Appendices

APPENDIX 1 - CORRECTIONS, CLARIFICATIONS AND CHANGES

INTRODUCTION

This Appendix serves three purposes. First, it points out four misprints in the syllabus document. Secondly, it aims to clarify the status (included or excluded) of some topics or techniques. Thirdly, it summarises the changes from the 1987 syllabus and indicates the status of some topics about which there have been questions.

CORRECTIONS

The following misprints occur in the syllabus document:

1. On p. 9, in "Number systems" paragraph 4, in the list of rules for indices: (ap)aq
   should read: (ap)q
2. On p.11, in "Algebra" paragraph 5, the equation (a/bx+c)+...+(p/qx+r)=d/e
   should read (a/bx+c)+(p/qx+r)=d/e
   that is, manipulation is restricted to two terms of the given form.
3. On p. 21, in "Applied arithmetic and measure"paragraph 2,
   time(s)
   should read time (s)
4. On p. 21, in "Applied arithmetic and measure"paragraph 3, the final "note" in the right-hand column is badly positioned; it should be opposite the entry "Application to problems."

CLARIFICATIONS

No syllabus document can specify the exact scope of the content or depth of treatment intended (and liable to be tested in state examinations). For a syllabus that has undergone only minor revisions, the nuances of the previous syllabus are deemed to guide the interpretation of the present one except where otherwise stated. The intended changes are flagged in the table in the following section of this appendix. This table also aims to clarify the status of some topics which were not mentioned explicitly in the 1987 syllabus but were deemed to be included, and of other topics which were dropped from the syllabus when it was revised in 1987 but have continued to be taught to at least some students.

• An example of a topic not actually mentioned in the 1987 syllabus ­ or in its predecessor of 1973 ­ but deemed to be part of the intended syllabus throughout the period isperimeter. Perimeter is mentioned explicitly in the present syllabus, but this does not constitute a change of content.
• An example of an item excluded from the syllabus in 1987, but apparently still used in a number of classrooms, is the notation N0for the set of natural numbers excluding 0. The notation was used in the 1973 syllabus but was removed from the syllabus in 1987.Itis worth emphasising here, since international practicevaries, that the set N includes 0.
• Of course, the fact that a topic is excluded from the syllabus does not mean that it cannot be taught in the classroom. It does indicate, however, that knowledge of the topic is not necessary for answering questions in the Junior Certificate examination.

While the inclusion or exclusion of topics can be tabulated, the intended depth of coverage is not easily captured in a succinct tabular summary. The set of proposed sample assessment materials, drawn up and circulated to schools by the Department of Education and Science, provides one indicator of the required depth for various topics. Naturally, however, these only sample the topics and techniques which might be examined. Further instances are provided in Section 5.6, where some typical "question parts" are presented.

Additional guidance is offered with regard to the limits in the depth of coverage in a number of cases. The following examples are outside the scope of the syllabus.

1. The expression of repeating decimals as fractions.
2. The division of compound surds.
3. The formation of a quadratic equation from given roots.
4. Calculation of the volume of, say, a hut with a pitched roof or a bar of chocolate with triangular cross-section. (The syllabus limits problems on this topic to objects with rectangular cross-section.)
5. Uses of linear and quadratic graphs that require manipulation of the algebraic expressions in addition to graph-reading skills. For example, determine from a graph of a function f(x) the values of x for which f(x) > 6.

INS AND OUTS: SUMMARY OF SIGNIFICANT CHANGES AND CLARIFICATIONS

Material excluded shown in Roman script; material included shown in Italic script.

***Use of calculators included; use permitted in Junior Certificate examinations***

APPENDIX 2 - NOTES ON GEOMETRY

CONTEXT

The purpose of this appendix is to outline the logical structure of the geometry section of the course, in order to provide background information for teachers and to clarify what will be expected of Higher Level students in the examination, vis-à-vis proofs of theorems.

