The Discussion of Well Posed Problems

Problem Solving andProblem Solving

Models for K 12:Preliminary

ConsiderationsAbstract: Before we can solve problems in mathematics we must havemathematical problems to solve. Problems in mathematics must be well-posed. A well-posed problem is, above all a question where every term isprecisely defined. However, one should be aware that what has to beprecisely defined in the problem and what can be assumed to be understoodis very much dependent on context. A problem that is well-posed on ateacher's quiz wouldn't necessarily be well-posed on a standardized test.People coming from different cultures or backgrounds will often understandterms in different ways. Similarly, one cannot assume that students in thirdgrade have a precise understanding of the terms that a student in eighth gradewould. But in all cases precise understanding depends on knowingdefinitions. In a classroom where there are no definitions there can be nowell-posed questions, and hence no mathematics. While many educatorshave embraced the notion of problem solving and often quote the four stepsof problem solving outlined by G. Polya, his model assumed that problemswere necessarily well-posed before a student would see them, which is oftennot the case. Thus, it is necessary to extend Polya's model for problemsolving to include these issues, and the failure to do this previously hasresulted in gross misconceptions about what mathematics is and how it canbe successfully taught.

In the early 1940's, G. Polya wrote his classic book How to Solve It, that tried toexplain the quantifiable steps in problem solving in mathematics. The discussionthere could be summarized in his four step model:

(1) Understand the problem(2) Devise a plan(3) Carry out the plan(4) Look back.1

1 There is a wealth of insight in Polya's books, and for mathematicians who are involved in teachingthese courses, at least his just reprinted classic, How to Solve it, should be a basic reference. For

For his projected audience, college students typically math majors at StanfordUniversity and the ETH in Zurich - such a model with the accompanying discussionin his book was apt to be adequate. [This book was a preliminary study for his muchmore serious two volume work Mathematics and Plausible Reasoning, that, in turnwas meant to support his two volume work with his long-time collaborator, G. Szego,Aufgaben und Lehrsze aus der Analysis. In turn this last work was, for many yearsone of the staples for advanced graduate students in mathematics.]

To understand better the context in which this work was developed I quote from theintroduction to Mathematics and Plausible Reasoning2:

A serious student of mathematics, intending to make it his life'swork, must learn demonstrative reasoning; it is his profession and thedistinctive mark of his science. Yet for real success he must alsolearn plausible reasoning; this is the kind of reasoning on which hiscreative work will depend. The general or amateur student shouldalso get a taste of demonstrative reasoning: he may have littleopportunity to use it directly, but he should acquire a standard withhe can compare alleged evidence of all sorts aimed at him in modernlife. But in all his endeavors he will need plausible reasoning. Atany rate, an ambitious student of mathematics, whatever his furtherinterests may be, should try to learn both kinds of reasoning,demonstrative and plausible.

I do not believe that there is a foolproof method to learn guessing. Atany rate, if there is such a method, I do not know it, and quitecertainly I do not pretend to offer it on the following pages. Theefficient use of plausible reasoning is a practical skill and it islearned, as any other practical skill, by imitation and practice. I shalltry to do my best for the reader who is anxious to learn plausiblereasoning, but what I can offer are only examples for imitation andopportunity for practice.

Today, the situation has changed in that what was addressed to a very advanced

example, on page 9 there is the marvelous paragraph:We know, of course, that it is hard to have a good idea if we have littleknowledge of the subject, and impossible to have it if we have no knowledge.Good ideas are based on past experience and formerly acquired knowledge.Mere remembering is not enough for a good idea, but we cannot have any goodidea without recollecting some pertinent facts; materials alone are not enough forconstructing a house but we cannot construct a house witout collecting thenecessary materials. The materials necessary for solving a mathematicalproblem are certain relevant items of our formerly acquired mathematicalknowledge, as formerly solved problems, or formerly proved theorems. Thus, itis often appropriate to start the work with a question: Do you know a relatedproblem?

