Keywords

1 Introduction

The mathematical challenges that schoolchildren encounter most frequently are probably problems from textbooks, if only because it is with them that schoolchildren in most cases have to deal. Some authors of textbooks even include “challenge” subheadings to show the high level at which they are conducting instruction (meanwhile, the really challenging problems might turn out to be not the ones thus highlighted). Although textbook authors do usually try to make their textbooks a little easier, the word “challenge,” and even more so the words “challenging education,” have become in a way commendatory. This, of course, has not always been the case. Behind the change in attitude toward the words is the desire (even if often it is no more than rhetorical) to make the student an active participant in the process of instruction and to teach through problem solving. Let us repeat: this has not always been the case. Problems for students to solve on their own, even when they have appeared in textbooks, have played the most varied roles and consequently have been organized in different ways. By studying the history of problem sets in textbooks, we can better understand both how the process of mathematics teaching has developed, including teaching through problems, and how and to what degree students have been offered difficult and challenging assignments.

This article in some sense represents a continuation of the article Karp (2015), which is devoted to problem sets in old Russian mathematics textbooks. Here, as there, the discussion will mainly focus on problem sets from school textbooks. The process of problem solving in schools has been studied many times (we might mention, for example, the now classic work by Schoenfeld, 1985, in which not a few pages are devoted to what takes place in schools). On the other hand, the study of textbooks is gradually growing into a separate area of research, with its own conferences and published collections of articles (see, for example, Schubring et al., 2018). Strange as it may seem, problem sets from textbooks have not been studied sufficiently, in our opinion (for example, Donoghue, 2003 or Michailovicz & Howard, 2003 write about them, but their main focus is drawn to other aspects; one of the few exceptions is the paper by Ponte, 2014, devoted to Portuguese textbooks). Meanwhile, the principles on which they are constructed – whether recognized by the authors or applied unconsciously – constitute an important characteristic of the process of mathematics instruction as it actually exists in practice.

As a rule, in the process of instruction, schoolchildren usually encounter not separate problems, but problem sets (although, to be sure, sometimes the principle on which such sets are constructed consists precisely in the absence of any strategy). Mathematics educators who are concerned with what is being done in schools must inevitably think about how problem sets should be constructed and used, and how future teachers are taught to do this. As a step in this direction, we believe it is important to investigate how problem sets are actually constructed in practice, how their construction has changed in history, and what theoretical possibilities for their construction exist.

One can talk as long as one likes about the importance of reasoning and proving, creative or critical thinking, the ability to generalize and apply, and other fine things, but if a student never has to do this in practice, then all of these beautiful pronouncements remain empty and, if anything, merely confuse the working teacher, which is in fact what we see happening in some measure with the use of the expression “problem solving” – everyone knows that this is something good, but what it is exactly is understood in utterly different fashions.

It should be said at once that the author of this article seeks as much as possible to eschew an evaluative approach: the article’s purpose is not to say, “Would you look at what kind of garbage is being published!” but to show what processes are taking place – the article is largely historical in nature (which does not mean, however, that the author sees anything objectionable in methodological criticism aimed specifically at reviewing today’s production and analyzing what works, in the reviewer’s opinion, and what doesn’t. Methodological criticism must exist just as theatrical or literary criticism exist, helping to understand and to assess what is being put before us, which are necessarily more subjective than, say, scholarly studies in philology or theater – although the borderline here is not always rigid). This remark necessitates a methodological discussion, which will be undertaken below.

We should say at once that ideally the analysis of problem sets must be carried out with due consideration for context – general social-historical, cultural, and educational – if only because a textbook may be used in very many different ways. The present article must be considered to a certain degree preliminary because the analysis of textbooks in it is accompanied by the analysis of other sources (for example, those that shed light on how exactly textbooks were used in schools) only to a very limited degree. At the same time, a set of problems already by itself creates a certain context – and from this stems the key difference between analyzing an individual problem and analyzing a problem set.

For example, a problem’s level of difficulty, understood not in terms of the number of operations that must be performed to solve it or some similar characteristic, but as a psychological or statistical characteristic (roughly speaking, how many students can solve the problem under given conditions), is practically impossible to determine based on a single isolated problem that is substantive to any degree – we simply do not know the conditions: it is one thing if the problem is simply given as is, but quite another if it was prefaced by the teacher’s explanations or a series of preparatory problems (therefore, for example, assessments of the level of difficulty of problems on exams can hardly be made without additional sources of information). Discussing problem sets from textbooks is in this respect safer: the reader, as well as the researcher, sees, if not everything that the student using this textbook sees, then still a great part of it.

Let us repeat one more time that, naturally, there are differences between the textbook and what takes place in class. A set of problems in a textbook usually does not prescribe the order in which they should be solved rigidly (although there are techniques to emphasize it, for example, by regarding problems not in isolation from one another, but as items within the same problem). The sequence of the problems to be solved is usually prescribed by the teacher and different ways of sequencing can lead to different results. An effectively sequenced set of problems can make it easier to solve them by first solving a simple version of the problem and then moving on to a more difficult version. The effect achieved by the teacher who has effectively sequenced the problems to be solved can, however, be far more powerful and complex than that (Karp, 2004; see also the introduction to this section of the book). The role of emotions in mathematics education is now often discussed (Schukajlow et al., 2017), but often it somehow turns out that students’ positive emotions can be triggered only by nonmathematical details – this, of course, is by no means the case. The sequencing of problems can be based on different principles and produce different effects – including emotional ones.

