## ICE-EM mathematics textbooks review

The International Centre of Excellence for Education in Mathematics (ICE-EM) has recently completed a full set of mathematics textbooks and related teaching resources (homework sheets, CD-ROMs, etc.), entitled “ICE-EM Mathematics“, that covers the transition period between primary and secondary education in Australia (or more precisely, from Upper Primary to Year 10). This package has been designed by professional mathematicians (including my former undergraduate advisor, Garth Gaudry) in collaboration with experienced teachers in primary and secondary mathematics, and in accordance with Australian state and territory maths curriculum requirements. Some more information about this package, including sample chapters and homework sheets, can be found here.

Recently, I was given a copy of the textbooks (there are six two-volume books in all, one for each year of schooling) to review for ICE-EM. The entire package spans about 5,000 pages in 12 volumes; my review focuses on three representative volumes, Transition 1A (that covers the first half of Year 5), Secondary 2B (that covers the second half of Year 8), and Secondary 4B (that covers the last half of Year 10).

Disclaimer: Of course, I am reviewing these books not as a primary or secondary school educator, but instead as a professional academic mathematician and tertiary mathematics educator. Nevertheless, in my experience with students at the tertiary level, I have certainly seen how any gaps or deficiencies in primary or secondary maths education can show up to cause significant conceptual difficulties at the tertiary level, and it is with this perspective that I am approaching my review of these texts.

— Transition 1A —

This text, aimed at the Year 5 level (i.e. ages 10-11) covers the fundamentals of arithmetic and geometry: it begins by reviewing place number notation for natural numbers, goes through the four basic arithmetic operations (addition, subtraction, multiplication, division) for natural numbers (focusing on numbers with two to four digits), and then applies these operations to compute lengths, areas, and volumes. Finally, the text introduces the concepts of fractions and decimals. (The 1B volume follows up on these topics by introducing other geometric concepts, such as angles and coordinates, as well as how to do arithmetic on fractions and decimals.)

In short, this text covers the bread-and-butter topics of primary-school mathematics; not the most exciting part of the subject, admittedly, but an incredibly important foundational component nevertheless, both for real-world applications (the typical student here will use basic arithmetic and geometry in adult life far more often than algebra, trigonometry, or calculus), but also for building a strong intuition about numbers, measurement, and geometry that is absolutely essential for preventing one’s high-school maths education from degenerating into a confusing list of mysterious rules, formulae, and algorithms. So it is rather important to teach this stage of the subject properly and thoroughly.

The 1A textbook, in my opinion, achieves this goal excellently. It takes a serious, no-nonsense, and practical approach to these basic concepts; there are more light hearted side topics, such as number systems other than Hindu-Arabic (Roman, Egyptian, Mayan, binary), or mathematical games (such as the classic “fizz buzz“, which I myself played as a child), but (as one would expect from a core textbook in a foundational subject), it does not let these diversions (or any other overly fancy teaching methodology) distract from the basic mission of the text, which is to convey the concepts of number and measurement in such a way that the students have both an intuitive and grasp of these concepts, as well as a variety of tools with which to manipulate these concepts quantitatively.

One thing I particularly liked about the text is that it consistently emphasised the multiplicity of approaches to any given problem. For instance, a subtraction problem 34-16 is viewed both as taking away 16 from 34, or from finding the number which, when added to 16, yields 34, and many different ways to compute such a subtraction are given, ranging from the standard digit-borrowing method to ad hoc methods such as subtracting 10 from 34 first, and then subtracting another 6, or subtracting 20 and then adding 4, etc., or trading one of the three tens in 34 into ten ones, etc. Similarly, place notation is expressed using standard decimal notation, but also using an abacus or historical number systems, geometrically using a number line, or physically using base-10 blocks (which I fondly remember from my own childhood as expressing the decimal system in a particularly clear manner). At the tertiary level, I have seen a regrettable tendency among some students to economise mental resources by remembering only one algorithm or method for each problem type and closing one’s mind off to all other approaches, thus leading eventually to a very fragile grasp of mathematics; it is thus important to reinforce the message that there are multiple approaches to any given problem (and that one needs to actually think in order to decide which strategy are likely to be superior in any given situation) as early as possible.