2 It is the distinction of these two different kinds of reasoning that is the core of Polya's work here.He describes them thus: We secure our mathematical knowledge by demonstrative reasoning, butwe support our conjectures by plausible reasoning. A mathematical proof is demonstrativereasoning, but the inductive evidence of the physicist, the circumstantial evidence of the lawyer,the documentary evidence of the historian, and the statistical evidence of the economist belong toplausible reasoning.

audience 45 years back is now needed much more broadly. Problem solving hasbecome a critical skill that needs to be taught in K 12 to the extent that it can be, andneeds to be accessible to all students. For the audience of K 8 teachers andultimately students, Polya's model is not nearly robust enough. At a minimum, whatare missing are two preliminary steps that tend to be taken for granted by Polya'saudience. The first is the query, Is the question a problem in mathematics? and thesecond is If the question is not a problem in mathematics, can it be sensibly madeso?

For the first query, we and the students we teach should understand what a problem inmathematics is. There are different kinds of problems that we see. Perhaps the mostcommon types are the set problems that demand little more than plugging explicitnumbers into given equations or procedures. These are typically meant to illustrateand provide practice with newly presented material, or to provide review of suchmaterial. It often seems that people tend to equate such problems with mathematics,hence the terms drill and kill, or drill and skill, that naturally associate to suchquestions. What is not as well understood as it should be is that such problems aremore properly regarded as necessary preparation for attacking actual mathematicalproblems. Actual problems in mathematics typically have the property that thesolution or the goal is not immediately achieved and there is no clear algorithm to usein order to arrive at the solution. But most questions of this kind do not come to us asactual questions in mathematics. We should understand that questions in mathematicsmust be well-posed, but all too often these questions are not. As Wu explains:

As teachers, we want to convey the clear message to our studentsthat in mathematics, no guesswork is needed for its mastery. Wewant to let them know that it is an open book that everybody canread. Among all branches of knowledge, mathematics ischaracterized by its WYSIWYG quality --- what you see is whatyou get --- and you have no need to assume anything that is notalready explicitly stated. This is another way of expressing the factthat every conclusion we draw in mathematics depends completelyon what is stated explictly up front. In order to convey thismessage clearly, it is absolutely essential that every problem wegive can be solved strictly by using the information within theproblem, no more and no less. No hidden agenda.

We will be more precise later. For now, Wu's comments aptly describe most of thekey steps that are necessary to recognize when a problem is well-posed.

In order to best understand what is meant by well-posed, we will start by showingexamples of problems that are not well-posed with a detailed explanation followingeach one.

1. Here is a classic example of a non-well-posed problem: Two friends are indifferent third grade classes that meet in different rooms. They want to know whichroom is bigger. How do they decide? In this case, the term bigger is not defined.It could mean anything.3

3 Compare the discussion on pages 6 and 7 of Working With Story Problems.

2. Here is an example of a question that appeared in the 1992 California MathematicsFramework as from the authors perspective- a serious question in mathematics:

The 20% of California families with the lowest annual earningspay an average of 14.1% in state and local taxes, and the middle20% pay only 8.8%. What does that difference mean? Do youthink it is fair? What additional questions do you have?

Let us not, at this stage, try to analyze the difficulties with this problem. Hopefully,the problems with it will be evident by the end of this essay, but one issue that we canindicate right away is the term fair. What does this mean in the context of thisproblem? Has it been specified? Can we solve the problem without understandingwhat it means?

3. Here is a problem from an early version of the MARS assessment.

The picture below shows a 5 by 5 section of an array of lockerswith only the 3 by 3 center group numbered.

11 12 1320 21 2229 30 31

Then the question is stated as Some of the numbers havefallen off the doors of some old lockers. Figure out themissing numbers and describe the number pattern.

Once more there are serious difficulties with this question that relate to the twoqueries above that are needed to make Polya's problem solving model sufficientlyrobust that it can be used in K - 12. But here the issue is slightly more subtle. Wemight presume that students would assume that the lockers are numberedlexigraphically from left to right and from top to bottom, and that the lockers form arectangular array. But nowhere in the statement has this been indicated. If thelockers do form a rectangular array and the numbering is sequential, there is anumambiguous answer, but the problem is not well-posed without these or otherassumptions being given and any number of answers will be possible. This is anexample of a problem with hidden assumptions. In an informal, in-class situationsuch a problem might be acceptable, given that the unstated assumptions would belikely to be understood from the context of previous discussions. However, on aproposed national assessment it is hard to imagine any way in which this questioncould be thought of as well-posed.