Textbook problems can be used in different ways; therefore, let us repeat, their analysis here is by definition incomplete. It is important, however, to ask whether the textbook offers problems in a sequence that is meaningful, whether such a sequence can be constructed on the basis of what is given in the textbook, and whether constructing such a sequence is easy.

This article begins with a brief historical note, proceeds to a discussion of methodology, goes on to analyze several problem sets from textbooks, and in conclusion relates the observations that have been made to one another and to certain theoretical and practical problems and theses.

2 On Problem Solving in Schools: A Historical Observation

Contrary to conventional wisdom, problem solving in schools is a relatively recent phenomenon. The frequently cited stories about various tournaments between solvers (the most popular of which seem to be about Fibonacci, who defeated all rivals in front of Emperor Frederick II) have nothing to do with this matter – we are talking specifically about schools. Students in schools usually solved a few problems, and in saying this, we use the word “problems” to mean not only assignments that required a nonstandard approach, but all assignments. This was noted explicitly in a report by the American subcommission of ICMI:

In the textbooks on algebra 50 years ago much more stress was placed on logical exposition than on the solution of problems. The development of arithmetic, as followed in the textbooks of the elementary school, was faithfully imitated in algebra, and various “operations” (some, like the square and cube root of polynomials, having no conceivable use, and other mere pedantic elaborations of methods that in simpler form were well worthwhile) were laboriously discussed and exploited before the use of equations in discussing problems was entered upon. (Committee III and IV, 1911, p. 23)

The United States was no exception. For example, many surviving materials demonstrate how mathematics was taught to future Russian tsars and grand dukes (Karp, 2020) – at the beginning of the nineteenth century, they did solve problems, although not very many, but this was clearly seen as a direct continuation or even part of theoretical knowledge. Certain assertions in geometry were called theorems; others, which required students to do something – usually, to construct something – were called problems. In algebra, that which was solved usually served as an example of a learned rule or algorithm. A story has survived about Grand Duke Konstantin (the son of Nicholas I) taking an exam in mathematics: “The questions, as always, were posed by the Sovereign, as tests of ingenuity. In algebra, there were two: one, a first-degree equation; the other, a quadratic equation. In geometry, the construction of a regular 15-gon on a given line.” (Golovnin, n.d., p. 22). Konstantin (1827–1892) was undoubtedly a remarkable individual; nonetheless, there can be no doubt that he was not required to invent an algorithm for constructing a regular 15-gon on the exam, nor does solving linear and quadratic equations offer very many opportunities for displays of ingenuity.

“Good” teaching (that is, teaching in a school of a high level) consisted precisely in the “pedantic elaboration” of various propositions (which were, to be sure, not always very precise by the standards of today’s higher mathematics).

3 Analysis of Problem Sets: Certain Methodological Considerations

We mentioned above that sometimes the principle on which a problem set is built is precisely the absence of any principle. In reality, this is, of course, a simplification – the most superficial arranger of a problem set must make some choices in any event, in the first place by determining how many problems should be included. In fact, this simple and almost indisputable characteristic of a problem set already carries considerable information – in certain textbooks, there are almost no problems – they were unnecessary, in the writer’s opinion (naturally, such cases must be distinguished from situations in which a textbook is accompanied by a separate problem book). The goal of instruction was seen to consist of something else, and a simply determinable number helps us to understand this.

There are certain other characteristics that it is desirable to determine in analyzing problem sets; for example, thematic distribution within a problem set: how many of the problems given involve the application of a formula directly, and how many of them involve applying it in one or another version – independently of what the arranger of the problem set planned to achieve (or of whether anything was planned at all), this is an important characteristic of the activity expected of the students.

When talking about modern textbooks, it is natural to consider a breakdown of their use of different forms, for example, how many multiple choice tasks, short answer tasks, and essay questions they contain (in old textbooks, such diversity was practically unknown). Moreover, it is natural to consider the type of activity suggested by a problem. Of course, to make judgements about this on the basis of a keyword in a problem – solve, draw, check, prove, compare, etc. – is somewhat of a simplification; nonetheless, analyzing problems from this (and even more so, from a more detail-oriented) perspective tells us about the conceptions of the learning process espoused by the problem set’s arranger (and let us repeat one more time: whether these conceptions were conscious or not).

But such data alone are not enough. We are confronted with having to analyze a text, and just as in the case of a literary text – in particular, a poetic text – not everything here is expressed in numbers. As we have already noted, in the history of mathematics education, as in other historical disciplines, often (and even usually) use is made of what is known as the historical-philological method, which is based on the attentive reading of texts and their comparison and juxtaposition with one another. With one additional feature, however, the analyzed texts can be mathematical ones (Karp, 2014). In different cases, the historical-philological method may be applied in different ways. (Note that proposals to use various elements of the methodology used in the philological disciplines have been made for a long time – see Schubring, 1987; Karp, 2004 – just as methodological parallels between pedagogical studies and art have also been drawn – see Lawrence-Lightfoot, 1997).

We will confine ourselves to citing a researcher whose name is recognized across different fields: Vygotsky (1971) analyzed classic Russian works of literature, in the vein of the studies of the so-called Formalist School, which was contemporary to him. Comparing the content and meaning of a work that are seemingly straightforwardly and openly communicated by its author, on the one hand, and its structure, the order in which events in the plot are presented, and the literary techniques it uses (first and foremost, verbal peculiarities), on the other, he showed that in reality, the author tells us far more than he or she seemingly promised (or sometimes something completely different). It is precisely the structure, organization, and language of a work of art that make it truly deep and substantive.

