And now put aside your pens and calculators: on mental problem solving in the high school mathematics classroom.

The aim of this article is to draw attention to the use of mental mathematics--to promote its use, and to pose several questions connected with it. Unfortunately, the psychological problems associated with the reception of teaching materials, the teaching materials themselves, and the concrete instructional methodology used in their presentation are frequently studied independently of one another. Yet, what working teachers often need in order to solve problems that they encounter is precisely a comprehensive analysis of the details and specifics of one or another methodological approach. Researching on the use of mental mathematics is going to take us at the intersection of several important fields currently being studied by mathematics educators, including problem solving, classroom management, and the problems of mathematical communication (Karp, 2004). This article presents the results of certain observations and analyzes them.

Mental mathematics is often mentioned in connection with elementary and middle school (Beishuizen & Anghileri, 1998; Hope & Sherill, 1987; Reys, Reys & Hope, 1993; Sowder, 1990). Mental calculations are often welcomed, in this context, as a means of encouraging and stimulating students to develop their own computational strategies, as a counterweight to the memorized "paper and pencil algorithm." Stressing the importance of such self-developed procedures, Sowder, following Plunkett (1979), lists their main positive characteristics. She notes that they are variable, flexible, active, holistic, and constructive. Most important of all, probably, is the fact that they "require understanding all along" (Sowder, 1990, p.20). The student who multiplies 35 by 4 without using paper or a calculator indeed possesses a much better number sense than the student who mindlessly computes the product using the paper and pencil technique. As Sowder rightly notes, "One topic that cannot be replaced by the calculator is the development of understanding of what numbers are" (p.20). Mental calculations stimulate the development of such an understanding and therefore remain important in an age when the use of technology is pervasive.

Much of what is taught in high school can also be effortlessly accomplished using a calculator (just a slightly more expensive calculator than the one needed to solve elementary school problems). However, discussions of what is essential in the traditional curriculum, and which parts of it will persist into the age of technology, rarely touch on the role of mental mathematics.

One of the very few publications devoted to mental mathematics in high school (Rubenstein, 2001) tellingly contains three question marks in its title: mental mathematics beyond the middle school level appears as something quite unexpected. The author of the article, however, lists a number of reasons to show why mental mathematics is useful at high school level as well. As Rubenstein argues, mental mathematics

* is useful for workers, consumers, and citizens;

* facilitates learning many structural topics;

* allows students to become less calculator-dependant;

* is rewarding because students are challenged by the problems and feel proud of their accomplishments. (pp. 442-443)

It is evident that the use of mental mathematics problems can be analyzed from different viewpoints--in terms of purely methodological as well as social-pedagogical frameworks. The following crucial questions deserve further attention:

* Which problems can be used as mental mathematics problems in high school? How can these problems be useful in class? What guidelines must teachers use in selecting (or constructing) mental mathematics problems?

* What forms of lesson organization permit teachers to make use of mental mathematics?

* How does the use of mental mathematics problems develop the students' communication skills? What peculiarities of students' mathematical language emerge during work on mental mathematics problems?

Below, the attempt is made to answer some of these questions.

What is a mental mathematics problem. Reys et al. (1993) write: "Mental computation is the ability to calculate exact numerical answers without the aid of calculating or recording devices" (p.306). This definition can be applied not only to computations, but to other types of problems as well: Mental mathematics problems are problems whose solutions do not require the use of pen or pencil or any other calculating or recording devices.

It is worth taking a closer look at the one of the problems offered by Rubenstein (2001): "Solve: 3[x.sup.3] = -24" (p. 442). This problem can undoubtedly be solved without writing anything down, but it can also be solved on paper, and its written solution would actually take up several lines. In many textbooks, such a problem will be given among many others without any additional instructions. Similarly, the arithmetical problem given above--to find the product of 4 and 35--can clearly be solved on paper as well. Put simply, a problem does not say whether it is a mental math problem or not. The same problem can be used in two different contexts. Consequently, the statement that a problem is suited for mental mathematics implies, in the first place, a methodological decision to use the problem in such a capacity, and only secondarily does it characterize the mathematical side of the problem as one that allows it to be utilized in this fashion.