On the other hand, one can err too much in the opposite direction. There is the wellknown story of Feynmann complaining about the school math problem that asksstudents to color the picture of a ball red. The old form Color the ball red, isincorrect as stated if all the student has to work with is a picture of a ball, but it is notambiguous. The attempt to make the problem precise results in a worse question thanthe original.

4. Here is a recent article from the Boston Globe illustrating very well the totalconfusion that results from ill posed problems.

Dinner party math

A 10th-grade student took a pattern used in computers and found a completely differentway to look at it. After explaining what she saw to MCAS officials, they gave her, and manyother students, credit for answering the question correctly. QUESTION:

Computers are designed around off/onswitches that are used to representnumbers. In the following pattern,which represents the numbers 0 to 10,

represents a switch that is on and

represents a switch that is off.

= 0

= 1

= 2

= 3

= 4

= 5

= 6

= 7

= 8

= 9

= 10

Which of the following represents thenumber 11?

A:

B:

C:

D: 'CORRECT' ANSWER:

The pattern is based on binary code.The four switches represent 8, 4, 2,and 1 from left to right. If a switch ison, it is added to the other switchesthat are turned on. The sum is thenumber.

2 + 1 = 3

4 + 2 = 6

4 + 2 + 1 =7

C: 8 + 2 + 1 =11

ALTERNATE ANSWER:

Jennifer Mueller found a pattern of white dots having dinner together.

= 0 You start with zero.

= 1 The first dot has dinner alone.

= 2 The second dot has dinner alone.

= 3 The first two meet and eat together.

= 4 The third dot has to have dinner alone before hemeets the other two.

= 5 The third dot has dinner with the first.

= 6 Then he has dinner with the second.

= 7 Now the first, second, and third dots can all eattogether.

= 8 The fourth dot has to eat alone before he meets theothers.

= 9 Then he has dinner with the first.

= 10 Then the second.

= 11 Then the third.

Graphic: Globe Staff / Christopher Melchiondo

Actually, since no rule is specified in the problem, any answer would be correct.Prof. Ralph Raimi gives a very eloquent discussion of this below:

The fault is in the whole idea of expecting an answer to a problem ofthe form "continue this pattern". There are no incorrect answers tosuch a problem, for a pattern can be continued any way one likes withequal justice -- unless the RULE for the pattern is made explicit andunambiguous.

So, if one begins a pattern 1,2,3,4,5 and asks for the next entry myanswer might be 1, on the grounds that the pattern I have in mind is1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,.... (continued by the rule: Repeat 1through 5 in order, endlessly). However, if one begins a pattern2,4,6,8 AND says it is to be an arithmetic progression, the next entrymust be 10 of course. In the first case the proposer of the patternbeginning 1,2,3,4,5 neglected to say "arithmetic sequence" and so didnot get his expected 6 from me.

In the case of the present problem, the proposer did not give a rule.The rule he had in mind was of course the binary system of denotingwhole numbers. If he had said this was the pattern, (c) would havebeen the only correct answer. But of course he filled the air withcomputers and switches, to make the problem sound "real-life" ofcourse, and neglected to ask a mathematically well-defined questionHe therefore got what he deserved.

The current orthodoxy of educational psychology says, doesn'teveryone know that 11 follows 10 in the sequence that begins1,2,3,4,5,6,7,8,9,10? Well, the answer to that is -- no. This time theygot the binary expression for 12 instead of the binary expression for11, with a rather cute justification. I believe they should grade all theother choices correct as well, as a rationale could be found for any ofthose without too much thought, even without inventing dinnercompanions.

Are these isolated cases? Not at all. We now give a series of sample problems takenfrom just two current state assessment exams in this country, an eighth grade examwith a very solid reputation, and a fifth grade exam from a different state that also hasa very solid reputation. All of the questions in the first group below have missing orhidden and in one case inconsistent - assumptions that are critical to solving theproblems.