Difficulties have been experienced in the past, owing to the fact that the geometry on the 1987 syllabus consists of a mixture of a transformation-based approach with a traditional one based on congruence. The revised syllabus eliminates this dual approach from the formal treatment of geometry. Transformations are removed from the formal treatment, so that the system adopted is a congruence-based one. It is largely built upon the same ideas as those used by Euclid, but supplemented, as is common in modern treatments, by the use of measure (length/distance, angle measure, and area).

It should be noted that the geometry section of the syllabus is the vehicle by which students are first introduced the ideas of formal deductive reasoning. It is therefore important that by the time they finish the course, they will have gained an appreciation of the manner in which results are built up in a coherent and logical way within a formal system. It is hoped that the revised version of the syllabus facilitates this to a greater extent than its predecessor.

Many people rightly express enormous appreciation for the monument that Euclid left to humankind in producing TheElements. That it was one of the greatest triumphs in the development of mathematical thinking, however, does not necessarily imply that it is the most suitable course for second level students to follow. By starting with a sparse set of assumed results (axioms), Euclid had a long task ahead in building up logically to the geometric results familiar to us all. Pythagoras' Theorem, for example, is Proposition 47 in his first book, and the ratio results for triangles do not appear until Book 6. In providing a reasonable modern course for second level students, it is desirable to reach some interesting, familiar and applicable results within a reasonable timeframe. After all, along with the formal proofs of theorems, we want the students to be able to demonstrate the ability to apply the results to new problems. This is best achieved when the students have available to them a varied battery of interesting theorems with which to work. Because of this, the geometry course adopted is one involving a highly redundant axiom system.

In other words, we assume far more results than are strictly necessary. We still demand, however, the intellectual rigour required in using these results and these results only (along with some assumed general properties) to build up towards the remaining results on the course.

The syllabus lists a number of geometric statements as "facts". The word "fact" is to be taken here as synonymous with the word axiom. It was decided that an appreciation of the exact meaning of the term axiom in a formal system is not accessible to all students at this level. The term "fact" reflects the idea that it is sufficient for the students to conceive of these statements as ones which we accept as being true without requiring any proof. It should of course be possible for students to appreciate that since all proofs rely only on results already proved, then one cannot get started at all if one is not prepared to assume something.

As mentioned elsewhere, it is most important that students encounter the proofs of all of the theorems, including those that are not examinable. Otherwise, there is a great risk of failure to appreciate the building up of results in a logically sound way.

NOTES ON THE PRELIMINARY CONCEPTS AND ASSUMED PROPERTIES

This section provides supplementary material to that in the syllabus, in order to clarify the intended definitions and assumed properties of the geometric objects considered. The extent to which teachers address the details explicitly in class is a matter for their own discretion. Note that theterminology and detail below are not designed for studentconsumption; many teachers, however, want clarity of definitions for their own benefit. Whether or not they are addressed explicitly in class, these properties nonetheless constitute those that it is legitimate to assume in presenting proofs at examination.

No attempt is made to define the terms point, line, plane. It is expected that an intuitive understanding of these will be acquired by analogy and in other ways (for example, "the plane is like a page that stretches on for ever in all directions"). It should of course be noted that the plane is an infinite set of points, that lines are subsets of the plane, and so forth. Students need to be aware that a line goes on for ever in both directions. Teachers generally have little difficulty in making clear the distinction between lines, segments and half-lines, and an intuitive understanding of terms such as "between" and "on the same side as" can be assumed in order to define these if required. For the present, lower case letters will continue to be used to denote points and upper case letters to denote lines and circles. (Lines of course may also be referred to by giving two points.) Notation for segments and half-lines also remains the same as before. Thus: line ab, line segment [ab], and half-line [ab. Three or more points that lie on the same line are called collinear.

It should be noted that, unlike Euclid, we are in a position to take advantage of the power of real numbers, the properties of which have been established on a solid logical foundation independent of geometry. Accordingly, we have a concept called distance or length and a concept called angle measure. The properties associated with these concepts are as follows.

