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Compressed units of mathematical thought
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JOURNAL OF MATHEMATICAL BEHAVIOR, 17 (4), 401± 404 ISSN 0364-0213. Copyright C 1999 Elsevier Science Inc. All rights of reproduction in any form reserved.
Compressed Units of Mathematical Thought Tony Barnard University of London, London, UK
This article discusses the phenomenon in mathematical thinking in which a section of mathematical structure is mentally compressed into a single unit, small enough to fit into the conscious focus of attention at a given time, and possessing an interiority which is able to both guide manipulation of the unit and also be subsequently expanded without loss of detail.
Mathematics has developed into an extensive hierarchy or network of concepts, each more abstract than, and dependent upon, those feeding into it. For example, in the terminology of Skemp (1971), a pair of objects would be a primary concept, two a secondary concept, number a tertiary concept, addition a fourth-degree concept, and so on. However, the amount that can be held in the mind and attended to at any one time is very limited and, in order to minimize the consequent constraints on thinking, various strategies are adopted for reducing the mental load of data to be considered. Such strategies include, for example, grouping items and considering them in sequence, labeling items and groups of items with words or symbols, writing things down in the form of abbreviated notes. When the data being attended to are of the kind that are part of a hierarchical network spanning several layers of abstraction with various interconnections, the reduction in mental load can also be achieved by a compression which results in linked items becoming associated with a single entity. A collection of related items, which could be, say, processes, sentences, representations, objects, properties, steps of logical deduction, become mentally compressed into one single entity which can then be easily manipulated using a minimum of thinking space and subsequently unpacked whenever needed. Such a piece of cognitive structure that can be held in the short-term focus of attention all at one time will be called a cognitive unit (Barnard & Tall, 1997). As well as being small enough to hold in the mind at the time of being used, a cognitive unit plays the role of an operative label. More than just saving mental space by being a shorthand in place of a collection of items, it actually carries with it, just beneath the surface, the structure of the collection and is operative in the sense that the live connection with this structure is able to guide the manipulation of the single compressed cognitive unit. For example, when the relationships implicit in Fig. 1 are conceived as a cognitive Direct all correspondence to: Tony Barnard, King's College, University of London, London, UK; E-Mail: [email protected]
401
402
BARNARD
FIGURE 1. The `canonical' equilateral triangle.
unit, they are not just replaced by a mental label, as if to enter a passive mode of simply being available when called upon. On the contrary, they are compressed into a mental entity which leaves space in the mind for bringing in other ideas, and actively influences this thinking by pushing to the fore elements pstructure of the unit such as p of the internal ``Pythagoras Theorem,'' ``the square of 3 is 3,'' ``12 32 22 ,'' ``if two sides of a triangle are equal then the base angles are equal.'' Discussions of ideas and phenomena related to the above can be found in Krutetskii (1976) (curtailed structures), Skemp (1979) (varifocal theory) and the process/object theories of Davis (1984) (noun status), Dubinsky (1991) (encapsulation), Sfard (1991) (reification), Gray and Tall (1994) (procepts). The thing which distinguishes the notion of ``compressing to a cognitive unit'' from the notion of ``chunking'' (Miller, 1956) is the feature of ``active influence.'' As if it were sending out radar signals, the awareness of the specific elements of the internal structure is able to guide further manipulation of the cognitive unit and help in the search for ideas during problem solving. The following examples illustrate the range of phenomena that can be thought of in this way. (1) Initially an individual may see the equations P QR;
P R; Q
P Q R
as three separate items, together with links of equivalence between them. Later this perception may become compressed into that of a single relationship between the quantities involved. The individual is then able to mentally manipulate with this as a single item without having in mind any particular representation, in much the same way as one can think about a good film without remembering if it had subtitles, or whether it was in black and white or in color. (2) The notion of a fraction as a procept, or as a process encapsulated as an object, has been discussed many times elsewhere, but the salient features of this example are worth repeating in the current context. Take 78, for example. This carries with it, just beneath the surface, and ready if needed, the information that it is 7 divided by 8 (the result of the process of dividing 7 by 8). It also carries with it, just beneath the surface, the information that it is 7 times 18 (the result of the process of multiplying 18 by 7). Once someone has compressed these processes and the concept of 78 as a single object, they are no longer weighed down by holding in their mind the complexity of the different representations. They can then operate