Naturally, computing, the integral [4.[integral].2][dx/(x - 1)(x - 5)] cannot be offered as a mental mathematics problem. Even though it would be easy and by no means time-consuming to make use of the equality 1/(x - 1)(x - 5) = 1/4. ([1/[x - 5]] - [1/[x - 1]]), to integrate the individual terms, and to substitute the limits of integration, it would make absolutely no sense to ask students to do this without writing anything down. On the other hand, computing the integral [1.[integral].-1][square root of (1 - [x.sup.2])]dx makes much more sense as a mental mathematics problem. Not only because virtually no computations have to be kept in mind simultaneously, but principally because, simply by posing such a problem, the teacher gives the students a kind of metacognitive hint. The students are in effect told that they should consider how it might be possible to avoid routine calculations. Students who notice that the problem requires them to find the area of a semicircle with radius 1 will encounter practically no further difficulties with its solution, and the chosen oral form of the problem is precisely what will help them make this observation almost on their own.

Consequently, the very definition of the concept "mental mathematics problem" is equivocal and based on certain methodological assumptions. Mental mathematics problems can be spoken of only in connection with classroom activities that involve "collective thinking and discussion," in which attention is focused on content and reasoning rather than on the technical aspect of the tasks at hand (and thus not on written computations). It is impossible to discuss the problems that are suitable for this type of activity without discussing the place of this activity within the general structure of the lesson and its role in the students' development, their comprehension of the subject matter, and so on.

At the same time, it is essential to stress the difference between "mental mathematics problems" and "easy problems." Above, a rather difficult problem was intentionally offered as an example of an exercise that could be used as a mental mathematics problem in class. Exercises that are used as mental mathematics problems can be drawn from any area of mathematics, including such advanced subjects as calculus, and they can demand substantial intellectual exertions, provided that these exertions require no written notations or computations. On the other hand, students find a problem easy if the steps that must be taken to arrive at its solution are obvious to them, and if they present no technical difficulties in themselves--and this is true for "mental" and "written" problems alike.

One more aspect of the definition of the concept "mental mathematics problem" must be made more precise. Not every mental activity and subsequent classroom discussion are connected with solving mathematical problems (i.e., with the search for exact answers to a precisely formulated question). The character of the mathematical problem can remain quite open--it does not need to be an exercise with a numerical solution. And there need not be only one right answer (for example, students might be asked to "come up with a similar problem"). Still, the present article is concerned with problems that involve a rather rigidly structured form of activity (which includes time constraints). For example, asking the students to discuss a sketch cannot be considered a mental mathematics problem (while asking them to find three mistakes in it in 10 minutes can be).

The examples analyzed below. Without making any claims to an exhaustive description of the topic, the discussion below confines itself to certain examples of the systematic use of mental mathematics problems in high school lessons, with an analysis of how and why they were used. Some of these examples are drawn from lessons observed by the author in Russia or described in the Russian literature on mathematical methodology. Russian mathematics education, in general, enjoys an excellent reputation. At the same time, it is impossible not to agree with the view (Froumin, 1996) that it is usually described in rather imprecise ways and with a substantial simplification of what actually goes on. Without going into the complex question about the strong and weak sides of Russian education, it will do well to note that the Russian experience seems particularly relevant when the discussion concerns the relation between the structure of the lesson and its mathematical content. Researchers (Wilson, Andrew, & Sourikova, 2001) have noted that Russian lessons can be "characterized as a focused series of linked tasks, each of which comprises a blend of oral work, pupil demonstration, written recording, and teacher questioning and explaining". (pp.41-42). Solving mental mathematics problems is an important component in this complex structure. Consequently, Russian education can provide numerous examples whose analysis can be of use to mathematics educators in other countries.

What mental mathematics problems can be used for. Several simple examples. N. Rybkin's problem book (Kiselev and Rybkin, 1995), which has gone through dozens of editions in Russia and was universally used for almost the entire Soviet period, provides useful examples of mental mathematics problems. Some of the problems in this book have a note next to them: "solve mentally." In this way, the teacher was given instructions about the specific way in which a given problem should be used (this was possible due to the considerable centralization of the teaching system). For example, problem 8.1 states: "The surface of a cube is equal to 24 [m.sup.2]. Find the length of its edge." The student must, say, picture a cube; figure out (recall) that is has 6 faces, which are squares; conclude that the area of one face is 4 [m.sup.2]; and finally give the answer that the length of an edge is 2 m. It can be supposed that, in posing this problem, the author considered it necessary to make the students imagine this cube, rather than using some model, drawing, etc. Likewise, students had to be capable of carrying out a rather long line of reasoning in their heads. Mental mathematics problems are useful in this context as a means for developing a specific technical skill--the skill of mental work.