5. This is question #4 from the eighth grade exam.

The difficulty here is that no rule is specified for generating the next term. Thequestion is better than average in that the meaning of n is specified. In manyquestions of this type we are even required to speculate on whether n is the inputvalue or whether it is the position of the element in the sequence. However, since norule is given we are required to guess that the actual rule is contained in the list ofpossible answers. The question could be fixed by rephrasing it: The chart abovegives the first four input, output pairs for one of the rules below. If the input is n,which rule is it?

6. And here is the sixth problem from the same eighth grade exam.

This has exactly the same difficulties as the problem above but it is even poorer. Thistime the variable n, the ordinal number above, has been vaguely indicated as theinput, but absolutely no idea of the way in which the pattern is to be generated isgiven, and the problem is a short response question, which means that there is no helpto be had from a list of possible answer choices. Students who have been trained torespond to this kind of question correctly will get credit. Students who think morecritically will get no credit.

7. The 12th problem on this eighth grade test is an open response item:

Here c is incomprehensible in terms of the assumptions given. Unless the eighth-grade class has only four members and we are not told this, so we should notassume it then the assumptions are that the class is decomposed into four disjointsubsets, and these subsets are to be assigned numbers between 1 and 4 in a randomway. In (a) it is not to big a jump to assume that her act means the act to beperformed by the subset of the class to which she belongs, and the same for (b). But(c ) is impossible to reconcile with the assumptions. Are we to assume that the threestudents are all in different subsets? But why?

The next two problems come from the fifth grade exam for another state. This examwas rated much better than average.

8. This problem from the fifth grade exam represents another aspect of the difficultieswith examples 3, 4, 5, and 6. We do not know what continued means.

9. This is another problem from the fifth grade exam.

Actually, the comment is not sufficiently precise. What was meant was that any linesegment in the plane is the hypotenuse of infinitely many right triangles.

In the next two problems from the eighth grade test we see an all to common error ofassuming that one can actually read off the coordinates of unlabeled points on agraph.

10. What follows is problem #3 from the eighth grade exam.

The difficulty with this problem is that the graphs are unlabeled. What is true is thatonly one of the graphs is LIKELY to represent a correct response, (B), presuming thatthe graphs represent continuous functions, but since no points on the graphs areactually identified, since we cannot actually see individual points, and since we havenot been given any properties of these graphs, it is impossible to KNOW that (B)actually contains the specified points.

The graphs could be identified by asserting that they all represent straight lines and bygiving two or more points on each one, or the problem could be rephrased by sayingsomething like The graphs below are all graphs of straight lines. One of these graphscontains the points in the table. Which one?

11. Next, consider problem #18 from this eighth grade exam:

Here we have the same sort of difficulty that we had with problem #3 in the previousexample. The points A, B, C, and D on the graph are not given as explicitcoordinates. We are asked to infer that they are at integer coordinates, a hiddenassumption.

12. The final example also comes from the fifth grade exam above. But it representsan extremely common and insidious error that occurs in texts as well as in questions.Wu reports that all K 6 textbooks that he has seen in this country use the wordname as a synonym for define. This leads to confusions like the following,which represents an imprecision in vocabulary that renders the question meaningless.

It is relatively easy to fix this question What kind of triangle is this? - will suffice.

Ultimately, the difficulty with all of these questions is that the authors either did notunderstand that a question in mathematics must be well-posed, or they did notunderstand what is required in order that a problem be well-posed.

The primary requirement in a well-posed problem is that every term be preciselydefined. In the first 11 examples certain terms were simply not defined or thedefinitions were hidden. The 12th example is subtly different in that all the terms weredefined, but the key term name in its standard meaning gives us a well posedquestion that cannot be answered with the data given.

Two related issues

Consistency is essential when producing problems. It cannot be good pedagogy togive students sequences of problems, some of which are well-posed and some notunless detecting the ill-posed problems and discussing them is part of the objective.