Of course, we would hardly seek for hidden depths on every page of every problem book – or claim that something will be inevitably revealed to those who solve problem No. 23 immediately after solving problem No. 22 – but the sequencing of problems and the structure of a problem set can carry a meaning that cannot be reduced to the meaning and content of each of the individual problems in it.

Here, the question arises: how did Vygotsky (and many other researchers before and after him) achieve what Schoenfeld (2007) calls trustworthiness, and what is it exactly that makes a work scientific? In effect, the practice of these researchers has been briefly to paraphrase the text, in a specially organized fashion, with quotations and examples, all intended to underscore those distinctive features of the text to which the researcher wished to direct the reader’s attention.

Just as any other method, this method can be criticized for a possible lack of objectivity – it is easy to select some text, and pick and choose quotes from it, to completely distort its meaning if one wishes to do so. Such distortions, however, can be disproven with other quotes, and the main thing that must be noted is that the described “philological” method, contrary to conceptions formed under the influence of generally justified respect for quantitative methods, may be every bit as fruitful as any “mathematical” (let’s call it that) method, in which some mathematical model is somehow constructed.Footnote 1

In what follows, we will describe the content of each analyzed problem set and offer examples.

Let us make a few more observations – in analyzing, one must distinguish between identical and nonidentical problems. The decision here depends on the reader’s viewpoint. Unquestionably, for a mathematician the equations.

$$ {x}^2-4x+3=0\ \textrm{and}\ {x}^2+3=4x. $$

are identical.

That is not the case for the schoolchild who is just beginning to study quadratic equations. We strive to pick up on such differences.

In conclusion, it must be noted that the selection of textbooks which will be discussed, and the sections in them that will be analyzed, is quite arbitrary – those textbooks were selected which were widely used in the United States, while the section chosen – in cases where there were different sections to choose from – was always the same, quadratic equations, unquestionably an important section. However, there were quite many widely used textbooks, and there were quite many important sections of the course as well. It should be pointed out that the author of this paper assumes a certain homogeneity both individually within each textbook and in general among the totality of textbooks circulating at any one time. Differences exist, of course, but radical differences of any kind (some of the textbooks were printed in color, the rest in black-and-white, or some of the textbooks contain many problems, the rest very few, and so on) seem not very likely, especially if these differences were not mentioned in the press or other literature. At the same time, it needs to be said that the author does not aspire to any all-encompassing generalizations: the aim of this paper is not to formulate general assertions concerning all textbooks of a certain time, but merely to note what happened at different times; whether it happened always, and why it happened, are important questions, which the author hopes to answer in other studies.

4 Robinson’s Geometry Textbook

Horatio Nelson Robinson (1806–1867) was a prominent writer whose textbooks were widely used for a long time. We will focus on a book with the long title – as was customary at the time – Elements of Geometry and Plane and Spherical Trigonometry with Numerous Practical Problems (Robinson, 1867). In his preface, the author again boasts that his book contains a “full collection of carefully selected Practical Problems” “given to exercise the powers and test the proficiency of the pupil.” The word “problem” is found several times in the table of contents of the geometrical sections – it appears in the title “Book IV: Problems in the construction of figures in plane geometry,” as well as in Book V and Book VII. Book IV provides solutions to a number of problems (the vast majority of them construction problems), beginning with constructing a perpendicular bisector and the bisector of an angle. The style of exposition here is practically identical to that of the other, “theoretical” sections. In the two other cases, judging by their subheadings, readers are offered practical problems specifically, just as promised – in Book V, in plane geometry; and in Book VII, in solid geometry. We will focus on the problems in plane geometry.

Thirty-nine problems are offered in all. Let us note at once that the word “practical” merely means that students will practice, or as the author writes, that these problems will “exercise students’ power”: one can detect in them no particular connection with real-world questions – they are ordinary geometrical problems about triangles, circles, trapezoids, and so on. From today’s point of view, for all of plane geometry, the number of problems is very small, but as has already been said, the gist of a subject was not seen to consist in solving problems. From today’s point of view, other criticisms can be voiced as well – for example, in problem 16 students are asked to find the area of a triangle with a given base and two adjacent angles of 800 and 700. In the answer, however, the author gives the sides of the constructed triangle, further noting that they cannot be determined exactly without using trigonometry, and trigonometry has not been covered yet, for which reason it is necessary to proceed by approximating – “we must be content with the approximate solutions obtained by the constructions and measurements” (p. 146). What we are interested in, however, is not this, but how the given problem set is organized.

The author himself says nothing about this. And he does so, even though certain problems are accompanied by solutions (so that it is possible to “exercise students’ power” only if the students do not have the textbook) and even methodological comments of sorts, which confirms that the book often functioned not merely as a textbook, but at least to some extent also as a teacher’s manual. It is not even entirely clear how the author envisioned the use of the problems that were offered – as a unified text, or one by one, as the corresponding topics were covered. The former seems far more likely – as attested to by the remarks with which the author opens the section, observing that what has been covered has been covered, and certain other topics still lie ahead, but before proceeding further, it is necessary to do some problem solving. However, it is still impossible altogether to rule out the possibility that students were given certain problems one by one as they progressed through the book.