It was noted above that carrying out complicated computations mentally, even though this is feasible, seems quite pointless today (unless the goal is to create a human calculator for public performances). On the other hand, the ability to picture a geometrical object, as well as the ability to reason mentally, are unlikely to lose their importance. Therefore, the exercise described above is unlikely to become outdated.

Another simple example of the use of mental mathematics problems is given by Rubenstein (2001). Students are given the problem: "Evaluate [sin.sup.[[pi].bar]]" (p. 442). In essence, the point here is to check how well they have mastered the equalities they have studied earlier. In such problems, the teacher can offer the students a series of formulas that they have already learned or conversely a series of objects for which formulas must be derived (say, a square, a rectangle, a parallelogram, etc., with the lengths of their sides marked, so that the students must find their areas). Such problems can be given at the beginning of the lesson, with a view to using subsequently the freshly revisited formulas in more substantive problems; and they can also be used during other parts of the lesson, as independent exercises, with the aim of preventing the students from forgetting these formulas and facts.

Somewhat analogous to the problems just described are problems involving object or concept recognition. An example of such a problem is an exercise that requires students to indicate which among various given graphs can be the graphs of various functions, and to explain how they reached their conclusion.

In all of the examples described, classroom activity follows more or less the same scheme: the teacher gives the students a list of questions, gives them (a little) time to think, and then asks them to begin the discussion. It would be easy to give examples in which analogous schemes are used to go over what has been learned earlier, and to check how well students have mastered what they have studied, and to prepare the students for new material, etc. Clearly, it is useful for the whole class to focus on a problem and to discuss it collectively. Why, then, is the "mental" character of the problem so important here? What would be different if students were allowed to make supplementary drawings or written calculations?

The "mental" character of the problem is useful not only because it develops the students' ability to engage in mental reasoning. By giving students mental mathematics problems and thus encouraging them to rely not on commonly used forms of notation, but on their own conceptions, teachers can avoid fostering their own notions on the students (or at least do so to a lesser degree than usual). In solving the problem about the edge of the cube, for example, the student can imagine not a cube but a piece of paper that is folded into a cube. This will make the problem even easier to solve. Students can become acquainted with still other approaches to solving the problem in the course of the class discussion. If, on the contrary, this problem about the area of the surface of a cube called for a "written" solution, then students would certainly be required to construct a sketch of the cube, which might involve them in much greater difficulties than the actual solution of the problem.

Educational literature (Goldin & Shteingold, 2001) commonly juxtaposes "external" and "internal" forms of mathematical representation. External forms of representation include, for example, the "conventional symbol systems of mathematics"--formal algebraic notation, graphs, or sketches. "Internal systems include students' personal symbolization constructs and assignments of meaning to mathematical notations," etc. (Goldin & Shteingold, p. 2). Undoubtedly, both systems are important (the author of the present article in no way wishes to imply that the ability to sketch a cube, for example, is not important). However, by focusing only on "written" problems, the teacher compels the students to use external representations from the start. It is true that mental mathematics problems are also ineluctably connected with external representations (at the very least because these problems are stated in a common language, even though an individual student might not know, for example, what is meant by a "cube"). Nonetheless, mental mathematics problems leave the students an incomparably greater amount of room for individual comprehension and modeling. In addition, students express their solutions in an oral form, that is, once again, in a form that is convenient for them. In talking through the solution, the students disclose the details of their thought process to the teacher. After this, the transition to using external representations along with internal ones becomes easier.

Silver (1987) writes that "the differences and interactions between the external task environment and the internalized problem space, or problem representation, are seen as critical to the solution of a problem" (p. 43). Recognizing the importance of such interactions, it is natural to reach the conclusion that what is needed are problems that provide practice for students in a given task, and at the same time help teachers to diagnose what the students have achieved. It appears that mental mathematics problems may serve as a kind of bridge between the internal and external realms, helping to set up the interactions between them.

