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Mathematics is one of the most important subjects. Mathematics is a subject of numbers, shapes, data, measurements and also logical activities. It has a huge scope in every field of our life, such as medicine, engineering, finance, natural science, economics, etc. We are all surrounded by a mathematical world.
The concepts, theories and formulas that we learn in Maths books have huge applications in real-life. To find the solutions for various problems we need to learn the formulas and concepts. Therefore, it is important to learn this subject to understand its various applications and significance.
There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics “the science that draws necessary conclusions”. Albert Einstein, on the other hand, stated that “as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”
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Newton and Leibniz
The essential insight of Newton and Leibniz was to use Cartesian algebra to synthesize the earlier results and to develop algorithms that could be applied uniformly to a wide class of problems. The formative period of Newton’s researches was from 1665 to 1670, while Leibniz worked a few years later, in the 1670s. Their contributions differ in origin, development, and influence, and it is necessary to consider each man separately.
Newton, the son of an English farmer, became in 1669 the Lucasian Professor of Mathematics at the University of Cambridge. Newton’s earliest researches in mathematics grew in 1665 from his study of van Schooten’s edition of La Géométrie and Wallis’s Arithmetica Infinitorum. Using the Cartesian equation of the curve, he reformulated Wallis’s results, introducing for this purpose infinite sums in the powers of an unknown x, now known as infinite series. Possibly under the influence of Barrow, he used infinitesimals to establish for various curves the inverse relationship of tangents and areas. The operations of differentiation and integration emerged in his work as analytic processes that could be applied generally to investigate curves.
Unusually sensitive to questions of rigour, Newton at a fairly early stage tried to establish his new method on a sound foundation using ideas from kinematics. A variable was regarded as a “fluent,” a magnitude that flows with time; its derivative or rate of change with respect to time was called a “fluxion,” denoted by the given variable with a dot above it. The basic problem of the calculus was to investigate relations among fluents and their fluxions. Newton finished a treatise on the method of fluxions as early as 1671, although it was not published until 1736. In the 18th century this method became the preferred approach to the calculus among British mathematicians, especially after the appearance in 1742 of Colin Maclaurin’s influential Treatise of Fluxions.
Newton first published the calculus in Book I of his great Philosophiae Naturalis Principia Mathematica (1687; Mathematical Principles of Natural Philosophy). Originating as a treatise on the dynamics of particles, the Principia presented an inertial physics that combined Galileo’s mechanics and Kepler’s planetary astronomy. It was written in the early 1680s at a time when Newton was reacting against Descartes’s science and mathematics. Setting aside the analytic method of fluxions, Newton introduced in 11 introductory lemmas his calculus of first and last ratios, a geometric theory of limits that provided the mathematical basis of his dynamics.
Newton’s use of the calculus in the Principia is illustrated by proposition 11 of Book I: if the orbit of a particle moving under a centripetal force is an ellipse with the centre of force at one focus, then the force is inversely proportional to the square of the distance from the centre. Because the planets were known by Kepler’s laws to move in ellipses with the Sun at one focus, this result supported his inverse square law of gravitation. To establish the proposition, Newton derived an approximate measure for the force by using small lines defined in terms of the radius (the line from the force centre to the particle) and the tangent to the curve at a point. This result expressed geometrically the proportionality of force to vector acceleration. Using properties of the ellipse known from classical geometry, Newton calculated the limit of this measure and showed that it was equal to a constant times 1 over the square of the radius.
Newton avoided analytic processes in the Principia by expressing magnitudes and ratios directly in terms of geometric quantities, both finite and infinitesimal. His decision to eschew analysis constituted a striking rejection of the algebraic methods that had been important in his own early researches on the calculus. Although the Principia was of inestimable value for later mechanics, it would be reworked by researchers on the Continent and expressed in the mathematical idiom of the Leibnizian calculus.
Leibniz’s interest in mathematics was aroused in 1672 during a visit to Paris, where the Dutch mathematician Christiaan Huygens introduced him to his work on the theory of curves. Under Huygens’s tutelage Leibniz immersed himself for the next several years in the study of mathematics. He investigated relationships between the summing and differencing of finite and infinite sequences of numbers. Having read Barrow’s geometric lectures, he devised a transformation rule to calculate quadratures, obtaining the famous infinite series for π/4:Error! Filename not specified.
Leibniz was interested in questions of logic and notation, of how to construct a characteristica universalis for rational investigation. After considerable experimentation he arrived by the late 1670s at an algorithm based on the symbols d and ∫. He first published his research on differential calculus in 1684 in an article in the Acta Eruditorum, “Nova Methodus pro Maximis et Minimis, Itemque Tangentibus, qua nec Fractas nec Irrationales Quantitates Moratur, et Singulare pro illi Calculi Genus” (“A New Method for Maxima and Minima as Well as Tangents, Which Is Impeded Neither by Fractional nor by Irrational Quantities, and a Remarkable Type of Calculus for This”). In this article he introduced the differential dx satisfying the rules d(x + y) = dx + dy and d(xy) = xdy + ydx and illustrated his calculus with a few examples. Two years later he published a second article, “On a Deeply Hidden Geometry,” in which he introduced and explained the symbol ∫ for integration. He stressed the power of his calculus to investigate transcendental curves, the very class of “mechanical” objects Descartes had believed lay beyond the power of analysis, and derived a simple analytic formula for the cycloid.
Leibniz continued to publish results on the new calculus in the Acta Eruditorum and began to explore his ideas in extensive correspondence with other scholars. Within a few years he had attracted a group of researchers to promulgate his methods, including the brothers Johann Bernoulli and Jakob Bernoulli in Basel and the priest Pierre Varignon and Guillaume-François-Antoine de L’Hospital in Paris. In 1700 he persuaded Frederick William I of Prussia to establish the Brandenburg Society of Sciences (later renamed the Berlin Academy of Sciences), with himself appointed president for life.