Also, the issue of how a problem will be read by different audiences is very delicate.Prof. Nancy Gonzales, a recently retired University of New Mexico mathematicseducator, focused almost all her research on these kinds of issues. How can we writequestions that are ethnically unbiased? How can we write questions that can beclearly understood by the entire audience? As an extreme example, how will a blindperson be able to answer a question that is presented as a picture with noaccompanying words?

Problems where psychology affects the outcome.Related to the issues in the paragraph above, there are often extraneous factors inproblems that distort our perceptions of what the problems are asking.

A classic problem that appeared on the proposed Clinton National Eighth Grade examis

MARCY'S DOTS

A pattern of dots is shown below. At each step more dots areadded to the pattern. The number of dots added at each step ismore than is added in the previous step. The pattern continuesindefinitely.

Marcy has to determine the number of dots at the 20th step but shedoes not want to draw all 20 pictures and then count the dots.

Explain or show how she can do this and give the answer thatMarcy should get for the number of dots.

Oddly enough, this problem is well-posed, but the picture is completely misleading.Even professionals tend to reach the wrong conclusion influenced by the picture. Theactual data that is given simply says that the number of dots at the nth stage, C(n),satisfies C(n + 1) > C(n), and (C(n+1) C(n) > C(n) C(n-1)) with C(3) = 12, andC(3) C(2) = 6. From this one can only conclude that C(20) is at least 267, but thatany whole number greater than 266 is possible for C(20). Indeed, the smallestincrease possible between C(3) and C(4) is 7, so C(4) > 18, between C(4) and C(5) is8, so C(5) > 26, and in general an easy induction shows that

C(n) > (n+3)(n+4)/2 10.

It is quite important that prospective teachers see such problems, but they must becarefully discussed with them.

Here is another problem that illustrates the effect preconceptions have on problemsolving. This problem comes from Russia, and is generally regarded as appropriatefor seventh or eighth grade students there.

Two ladies started walking at sunrise each from her village tothe other's village. They met at noon. The first lady arrived inthe second's village at 4:00PM, while the second lady arrived atthe first lady's village at 9:00PM. They walked at constant rates.What time was sunrise?

One has a tendency to expect that there is missing data in this problem. But theproblem is well-posed, so, before actually stating that the problem is not solvable,teachers should be encouraged to attempt the solution. In fact, if d is the distancebetween the villages, r1 is the rate at which the first lady walks, r2 is the rate thesecond lady walks, s is the time of sunrise, and L - the time spent to arrive at themeeting point - is L = 12 s, then the data in the problem translates into threeequations:

d = L(r1 + r2)d = (L + 4)r1d = (L + 9)r2.

Subtracting the second from the first gives 4r1 = Lr2, subtracting the third from thefirst give 9r2 = Lr1. Hence L/4 = r1/r2, while L/9 = r2/r1. Multiplying gives L2/36 = 1,and, since L is non-negative, L = 6, so sunrise was at 6:00AM.

Wu pointed out another example that appears to run along the same lines theanswer is not what one would expect:

Fresh cucumbers contain 99% water by weight. 300 lbs. ofcucumbers are placed in storage, but by the time they arebrought to market, it is found that they contain only 98% ofwater by weight. How much do these cucumbers weigh at thetime they are brought to market?

What the first problem really illustrates is the non-intuitive nature of non-linearproblems, especially for students who tend to have only seen linear problemspreviously. The second problem could be thought of as linear but there is anassumption being made that the only part of the weight that changed is the weight ofthe water in the cucumbers. This is, strictly speaking, not quite true, but is closeenough to true for us to routinely assume it. However, we should be aware that wedo make such assumptions, since prospective teachers may not, and some of thestudents they will teach may not. Note that without such an assumption, the problemcannot be solved, so this is an example of a problem that is not well-posed but can bemade into a well-posed problem with reasonable assumptions.

This brings us to other aspects of problem construction that need to become part ofthe awareness of prospective teachers.

Other Issues:

Construction of problems and recognition of good and bad problems is a veryimportant maybe critical part of the skill set that a K 8 teacher has to have. Wehave discussed what we mean by well-posed above. It remains to mention that anaspect of this is also that problems should not be ethnically biased. Social groupstend to have certain kinds of conventions that are taken for granted, but theseconventions differ across groups. This makes it even more important that teachersbecome sensitive to exactly what is being implicitly assumed in the statements ofproblems. Of course, the best approach is to be sure that there are no implicitassumptions.