It can be seen that certain problems directly support the course presented in the book – for example, in problem 14 students are shown a triangle with three given sides and asked to find the lengths of the segments into which the bisector of one of the angles will divide the opposite side. So that the reader (teacher?) should have no doubt as to how this problem should be solved, it is immediately accompanied by the indication, “see Theorem 24, Book II” (p, 145) – and this theorem answers precisely the question posed, in a general way. Far from all, problems are connected with the covered material in such a direct way, however, and conversely, by no means every theorem comes accompanied with a numerical example.

We can see several mini-sets in which problems are interconnected, for example, by focusing on the same geometrical situation – thus, problems 4, 5, and 6 are devoted to two parallel lines, the distances between them, and related questions. However, problem 15, in which three parallel lines are examined, and for which the ideas examined in problems 4–6 are useful, stands alone. This is not the only case. In problem 2, students are given a right triangle with a 300 angle and a shortest side of length 12 and asked to find the length of the hypotenuse. In problem 22, they are given a right triangle with a leg whose length is 320 and an adjacent angle of 60°, and asked to find the lengths of the remaining two sides (the Pythagorean Theorem was used in many problems at the beginning of the set). Practically identical problems have been placed at different ends.

From a didactic point of view, the problem set, of course, is not diverse. There are no problems aimed at reinforcement – the idea that a student who couldn’t do something the first time might be able to do it the second or third time around is not made use of in any way, and there is no evidence that it occurred to the author. The author distinguishes between difficult and easy problems (for the difficult ones, he provides solutions), but he does not prepare students for solving the difficult ones in any way – there is no movement from the simple to the complicated. The set begins with a problem that indicates that the base of an isosceles triangle has a length of 6, while the angle opposite the base is equal to 600; students are then asked to find the sides’ lengths. The problem is not difficult, although its solution does involve several steps. But, for example, problem 33, in which students are given an isosceles triangle with sides whose lengths are 20, 20 and 12, and asked to find the length of the altitude drawn to the base, is followed by problem 34, in which students are asked to construct a right triangle, given the length of its hypotenuse and the difference between the lengths of its legs, that is, something noticeably more difficult and not prepared in any way.

We have noted similar things in Russian textbooks also. Thus, Davidov’s geometry textbook (1864) contained far more problems, but, for example, problems accompanying the section on the areas of polygons began with a problem in which students were asked to find the locus of the vertices of all triangles that are equal in area and have a common base – a problem that one could hardly expect to be solved by a student who did not possess that which Schoenfeld (1985) has referred to as “resources.” How were these problems solved, then? We possess a sufficiently large amount of information about Russian gymnasium students of that time to be able to assert that at least one way was to solve problems at home with a tutor, and then to commit the solution to memory (Karp, 2018). From a certain point on, collections of solutions to problems from Davidov’s textbooks began to be sold. The social background and consequently the habits and demeanor of students who were taught using Robinson’s text and students at Russian gymnasia differed; identifying historical evidence indicating how problems were solved in American schools at that time appears to us an interesting question for research.

In any case, we can assert that an analysis of Robinson’s text confirms that little attention was paid to problem solving at that time, and that problem sets were characterized by their didactic and methodological poverty; at the same time, the book contains problems that are grouped on the basis of some connection between them – for example, problems that pertain to similar geometrical situations. To what extent this was done deliberately, however, is not clear.

5 Algebra by Robinson

Both in terms of the history of its formation as a school subject, and in terms of its nature, algebra is, of course, different from geometry. Therefore, it is useful to examine an algebra textbook from that time as well – we will choose a textbook by the same Robinson for this purpose, A Traditional and Practical Treatise on Algebra (1848). It contains more problems to solve independently than the geometry textbook examined above; the textbook is divided into “Sections,” and each of them contains problems. We will focus briefly on Chapter 1 of Section IV Quadratic equations, which is devoted to quadratic equations in one variable.

Here, we will also confine ourselves to two sets of “Examples for Practice” or simply “Examples.” In addition to them, there are several other sets: a set devoted not to solving quadratic equations, but to the concept of the perfect square; a set devoted to equations of higher powers, which are solved by using quadratic equations; a set in which the author advocates solving certain problems with numerical coefficients by replacing them with letter coefficients, which allegedly makes calculations easier; and finally, a set that represents a kind of conclusion. Although all of this, along with the author’s mathematical inaccuracies, is of interest, we will not dwell on it here (although it should be noted that some of these sets are evidently preparatory to some extent, while others conversely offer something like the application of what will be discussed later on).

The first set we will examine contains 16 problems. All of them are more or less of the same type: students are asked to solve a quadratic equation. The first 10 are devoted to equations with a leading coefficient of 1, and the last three of these have irrational roots. The remaining six problems have different coefficients, which the author deliberately emphasizes, and by analyzing problem No. 11, demonstrates how to reduce such equations to an equation with a leading coefficient of 1 (it is noteworthy that this is done not by means of simple division, but by means of a substitution, which the author deliberately thanks a certain professor for pointing out – the author avoids fractions). Problem No. 16 is again analyzed since in it the second coefficient is odd – students are advised to multiply the equation by two.