COMPRESSED UNITS OF MATHEMATICAL THOUGHT
403
arithmetically with 78 with no more difficulty than they would have with any integer and yet, at the same time, they have not lost awareness of the related processes. (3) In the last example, the compressed entity was associated with a natural visual label, namely 78. However in other instances, such as in the compression of a chain of steps of manipulation, the mental image, if there is one, may be more vague. For example, in seeing that a linear combination of a linear combination is a linear combination, the essence of the rearrangement of terms in the equation a rx sy b ux vy ar bux as bvy Pn Pm Pn Pm (or more generally i1 ai j1 rj xij i1 j1 ai rj xij ), might be conceived as a single `unit of thought' without there being a clearly defined label. (4) Pupils are sometimes encouraged to remember the trigonometric ratios sine, cosine and tangent via an image of a specifically labeled right-angled triangle, or recall of sin
opposite ; hypotenuse
cos
adjacent ; hypotenuse
tan
opposite ; adjacent
possibly aided by a mnemonic like ``SOHCAHTOA.'' In order to use such a representation for any given right-angled triangle, there is an initial step of matching the words ``opposite,'' ``adjacent,'' ``hypotenuse'' to the sides of the given triangle. However a pupil who mentally holds the trigonometric ratios as a compressed entity, perhaps accompanied by an image of an unlabelled right-angled triangle, is able to bypass this step and operate on the given triangle directly, unpacking the compressed entity only at the point of application. (5) Consider the equation 3x ÿ 2 x ÿ 5 1 ÿ 2x : 10 3 5 There are pupils who can follow, and understand, a demonstration by the teacher of the method of solving this which begins by multiplying throughout by 30, but for whom the steps of the process of clearing denominators have not been compressed into a single mental entity. Such pupils would therefore not have this available to them for use as an intermediate stage in formulating their own solution and so would probably begin differently, perhaps by adding the fractions on the left-hand side of the equation. (6) For integers a and d, ``d divides a,'' ``a = ds for some integer s'' and ``a is a multiple of d'' are three equivalent statements. There are some individuals who conceive these as three separate statements, each of which can be transformed to either of the other two, and there are others who are able to conceive them as a single cognitive unit. Consider the role of this cognitive unit in understanding the deduction, xa yb h and d divides both a and b ) d divides h: A student who has compressed ``d divides a,'' ``a = ds for some s'' and ``a is a multiple of d'' as a single entity, will make the deduction immediately. Moreover the deduction itself will become a thing for the student, something that can be held in the mind or called upon when needed. On the other hand, students for whom these mathematically equivalent relationships between d and a are separate mental units, will proceed to write down ``a =
404
BARNARD
ds, b = dt'' and substitute these into the equation ``xa + yb = h.'' Because they are having to hold more in their minds, such students are more likely to lose the thread of the argument of which this deduction might have been a part, or even make a technical slip along the way and thereby get totally lost. Moreover, if such a student was trying to formulate, rather than simply follow, an argument of which this deduction was an intermediate step, his/her handicap would be even more severe. For if the step, ``if xa + yb = h and d divides both a and b, then d must divide h,'' did not exist as a thing for such a student, it simply would not be available for use. Although each of the components of this step might be available for use, none of these separate pieces would have the guiding influence embodied in the synthesized whole. The important features then of a cognitive unit are its small size and its living interiority. One might say that its value in mathematical thinking lies in it being a whole which is both smaller and greater than the sum of its parts Ð smaller in the sense of being able to fit into the short term focus of attention, and greater in the sense of having holistic characteristics which are able to guide its manipulation. REFERENCES Barnard, Tony, & Tall, David O. (1997). Cognitive units, connections and mathematical proof. Proceedings of the 21st International Conference for the Psychology of Mathematics Education, 2, 41± 48. Davis, Robert B. (1984). Learning mathematics: The cognitive science approach to mathematics education. Norwood, NJ: Ablex. Dubinsky, Ed. (1991). Reflective abstraction in advanced mathematical thinking. In D.O. Tall (Ed.), Advanced mathematical thinking (pp. 95 ±123). Dordrecht: Kluwer. Gray, Eddie M., & Tall, David O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115±141. Krutetskii, Vadim A. (1976). J. Kilpatrick & I. Wirzup. (Eds.), The psychology of mathematical abilities in schoolchildren (Joan Teller, Trans.). Chicago IL: University of Chicago. Miller, George. (1956). The magic number 72. Psychological Review, 63, 81± 97. Sfard, Anna. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1± 36. Skemp, Richard R. (1971). The psychology of learning mathematics. London: Penguin Books. Skemp, Richard R. (1979). Intelligence, learning and action. London: Wiley.