Leibniz’s vigorous espousal of the new calculus, the didactic spirit of his writings, and his ability to attract a community of researchers contributed to his enormous influence on subsequent mathematics. In contrast, Newton’s slowness to publish and his personal reticence resulted in a reduced presence within European mathematics. Although the British school in the 18th century included capable researchers, Abraham de Moivre, James Stirling, Brook Taylor, and Maclaurin among them, they failed to establish a program of research comparable to that established by Leibniz’s followers on the Continent. There is a certain tragedy in Newton’s isolation and his reluctance to acknowledge the superiority of continental analysis. As the historian Michael Mahoney observed:
Whatever the revolutionary influence of the Principia, mathematics would have looked much the same if Newton had never existed. In that endeavour he belonged to a community, and he was far from indispensable to it.
The 18th century
Institutional background
After 1700 a movement to found learned societies on the model of Paris and London spread throughout Europe and the American colonies. The academy was the predominant institution of science until it was displaced by the university in the 19th century. The leading mathematicians of the period, such as Leonhard Euler, Jean Le Rond d’Alembert, and Joseph-Louis Lagrange, pursued academic careers at St. Petersburg, Paris, and London.
The French Academy of Sciences (Paris) provides an informative study of the 18th-century learned society. The academy was divided into six sections, three for the mathematical and three for the physical sciences. The mathematical sections were for geometry, astronomy, and mechanics, the physical sections for chemistry, anatomy, and botany. Membership in the academy was divided by section, with each section contributing three pensionnaires, two associates, and two adjuncts. There was also a group of free associates, distinguished men of science from the provinces, and foreign associates, eminent international figures in the field. A larger group of 70 corresponding members had partial privileges, including the right to communicate reports to the academy. The administrative core consisted of a permanent secretary, treasurer, president, and vice president. In a given year the average total membership in the academy was 153.
Prominent characteristics of the academy included its small and elite membership, made up heavily of men from the middle class, and its emphasis on the mathematical sciences. In addition to holding regular meetings and publishing memoirs, the academy organized scientific expeditions and administered prize competitions on important mathematical and scientific questions.
The historian Roger Hahn noted that the academy in the 18th century allowed “the coupling of relative doctrinal freedom on scientific questions with rigorous evaluations by peers,” an important characteristic of modern professional science. Academic mathematics and science did, however, foster a stronger individualistic ethos than is usual today. A determined individual such as Euler or Lagrange could emphasize a given program of research through his own work, the publications of the academy, and the setting of the prize competitions. The academy as an institution may have been more conducive to the solitary patterns of research in a theoretical subject like mathematics than it was to the experimental sciences. The separation of research from teaching is perhaps the most striking characteristic that distinguished the academy from the model of university-based science that developed in the 19th century.
Analysis and mechanics
The scientific revolution had bequeathed to mathematics a major program of research in analysis and mechanics. The period from 1700 to 1800, “the century of analysis,” witnessed the consolidation of the calculus and its extensive application to mechanics. With expansion came specialization as different parts of the subject acquired their own identity: ordinary and partial differential equations, calculus of variations, infinite series, and differential geometry. The applications of analysis were also varied, including the theory of the vibrating string, particle dynamics, the theory of rigid bodies, the mechanics of flexible and elastic media, and the theory of compressible and incompressible fluids. Analysis and mechanics developed in close association, with problems in one giving rise to concepts and techniques in the other, and all the leading mathematicians of the period made important contributions to mechanics.
The close relationship between mathematics and mechanics in the 18th century had roots extending deep into Enlightenment thought. In the organizational chart of knowledge at the beginning of the preliminary discourse to the Encyclopédie, Jean Le Rond d’Alembert distinguished between “pure” mathematics (geometry, arithmetic, algebra, calculus) and “mixed” mathematics (mechanics, geometric astronomy, optics, art of conjecturing). Mathematics generally was classified as a “science of nature” and separated from logic, a “science of man.” The modern disciplinary division between physics and mathematics and the association of the latter with logic had not yet developed.
Mathematical mechanics itself as it was practiced in the 18th century differed in important respects from later physics. The goal of modern physics is to explore the ultimate particulate structure of matter and to arrive at fundamental laws of nature to explain physical phenomena. The character of applied investigation in the 18th century was rather different. The material parts of a given system and their interrelationship were idealized for the purposes of analysis. A material object could be treated as a point-mass (a mathematical point at which it is assumed all the mass of the object is concentrated), as a rigid body, as a continuously deformable medium, and so on. The intent was to obtain a mathematical description of the macroscopic behaviour of the system rather than to ascertain the ultimate physical basis of the phenomena. In this respect the 18th-century viewpoint is closer to modern mathematical engineering than it is to physics.
Mathematical research in the 18th century was coordinated by the Paris, Berlin, and St. Petersburg academies, as well as by several smaller provincial scientific academies and societies. Although England and Scotland were important centres early in the century, with Maclaurin’s death in 1746 the British flame was all but extinguished.
History of analysis
The history of analysis in the 18th century can be followed in the official memoirs of the academies and in independently published expository treatises. In the first decades of the century the calculus was cultivated in an atmosphere of intellectual excitement as mathematicians applied the new methods to a range of problems in the geometry of curves. The brothers Johann and Jakob Bernoulli showed that the shape of a smooth wire along which a particle descends in the least time is the cycloid, a transcendental curve much studied in the previous century. Working in a spirit of keen rivalry, the two brothers arrived at ideas that would later develop into the calculus of variations. In his study of the rectification of the lemniscate, a ribbon-shaped curve discovered by Jakob Bernoulli in 1694, Giulio Carlo Fagnano (1682–1766) introduced ingenious analytic transformations that laid the foundation for the theory of elliptic integrals. Nikolaus I Bernoulli (1687–1759), the nephew of Johann and Jakob, proved the equality of mixed second-order partial derivatives and made important contributions to differential equations by the construction of orthogonal trajectories to families of curves. Pierre Varignon (1654–1722), Johann Bernoulli, and Jakob Hermann (1678–1733) continued to develop analytic dynamics as they adapted Leibniz’s calculus to the inertial mechanics of Newton’s Principia.
Geometric conceptions and problems predominated in the early calculus. This emphasis on the curve as the object of study provided coherence to what was otherwise a disparate collection of analytic techniques. With its continued development, the calculus gradually became removed from its origins in the geometry of curves, and a movement emerged to establish the subject on a purely analytic basis. In a series of textbooks published in the middle of the century, the Swiss mathematician Leonhard Euler systematically accomplished the separation of the calculus from geometry. In his Introductio in Analysin Infinitorum (1748; Introduction to the Analysis of the Infinite), he made the notion of function the central organizing concept of analysis:
A function of a variable quantity is an analytical expression composed in any way from the variable and from numbers or constant quantities.