Given any two points a and b, there is a real number called the distance from a to b, or the length of the line segment [ab]. By observation, distance has the following properties (for any a,b,c).

• |ab| = |ba|.
• If a = b then |ab| = 0. Otherwise |ab| > 0.
• If b lies on [ac] then |ac| = |ab| + |bc|.
• If b does not lie on [ac] then |ac| < |ab| + |bc|.
• Given any half-line [ab and any positive real numberk, there is a unique point c on [ab such that |ac| = k.

Without ever making reference to these properties, note that students will certainly assume the first three of them without even thinking about them, as they already have an intuitive understanding of length. The fourth property is perhaps not so immediate and hence is listed as a "fact" in the syllabus document (third "fact" on page 13). The fifth property simply states that one can measure and mark off a certain distance along a line from any point; once again students assume it unquestioningly.

By observation, we note the following basic properties of angles and angle measure.

• When two half-lines [ab and [ac have the same initial point a, two angles are formed. (It may be helpful to think of the two angles as the two [closed] regions of the plane.)
• Every angle has a measure, which is a real number of degrees in the range [0°, 360°].
• The two angles formed by [ab and [ac have measures that sum to 360°.
• If a is between b and c, the two angles are called straight angles, and the measure of each is 180°. Otherwise, one has measure less than 180°; this angle is referred to as bac. The other has measure greater than 180° and is referred to as bac reflex. (In the case of straight angles, the context is usually sufficient to determine which angle is being referred to as bac.)
• If the angle bac contains the half-line [ad, then |bac| = | bad| + | dac|. (This can also be formulated with more case-by-case detail to cover reflex angles in the required way.)
• Given any half-line [ab and any real number kin the range [0°,180°), there exists a unique half-line [ac on each side of ab such that | bac| = k°.
• If the angle bac contains the half-line [adand if | bac| < 180° then [ad intersects [bc].

Note that if students have an intuitive understanding of angles and their measurement, then the above properties are already known and used, although not verbalised. There is no need to address them unless the students raise the issue.

Note also that one of these properties, the fact that a straight angle measures 180°, is listed in the syllabus (as the first "fact"). This is done so that the first theorem can refer to it directly, to assist in developing the idea of building logically from assumed facts to theorems. Teachers may wish to discuss the rationale behind this "fact", referring to the idea that some unit has to be chosen to measure angles and that historically somebody decided to do this by dividing a full angle into 360 parts. This provides a nice opening into the possibility of some investigation of the history involved. It also lends itself well to the discussion about simply having to agree to start somewhere. There is the possibility too of investigating other angle measures with the calculator (gradient and radian).

Having sorted out the properties of angle measure, acute, right and obtuse angles are definable accordingly.

We take the following assumed properties of lines, points and circles and the following definitions. (Once again, the students will assume these properties anyway).

• Through any two distinct points there exists a unique line.
• Any two distinct lines have either one or no point in common.
• Two lines that have no point in common are calledparallel. Also, for any line L we take L|| L.
• · Given any point p and any line L, there is exactly one line through p parallel to L and exactly one line through p perpendicular to L. (Perpendicular is defined via angle measure. The distance from p to L is defined as the distance fromp to the point at which L intersects the perpendicular to L through p.)
• The definitions of parallel and perpendicular are extended in the obvious way to cover half-lines and line segments also, so that we may refer to line segments being parallel to each other, and so forth.
• Any given line and any given circle have 0, 1 or 2 points in common. (A suitable definition of circle is given below.)
• If a line and a circle have exactly one point in common, the line is said to be a tangent to the circle, and the point is called the point of contact.