The second set contains 17 problems, the first of which is the following equation:

$$ {\left(x+12\right)}^{1/2}+{\left(x+12\right)}^{1/4}=6. $$

In the preceding theoretical paragraph, the author developed the idea that if one exponent is twice as large as another while their bases are equal, then it is possible to transform the equation into a quadratic equation. This problem is solved following this model, as are the subsequent 11 problems. Probably the most difficult of them is problem No. 12, which requires students to solve the equation

$$ {x}^2-2x+6{\left({x}^2-2x+5\right)}^2=11, $$

in which the technique is slightly disguised. It is noteworthy that No. 13 is far easier – here, students must solve the equation

$$ \frac{x^2}{361}-\frac{12x}{19}=-32, $$

The author, however, apparently does not notice that this equation can be solved without any special stratagems, without introducing a new variable y = x/19 (or else does notice it, but fears computational difficulties). Moreover, he makes the following observation: “If much difficulty is found in resolving this 13th example, the pupil can observe the 9th example” (p. 167). This ninth equation is as follows:

$$ {x}^{6/5}+{x}^{3/5}=756, $$

and how exactly it is supposed to help (apart from increasing students’ experience with working with comparatively large numbers) is not entirely clear. In the remaining cases, such observations are not made, but this observation serves as proof of the fact that the author recognized the importance of sequencing in solving problems. Nos. 14–16 do not resemble the problems that precede them at all; for example, in No. 14 students are offered the following equation:

$$ 81{x}^2+17+\frac{1}{x^2}=99. $$

The author gives a hint here, pointing out that the first and third terms of this expression are squares and referring students to the section on perfect squares. Indeed,

$$ 81{x}^2+18+\frac{1}{x^2}={\left(9x+\frac{1}{x}\right)}^2, $$

and using this fact, it is not difficult to solve the equation. But it is difficult to understand why these problems are offered in this set rather than the preceding one. As for the last problem, No. 17, this is an equation,

$$ \frac{4{x}^2}{49}+\frac{8x}{21}=6\frac{2}{3}, $$

that may be converted into the form:

$$ {\left(\frac{2x}{7}+\frac{2}{3}\right)}^2=\frac{64}{9}, $$

but which can also be solved without any special tricks.

Summing up, we would say that in the algebra textbook the connection between problems is felt by the author more strongly than in the geometry textbook. But this connection usually amounts to the author offering several problems in a row that focus on the same rule, even if he makes some attempt to group together problems that resemble each other in other ways. Also, the number of problems given is clearly too small, if one considers their use from today’s perspective.

6 Textbook by William Hart

Let us examine a more recent textbook by Hart (1934). This is a textbook in algebra that is radically different from the one examined above in terms of the number of problems in it – there are very many of them. As the author himself explains in his introduction, “The practice examples conform to the principle that learners profit more from doing successfully many easy examples than from relatively futile efforts to solve complicated examples. Each topic is accompanied by an unusually large number of easy examples” (p. V). We will confine ourselves to one section, XIV Quadratics. This section includes theoretical subsections 263 through 281 (which contain many examples), numerous exercises, on which we will focus, as well as sections entitled “Chapter Mastery Test” and “Written Review,” which also offer problem sets. It is important to note at once that both in the theoretical subsections and in the problems, the author uses the signs X and Y to distinguish certain sections, with the recommendation that “the study of such material be required only of the abler pupils” (p. V).

The material is organized as follows. The author begins with the theoretical subsections, which present quadratic equations and incomplete quadratic equations, followed by subsection 265, “Solving an incomplete quadratic equation,” which provides examples of solving such equations, and as a conclusion, he formulates a rule about what needs to be done to solve such an equation. This is followed by Exercise 201, a set of problems devoted precisely to these equations. It contains 48 problems. Only the first 18 may be considered repetitions of what was required in the previously examined examples (and even this is a stretch – problem 2 already asks students to solve the equation

$$ {y}^2-4=21, $$

which is not actually an incomplete quadratic equation, but one that can be brought into the standard form). In problems 19 through 32, students must convert equations to incomplete quadratic equations in more and more intricate and technically complicated ways (this should not be taken to mean that each successive problem is invariably more difficult than the preceding one in this respect, but the tendency is obvious – problem 19 asks students to solve the equation

$$ 5{a}^2-125=3{a}^2-27, $$

while problem 32 asks them to solve the equation

$$ \frac{y+3}{y-3}+\frac{y-3}{y+3}=\frac{17}{4}, $$

which is obviously more technically complicated, if only in terms of the number of operations that must be carried out).

This is followed by problems 33–41, in which students are asked to solve equations with letter coefficients, for example, No. 41:

$$ \frac{a{x}^2}{b}-\frac{c}{b}=1. $$

Lastly, problems 42–48 are equations with letter coefficients that represent relationships with which the students are familiar, for example, No. 43:

$$ A=\pi {r}^2. $$

The next subsection is devoted to the Pythagorean Theorem. It is followed by Exercise 202, which contains 10 problems – the first five are purely algebraic. For example, No. 5:

  • Solve the formula [a2 + b2 = c2] for c in terms of a and b.

The geometric problems that follow are more or less of the same type, for example, problem No. 10, the last and most difficult of them because magnitudes are given in letter form: “If the equal sides of an isosceles triangle are each m in long, and the base is 2n in long, find the length of the altitude.”

Then follows a subsection about solving “a complete quadratic graphically,” which introduces the parabola (constructed based on points), and then in Exercise 203 offers 12 problems that in effect repeat the example analyzed in the theoretical subsection, but in which different parabolas must be constructed.