Euler’s analytic approach is illustrated by his introduction of the sine and cosine functions. Trigonometry tables had existed since antiquity, and the relations between sines and cosines were commonly used in mathematical astronomy. In the early calculus mathematicians had derived in their study of periodic mechanical phenomena the differential equationError! Filename not specified.and they were able to interpret its solution geometrically in terms of lines and angles in the circle. Euler was the first to introduce the sine and cosine functions as quantities whose relation to other quantities could be studied independently of any geometric diagram.
Euler’s analytic approach to the calculus received support from his younger contemporary Joseph-Louis Lagrange, who, following Euler’s death in 1783, replaced him as the leader of European mathematics. In 1755 the 19-year-old Lagrange wrote to Euler to announce the discovery of a new algorithm in the calculus of variations, a subject to which Euler had devoted an important treatise 11 years earlier. Euler had used geometric ideas extensively and had emphasized the need for analytic methods. Lagrange’s idea was to introduce the new symbol δ into the calculus and to experiment formally until he had devised an algorithm to obtain the variational equations. Mathematically quite distinct from Euler’s procedure, his method required no reference to the geometric configuration. Euler immediately adopted Lagrange’s idea, and in the next several years the two men systematically revised the subject using the new techniques.
In 1766 Lagrange was invited by the Prussian king, Frederick the Great, to become mathematics director of the Berlin Academy. During the next two decades he wrote important memoirs on nearly all of the major areas of mathematics. In 1788 he published his famous Mécanique analytique, a treatise that used variational ideas to present mechanics from a unified analytic viewpoint. In the preface Lagrange wrote:
One will find no Figures in this Work. The methods that I present require neither constructions nor geometrical or mechanical reasonings, but only algebraic operations, subject to a regular and uniform course. Those who admire Analysis, will with pleasure see Mechanics become a new branch of it, and will be grateful to me for having extended its domain.
Following the death of Frederick the Great, Lagrange traveled to Paris to become a pensionnaire of the Academy of Sciences. With the establishment of the École Polytechnique (French: “Polytechnic School”) in 1794, he was asked to deliver the lectures on mathematics. There was a concern in European mathematics at the time to place the calculus on a sound basis, and Lagrange used the occasion to develop his ideas for an algebraic foundation of the subject. The lectures were published in 1797 under the title Théorie des fonctions analytiques (“Theory of Analytical Functions”), a treatise whose contents were summarized in its longer title, “Containing the Principles of the Differential Calculus Disengaged from All Consideration of Infinitesimals, Vanishing Limits, or Fluxions and Reduced to the Algebraic Analysis of Finite Quantities.” Lagrange published a second treatise on the subject in 1801, a work that appeared in a revised and expanded form in 1806.
The range of subjects presented and the consistency of style distinguished Lagrange’s didactic writings from other contemporary expositions of the calculus. He began with Euler’s notion of a function as an analytic expression composed of variables and constants. He defined the “derived function,” or derivative f′(x) of f(x), to be the coefficient of i in the Taylor expansion of f(x + i). Assuming the general possibility of such expansions, he attempted a rather complete theory of the differential and integral calculus, including extensive applications to geometry and mechanics. Lagrange’s lectures represented the most advanced development of the 18th-century analytic conception of the calculus.
Beginning with Baron Cauchy in the 1820s, later mathematicians used the concept of limit to establish the calculus on an arithmetic basis. The algebraic viewpoint of Euler and Lagrange was rejected. To arrive at a proper historical appreciation of their work, it is necessary to reflect on the meaning of analysis in the 18th century. Since Viète, analysis had referred generally to mathematical methods that employed equations, variables, and constants. With the extensive development of the calculus by Leibniz and his school, analysis became identified with all calculus-related subjects. In addition to this historical association, there was a deeper sense in which analytic methods were fundamental to the new mathematics. An analytic equation implied the existence of a relation that remained valid as the variables changed continuously in magnitude. Analytic algorithms and transformations presupposed a correspondence between local and global change, the basic concern of the calculus. It is this aspect of analysis that fascinated Euler and Lagrange and caused them to see in it the “true metaphysics” of the calculus.
Other developments
During the period 1600–1800 significant advances occurred in the theory of equations, foundations of Euclidean geometry, number theory, projective geometry, and probability theory. These subjects, which became mature branches of mathematics only in the 19th century, never rivaled analysis and mechanics as programs of research.
Theory of equations
After the dramatic successes of Niccolò Fontana Tartaglia and Lodovico Ferrari in the 16th century, the theory of equations developed slowly, as problems resisted solution by known techniques. In the later 18th century the subject experienced an infusion of new ideas. Interest was concentrated on two problems. The first was to establish the existence of a root of the general polynomial equation of degree n. The second was to express the roots as algebraic functions of the coefficients or to show why it was not, in general, possible to do so.
The proposition that the general polynomial with real coefficients has a root of the form a + bSquare root of√−1 became known later as the fundamental theorem of algebra. By 1742 Euler had recognized that roots appear in conjugate pairs; if a + bSquare root of√−1 is a root, then so is a − bSquare root of√−1. Thus, if a + bSquare root of√−1 is a root of f(x) = 0, then f(x) = (x2 − 2ax − a2 − b2)g(x). The fundamental theorem was therefore equivalent to asserting that a polynomial may be decomposed into linear and quadratic factors. This result was of considerable importance for the theory of integration, since by the method of partial fractions it ensured that a rational function, the quotient of two polynomials, could always be integrated in terms of algebraic and elementary transcendental functions.
Although d’Alembert, Euler, and Lagrange worked on the fundamental theorem, the first successful proof was developed by Carl Friedrich Gauss in his doctoral dissertation of 1799. Earlier researchers had investigated special cases or had concentrated on showing that all possible roots were of the form a ± bSquare root of√−1. Gauss tackled the problem of existence directly. Expressing the unknown in terms of the polar coordinate variables r and θ, he showed that a solution of the polynomial would lie at the intersection of two curves of the form T(r, θ) = 0 and U(r, θ) = 0. By a careful and rigorous investigation he proved that the two curves intersect.