DEFINING THE PLANE FIGURES

The rectilinear figures listed in the syllabus could be defined precisely as various intersections of half-planes. This is probably not be the most suitable approach with students, so non-rigorous definitions will suffice (for example, a triangle as any three-sided figure; a quadrilateral as any four-sided figure; a convex quadrilateral as one whose interior angles all measure less than 180°.) As convex is not a term in use at this level heretofore, teachers may wish to note that a region in the plane is called convex if, given any two points in the region, the line segment joining them is contained in the region. Intuitively, however, it is sufficient to understand that a convex quadrilateral is a quadrilateral that does not have any corners "sticking inwards". A few examples and counter-examples will clarify this with ease for students. Note that re-entrant (i.e. non-convex) quadrilaterals are not on the course.

Unlike in the classical world, modern definitions of geometric shapes are usually inclusive. In other words, a square is a rectangle, a rectangle is a parallelogram, and a parallelogram is a quadrilateral. A square is also a rhombus. The following definitions could be used:

A parallelogram is a quadrilateral in which opposite sides are parallel.

A rhombus is a parallelogram with all its sides equal in length.

A rectangle is a parallelogram with a right angle at each vertex.

A square is a rectangle with all its sides equal in length.

Note also that triangles and quadrilaterals are regions. Hence they have areas (see below). A circle, on the other hand, is not a region but a curve. A circle does not have an area (although it encloses a disc that has an area). Hence, one can (somewhat loosely) say: "the area enclosed by the circle" but not "the area of the circle". Note also that the length of a circle is the distance around the circle, whereas the length of a rectangle is not the distance around the edge of the rectangle, but rather the length of one of its sides.

A circle is the set of all points that are a given distance from a given point (the centre). Any line segment joining the centre to a point of the circle is called a radius. The given distance (the common length of all the radii of a circle) is called the radius-length. Where confusion would not arise, radius may be used instead of radius-length, as is common practice (e.g. "a circle of radius 5 cm").

AREA

Note that in this respect the syllabus differs significantly from the Leaving Certificate Ordinary Level syllabus introduced in 1992. In the latter syllabus, area is not an assumed concept. Rather, the theorems on that course build towards, among other things, establishing a definition of area. On this syllabus, however, area is taken to be an assumed concept; i.e. it is assumed that a plane figure has such a thing as an area and that this idea of area has certain properties. The properties assumed for area are as follows.

• Each rectilinear figure (triangle, quadrilateral, ...) has an area. The area is a positive real number. (We do not consider degenerate cases, so area is not zero.)
• Congruent triangles have equal area.
• If a rectilinear figure can be decomposed into two nonoverlapping rectilinear figures, then its area is equal to the sum of their two areas. (Non-overlapping means that their intersection consists of, at most, boundary line segments.)
• The area of a rectangle is equal to its length multiplied by its breadth.

The last two of these properties are listed as "facts" in the syllabus.

It is recognised that this approach to area via rectangles introduces a potential inconsistency into the system (which does not in fact materialise), and that the Leaving Certificate approach is logically more satisfactory. However, the approach adopted here facilitates to a greater extent the transfer of the understanding of area developed in the measure section. The conceptual development of area in that section usually involves investigations that include counting squares inside a region and similar activities; it thus relies heavily on rectangles.

LEGITIMATE PROOFS OF THEOREMS

A proof cannot be seen in isolation from the assumptions upon which it depends. When proofs of theorems are required in the examination, the context is the system of geometry as laid out in the syllabus and expanded upon here. Accordingly, proofs must rely only on the assumed properties, the facts listed in the syllabus and any theorems listed earlier in the syllabus than the one being proved.

In particular, teachers should note the following.