A simple example is offered by the problem described above, which requires students to make use of a known equality. It encourages students to use only their internal resources (long term memory), without relying on external ones. The problem that deals with object recognition that was also mentioned above involves a search for interconnections between an external representation (for example, a graph) and internal representations (for example, about a function). By asking students to answer the question, "Is the number [3.sup.3] x [5.sup.2] x 7 divisible by 7?" orally, the teacher can diagnose the possible difficulties connected with their understanding of what is meant by "divisible by" (Zazkis and Gadowsky, 2001, note that many students can solve this problem only by multiplying the numbers on a calculator and then dividing by 7). On the other hand, a class discussion of such a problem trains the students and facilitates the development of essential representations.

By using mental mathematics problems, the teacher can "adjust the dosage" of different kinds of representations, at first concentrating mainly on substantive ("mental", "internal") components, and only later on the technical difficulties of the external representations.

Carpenter and Lehrer (1999) propose that there are "five forms of mental activity from which mathematical understanding emerges" (p. 20). Among these are "constructing relationships", "extending and applying mathematical knowledge", "reflecting about experiences", "articulating what one knows", and "making mathematical knowledge one's own" (p. 20). Mental mathematics problems, it would seem, can be useful in all of these respects.

The organization of lessons that use mental mathematics problems: Some more complex examples. By giving mental mathematics problems, the teacher deliberately limits the students' choices, excluding certain forms of activity. This immediately presupposes a rather rigid form of lesson organization, in which, for example, such restrictions are understood and accepted by the students. It should be stressed at once that a rigid form of lesson organization must not be taken to mean that the class must be teacher-centered. To a certain extent, the opposite is true. The teacher systematically and methodically plans various teaching activities, but within the established bounds it is the student who is at the center of these planned activities, not the teacher.

The term "structured problem-solving," suggested by Stigler and Hiebert (1999) to describe lessons in Japan, felicitously characterizes lessons that make successful use of mental mathematics problems. Within the framework of such a structured lesson, the lesson's stages (including the stage devoted to mental mathematics) acquire an additional meaning due to their interaction and interplay with one another. In this context, mental mathematics problems turn out to be useful not only for the development of internal and external representations, but also for an in-depth comprehension of the subject matter. Several examples will suffice.

A lesson that investigated the definition of a function began with a series of mental mathematics problems, including the following ones:

1. Name some value of x for which the following operations can be performed, and some other value of x for which the following operation cannot be performed:

a) 1/x; b) [square root of x]; c) 1/[square root of x]

2. Indicate the domains on which the following functions are defined:

a) y = 1/x; b) y = [square root of x]; c) y = 1/[square root of x]

3. Offer some explanation for why graphs a-c (Fig. 1) cannot be the graphs of functions a-c, respectively, given above.

[FIGURE 1 OMITTED]

These problems made it possible to discuss the concept of a domain on which a function is defined, which had been introduced in the previous lesson, by connecting it both with an operational approach as well as a graphic one. In certain cases, the students' answers made it possible to identify certain misconceptions that had been formed (for example, x=3 was offered as an answer to question lb, since "no root can be extracted from 3").

Students were then offered written problems, which may be divided into two sets. The first set contained problems that require students to make direct use of the results obtained in the mental mathematics problems (for example, x = [t.sup.2] - 4; indicate all t for which the function y = [square root of x] is defined). The second set contained problems that were similar to the mental mathematics ones but more technically complex.

At the conclusion of the lesson, students were again given a mental mathematics problem.

4. Give an example of some other functions with the same domain as in problems 2(a-c).

5. Give examples of some functions whose domain is

a) {x: x [not equal to] 1}, b) {x: x [greater than or equal to] 1}, c) {x: x > 1}, d) {x: x < 0, x > 1}.

In this way, the whole lesson was structured around the interaction of "oral" and "written" problems. The mental mathematics problems

* helped clear up the students' understanding of the subject matter;

* modeled the contents of the entire lesson on technically simple material;

* provided materials that were subsequently used in technical problems;

* provided materials for subsequent oral work at a higher level of creativity.

It is fair to say that the mental mathematics questions in this lesson were devoted to mathematical ideas in their pure form, anticipating problems whose mathematical essence was somewhat obscured by technical intricacies.

In another lesson, students were first asked to solve a series of trigonometric equations using various methods, as a continuation of the preceding lessons of the unit, and, more globally, of the whole large section in which power, exponential, and logarithmic functions and equations were introduced. Students were then given mental mathematics problems, beginning with the following ones:

1. Explain what the following equations have in common:

[sin.sup.2] x - 4 sin x + 3 = 0; [x.sup.4] - 4[x.sup.2] + 3 = 0 and [4.sup.x] - [2.sup.x+2] + 3=0.