Gauss’s demonstration of the fundamental theorem initiated a new approach to the question of mathematical existence. In the 18th century mathematicians were interested in the nature of particular analytic processes or the form that given solutions should take. Mathematical entities were regarded as things that were given, not as things whose existence needed to be established. Because analysis was applied in geometry and mechanics, the formalism seemed to possess a clear interpretation that obviated any need to consider questions of existence. Gauss’s demonstration was the beginning of a change of attitude in mathematics, of a shift to the rigorous, internal development of the subject.
The problem of expressing the roots of a polynomial as functions of the coefficients was addressed by several mathematicians independently about 1770. The Cambridge mathematician Edward Waring published treatises in 1762 and 1770 on the theory of equations. In 1770 Lagrange presented a long expository memoir on the subject to the Berlin Academy, and in 1771 Alexandre Vandermonde submitted a paper to the French Academy of Sciences. Although the ideas of the three men were related, Lagrange’s memoir was the most extensive and most influential historically.
Lagrange presented a detailed analysis of the solution by radicals of second-, third-, and fourth-degree equations and investigated why these solutions failed when the degree was greater than or equal to five. He introduced the novel idea of considering functions of the roots and examining the values they assumed as the roots were permuted. He was able to show that the solution of an equation depends on the construction of a second resolvent equation, but he was unable to provide a general procedure for solving the resolvent when the degree of the original equation was greater than four. Although his theory left the subject in an unfinished condition, it provided a solid basis for future work. The search for a general solution to the polynomial equation would provide the greatest single impetus for the transformation of algebra in the 19th century.
The efforts of Lagrange, Vandermonde, and Waring illustrate how difficult it was to develop new concepts in algebra. The history of the theory of equations belies the view that mathematics is subject to almost automatic technical development. Much of the later algebraic work would be devoted to devising terminology, concepts, and methods necessary to advance the subject.
Foundations of geometry
Although the emphasis of mathematics after 1650 was increasingly on analysis, foundational questions in classical geometry continued to arouse interest. Attention centred on the fifth postulate of Book I of the Elements, which Euclid had used to prove the existence of a unique parallel through a point to a given line. Since antiquity, Greek, Islamic, and European geometers had attempted unsuccessfully to show that the parallel postulate need not be a postulate but could instead be deduced from the other postulates of Euclidean geometry. During the period 1600–1800 mathematicians continued these efforts by trying to show that the postulate was equivalent to some result that was considered self-evident. Although the decisive breakthrough to non-Euclidean geometry would not occur until the 19th century, researchers did achieve a deeper and more systematic understanding of the classical properties of space.
Interest in the parallel postulate developed in the 16th century after the recovery and Latin translation of Proclus’s commentary on Euclid’s Elements. The Italian researchers Christopher Clavius in 1574 and Giordano Vitale in 1680 showed that the postulate is equivalent to asserting that the line equidistant from a straight line is a straight line. In 1693 John Wallis, Savilian Professor of Geometry at Oxford, attempted a different demonstration, proving that the axiom follows from the assumption that to every figure there exists a similar figure of arbitrary magnitude.
In 1733 the Italian Girolamo Saccheri published his Euclides ab Omni Naevo Vindicatus (“Euclid Cleared of Every Flaw”). This was an important work of synthesis in which he provided a complete analysis of the problem of parallels in terms of Omar Khayyam’s quadrilateral (see the figure). Using the Euclidean assumption that straight lines do not enclose an area, he was able to exclude geometries that contain no parallels. It remained to prove the existence of a unique parallel through a point to a given line. To do this, Saccheri adopted the procedure of reductio ad absurdum; he assumed the existence of more than one parallel and attempted to derive a contradiction. After a long and detailed investigation, he was able to convince himself (mistakenly) that he had found the desired contradiction.
In 1766 Johann Heinrich Lambert of the Berlin Academy composed Die Theorie der Parallellinien (“The Theory of Parallel Lines”; published 1786), a penetrating study of the fifth postulate in Euclidean geometry. Among other theorems Lambert proved is that the parallel axiom is equivalent to the assertion that the sum of the angles of a triangle is equal to two right angles. He combined this fact with Wallis’s result to arrive at an unexpected characterization of classical space. According to Lambert, if the parallel postulate is rejected, it follows that for every angle θ less than 2R/3 (R is a right angle) an equilateral triangle can be constructed with corner angle θ. By Wallis’s result any triangle similar to this triangle must be congruent to it. It is therefore possible to associate with every angle a definite length, the side of the corresponding equilateral triangle. Since the measurement of angles is absolute, independent of any convention concerning the selection of units, it follows that an absolute unit of length exists. Hence, to accept the parallel postulate is to deny the possibility of an absolute concept of length.
The final 18th-century contribution to the theory of parallels was Adrien-Marie Legendre’s textbook Éléments de géométrie (Elements of Geometry and Trigonometry), the first edition of which appeared in 1794. Legendre presented an elegant demonstration that purported to show that the sum of the angles of a triangle is equal to two right angles. He believed that he had conclusively established the validity of the parallel postulate. His work attracted a large audience and was influential in informing readers of the new ideas in geometry.
The 18th-century failure to develop a non-Euclidean geometry was rooted in deeply held philosophical beliefs. In his Critique of Pure Reason (1781), Immanuel Kant had emphasized the synthetic a priori character of mathematical judgments. From this standpoint, statements of geometry and arithmetic were necessarily true propositions with definite empirical content. The existence of similar figures of different size, or the conventional character of units of length, appeared self-evident to mathematicians of the period. As late as 1824 Pierre-Simon, marquis de Laplace, wrote:
Thus the notion of space includes a special property, self-evident, without which the properties of parallels cannot be rigorously established. The idea of a bounded region, e.g., the circle, contains nothing which depends on its absolute magnitude. But if we imagine its radius to diminish, we are brought without fail to the diminution in the same ratio of its circumference and the sides of all the inscribed figures. This proportionality appears to me a more natural postulate than that of Euclid, and it is worthy of note that it is discovered afresh in the results of the theory of universal gravitation.