In a similar spirit, the problems in the text have a distinctly heterogeneous flavour, mixing together symbolic computation, word problems, experiments and group activities, and conceptual questions (e.g. what is the difference between a factor and a multiple?) together. This guards against the student applying a memorised technique to attack these problems based on key words or patterns discerned in the problem text (another unfortunate habit of some students at the tertiary level). Indeed the emphasis throughout the text is on strategy and conceptualisation rather than memorisation; the material unfolds naturally, rather than being chopped up into a series of boxed “Definitions”, “Worked examples”, and “Key facts” (which, unfortunately, often gives the unintended impression that all the material between these boxes can be safely forgotten by the student). [Of course, there still are important definitions, examples, and formulae appearing in the text, as well as summaries of extended discussions, but they are not highlighted to the extent that all the other exposition of the material suffers as a consequence.]

The sections on measurement (of lengths, areas, volumes, etc.) are well balanced between both quantitative (exact) measurement, and qualitative (approximate) measurement; of course, both types of measurement are vitally important in real life applications, and there is a healthy interplay between visual, physical, numerical, and formulaic approaches to measurement throughout the text. The metric system of length measurement – which, of course, will be ubiquitous in one’s adult life – is covered thoroughly and clearly.

The text totally avoids any use of calculators at this primary level stage, which I think is quite wise. At this stage, the students are just beginning to form a broad base of numerical intuition through direct calculations, and to short-circuit their development of this intuition by encouraging the use of calculators would be quite damaging. Of course, at the secondary level in which the subject matter has shifted to non-numerical topics such as algebra, the use of calculators to handle the now-routine numerical computations makes more sense, though even there the text only encourages the use of calculators when it is particularly necessary (e.g. to compute trigonometric or exponential functions, or to manipulate decimals).

As can of course be expected from a mathematics text written in part by professional mathematicians, each mathematical concept is introduced in a correct and clear way. For instance, when covering fractions, it is made clear (and illustrated through the visual device of folding a strip of paper) that equivalent fractions such as 1/2 and 3/6 are indeed equal to each other in value, even though they differ in form; remarkably, I have seen high school mathematics teachers in the US quite confused on this very point!.

— Secondary 2B —

This text, aimed at the Year 8 level (ages 13-14), is now well into the secondary mathematics curriculum; now that all aspects of integer and rational arithmetic, as well as basic geometry have been covered in previous volumes, the text now covers ratios and intermediate-level algebra (e.g. quadratic factorisation), introduces students to graphs, tables, charts, statistics, and probability, and continues the coverage of geometry with such topics as congruence and proofs in Euclidean triangle and circle geometry, and areas and volumes of shapes such as parallelograms, cylinders, and triangles.

The approach of the text is similar to that of the primary level text reviewed above, though there is necessarily of course many more formulae now, some of which do unfortunately have to be memorised, being difficult to motivate by other means at this level of mathematical understanding. The text does identify the formulae of this type, but this is done sparingly, and much of the discussion remains conceptual rather than prescriptive. (For instance, the formula $\pi r^2$ for the area of a circle – which is one of the few formulae that is recommended to be memorised – is heuristically derived from the formula $2\pi r$ for the circumference of that circle by slicing and rearranging the circle into a near-rectangle, which is a visually convincing argument that can eventually be made perfectly rigorous once one knows the more advanced concept of a limit.)

As before, the emphasis is on multiple perspectives and methods for dealing with any given concept. For instance, the concept of a ratio is illustrated through a variety of means using percentages, per-unit prices, monetary exchange rates, speeds and other rates, and scales, as well as through a diverse set of problems (e.g. to estimate the height of a building in a photograph, based on ratios between that building and people pictured in the photograph).

The graphs and statistics section is quite thorough, with an emphasis on data interpretation using all sorts of numerical, tabular, and graphical formats, and the explanation of key concepts (such as that of an outlier in a data set, illustrated by completion times for a game of sudoku) is clear.

The proof-type problems in the sections on Euclidean and triangle geometry are done quite traditionally, with an organised line of deductions from hypotheses to conclusion, justified at each step. This is of course the classic way in which students are first exposed to rigorous mathematical thinking – I myself was taught this way – but sometimes the old ways are still the best. (By the tertiary level, one would have to deal with more complex and free-form arguments, that are often best phrased using a blend of mathematical equations and natural language sentences, rather than the classically elegant structure of a Euclidean geometry proof; but the Year 8 maths curriculum is hardly the place to deal with these more sophisticated examples of mathematical reasoning. By means of comparison, proofs of any sort – Euclidean or otherwise – are rapidly disappearing from the high school curricula in the US, or reserved for advanced courses only.)

Probability is a notoriously subtle subject, full of pitfalls from sloppy reasoning or loose wording, but the authors have been particularly careful here to make the material here correct, unambiguous, and clear, in particular relying on visual aids such as arrays and trees to facilitate intuition. (At the tertiary level, one would want to challenge the students with the (many) probability paradoxes and other subtleties in order to test their grasp of the subject, but one should of course avoid this for more junior students who are only just encountering the subject for the first time.)