• Transformation techniques are not valid, since the definitions and properties of transformations are not in the system. Hence, one cannot use an axial symmetry to prove the isosceles triangle theorem, nor use an isometry in the similar triangles theorem.
• Results not on the syllabus cannot be parachuted in. For example, the proof ­ usually used in the Leaving Certificate course ­ that a line drawn parallel to one side of a triangle divides the other two sides in the same ratio involves drawing a set of parallel lines and relies on another result concerning transversals and parallels. This latter result is not on the Junior Certificate course, and hence that proof is not valid here. (Conversely, the envisaged proof on this course, which involves areas of triangles, is not valid in the context of the Leaving Certificate course.)
• Proofs need to be laid out in a clear and logical fashion. Proofs should be illustrated by well-labelled diagrams but such illustrations are not a substitute for written lines in a proof. In particular, they do not convey the sequence of the assertions, and are often not as precise in their meaning as the written form.
• Where practicable, each assertion made in a proof should have an associated reason given (reference to a fact or previous theorem or to how the earlier lines are being used). Words and logical connectives should be used as appropriate, so that when the written proof is verbalised, it makes sense as piece of language. Commonly used abbreviations are of course legitimate, for example, SAS for the Side Angle Side congruence rule, and so forth.

Consider, for example, the following two versions of the same proof of the isosceles triangle theorem.

Prove that if two sides of a triangle are equal in measure, then the angles opposite these sides are equal in measure.

Given: abc, |ab| = |ac|

To Prove: |abc| = |acb|

Construction: [ad]

Proof:

|ab| = |ac|
|ad| = |ad|
|db| = |dc|
SSS
| abc| = |acb|

Given: abc, with |ab| = |ac|

To Prove: | abc| = | acb|

Construction:Construct [ad], where d is the midpoint of [bc]

Proof: |ab| = |ac| (given). |ad| = |ad| (since they are the same segment). |db| = |dc| (since d is the midpoint of [bc]). abd is congruent to acd (by the SSS rule) | abc| = | acb| (These are corresponding parts of the congruent triangles.)

The second version is more complete and is a more effectively communicated argument, and therefore has more merit. There are, of course, other valid proofs of this theorem.

FINAL NOTE

It is important to appreciate that this appendix has concerned itself with the logical detail of the system from the perspective of formal proofs. Many other arguments, justifications and explorations of the results have a valuable role to play in teaching and learning of the material, even though they are not valid proofs for examination. For

example, as detailed in Section 4, justification of Pythagoras' Theorem by a variety of dissection methods is both beneficial and interesting. Explorations and informal reasoning based on paper-folding and on transformations are similarly to be encouraged. However, in proving the results in examination, candidates must be able to operate within the logical system.

APPENDIX 3 - RESOURCES

MATHEMATICS TEACHERS' ASSOCIATIONS

Irish Mathematics Teachers Association Information can be found at http://www.imta.ie

Other associations National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 20191-1593; email nctm@nctm.org, website address http://www.nctm.org

Mathematical Association, 259 London Road, Leicester, LE2 3BE, England

Association of Teachers of Mathematics, 7 Shaftesbury Street, Derby, DE23 8YB, England

These three associations are among the best sources of teaching materials, posters, and so forth.

MATHEMATICAL WEBSITES

The following is a sample selection of mathematical websites. Each is accompanied by a short review.

Cornell Theory Center Math and Science Gateway

Website address: http://www.tc.cornell.edu/Edu/MathSciGateway/math.html

Review:

A very comprehensive gateway site (a site that categorises and links to lots of other sites). It contains sections on

• General Topics
• Geometry
• Fractals
• History of Mathematics
• Tables, Constants and Definitions
• Mathematical Software.

Mathematics Teaching Resource Centre

Website address: http://www.qesn.meq.gouv.qc.ca/mapco/index.htm

Review:

This Canadian site has a notice board which tells what is new to the site, a framework for "Improving Student Performance in Mathematics", a resource toolkit, a staff lounge and many other mathematical links. To read most of the material on this site, a programme called Acrobat

Reader is required. It can be downloaded from the site, but be prepared to wait a while.

Great Math Programs

Website address: http://xahlee.org/PageTwo_dir/MathPrograms_dir/mathPrograms.html

Review:

This site gives a description of a wide range of recreational and educational mathematical software available mostly for the Macintosh platform rather than for PCs.