Craig G. Fraser
Mathematics in the 19th century
Most of the powerful abstract mathematical theories in use today originated in the 19th century, so any historical account of the period should be supplemented by reference to detailed treatments of these topics. Yet mathematics grew so much during this period that any account must necessarily be selective. Nonetheless, some broad features stand out. The growth of mathematics as a profession was accompanied by a sharpening division between mathematics and the physical sciences, and contact between the two subjects takes place today across a clear professional boundary. One result of this separation has been that mathematics, no longer able to rely on its scientific import for its validity, developed markedly higher standards of rigour. It was also freed to develop in directions that had little to do with applicability. Some of these pure creations have turned out to be surprisingly applicable, while the attention to rigour has led to a wholly novel conception of the nature of mathematics and logic. Moreover, many outstanding questions in mathematics yielded to the more conceptual approaches that came into vogue.
Projective geometry
The French Revolution provoked a radical rethinking of education in France, and mathematics was given a prominent role. The École Polytechnique was established in 1794 with the ambitious task of preparing all candidates for the specialist civil and military engineering schools of the republic. Mathematicians of the highest calibre were involved; the result was a rapid and sustained development of the subject. The inspiration for the École was that of Gaspard Monge, who believed strongly that mathematics should serve the scientific and technical needs of the state. To that end he devised a syllabus that promoted his own descriptive geometry, which was useful in the design of forts, gun emplacements, and machines and which was employed to great effect in the Napoleonic survey of Egyptian historical sites.
In Monge’s descriptive geometry, three-dimensional objects are described by their orthogonal projections onto a horizontal and a vertical plane, the plan and elevation of the object. A pupil of Monge, Jean-Victor Poncelet, was taken prisoner during Napoleon’s retreat from Moscow and sought to keep up his spirits while in jail in Saratov by thinking over the geometry he had learned. He dispensed with the restriction to orthogonal projections and decided to investigate what properties figures have in common with their shadows. There are several of these properties: a straight line casts a straight shadow, and a tangent to a curve casts a shadow that is tangent to the shadow of the curve. But some properties are lost: the lengths and angles of a figure bear no relation to the lengths and angles of its shadow. Poncelet felt that the properties that survive are worthy of study, and, by considering only those properties that a figure shares with all its shadows, Poncelet hoped to put truly geometric reasoning on a par with algebraic geometry.
In 1822 Poncelet published the Traité des propriétés projectives des figures (“Treatise on the Projective Properties of Figures”). From his standpoint every conic section is equivalent to a circle, so his treatise contained a unified treatment of the theory of conic sections. It also established several new results. Geometers who took up his work divided into two groups: those who accepted his terms and those who, finding them obscure, reformulated his ideas in the spirit of algebraic geometry. On the algebraic side it was taken up in Germany by August Ferdinand Möbius, who seems to have come to his ideas independently of Poncelet, and then by Julius Plücker. They showed how rich was the projective geometry of curves defined by algebraic equations and thereby gave an enormous boost to the algebraic study of curves, comparable to the original impetus provided by Descartes. Germany also produced synthetic projective geometers, notably Jakob Steiner (born in Switzerland but educated in Germany) and Karl Georg Christian von Staudt, who emphasized what can be understood about a figure from a careful consideration of all its transformations.
dualityDuality associates with the point P the line RS, and vice versa.
Within the debates about projective geometry emerged one of the few synthetic ideas to be discovered since the days of Euclid, that of duality. This associates with each point a line and with each line a point, in such a way that (1) three points lying in a line give rise to three lines meeting in a point and, conversely, three lines meeting in a point give rise to three points lying on a line and (2) if one starts with a point (or a line) and passes to the associated line (point) and then repeats the process, one returns to the original point (line). One way of using duality (presented by Poncelet) is to pick an arbitrary conic and then to associate with a point P lying outside the conic the line that joins the points R and S at which the tangents through P to the conic touch the conic. A second method is needed for points on or inside the conic. The feature of duality that makes it so exciting is that one can apply it mechanically to every proof in geometry, interchanging “point” and line” and “collinear” and “concurrent” throughout, and so obtain a new result. Sometimes a result turns out to be equivalent to the original, sometimes to its converse, but at a single stroke the number of theorems was more or less doubled.
Making the calculus rigorous
Monge’s educational ideas were opposed by Joseph-Louis Lagrange, who favoured a more traditional and theoretical diet of advanced calculus and rational mechanics (the application of the calculus to the study of the motion of solids and liquids). Eventually Lagrange won, and the vision of mathematics that was presented to the world was that of an autonomous subject that was also applicable to a broad range of phenomena by virtue of its great generality, a view that has persisted to the present day.
During the 1820s Augustin-Louis, Baron Cauchy, lectured at the École Polytechnique on the foundations of the calculus. Since its invention it had been generally agreed that the calculus gave correct answers, but no one had been able to give a satisfactory explanation of why this was so. Cauchy rejected Lagrange’s algebraic approach and proved that Lagrange’s basic assumption that every function has a power series expansion is in fact false. Newton had suggested a geometric or dynamic basis for calculus, but this ran the risk of introducing a vicious circle when the calculus was applied to mechanical or geometric problems. Cauchy proposed basing the calculus on a sophisticated and difficult interpretation of the idea of two points or numbers being arbitrarily close together. Although his students disliked the new approach, and Cauchy was ordered to teach material that the students could actually understand and use, his methods gradually became established and refined to form the core of the modern rigorous calculus, a subject now called mathematical analysis.
Traditionally, the calculus had been concerned with the two processes of differentiation and integration and the reciprocal relation that exists between them. Cauchy provided a novel underpinning by stressing the importance of the concept of continuity, which is more basic than either. He showed that, once the concepts of a continuous function and limit are defined, the concepts of a differentiable function and an integrable function can
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Mathematics is a universal language that helps us to understand the world, and it is a core part of the curriculum. As well as teaching about numbers, shapes, statistics and patterns, it provides important tools for work in areas such as physics, architecture, medicine and business. It helps learners to develop logical and methodical thinking, to focus and to solve a wide range of mathematical problems. Success can be ‘coordinated’ when leaders ensure all the elements of pupils’ mathematics education are supporting each pupil’s progress in the subject. Success in mathematics leads to many opportunities for further study and employment. Mathematics is a core subject of the national curriculum. It is an important entitlement for all pupils in England’s schools: academies, free schools and maintained schools.