2. Explain what the following equations have in common:

(sin x - 1) cosx = 0; ([3.sup.x] - 1) ([4.sup.x]-[4.sup.2-x]) = 0 and (x - 1) ([x.sup.2] - 4) = 0.

In the preceding lessons, students had been immersed in technical manipulations connected with solving equations by chunking or using the fact that the product is equal to zero only if at least one of its factors is equal to zero. The mental mathematics problems made it possible to pass to a discussion of the fact that the techniques studied earlier were general ones--applicable in different fields. Here, again, the mental figured as a synonym for the conceptual, but in this instance it did not precede the technical, but followed and generalized it.

Mental mathematics problems can play all kinds of different roles in a lesson. It would be incorrect to say that they must necessarily be the conceptual focal point of the lesson--any other problem can obviously play that role as well. Nonetheless, it may be argued that the interaction of the oral and the written often makes it possible to present the idea being studied more vividly.

For example, here is the brief description of a part of a lesson devoted to deriving the formula for the roots of a quadratic equation. In the beginning of the lesson, the students were asked to solve mentally several quadratic equations that could be solved by a simple factoring or direct extraction of a square root. The students easily solved these problems. After this, they were asked to solve the equation [x.sup.2] + 2x - 1 = 0 within the same time limits. Naturally, none of the answers they offered passed muster. Some students declared that there were no roots. The teacher suggested that they check for themselves, on paper, to see that the number [square root of 2] - 1 is in fact the root of the equation. And this was what initiated the discussion of how the root might have been guessed.

Mental mathematics problems here figured as a synonym for the commonly known, while it was the new idea that appeared in written form. However, the resulting "cognitive conflict"--the students' recognition of the inadequacy of the old methods--was made more salient through the interaction of the oral form with the written one.

What kinds of problems make good mental mathematics problems. Earlier, a problem dealing with an integral was cited as one that was unsuited for an oral format. Suitable problems are ones that can be solved in a reasonable number of steps and make use of a reasonable number of calculations, pictures (imaginary ones), and so on. This "reasonable" number in and of itself varies from individual to individual. Psychologists (Krutetskii, 1976) have noted that gifted students possess, for example, a greater than average mathematical memory, and also establish connections between different areas of the subject matter more easily. Thus, many assignments that can be given as mental mathematics problems in a class for gifted students would be unsuitable for this purpose in an ordinary classroom.

Here is an example. In a lesson for a class of mathematically gifted students, students were asked to solve mentally the equation [2.sup.x] + [3.sup.x] = 5. The fact that x=1 is a root of the equation was immediately obvious to the students, but it was also clear to them that the solution required them to show that no other roots exist. They could do this because they

a) remembered that y = [2.sup.x] and y = [3.sup.x] are increasing functions;

b) easily concluded that, therefore, the sum of these functions y = [2.sup.x] + [3.sup.x] is also an increasing function;

c) could relate the monotonicity of the function corresponding to the left-hand side of the equation to its solution--it was clear to them from this monotonicity that the function cannot take on any value more than one.

For comparison, students in an ordinary classroom immediately offered the answer x=1, but it was not clear to them that they had to show anything else in addition to this. Only after a discussion with the teacher did they recognize that it was necessary to check for the absence of other solutions, which required considerable effort (many students performed various algebraic transformations and became persuaded that the function takes on the value 5 only once after the teacher suggested that they use a graphing calculator).

Thus, the mental mathematics format of this problem only sent a misleading signal to the students when it was posed in an ordinary classroom. On the other hand, in the class for the gifted it helped the students to organize their conceptual activity more precisely, leading them to reject all patently non-mental solutions. The accessibility of the mental solution in the sense described above is obviously the main requirement for all mental mathematics problems. But based on the analysis above, two additional practical demands can be formulated.

The first of them derives from the nature of mental mathematics problems as a bridge between external and internal representations. Mental mathematics problems are especially useful when they are specifically devoted to the task of representing mathematical objects. They are particularly useful for clarifying how an object is understood by the students and giving them a possibility to practice different ways of understanding it. Problems of this kind include ones that require students to describe an object in a different language, to recognize that an object fits a given description (definition), and so on.