BBC Education

Website address: http://www.bbc.co.uk/education/megamaths/

Review:

Some fun numeracy games here can be played online. A good way to practice tables against the clock!

Internet Mathematics Library

Website address: http://forum.swarthmore.edu/library/

Review:

This is part of one of the most famous sites for mathematics education: the Math Forum at Swarthmore. There are many categorised links to other pages.

Mathematics with Alice

Website address: http://library.thinkquest.org/10977/

Review:

This is a quirky site based on the Lewis Carroll stories.

Yahoo Mathematics

Website address: http://dir.yahoo.com/science/mathematics/

Review:

This is a search engine and yields Yahoo's categorisation of many mathematical sites.

Center of Excellence for Science and Mathematics Education

Website address: http://cesme.utm.edu/MathLinks/mathlinks.htm

Review:

This is another resource with many links to other mathematical sites. Following some of the links brings the surfer to a whole page of links dedicated to helping students with solving word problems.

A selection of additional web-site addresses

Overview of the history of mathematics, with links to many interesting topics:

http://www-history.mcs.st-and.ac.uk/history/HistTopics/History_overview.html

Development of the symbols we use for numbers: http://www.islam.org/Mosque/ihame/Ref6.htm

Fibonacci numbers and the Golden Ratio: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.htmlhttp://math.holycross.edu/~davids/fibonacci/fibonacci.html

The Four-Colour Problem: http://129.237.247.243/wizard/pages/01391.html

Search for previously unknown Mersenne primes: http://www.mersenne.org/prime.htm

Biographies of female mathematicians (note the members of the Boole family): http://www.agnesscott.edu/lriddle/women/women.htm

The American Mathematical Society's pages on what is new in mathematics (rather advanced, but may be of interest): http://www.ams.org/new-in-math/ (see for example a reference to William Rowan Hamilton: http://www.ams.org/new-in-math/cover/dna-abc2.html )

Logo: http://www.terrapinlogo.com/ (for instance, follow the route: FOR EDUCATORS/Why use Logo?) http://www.softronix.com (and look at information and on-line books on MSW Logo) http://www.lcsi.ca/ (and for instance look at "links")

Many enjoyable activities and puzzles: http://www.mathsyear2000.org/

This is just a short selection of some of the sites to give mathematics teachers a flavour of what is available. They contain, among other things, class plans, chat groups, mathematical games and puzzles, free software, reviews of programs, interesting projects and links with schools. See also http://www.scoilnet.ie for ideas and references submitted by Irish teachers.

BOOKS

Lesson ideas

The books in this category contain ideas for teaching mathematics (in some cases alongside discussion of mathematics or mathematics education).

Bolt, B. Mathematical Activities: a Resource Book forTeachers. Cambridge: Cambridge University Press, 1982 [and subsequent books of activities by Bolt].

Brophy, Tim. Mathematics and Science with MathView. John F Marshall, 1997.

Cotton, David. Mathematics Lessons at a Moment's Notice. London: Foulsham, 1986.

Creative Publications. Algebra with Pizazz and MiddleSchool Math with Pizazz. [See the website: http://www.creativepublications.com]

Jones, Lesley, ed. Teaching Mathematics and Art. Cheltenham: Stanley Thornes, 1991.

Kjartan, Poskitt. Murderous Maths. Hippo Publications, 1997.

Mitchell, Merle. Mathematical History: Activities, Puzzles,Stories and Games. Reston, VA: NCTM, 1978.

Pappas, Theoni. Mathematics Appreciation. San Carlos, CA: Wide World Publishing / Tetra, 1987. [Postal address: PO Box 476, San Carlos, CA 94070]

Sawyer, W. W. Vision in Elementary Mathematics. Harmondsworth: Penguin, 1964.

Shan, Sharan-Jeet, and Bailey, Peter. Multiple Factors:Classroom Mathematics for Equality and Justice. Stoke-onTrent, England: Trentham Books, 1991. [This book contains lesson ideas from many cultures.]