A second issue is that of real world problems.

We have to deal with the issue that problems that are brought to mathematicians fromoutside of mathematics are usually not well-posed. Since this is really the way inwhich mathematics interacts with real world issues, we have to convey to pre-serviceteachers, in-service teachers, and students some of the issues involved in convertingsuch problems into meaningful problems in mathematics.

It is unclear what is meant by the term real world problems in the usualterminology of K 12 discussions. The second example, the problem from the 1992California Mathematics Framework on fairness was meant to be a real-worldproblem. However, in absolute terms, we can understand real world to mean the kindsof problems that mathematicians are asked to help solve outside of mathematics.Typically, such problems are not well-posed. Here is one that that came up inrobotics. Automated vehicles on factory floors usually are guided by wiresembedded in the floor. How does one program these vehicles to avoid one anotherand get to where they are supposed to go as efficiently as possible? As stated, theproblem is far too vague. What do we mean by vehicle, guided, efficiently?One has to start making assumptions, checking them for reasonableness, andgradually create one or more well-posed problems that are sufficiently precise to bemathematical in nature, but are still sufficiently related to the original question thatthe answers will be useful.

Sometimes, as the fairness example illustrates, it will not be possible to convert thequestion into a problem in mathematics that has a sensible relationship with theoriginal. It is not possible to quantify fairness in a situation this general.

At present, applying mathematical thinking to non-mathematical problems is still anart-form. In theory it should be possible to state general principles that will often beuseful in such a setting, but it is difficult to believe that we have anything useful tosay at this time except that when we discuss problem solving, we should repeat,almost as a mantra, the sentence of the paragraph above

One has to start making assumptions, checking them forreasonableness, and gradually create one or more well-posed problemsthat are sufficiently precise to be mathematical in nature, but are stillsufficiently related to the original question that the answers will beuseful.

When working with pre-service and in-service teachers, it seems to be vital that thestep of creating well-posed problems from ill-posed questions must be carefullyseparated from the issues involved in recognizing when a problem is well-posed.

A Real World Problem

In our general discussion the following remarkable e-mail exchange occurred recentlybetween two mathematicians, A and B.

A's question:

Why do you multiply 26x26 to find how many 2-letter acronyms there are.Teachers should be able to explain such things very clearly.

B's first response:

Just want to make sure you didn't have any other answers in mind other than thefact that adding 26 to itself 26 times takes too long.

A's reply:

No, I have something much more fundamental in mind: why don't you add 26 +26, for example? When I give students this problem I ask students to use thedefinition of multiplication to explain why we multiply.

B's response:

Forgive me for being dense, but I am more confused than before. I am lost as towhat you had in mind about

to use the definition of multiplication to explain why we multiply.

And why is "not adding 26+26" more fundamental than "not adding 26+26+..+26(26 times)" ? Aren't they the same principle?

A's detailed response:

The point I want to make is that teachers should be able to explain whymultiplication is an appropriate operation to use to solve a problem (assuming weare considering a problem that can be solved by multiplying). How do you knowthat multiplication can be used in the case of the 2-letter acronyms? I thinkteachers should be able to justify this, and I hope that they will do this with theirstudents too. In other words, teachers shouldn't just say: "you solve this problemby multiplying" but they should be able to discuss, and hopefully help childrenunderstand why we can multiply to solve a problem.4

You need a definition, or some fundamental result about what multiplication is inorder to do this. The definition of multiplication that I use is: AxB (for non-negative A, B) means the total number of objects in A groups if there are B objectsin each group.

To use this definition, students must explain that there are 26 groups of 2-letteracronyms, with 26 acronyms in each group. This justifies multiplying. If youdefined multiplication in terms of repeated addition, then the same argument canbe used. If you defined multiplication in terms of areas of rectangles, then you'dprobably want to establish first an intermediate result relating it to groups ofthings.