The next subsection is entitled “Solution by completing the square” (in the case of an equation with a leading coefficient equal to 1). Here, we find preparatory exercises that help to elucidate the idea, after which a numerical example is employed to show how to solve an equation by using the explained method. Exercise 204 is a set of 30 problems, the first 19 of which are equations with integer roots, with No. 11 being practically fully solved in the text of the textbook. Problems No. 20–30 contain equations with irrational roots which must be found approximately. Problem No. 20 is again fully solved.

Then the text examines the case of a quadratic equation with a leading coefficient not equal to 1, and the set Exercise 205 follows, which repeats the structure of the set Exercise 204, but now for such equations. The theoretical subsection here is marked with an X – indicating heightened difficulty – while the set of exercises is not marked with such a sign. Finally, the formula for the roots of a quadratic equation is derived, about which the students are told: “take three minutes now and memorize this formula” (p. 345); and then follow two sets, Exercise 206 and Exercise 207 (both they and the corresponding theoretical paragraphs are marked with an X), in which students are required to solve equations using the formula and by factoring, if possible. It is easy to distinguish groups of problems: those in standard form and easily solvable by factoring; those with expressions on both sides of the equation; those with fractions; those with irrational roots, which must be solved approximately; those with quite complicated expressions that must be transformed in order to convert them into standard form. For example, these exercises are concluded by No. 34:

$$ \frac{3r-1}{7-r}-\frac{5-4r}{2r+1}=3. $$

Exercise 208 is devoted to word problems. The first three problems repeat almost verbatim the example analyzed in the theoretical part. Then come certain changes – in the theoretical part, a problem was analyzed in which students were asked to find two consecutive integers whose product was equal to 20. No. 1 is the same problem, but the product is equal to 72; No. 3 is about the sum of the squares of two consecutive integers; No. 8 gives the sum and product. Problems Nos. 10–14 have a geometrical content – for example, students are given the perimeter and area of a rectangle and asked to find its dimensions. Then come various problems both about numbers and geometrical questions, which are slightly more difficult than the ones previously analyzed. Exercise 209 is once again devoted to word problems, but now students are advised in certain cases to “draw a figure,” and most importantly the whole set is marked with an X. And indeed, the problems here are somewhat more difficult – they also come in groups. A typical problem from the first group is No. 5: “A picture is 10 inches long and 5 inches wide. The area of the picture and its frame is 84 square inches. How wide is the frame?” Then come problems about motion – for example, No. 14: “An airplane flew 90 miles and returned in a total time of \( 2\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right. \) h. The rate of the wind was 5 miles per hour. At what rate in calm air was the airplane flying?”

This section continues, and various other exercises are given, but we will stop our description here.

The difference from the previously analyzed textbooks consists not only in the quantity of problems but also in the clear understanding of the role of grouping and sequencing problems. The classic phrase that a problem can be reduced to the previous problem speaks specifically of the previous problem, not of one given 20 problems ago. The author proceeds in small and carefully thought out steps. Furthermore, the problems in Hart’s textbooks are in general easier than many of those in Robinson’s textbook – or more precisely, would have been easier if they had to be solved in isolation, but the problems in Robinson’s textbook are not solved in isolation – the idea is presented in the theoretical section, and then in effect must be memorized by rote by being applied several times in a row. Hart’s textbook contains relatively complicated problems, which students come to solve on their own by solving problems that precede them, and not simply by applying what was done in the theoretical section to a problem with different numerical values. But usually, the increase in the level of difficulty is technical – first, we apply the formula directly; next, we perform some algebraic operation, arrive at the standard form, and then apply the formula; and after that, we perform five algebraic operations, some of which are themselves not simple, and only at the very end apply the formula.

One can ask what spurred the authors of textbooks to organize problem sets better – and one can try to understand to what extent they were acting consciously, and to what extent they were, for example, copying other textbooks. It appears natural to think that a large role was played by the increase in the number of those taught and those teaching. It was precisely during the years when Hart’s textbooks were being used, and those preceding them, that rapid growth in these numbers occurred – the textbook that won in competition with other textbooks was the one that taught students quadratic equations in a way that was more simple and effective, and hence it came about that more attention began to be paid to didactic principles.

The Progressive Era brought not only an increase in the number of high schools, but also a change in the understanding of the purposes and goals of mathematics education, including an increase in attention paid to the practical applications of mathematics, developing social efficacy, and so forth (Kilpatrick, 2009). In those parts of Hart’s textbook which were analyzed above, the demand to make the course more practically oriented is difficult to discern – the word problems examined above were clearly formulated with other aims in mind. This does not mean, of course, that demands for a practical orientation exerted no influence on the teaching of mathematics in schools – they exerted such an influence if only because many students were left without a course in algebra – but even with all the will in the world (which was, of course, by no means necessarily shared by all authors of textbooks at that time), putting the exhortation to increase the practical orientation of the course into practice was far more difficult than merely formulating it.

On the other hand, what is clearly on display in the textbook is an individualization of the approach, which was also in line with the demand for social efficacy – teaching every intricacy to those who were “incapable” seemed unnecessary, and among other things, it was recommended not to insist that everyone learn to solve quadratic equations using the formula or to solve word problems of any degree of difficulty, which were undoubtedly identified and separated out by the experienced teacher.

The section from Hart’s textbook analyzed above unquestionably contains more graphical and geometrical content than Robinson’s algebra, but also not a great deal. In general, despite the quantity of problems, one does not see very much diversity among them: as we have seen, the absolute majority of them are “solve the equation” problems. Problems that require students to prove something, to verify something, to compare something, to invent something, and so on, are completely absent.