The second practical demand derives from the role that mental mathematics problems play in a lesson, as described above. Mental mathematics problems can be given in blocks, with their structure and interconnections, and these blocks can be made to reflect the structure and interconnections of the contents of the lesson. Structuring the problems in this way in general makes them more effective educational instruments.

What students say when they solve mental mathematics problems. It is universally recognized that the language students use in doing mathematics orally is different from the language they use in doing mathematics in written form (Pimm, 1987). The way students speak when solving mental mathematics problems calls for further in-depth investigation. The present section of this article does not aim to reach any specific conclusions on this matter (except to stress that understanding the nature of students' speech and why they make mistakes of one kind or another seems extremely important for the understanding of how they think and learn). It merely seeks to indicate questions that arise in the process of observation, not to give answers to them.

Students engaged in solving oral problems do not have to apply mathematical symbolism, although they often still have to understand it (for example, if it is contained in the formulation of the problem). This immediately distinguishes the oral speech of the problem-solvers from their written language. Moreover, the speech of students who are solving oral problems is often conspicuously informal, in the sense that it does not reflect conventional speaking standards ("speaking for others," as this is called by Pimm, 1987), but their own images. Sometimes students even express themselves using gestures instead of words. Thus, for example, in solving the problem involving the solution of equation [2.sup.x] + [3.sup.x] = 5, some students simply traced a curve in the air and said: "This way on the left side." As has already been noted, the possibility to see in this way what specifically a student has in mind is very valuable in itself, and constitutes a sufficient motive for using mental mathematics problems.

The connection between speech and representation is, naturally, far from simple. Researchers and working teachers alike often have occasion to wonder whether an imprecise (or simply meaningless) statement made by a child reflects a mistaken understanding or an unfortunate formulation. For example, the ellipsis that is characteristic for ordinary speech also appears in oral solutions. In ordinary speech a person stepping on a bus might say, simply, "Is this the 6?" instead of "Is this the bus that goes along route number 6?" Students tend to abbreviate their speech in a similar fashion. Thus, for example, the rather frequently heard phrase "This triangle is congruent"--which sometimes leads the teacher to cut the student off and to start explaining that a triangle can be congruent only to another triangle, etc.--does not necessarily imply that the student is under an actual misconception. Moreover, students who make such mistakes are often capable of discovering them for themselves. It would seem that oral discussions of problems and solutions can help to figure out what specifically is going on in each particular case.

The ability to use formal oral speech (speaking for others), which teachers generally demand from their students, watching for both the precision of the words they use (Zaskis, 2001) and the completeness and comprehensibility of their sentences, is an important element in the formation of the grammatical written idiom. Pimm (1987) rightly notes that written propositions need not be symbolic in form; for example, they can be expressed in words. Moreover, the transition to a written symbolic language, which in a sense has the same syntax but a new vocabulary, is clearly made easier if the students are already capable of writing down what they have to say in words. The progressive stages in the development of mathematical writing (informal mathematical speech--formal mathematical speech--verbal mathematical notation--symbolic mathematical notation), the typical difficulties encountered at each stage by the students, and the successful strategies used by teachers, all deserve further investigation. Observing students in the process of solving mental mathematics problems would be highly useful in this respect.

Conclusion. Mental mathematics questions can be useful both in terms of solving the practical problems that teachers face, and in terms of investigating the teaching process. Naturally, the use of mental mathematics problems in turn raises new questions of both a psychological and a methodological nature. The individual limits of oral work were brought up above. Are students' capacities for carrying out mental calculations and their ability to reason out loud connected, and if so, in what way? Are students' capacities for mental reasoning connected with their ability to picture graphic images? In general, how is the ability to solve mental mathematics problems connected with the ability to solve written ones? What role is played by age in this regard? How is the ability to solve problems mentally connected to teaching, and what are the best methods for developing this ability? The list of open questions could easily be continued.

What is abundantly clear is that mental mathematics problems help the teacher, revealing new sides of the students and helping them to acquire a better grasp of mathematics and its characteristic ways of thought. As Sowder (1990) and Plunkett (1979) have noted, mental mathematics can be variable, flexible, active, holistic, and constructive. Mental mathematics problems "require understanding all along" (Sowder, p. 20). Mental mathematics possesses all of these characteristics in high school as well.

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Alexander Karp

Columbia University