The text ends with the beginnings of analytic geometry – in particular, showing one can combine algebraic manipulation of equations such as $y = mx+b$ with the geometric manipulation of objects such as lines, triangles, and rectangles to gain a deeper understanding of both subjects. This subject is of course the gateway to calculus and a large fraction of higher mathematics, and the crucial connection between algebra and geometry that is forged here (and persists all the way through to the highest levels of mathematics) is appropriately emphasised.

— Secondary 4B —

This is the last volume in the series, covering the second half of Year 10 (ages 15-16), and dealing with the standard precalculus topics, such as combinatorics, advanced trigonometry, logarithms, polynomials, analytic geometry, and the general theory of functions and variables, as well as a number of other topics, such as standard deviation in statistics and advanced circle geometry, as well as some further topics in probability (such as conditional expectation and independence).

Trigonometry is a particularly formula-intensive subject, but the authors here have managed to restrict attention to the most important formulae (from which the others can be easily derived), as well as giving brief but accurate derivations of each of these. Some important subtleties (for instance, the fact that the sine rule sometimes gives an ambiguous answer when reconstructing a triangle from two sides and a side angle) are also carefully pointed out.

The subject of combinatorics is often the first place where students are exposed to conceptual mathematical reasoning; the actual concepts involved in counting various types of combinations and permutations are not difficult, but applying them correctly requires clear and careful thought, and one often needs to argue from first principles rather than rely on memorised formulae. In this subject, the role of the homework problems is particularly important, and the problems here are carefully worded and require real thought on the part of the student. This section also meshes well with the subsequent section on probability, where the problems are similarly challenging.

The subject of polynomials is considered in some depth, both in its own right and as a major motivation for the more general theory of functions, which is the final topic of the book and is of course the launching point for calculus. I especially liked how division of polynomials was explained, by emphasising the analogy with long division of integers (while avoiding the abstract formalism of synthetic division, which can be a little mysterious if presented baldly). The treatment of functions focuses on foundational issues, such as the concepts of domain, range, and inverse of a function; I have seen many a calculus student flounder due to an imperfect understanding of these basic concepts, so I am glad to see that the text devotes a fair bit of time and space to these mundane but important aspects of the theory, in preparation for the calculus material in the final years of high school.

— Conclusions —

The textbooks are well organised both within volumes, and across volumes (though each year’s text is mostly self-contained, reviewing previous year’s topics when necessary). Topics are often introduced multiple times across the series, the first time round in a conceptual and simplified manner in which any subtleties are carefully avoided, with more detailed treatments occurring in later volumes. Related topics tend to be grouped together, and the many different branches of high school mathematics (algebra, geometry, discrete mathematics, etc.) remain connected to each other throughout; thus each section of material reinforces other sections, leading to a robust understanding of mathematics as a unified whole, rather than a disparate collection of niche topics.

The difficulty level of these texts is moderate: on the one hand there are few problems that are deliberately tricky or exceptionally challenging (though the challenge problems at the end of each section are certainly non-trivial), but on the other hand the text has not been “dumbed down” in any way, and students will still have to actually understand the material, in addition to thinking about strategy and memorising some key formulae and methods, in order to solve the problems given.

Students who are particularly gifted and interested in mathematics will probably need to supplement this textbook with more advanced or sophisticated material; the texts are focused on fundamentals and on the mathematical topics which will be of relevance to the majority of Australian primary and secondary school students, rather than dwelling on more specialist topics such as connections with higher mathematics, sophisticated real-world applications, or mathematical puzzles and paradoxes, which I myself would have found fascinating, but may perhaps not be suitable for a general text. In any event, such bright students can (and should) be seeking out more information on these topics by themselves, for instance through the library or the internet.

In summary, these are serious and substantial textbooks; the focus is on content, concepts, and computation, with “bells and whistles” in the presentation definitely being a secondary concern, particularly at the secondary level. But it covers all the important core topics in the Australian curriculum thoroughly, carefully, correctly, coherently, and efficiently, and an attentive student will certainly be able to obtain a broad and robust grounding in upper primary and early secondary school mathematics through this series, which is excellent preparation both for everyday life (and especially in any career that emphasises quantitative thinking), as well as in any higher maths education that a student may wish to pursue. In my opinion, they are eminently suitable for general use in Australian schools, and I endorse them for this purpose.

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