JOURNAL OF MATHEMATICAL BEHAVIOR, 17 (4), 401± 404 ISSN 0364-0213. Copyright C 1999 Elsevier Science Inc. All rights of reproduction in any form reserved.
Compressed Units of Mathematical Thought Tony Barnard University of London, London, UK
This article discusses the phenomenon in mathematical thinking in which a section of mathematical structure is mentally compressed into a single unit, small enough to fit into the conscious focus of attention at a given time, and possessing an interiority which is able to both guide manipulation of the unit and also be subsequently expanded without loss of detail.
Mathematics has developed into an extensive hierarchy or network of concepts, each more abstract than, and dependent upon, those feeding into it. For example, in the terminology of Skemp (1971), a pair of objects would be a primary concept, two a secondary concept, number a tertiary concept, addition a fourth-degree concept, and so on. However, the amount that can be held in the mind and attended to at any one time is very limited and, in order to minimize the consequent constraints on thinking, various strategies are adopted for reducing the mental load of data to be considered. Such strategies include, for example, grouping items and considering them in sequence, labeling items and groups of items with words or symbols, writing things down in the form of abbreviated notes. When the data being attended to are of the kind that are part of a hierarchical network spanning several layers of abstraction with various interconnections, the reduction in mental load can also be achieved by a compression which results in linked items becoming associated with a single entity. A collection of related items, which could be, say, processes, sentences, representations, objects, properties, steps of logical deduction, become mentally compressed into one single entity which can then be easily manipulated using a minimum of thinking space and subsequently unpacked whenever needed. Such a piece of cognitive structure that can be held in the short-term focus of attention all at one time will be called a cognitive unit (Barnard & Tall, 1997). As well as being small enough to hold in the mind at the time of being used, a cognitive unit plays the role of an operative label. More than just saving mental space by being a shorthand in place of a collection of items, it actually carries with it, just beneath the surface, the structure of the collection and is operative in the sense that the live connection with this structure is able to guide the manipulation of the single compressed cognitive unit. For example, when the relationships implicit in Fig. 1 are conceived as a cognitive Direct all correspondence to: Tony Barnard, King's College, University of London, London, UK; E-Mail: [email protected]
401
402
BARNARD
FIGURE 1. The `canonical' equilateral triangle.
unit, they are not just replaced by a mental label, as if to enter a passive mode of simply being available when called upon. On the contrary, they are compressed into a mental entity which leaves space in the mind for bringing in other ideas, and actively influences this thinking by pushing to the fore elements pstructure of the unit such as p of the internal ``Pythagoras Theorem,'' ``the square of 3 is 3,'' ``12 32 22 ,'' ``if two sides of a triangle are equal then the base angles are equal.'' Discussions of ideas and phenomena related to the above can be found in Krutetskii (1976) (curtailed structures), Skemp (1979) (varifocal theory) and the process/object theories of Davis (1984) (noun status), Dubinsky (1991) (encapsulation), Sfard (1991) (reification), Gray and Tall (1994) (procepts). The thing which distinguishes the notion of ``compressing to a cognitive unit'' from the notion of ``chunking'' (Miller, 1956) is the feature of ``active influence.'' As if it were sending out radar signals, the awareness of the specific elements of the internal structure is able to guide further manipulation of the cognitive unit and help in the search for ideas during problem solving. The following examples illustrate the range of phenomena that can be thought of in this way. (1) Initially an individual may see the equations P QR;
P R; Q
P Q R
as three separate items, together with links of equivalence between them. Later this perception may become compressed into that of a single relationship between the quantities involved. The individual is then able to mentally manipulate with this as a single item without having in mind any particular representation, in much the same way as one can think about a good film without remembering if it had subtitles, or whether it was in black and white or in color. (2) The notion of a fraction as a procept, or as a process encapsulated as an object, has been discussed many times elsewhere, but the salient features of this example are worth repeating in the current context. Take 78, for example. This carries with it, just beneath the surface, and ready if needed, the information that it is 7 divided by 8 (the result of the process of dividing 7 by 8). It also carries with it, just beneath the surface, the information that it is 7 times 18 (the result of the process of multiplying 18 by 7). Once someone has compressed these processes and the concept of 78 as a single object, they are no longer weighed down by holding in their mind the complexity of the different representations. They can then operate