This should be viewed as an issue in real world problem solving. The difficulty thatB had was that A's original question is not entirely well-posed in the context of the e-mail where it was posed.5 It is a question outside of mathematics proper, yet an aspectof it requires a precise, mathematical answer.

4 An observer made the following comment here that is worth repeating: In the exchange

between A and B, it is clear to me that B did not appreciate what A was saying when mentioning 26plus 26. I am not sure B really knows the depth of our problem and did not imagine that a studentwould give 52 as the answer. Many would, unfortunately. 5On the other hand, A pointed out that in the proper context, where multiplication has already beendefined, as has the notion of explain clearly, the problem will be well-posed. Thus, there is anelement of dependence on circumstances that is present in this discussion. What are we free toasssume when we ask a question? It clearly depends on the audience, but questions designed for alarge audience where we cannot assume common backgrounds must be very carefully constructed.

As a real world problem there are three terms that need to be precisely defined whenwe try to make it into appropriate well-posed questions in mathematics : explainclearly, 2-letter acronyms, and multiply. The second, 2-letter acronyms, will be takento mean ordered pairs of letters in the English alphabet. There are other alphabets, butthis is the meaning A had in mind. The third term, multiply, has been discussedadequately above. We choose one definition and work with it.

As stated, it is now clear that what is required is the reasoning that confirms theequality of two numbers: the number of ordered pairs of letters and 26 x 26. Since allthe terms involved are precise, this part of the problem is well-posed.

We still must deal with the term explain clearly. This causes real difficulties sincemuch of explain clearly is normally pedagogy, a topic that lies outside ofmathematics. It involves taking account of the audience and selecting the methodbest suited to helping them understand.

We have come to a key point. One should no more ask mathematicians to teachpedagogy than one should ask specialists in education to teach underlying concepts inmathematics. So within the context of what mathematicians should do in teachingmathematics to pre-service teachers, we should focus on purely mathematical aspectsof the issues.6 Thus, explain clearly might be translated into something like Use aminimum amount of terminology and definitions to give a precise mathematicalanswer to ..., or as A expanded, I basically think of explain clearly as prove, and Iwould say that most of explain clearly is getting a coherent, logical argument. AndA indicates that in the context where this question had originally been asked, it hadalready been explained to the students about the kinds of arguments expected, as itmust be whenever mathematicians work with teachers. If teachers are to show theirstudents that mathematics is more than a list of facts, they will have to develop theirskills in logical reasoning. Thus, if we are to expect teachers to give clear, logicalarguments about the mathematics they teach, they must have plenty of(mathematically correct) examples and plenty of opportunity to practice making sucharguments themselves.

It is likely that one of the things that A had in mind was that we should be takingpains to avoid presenting the mathematics in these courses as lists of facts to bememorized, but rather get to the underlying principles. A key aspect of this is dealingwith the issues of definitions, precise use of language, and the requirement that anyproblem that lies in mathematics must be well-posed. Indeed, these are among themost basic of the underlying principles in the subject.7

6 This regards content only. The teaching in these courses must be of the highest quality. Manyeducation schools tend to regard the mathematics departments as among the worst teachingdepartments in the university. To deal with this it is very important that the best teachers in thedepartment be involved in teaching and developing these courses. The pre-service teacherscoming into these courses will, typically, have very weak mathematics backgrounds and low self-confidence currently. This is often manifested by a very critical attitude in the class towards realor perceived pedagogical errors on the part of the instructor, and instructors that cannot handlethese issues appropriately would not be good choices to teach these courses.

7 A few years back Prof. Norman Gottlieb proposed a possible definition of mathematics as thestudy of precisely defined objects. It is unlikely that there can be a definition of mathematicswithin ordinary language, but this suggestion comes very close.

Suggested methodology.

In all four courses there should be repeated discussions of these issues, and problemsshould be regularly supplied that give examples of improperly phrased problems andask for discussions of what the difficulties are and how they could be fixed. Also,there should be examples of ill-posed real world type problems and discussions ofhow assumptions can be made that will result in the creation of well-posed problemswhose solutions will be useful in analyzing the original problems throughout thesequence. Such considerations should be part of assessment as well.