7 Textbook by Larson et al.

Let us look at problem sets about quadratic equations from a more modern textbook. The textbook by Larson et al. (2001) contains a chapter with the title “Quadratic Equations and Functions,” which will be discussed below. Its first section is titled “Solving quadratic equations by finding square roots.” This section provides the definitions of a root and of a quadratic equation and demonstrates how to solve the simplest quadratic equations. The authors provide an example that offers an algorithm for solving the following equation:

$$ 3{x}^2-48=0. $$

Numerous other examples are followed by a section with the title “Guided Practice,” which practically repeats the definitions given previously and lists the basic problems that students must know how to solve (like the equation formulated above). Then finally comes a section entitled “Practice and application,” which we will discuss in greater detail.

This section contains groups of problems unified under headings. The first four of these groups are devoted to square roots and expressions with them (each group contains eight or nine problems that are similar to one another). The next group is titled simply “Quadratic Equations”: there are 15 of them. There are differences among them: the equations involve different variables, they sometimes do and sometimes do not have a solution, sometimes they are given in standard form, sometimes certain operations are required in order to convert them to standard form. The group begins with the equation

$$ {x}^2=36, $$

and ends with the equation

$$ 7{x}^2-63=0. $$

The next group contains nine problems and requires the use of a calculator for finding solutions to the nearest hundredths. The final equation is as follows:

$$ 5{a}^2+10=20. $$

The next group has the title “Critical Thinking” and contains three problems: write an equation of the form x2 = d so that it has one solution, two solutions, and no solutions.

Finally, the next seven headings (from one to six problems below each of them) are devoted to applying what has been learned – the authors give a quadratic formula that describes some real-world process and pose questions about this process (one of them has the subheading: “Challenge”; another: “Critical Thinking”). For example, already in the theoretical sections, the authors presented the so-called failing object model, according to which the height h of an object falling from height s at time t is equal to:

$$ h=-16{t}^2+s. $$

Subsequently, in both the examples analyzed and in the problems for independent work, the following question, for example, is discussed: when is the height h = 0, given that a value for s is specified?

The sections that follow are devoted to radicals, the graph of a quadratic function, and solving quadratic equations by graphing. We will skip over them and proceed to section 9.5, “Solving quadratic equations by the quadratic formula.” This section gives the formula and offers examples of how it may be used to solve equations. The section “Practice and Applications” is organized in the same way as the section examined above. It begins with a subsection in which, in nine given equations, students must find the discriminant (this word, however, is introduced only in the next section – here, students are simply given the formula). There are certain technical differences among the problems – in some, the coefficients are fractions; in others, they are negative. The next set gives 12 quadratic equations in standard form – they must be solved using the formula. The differences among the problems again come from what kinds of coefficients they have. In the next set, which has nine problems, students must first convert equations into standard form and then solve them using the formula (all of them are polynomials, so all that is required is to move all terms over to the same side). The next group is devoted to finding “x-intercepts of the graph of the equation,” that is, again solving a quadratic equation (it is not stipulated, however, that this must be done using the formula). In the problems of the next group, students are given a choice between solving simply “by square roots” or by “using the quadratic formula” (unfortunately, the question of whether it is possible to use the quadratic formula in those cases where solutions can be found by square roots is not raised). Next, students are given several more problems oriented around applications, that is, problems in which students are asked to solve quadratic equations using given quadratic functions that model various real-world processes. The last problem (which appears under the heading “Challenge”) is of interest. It contains two parts – in the first, entitled “Visual thinking,” students are asked to use a graph to find the equation for the axis of symmetry of a quadratic function and notice that it is “halfway between the two x-intercepts.” In the second part, entitled “Writing,” students are asked to make sure that their answer to the preceding part is correct, using the formula for the roots of a quadratic equation.

8 Discussion

An obvious distinction of the newer textbook consists in the fact that students are offered problems that are far more simple technically than those in Hart’s textbook, let alone Robinson’s textbook. The authors see no need for technical intricacy. Hart clearly believed that it was pointless to make students memorize several types of difficult problems, which it would have been impossible to teach them to solve in a meaningful and independent way in any case (let us recall his words about the “relatively futile efforts to solve complicated examples”). But his textbook retained many-step problems, which even students who were not considered very capable could, in Hart’s opinion, be led up to solving. In the newer textbook, everything is simpler.

At the same time, connections between problems are undoubtedly acknowledged – the textbook includes material for reinforcing what has been learned, and as we have noted, there are certain differentiations within each group. On the other hand – without going into a discussion of how much technically difficult algebraic problems are needed in our computer age – we should note that what is disappearing (or at least noticeably decreasing) along with such problems is the incremental or many-step aspect of learning and reasoning in general. Broadly speaking, Robinson’s textbooks (this is particularly evident in his geometry) taught students problems each time as a kind of isolated phenomenon. With Hart, we see a certain movement, even if it is limited to increasing technical difficulty. Hart clearly gave some thought to these matters – in his introduction, he writes about the “spiral organization” of his book, in which problems go back to ideas that had already come up, but now on a new level. The content is thus richer than simply the sum of the separate problems – there is, additionally, movement from problem to problem (or at least the possibility of achieving such movement in class or in teaching practice). In the newer textbook, the technical simplification of the problems as a whole has led to a simplification of the whole structure of the set as well.