Sharp, R. M., and Metzner, S. The Sneaky Square and 113Other Math Activities for Kids. Blue Ridge Summit, PA: Tab Books, 1990.

Thyer, Dennis. Mathematical Enrichment Exercises: ATeacher's Guide. London: Cassell, 1993.

General interest

The books in this section are chiefly books about mathematics, and give scope for enjoyable browsing. Most are aimed at a general audience, but may be particularly interesting for teachers. Some books could be relevant for students also.

Bondi, Christine, ed. (for the Institute of Mathematics and its Applications). New Applications of Mathematics. Harmondsworth: Penguin, 1991.

Cosgrave, John B. A Prime for the Millennium. Roundstone: Folding Landscapes, 2000.

Davis, Philip J., and Hersh, Reuben. The MathematicalExperience. Boston: Birkhauser, 1980; also Harmondsworth: Pelican Books, 1983.

Devlin, Keith. Mathematics: the Science of Patterns. New York: Scientific American Library, 1994.

Eastway, Rob, and Wyndham, Jeremy. Why Do BusesCome in Threes? The Hidden Mathematics of EverydayLife. London: Robson Books, 1998, 1999.

Flannery, Sarah, with Flannery, David. In Code: aMathematical Journey. London: Profile Books, 2000.

Houston, Ken, ed. Creators of Mathematics: the IrishConnection. Dublin: UCD Press, 2000.

Jacobs, Harold R. Mathematics: a Human Endeavor: aBook for Those who Think They Don't Like the Subject. 2nd ed. San Francisco: Freeman, 1982.

Pappas, Theoni. The Joy of Mathematics: DiscoveringMathematics All Around You. San Carlos, CA: Wide World publishing / Tetra, 1986, 1989.

Pappas, Theoni. More Joy of Mathematics. San Carlos, CA: Wide World Publishing / Tetra, 1991 .

Singh, Simon. Fermat's Last Theorem. London: Fourth Estate, 1997 [paperback 1998].

Smullyan, Raymond. What is the Name of This Book? TheRiddle of Dracula and Other Logical Puzzles. Harmondsworth: Penguin, 1981.

Struik, Dirk J. A Concise History of Mathematics, 4th ed. Toronto: Dover Publications, 1987.

Vorderman, Carol. How Mathematics Works. London: Dorling Kindersley, 1996, 1998.

Wells, David. The Penguin Dictionary of Curious andInteresting Numbers. Harmondsworth: Penguin, 1986, 1987.

Mathematics education

The first book listed below is aimed chiefly at beginning teachers, but is mentioned here because its philosophy is so close to that of the one underlying the methodology discussed in these Guidelines. The second book is a very valuable resource containing many practical ideas presented against a background of research.

Backhouse, J., Haggarty, L., Pirie, S., and Stratton, J. Improving the Learning of Mathematics. Children, Teachers and Learning Series. London: Cassell, 1992.

National Council of Teachers of Mathematics. Principlesand Standards for School Mathematics. Reston, VA: NCTM, 2000.

SUGGESTED RESOURCES AND EQUIPMENT FOR A MATHEMATICS CLASSROOM

Active teaching and learning are often facilitated by the use of various kinds of material. Useful items include the following:

Many of these items can be bought in stationers' shops. References to others can be found in the journals or on the websites of mathematics teachers' associations.

• Large geometry compasses, protractor and setsquares.
• Set of scissors (the "blunt-nosed" variety!)
• Coloured paper or thin cardboard
• Paper fasteners
• Trundle wheel
• Dominoes
• Clinometers
• Boxes of 2D and 3D shapes
• Graduated cylinders
• Set of calculators
• Set of log tables
• The polydron kit
• Jim Marsden's activity kits
• Calculator hex board game
• Dart mathematics game
• Dice
• Counters
• Unifix cubes
• Mathematical posters
• Geoboards, pegboards and pegs