At GCSE, with the exception of results in 2020 and 2021, the proportion of pupils attaining a grade 4 or above has steadily increased over time. In 2019, 71.5% of pupils obtained a grade 4 or above, and 20.4% of pupils obtained a grade 7 or higher. These are increases of 9.1 and 3.3 percentage points since 2013 The results of national reference tests, introduced by Ofqual in 2017, also reflect a steady improvement in pupil performance between 2017 and 2020. Data from 2021 does not follow this trend, which is likely due to the impact of pandemic restrictions. However, national reference test results in 2022 suggest that attainment is improving again, although it has not yet reached the levels recorded in the 2020 test.
Mathematics is increasingly the most popular choice at A level. Within the top 15 subjects chosen by students, mathematics and further mathematics are the subjects that have the highest proportion of A* and A grades awarded. In addition, level 3 core mathematics is an increasingly popular post-16 course taken by pupils on top of 3 A levels, or equivalents.
Although pupils in England, on average, perform better in mathematics than pupils in many other countries, there is a large gap between the lowest and highest achievers, and between disadvantaged and advantaged pupils. Studies and media reports continue to show that there is a shortage of specialist mathematics teachers and that this is a longstanding issue found in other countries.
This report evaluates the common strengths and weaknesses of mathematics in the schools inspected and considers the challenges that mathematics education faces. The evidence was gathered by His Majesty’s Inspectors as part of routine inspections. The report follows our mathematics research review published in 2021, which sets out our idea of a high-quality mathematics education. The report is split into findings in primary schools and those in secondary schools, and includes evidence from Reception classes and sixth forms. Within each of these sections, we talk about:
aspects of the curriculum
pedagogy
assessment
the way schools are organised
the impact of this on what pupils learn
It is important to note that we evaluate schools against the criteria in the school inspection handbooks. Findings from this report will not be used as a ‘tick list’ by inspectors when they are inspecting schools: we know that there are many different ways that schools can put together and teach a high-quality mathematics curriculum.
During the period of evidence-gathering, schools were facing many challenges because of COVID-19. As other studies have shown, pupils have been affected by remote learning, lockdowns and national restrictions. Therefore, some of the evidence gathered for this report may not represent ‘business as usual’. However, by focusing on the curriculum and its implementation over time, we hope that this risk has been reduced.
Key terms used in this report
Knowledge in mathematics
Throughout this report, we use the same terminology for mathematics knowledge as we used in our mathematics research review. These are not necessarily terms that Ofsted would expect pupils or teachers to use:
declarative knowledge: facts, concepts, formulae
procedural knowledge: methods, procedures, algorithms
conditional knowledge: strategies formed from the combinations of facts and methods to reason and problem-solve
Main findings
Primary schools
In the last few years, a resounding, positive shift in mathematics education has taken place in primary schools. Curriculum is now at the heart of leaders’ decisions and actions. Generic approaches, such as the expectation that all teaching should always be differentiated, have dissipated. We now see high quality curriculums, collaborative support for teachers and a focus on mathematics teaching. Leaders intend that pupils ‘keep up, not catch up’. These approaches set out a better path to proficiency for pupils.
Teachers help pupils to understand new concepts. Networks of support, such as the Maths Hubs, provide regular and highly useful training. This helps teachers to adopt new and improved ways of explaining and modelling concepts. Often, teachers use physical resources and pictorial representations to help pupils see underlying mathematical structures. They also teach and model new vocabulary, regularly check pupils’ understanding and swiftly pick up misconceptions.
There are some deficiencies in the quality and quantity of practice that pupils undertake. Even when teachers teach with clarity and precision, it is likely that these deficiencies undermine pupils’ ability to remember important knowledge. For older pupils, these deficiencies affect their ability to attain procedural fluency (speed and accuracy).
Pupils’ gaps in knowledge tend to be centred around, but not limited to, addition facts in younger year groups. This was for some, but not all pupils. These early gaps in knowledge may not become apparent until a significant amount of time has elapsed. This is because it is possible, in the medium term, for pupils to understand what is being taught and then keep up with extra classroom support and slower calculation. However, this is at the expense of later ability to access the curriculum.
Accountability measures and wide spreads of attainment tend to influence leaders’ decision making and resource allocation for Year 6 cohorts. Allocating additional resources to year 6 leaves leaders with fewer resources to invest in pupils’ earlier education. Further, a goal of true proficiency is superseded by ‘age related expectations’ which roughly equates to 50% accuracy in end of key stage tests. As a result, many pupils aren’t as prepared for the rigours of secondary education as they could be.
Secondary schools
Notable improvements have taken place in mathematics education in recent years. Widespread weaknesses identified at the time of Ofsted’s last mathematics subject report, around deficiencies in curriculum guidance and weaknesses in ongoing professional development for staff, are now much less likely to be evident in schools. However, some weaknesses identified in that report persist and continue to limit pupils’ learning of mathematics. The teaching of disparate skills to enable pupils to pass examinations but not equip them for the next stage of education, work and life, and weaknesses in the teaching of mathematical problem solving, remain areas of weakness across many schools. These weaknesses are disproportionately likely to be evident in schools that struggle to recruit and retain specialist mathematics teachers.
Recruiting and retaining high-quality, specialist, maths teachers is a challenge for many schools. Leaders’ curriculum decisions are increasingly influenced by the need to cope with these difficulties. Some schools that identify this problem do not take steps to develop the subject knowledge and subject specific pedagogical knowledge of less experienced and non-specialist teachers, usually provide a weaker mathematics education to their pupils.
A high quality of education leads to strong pupil outcomes, but this is not necessarily true in reverse. Strong exam outcomes do not, necessarily, indicate a high-quality mathematics education because, in some schools, pupils are taught a narrowed curriculum that allows them to be successful in exams without securing the mathematical knowledge they need to be successful later. These decisions are made because leaders and teachers are acutely aware of the impact of pupils achieving certain threshold grades in terms of post-16 opportunities, and implications for school accountability.
