspect of advertising all right, but not the only thing to it. Through advertising we get a general inventory of what a store, a business, or an industry has to offer. Consider buying a car as an example. Many people spend considerable time surveying the literature before ever setting foot on a car lot. This may start out as simply noticing the new car ads in the newspaper. But by the time they seriously think about buying a new car they have done a lot more than that. Extensive information is available, and people use it.

      Rather than figuring out specifications of what one needs in a car, it usually makes more sense to see what is available and then to see what would best fit one’s needs - to be opportunistic rather than prescriptive, in other words. Similarly when Walmart or J. C. Penney’s puts out an advertisement in the newspaper they are expecting people to be opportunistic, to realize when they see an item that they need that item, or at least that they would benefit more by buying the item than not buying it.

      There are times, of course when the other perspective, the prescriptive perspective, is much more appropriate. If I need a part for my car, spark plugs for example, then I need to be able to prescribe exactly what I need before going to the store. Not just any spark plug will work in my car. Before I go to the auto parts store I need to look in my owners manual, or somewhere, and find out exactly what spark plugs I need. If I don’t find exactly what I need, then I won’t buy. The prescriptive perspective dominates here. There may still be room for a little opportunism. One brand of spark plug may be on sale, while another is not. I can be opportunistic by buying the brand on sale, but only if it fits the prescription I have in mind. The wrong spark plug, on sale or not, does me no good.

      Starting with a structure of knowledge (the math first approach) is like taking a preliminary inventory of the store in advance of needing to go to the store later. Starting with a particular problem (the PBL, or problem-first approach) is like paying no attention to the store until there is a need to get something particular from that store. Both perspectives have some value. But in different situations one or the other scenario may make more sense. But I would argue that for learning math the primary goal is to get an inventory of what is available. We need to build up a structure of knowledge of mathematical concepts and skills. Without that inventory of what is available one is reduced to learning by memorizing recipes. It is the teacher's job to insure that that inventory of what is available is in the student's mind. In other words, it is the teacher's job to make sure that the student builds up a structure of knowledge in the students' minds. It is not enough that the students learn how to solve specific problems.

      As teachers of math we are like guides to a new store. We are confident, or should be confident, that if we can give people a general inventory of the store, they will come. Not to give this inventory is like not advertising. A store that doesn’t advertise may have exactly what I need, but if I don’t know that then I won’t go there. A store needs to advertise, to let people know what’s there. An educational system, though we don’t think of it in terms of advertising, needs to do the same thing. By building a structure of knowledge we are letting people know what’s in the store. Then when they need something they can come and buy.

      I have a personal example to illustrate this. I think it was in the early eighties when personal computers began to become affordable. Several times, as I recall, I went into a computer store, or at least a store that sold computers, and talked with a salesman, trying to get some idea of what was available, at what price, and what a computer could do. It was that last question that seemed most relevant. What can a computer do? I didn’t know. Actually at that time I had taken a course in Fortran. In fact my first course in Fortran was in about 1969. Then in the mid seventies I had taken another course in computer science. This was at a time when you learned programming at a terminal, far away from the actual computer, and went in the next day to pick up the results of your program. But my computing experience didn't give me much of an idea of what a personal computer could do in my home.

      I don’t remember computer commercials at this time, but I do remember something of how computers were portrayed in the popular media. You could keep recipes on your computer. Throw out the old file box of recipes on cards! Just bring up the recipe you want on the screen of your computer. Not only was it instantly available that way, but you could tell the computer to halve the recipe, or double it. The computer would do the calculations for you. And of course you could keep your address book on the computer, and lots of other things like that. This didn’t make too much sense to me. I reasoned that before your computer could bring up the recipe you want, you had to input the recipe into the computer. This meant a lot of typing. And if you want your address book on your computer you had a lot of input, a lot of typing, to do. So for several years I couldn’t see much use for a computer.

      But still I had a nagging suspicion that I could probably find uses for a computer. The usual tactic of a computer salesman was to turn the question around. When I would ask, What can a computer do?” any salesman I talked to would say, “Well, what do you want a computer to do for you?” That's like a problem based approach. I didn’t know. I didn't have a problem in mind. I really didn’t want to put recipes on a computer, or our address book, so I didn’t buy a computer for a few years. But I did analyze the sales pitch. I decided very few potential customers could really answer that question. What I thought a salesman should do is set up a computer with a ten minute tutorial, then sit the potential customer down at the computer to get at least a little idea of what a computer could do. In other words the salesman should let the customer be opportunistic, not prescriptive.

      But what does this have to do with modeling? I said that subject based instruction, the math-first perspective, corresponds to getting a general inventory of a store in advance of needing anything in particular. Then the customer can be opportunistic, choosing what appears to be beneficial. Problem based learning corresponds to first defining a specific need, and then thinking about what store to go to. Here the customer must be prescriptive. My early computer shopping experiences were like trying to find out what’s in the store, and then being met by a gatekeeper at the door of the store who won’t let me browse, won’t let me be opportunistic, insisting that I be prescriptive, that I first tell him what I want and he will go get it for me. This wasn’t of much help to me.

      There are stores where you tell the clerk what you want and he goes and gets it for you. Automotive parts stores operate much this way. There is good reason for this, of course. But it’s also true, I think, that over the decades auto parts stores have put more and more merchandise on open shelves, letting the customer browse all they want. The customer must still be prescriptive of course. He he needs a fuel pump for his care he needs to know the exact information needed to get the right part.

      I think at the turn of the twentieth century the “general store” worked pretty much this way, and it was the most common way for products to get to the final user. You told the clerk what you wanted and he got it off the shelf for you. I believe I was very young when the idea of the “supermarket” was still new and exciting. You browsed the aisles and just loaded up what you needed in a shopping cart, and wasn’t it wonderful? Now you could be opportunistic, and both buyers and sellers discovered that this is a much better way to do things.

      People can certainly differ on the aims of education. But I would argue that in most anyone’s view one of the aims of education would be to develop a general inventory of knowledge, skills, and perspective in advance of any specific need. The child does not go to school on a given day with a shopping list of knowledge to acquire, facts to be picked off the shelf like cans of soup or boxes of cereal. We do not ask the child, when he arrives at school each morning, what knowledge he needs today in order to take home and apply to his tasks of today. Rather he goes to school with a willingness to accept what is given. And what is given, in the long run, is a survey, an inventory. I have discussed this idea in other contexts with the "opening of doors" idea. Taking a course opens the doors to that subject for the students, and that can be very important. Opening doors in math requires building up mathematical structures of knowledge. They are not necessarily complete structures of knowledge. Indeed it would be hard to say that any structure of knowledge could ever be totally complete. But we build up some of the basic structural framework of a subject, and then we can add on to as needed to apply that knowledge to specific problems and situations. A math based approach provides for an efficient process of systemically building a reasonably comprehensive structure of knowledge that is of long term benefit to the learner. A problem based approach is a poor substitute for this.