Here, however, there are certain exceptions. We noted above that the textbook includes a group of problems, following the solution of a quadratic equation, that require students to find the x-intercept of a quadratic trinomial. Before us is the development of the examined problem, yet not in the direction of increasing technical difficulty, but in the direction of a kind of translation or transfer to a different object. The same thing, but in a different way – about different concepts. We have not come across such problems in other textbooks. In general, the role of graphs has noticeably increased, and therefore a space has been opened up for interactions between the purely formula-based and the graphical.

Let us also note the appearance in the newer textbook of several assignments in which students are asked not only to solve something, but also to provide examples, draw a conclusion based on a figure, and even engage in reasoning. This is another way in which this textbook differs from the older ones – although, to be sure, very few such problems are given, and in a number of cases they are given under the heading “Challenge,” which hints that these problems must be assigned only in exceptional cases – and they are indeed difficult to prepare for by using the textbook.

Another obvious distinction of the textbook by Larson et al. (2001), which its authors themselves point out, is that it contains a large number of problems with content taken from the real world. These problems are, of course, altogether different from the classic ones about motion, areas, or finding numbers, which we mentioned when discussing the textbook by Hart. At the same time, they appear rather to advertise the importance of mathematics in the real world than to demonstrate this mathematics itself (in other places in the textbook, too, the authors never tire of repeating that mathematics is very necessary for people in various professions, with which, of course, one can only agree). The problems state that one or another mathematical model is being made use of, but the various models are discussed only later in a special section – in real life, the logic of reasoning is reversed. Although there are relatively many problems and they are devoted to a variety of objects, they are relatively unvarying in character.

Finally, let us touch on another aspect: Robinson did not worry about the individual approach in his textbooks – very few people used these textbooks in school, and the author clearly believed that if some of them failed to learn what they were taught, this was not anything to worry about. For Hart, differences in perception are important, and he methodically offers material for differentiation, emphasizing that he identifies the minimum necessary for everyone, and provides additional material for the gifted. For the more modern textbook, such an approach is unacceptable – if only due to the danger that the proclaimed minimum might become the maximum of what is possible for certain underserved groups of population. As a result, differentiation is provided for only by rare problems under the heading “Challenge,” plus recommendations to consult a website or some other source for additional problems.

9 Conclusion

As we stated at the very beginning, this article is preliminary in character: the examination of a greater number of American textbooks from different periods will help better to understand what was taking place in the country. The processes taking place in the country were connected with what was happening in the world, and therefore an analysis of textbooks from other countries from the same point of view would also be useful. In general, analyzing problem sets in textbooks is no less useful and informative than analyzing what might be called their presentation of theory. As has already been remarked, it is desirable to supplement the direct analysis of textbooks from this point of view with the analysis and collection of evidence indicating how problems from textbooks were solved in practice – in school, at home, with a private teacher, and so on. An understanding of the methodological changes taking place in the teaching of mathematics as part of broader social changes is precisely what we regard as the objective to be achieved, and it can be achieved only by combining the analysis of mathematical and methodological-mathematical texts with the analysis of all sorts of historical documents pertaining to everyday life.

The most recent of the textbooks examined by us above was published 20 years ago, that is, it, too, belongs to history. In the time that has passed since then, many textbooks have been published, and the textbook of these authors itself has gone through many changes, including changes touching on aspects that were discussed above. We should repeat that the present article is consciously historical. It would be interesting to juxtapose problem sets in textbooks published during the last decade. But historical analysis in itself compels us to think about the present period.

In the article Karp (2015), we reached the conclusion that approximately during the years 1880–1900, something like a methodological revolution took place in Russia – the system for and practice of working with problem sets in textbooks underwent a significant change. Among its causes, one might point to the growing scale of education, to foreign influences, and to the development of pedagogical thinking in general – further studies are unquestionably needed here. The present article confirms that such a methodological revolution occurred in the United States as well, even though no attempt is made here to assign to it a precise date.

A.R. Maizelis, an outstanding St. Petersburg teacher of mathematics, once jokingly told the author of this article that children somehow recognize the problems that were not in the famous problem book by Nikolay Rybkin (1861–1919) and reject them. A very high methodological art for working with problems was indeed attained at different periods in different countries, which made it possible to move very gradually and smoothly from problem to problem, increasing the technical as well as, sometimes, the conceptual level of difficulty and the substantive heft of each problem, while remaining accessible and comprehensible to children – developing them and coming up to the very limits of what is possible for them at each stage, but nonetheless staying within these limits.

It may be argued that we are living in a period of a new methodological revolution, or at least, a period when such a revolution is needed. The sensible and dynamic character of old textbooks is disappearing before our eyes, if only because many problems in today’s computerized world are losing their value, just as many technical skills are losing theirs. However, intellectual skills, including the ability to move from one problem to another, are not going anywhere. This is what gives rise to the need to develop these skills under new conditions in schools and textbooks for the general population.

It may be said, based among other things on what has been said above about the types of problems found in the relatively new textbook, that today it is relatively widely recognized that school mathematics does not amount to simply “solve,” “compute,” and even “prove,” but much else besides. This recognition must find expression in the problem sets in textbooks.

Simplicity in a textbook, which is worth striving for, does not preclude complexity in the organization of problem sets that are gripping to work on, in which each new success brings with it new feelings, giving meaning to new lines of reasoning. The experience of working with such sets will also help to develop teachers who will themselves take delight in mathematics, who will not treat it as medicine – useful, but unpleasant – and who will be able to transmit their feelings to students in class.