Curriculum planning around the teaching of mathematical facts and methods is usually strong. Where it is weaker, however, it tends to take less account of what pupils have learned previously and what they will study later. In these schools, pupils often are taught a series of disconnected mathematical methods and ‘tricks’ that apply only in specific circumstances.
Long term curriculum planning to develop pupils’ ability to use the facts and methods they have been taught to solve familiar, and unfamiliar, problems is uncommon. Curriculum decisions about problem solving are often left to individual class teachers. The quality of these decisions is variable. As a result, some pupils, particularly those who find learning mathematics more challenging and those taught by non-specialist teachers, are not effectively taught how to solve problems mathematically.
Maths leaders, and teachers, consistently emphasise the importance of clarity and technical accuracy in written, and spoken, mathematics. Where teachers model this accurately and explicitly, it is more likely to be present in pupils’ communication.
In most schools, exercises and activities are used by teachers, but in some schools, pupils are asked to undertake exercises and activities that are not carefully designed, or some pupils are moved on without having had sufficient practice to consolidate new learning.
Pupils who are learning mathematics more slowly than their peers frequently receive a mathematics education that does not meet their needs. They are often rushed through the study of new content, in order to ‘complete the course’, without securely learning what they are studying. This frequently results in pupils repeating content, in key stage 4 that they have already studied, but not learned, in key stage 3 (and 2). Often the curriculum for these pupils is narrowed with little teaching of how the facts and methods learned can be used to solve problems mathematically. Many of these pupils develop a negative view of mathematics.
Leaders’ application of GCSE grade thresholds to internal assessments gives false assurance about pupils’ learning. As a result of teachers and leaders accepting this level of achievement in internal assessments, some pupils are progressing through the mathematics curriculum with significant, and growing, gaps of knowledge.
In the vast majority of secondary schools, department meeting time is allocated to improving the quality of provision in mathematics as opposed to undertaking administrative tasks. Many leaders and teachers note that this had been a significant and positive change over the recent years.
Discussion of the findings
The overall picture of mathematics education in England is broadly healthy. This positive picture did not arise through chance but through the commitment of school leaders, teachers and members of the mathematics subject community.
Leaders prioritise creating or adopting a high-quality mathematics curriculum. They give careful and ongoing consideration to the effective teaching of that curriculum. Leaders make good use of support and resources from Maths Hubs, the National Centre for Excellence in the Teaching of Mathematics (NCTEM) and from commercial providers. As a result, many teachers receive high quality, subject-specific continual professional development (CPD) and there are flourishing formal and informal networks of teaching professionals.
This is a significant shift compared to when Ofsted’s last mathematics subject report, Made to Measure, noted that ‘very few schools provided curricular guidance for staff, underpinned by professional development that focused on enhancing subject knowledge and expertise in the teaching of mathematics, to ensure consistent implementation of approaches and policies’.
Given this very positive picture, this report explores the features of this widespread effective practice to support its replication. This report also considers the factors which might explain the continued weaknesses in outcomes for some children.
The phrase ‘coordinating mathematical success’ describes how effective schools make sure that curriculum plans, teaching approaches, pupil tasks, assessments and mechanisms for evolving these align well. When successful, each individual element is of high quality, and the elements work in harmony, together supporting pupils to learn effectively. It means setting out a path to proficiency in the subject, checking pupils are on that path and helping them to stay on that path.
Many of the features of the conception of quality, as outlined in our research review, are prevalent in the schools visited. For example, curriculum sequencing that includes ‘small steps’ approaches towards increasing mathematical proficiency, teaching that helps pupils to understand, and carefully curated opportunities to practise. In most schools, teachers routinely assess whether pupils have the necessary prior knowledge to undertake new learning. Primary school teachers quickly identify misconceptions, including through computerised tests.
However, it was when each element of mathematics education was of high quality, and those elements worked together, that pupils learnt most effectively. A centralised approach, with a carefully sequenced curriculum at its core, also supports schools faced with higher teacher turnover and/or teachers with less experience and subject knowledge. Even in schools without these challenges, teachers’ shared understanding of curriculum progression and high-quality teaching contributes to high quality mathematics education.
It is now common for teachers, in both primary and secondary schools, to receive regular subject-specific professional development. In primary schools, this support is often provided through Maths Hubs and is put in place by leaders who have a clear focus on developing teachers’ subject-specific teaching knowledge. Leaders’ strong understanding of high-quality mathematics education tends to be reflected in their monitoring foci. In secondary schools, it is much more common for professional development to take the form of departmental meetings that focus on curriculum design and effective curriculum practice.
In schools with less experienced or non-specialist teachers, there is a need to develop a shared understanding of curriculum progression and features of effective practice. Schools with more experienced subject-specialist staff engage in professional debates about how to teach aspects of the mathematics curriculum most effectively. Schools where mathematics provision is strongest ensure that other adults working with pupils, including teaching assistants and tutors, understand the curriculum and its implementation. Where these other adults do not have this shared understanding, the effectiveness of their support is limited.
There are examples in some schools of less successful practices. At primary level inspectors encountered curriculums that lack specified detail in the Reception Year, that allocate geometry to the summer term only or do not provide for enough learning of conditional knowledge. Sometimes questioning causes pupils to guess rather than recall. In other cases, multiple representations cause confusion rather than clarity.
In some schools, towards the end of primary and secondary phases, the focus of assessment shifts away from identifying pupils’ needs and moved towards exam or test preparation. This phenomenon is often coupled with an increase in resources to provide for interventions and, in primary schools, reduced class sizes. The need for significant ‘last-minute’ intervention in some schools suggests deficiencies in the curriculum, teaching or rehearsal earlier on in pupils’ mathematical education.
In most secondary schools visited, decisions about GCSE entry tiers are, in practice, taken at the end of Year 9, based on levels of attainment at that time rather than, more appropriately in Year 11. These early decisions about tiers of entry determine the curriculum pathway for these pupils in key stage 4 and this limits the mathematics that some pupils learn.
In some schools, the choice of mathematical methods taught is left to individual teachers’ preferences rather than taken as a broader curriculum decision. This means that the approaches that are taught may only be useful to answer questions of the precise type identified in the curriculum for a particular year or key stage. The opportunity to engineer success over time is lost and pupils in these schools typically develop what can be characterised as ‘disconnected pieces of knowledge’.