COMPRESSED UNITS OF MATHEMATICAL THOUGHT
403
arithmetically with 78 with no more difficulty than they would have with any integer and yet, at the same time, they have not lost awareness of the related processes. (3) In the last example, the compressed entity was associated with a natural visual label, namely 78. However in other instances, such as in the compression of a chain of steps of manipulation, the mental image, if there is one, may be more vague. For example, in seeing that a linear combination of a linear combination is a linear combination, the essence of the rearrangement of terms in the equation a rx sy b ux vy ar bux as bvy Pn Pm Pn Pm (or more generally i1 ai j1 rj xij i1 j1 ai rj xij ), might be conceived as a single `unit of thought' without there being a clearly defined label. (4) Pupils are sometimes encouraged to remember the trigonometric ratios sine, cosine and tangent via an image of a specifically labeled right-angled triangle, or recall of sin
opposite ; hypotenuse
cos
adjacent ; hypotenuse
tan
opposite ; adjacent
possibly aided by a mnemonic like ``SOHCAHTOA.'' In order to use such a representation for any given right-angled triangle, there is an initial step of matching the words ``opposite,'' ``adjacent,'' ``hypotenuse'' to the sides of the given triangle. However a pupil who mentally holds the trigonometric ratios as a compressed entity, perhaps accompanied by an image of an unlabelled right-angled triangle, is able to bypass this step and operate on the given triangle directly, unpacking the compressed entity only at the point of application. (5) Consider the equation 3x ÿ 2 x ÿ 5 1 ÿ 2x : 10 3 5 There are pupils who can follow, and understand, a demonstration by the teacher of the method of solving this which begins by multiplying throughout by 30, but for whom the steps of the process of clearing denominators have not been compressed into a single mental entity. Such pupils would therefore not have this available to them for use as an intermediate stage in formulating their own solution and so would probably begin differently, perhaps by adding the fractions on the left-hand side of the equation. (6) For integers a and d, ``d divides a,'' ``a = ds for some integer s'' and ``a is a multiple of d'' are three equivalent statements. There are some individuals who conceive these as three separate statements, each of which can be transformed to either of the other two, and there are others who are able to conceive them as a single cognitive unit. Consider the role of this cognitive unit in understanding the deduction, xa yb h and d divides both a and b ) d divides h: A student who has compressed ``d divides a,'' ``a = ds for some s'' and ``a is a multiple of d'' as a single entity, will make the deduction immediately. Moreover the deduction itself will become a thing for the student, something that can be held in the mind or called upon when needed. On the other hand, students for whom these mathematically equivalent relationships between d and a are separate mental units, will proceed to write down ``a =
404
BARNARD
ds, b = dt'' and substitute these into the equation ``xa + yb = h.'' Because they are having to hold more in their minds, such students are more likely to lose the thread of the argument of which this deduction might have been a part, or even make a technical slip along the way and thereby get totally lost. Moreover, if such a student was trying to formulate, rather than simply follow, an argument of which this deduction was an intermediate step, his/her handicap would be even more severe. For if the step, ``if xa + yb = h and d divides both a and b, then d must divide h,'' did not exist as a thing for such a student, it simply would not be available for use. Although each of the components of this step might be available for use, none of these separate pieces would have the guiding influence embodied in the synthesized whole. The important features then of a cognitive unit are its small size and its living interiority. One might say that its value in mathematical thinking lies in it being a whole which is both smaller and greater than the sum of its parts Ð smaller in the sense of being able to fit into the short term focus of attention, and greater in the sense of having holistic characteristics which are able to guide its manipulation. REFERENCES Barnard, Tony, & Tall, David O. (1997). Cognitive units, connections and mathematical proof. Proceedings of the 21st International Conference for the Psychology of Mathematics Education, 2, 41± 48. Davis, Robert B. (1984). Learning mathematics: The cognitive science approach to mathematics education. Norwood, NJ: Ablex. Dubinsky, Ed. (1991). Reflective abstraction in advanced mathematical thinking. In D.O. Tall (Ed.), Advanced mathematical thinking (pp. 95 ±123). Dordrecht: Kluwer. Gray, Eddie M., & Tall, David O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115±141. Krutetskii, Vadim A. (1976). J. Kilpatrick & I. Wirzup. (Eds.), The psychology of mathematical abilities in schoolchildren (Joan Teller, Trans.). Chicago IL: University of Chicago. Miller, George. (1956). The magic number 72. Psychological Review, 63, 81± 97. Sfard, Anna. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1± 36. Skemp, Richard R. (1971). The psychology of learning mathematics. London: Penguin Books. Skemp, Richard R. (1979). Intelligence, learning and action. London: Wiley.