Pupil practice is sometimes limited in quality and quantity in both primary and secondary schools. This happens when leaders see practice as an activity, rather than focusing on its outcomes – whether pupils have practised until they have learned, to automaticity, the intended mathematical knowledge. There is often no consensus among leaders about benchmarks for optimal quality and quantity of practice that gives assurance that pupils have learned what is intended.
In some secondary schools, lack of revisiting conditional knowledge and, in primary schools, less emphasis on procedural automaticity (which develops through practice) is likely to compound this situation. Leaders are increasingly identifying the need for more practice, but not the reasons for this need: deficiencies that might be occurring in curriculum quality, teaching, or in the quality and quantity of practice in pupils’ main lessons, for example.
An ambitious curriculum is one that maximises the mathematics that pupils learn. In some schools, teachers move on before ensuring pupils have learned important knowledge and committed that knowledge to long term memory. In schools where this is common, leaders focus on what pupils study, rather than on what pupils learn. Moving on when pupils are not mathematically ready gives the illusion of progress but creates ever greater gaps that will take more time to address in the future. Some leaders gain false assurance about the effectiveness of their curriculum design and practice through internal assessments that closely align with ‘expected performance thresholds’ of external assessments. This approach often leads to an acceptance of pupils moving through the mathematics curriculum with significant gaps in their knowledge and leaders failing to make necessary adjustments to their curriculums. In these schools, some pupils would be better served by studying less, but securely learning more.
Curriculum thinking about problem-solving and reasoning differs between primary and secondary schools. Pupils need to learn strategies and the most useful combinations of facts and methods to solve types of problem. Since it is not possible for pupils to encounter every possible problem, a suitable curriculum can identify strategies to solve an identified range of problem types. In some primary schools, pupils self-selected the problems they will solve and in many secondary schools there is a lack of curriculum planning to ensure all pupils encounter a range of problem types and have practice solving these problems. In some cases, pupils have little or no opportunity to use the mathematical knowledge they have to reason mathematically or solve problems. This is especially an issue for pupils who find learning mathematics more challenging.
Recommendations
Curriculum
All schools should make sure that:
curriculums emphasise secure learning of, rather than encountering, mathematical knowledge.
curriculum sequencing prepares pupils for transitions between key stages and phases
Primary schools should make sure that:
they identify and sequence small steps in the Reception Year curriculum
all pupils learn to apply facts and methods to wider problem-solving
geometry knowledge is sequenced throughout, rather than at the end of, each year’s curriculum
Secondary schools should make sure that:
the curriculum specifies the mathematical methods that leaders want all pupils to learn, and that these form a coherent, ‘forward-facing’, base of mathematical knowledge rather than a collection of disconnected algorithms and tricks
key stage 4 curriculums focus on maximising pupils’ learning, and that decisions on GCSE tier of entry are based on what pupils know and can do towards the end of key stage 4 rather than what pupils knew and could do at the end of key stage 3, or earlier
the curriculum plans how pupils will learn conditional (problem-solving) knowledge over time, for example by:
clearly identifying the range of problems to which pupils should be able to apply their new declarative (facts) and procedural (methods) knowledge
teaching pupils how their new facts and methods can be used to solve a range of problems
providing all pupils with sufficient practice in solving problems using newly learned facts and methods
ensuring that all pupils have enough opportunities to practise solving problems, after they have first being taught, and that these opportunities require pupils to make decisions about how best to solve these unfamiliar problems
Pedagogy and assessment
All schools should:
make certain that teachers routinely check whether pupils have secure knowledge and understanding of prerequisite mathematics and address any gaps identified, before moving on to the next stage of learning
make sure that teachers regularly connect new learning to what pupils have learned before, including showing pupils how it connects with learning in other subjects
make sure that all pupils practise and consolidate new learning through well-designed exercises and activities, including sequences of problem-solving
check that pupils are developing ‘procedural fluency’ (speed and accuracy of recall of methods) and address gaps in pupils’ procedural knowledge at the earliest possible opportunity
Primary schools should:
consider using routines, keeping noise levels low and making sure that pupils are facing the teacher when they are explaining new content and giving instructions, to help them focus on what is being taught
help younger pupils to learn their addition facts by heart and regularly check their recall of this knowledge
reflect on the extent to which additional afternoon practice is due to deficiencies in the early curriculum and its implementation
aim for pupils to become proficient and ready for Year 7, rather than just meet age related expectations for end of key stage tests
make sure that questioning helps all pupils to recall and make connections, rather than allowing pupils to guess
provide pre-teaching, additional teaching and extra practice for most pupils with special educational needs and/or disabilities (SEND)
Secondary schools should:
make sure that pupils have sufficient opportunities to practise reasoning, explaining and problem-solving using the facts and methods they have been taught
make sure that assessment information is used to reflect on the effectiveness of the curriculum and the way it is implemented, to guide future improvements
Systems at subject and school level
All schools should:
provide continuing professional development for teaching assistants, and other adults working with pupils, to help them to understand the intended school mathematics curriculum and the way it is put into practice
Primary schools should:
make sure that discussions with leaders about progress specifically address the needs of the lowest attaining younger pupils
aim to prioritise resourcing for younger year groups, to better engineer success from the start of a pupil’s mathematics journey
when leaders observe lessons, focus on pupils’ thinking and the quality and quantity of practice they undertake
Secondary schools should:
make sure that non-specialist teachers receive the necessary professional development, including subject knowledge and subject specific pedagogical knowledge, to teach mathematics effectively
Other organisations
Other organisations should:
provide sufficient quality and quantity of practice within scheme resources
develop and offer computerised tests of facts and methods that share information about progress and attainment with pupils, teachers and leaders. Benchmarks for attainment should be based on proficiency
extend the remit of Maths Hubs to make sure that most schools benefit from them
offer the Mastering Number programme to all schools as an example of good practice in early mathematics
Those responsible for recruiting teachers should:
make sure that all pupils are taught by teachers with appropriate levels of subject knowledge and subject teaching knowledge
The Department for Education, Ofqual and Awarding bodies should:
explore whether the current design of the mathematics GCSE, including the tiers of entry offered and current typical grade thresholds, contribute to practices in schools that are not in pupils’ best interests
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