3 Reviews of David B. Surowski’s “Advanced High-School Mathematics”

The first one is written by Rafiul and the second one is written by Borno. Any suggestion is welcome and will be incorporated into the text. The last one is by Nafia and her review sums up the entire picture. 

Review 1: General View of Surowski’s Advanced High-school Mathematics [Link] 

On the surface, David B. Surowski’s “Advanced High-School Mathematics” is just like another regular Math book. It talks about triangles, algorithm, inequality, Venn-diagram, limitation and Statistics. But it has some distinct features.

The proof of Pythagoras theorem is shown using rectangles and triangles. The congruent, the similarities in triangles are some fundamental things to learn Geometry which are taught in grade 8. It also shows advanced things of geometry which are sensed magnitude, Euclid’s theorem, Brief interlude, circle geometry and mass point geometry which is basically vector which are not introduced to A-level students. The division algorithm has been exposed to us a really small portion of it which is equivalent to the tip of an iceberg. The Diophantine equation, Chinese remainder theorem, primes, fundamental theorem of arithmetic and Fermat’s and Euler’s theorem things are not introduced to A-level students, yet the book converses about these topics elaborately enough for even a 10^th grader can even understand what it is about and has shown how these are helpful for daily life.

Kruskal’s algorithm is one of the most essential things which solve a lot of mathematical problem we have to deal with. There are problems with inequalities discussed which includes harmonic sequence, Cauchy-Schwarz inequality, Jensen’s inequality, Holder or graphical inequality and the discriminant of which most of them are learnt in O levels. The limitation is one of the most important mathematical issues we have to deal with but in English Medium system, it is only a part of the chapter which does not come in exams. So teachers also ignore it whereas it is taught in Bengali Medium and English Version really widely which covers two chapters of their Mathematics book. No wonder why, English Medium students lag behind a lot. The “Advanced Mathematics Book” elaborates the limitation in simple way which can be even understood by a 10^th grader.

The book ends with the most predominant thing for Mathematics, Statistics, which is used in almost every profession. This delivers us information not only of what we learnt in A levels, for example, mean, variance, normal distribution but also new terms as binomial distribution, densities, stimulation and hypothesis testing on means and proportions. The binomial distribution plays a really important role in understanding Statistics which are not yet taught by teachers as it is not included in syllabus. This book is really indispensable for university admission tests and also for SAT because it gives a broad idea on Mathematics which is necessary for any student. These should be introduced to A-levels applicants. The reason, it is not instituted is because the British Council thinks these are advanced materials for students to work with, not knowing these are the most needed things to clear our concepts and become a better learner. 

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Review 2: Structural Design and Special Topic Areas

The book consists of lucid presentations and observations of the components of advanced Euclidean geometry (classical constructions of triangles, circles and their intersections), basic combinatorics and graph theory, inequalities and extreme points, algebra, infinite series and differential equations and the constituents of probability theory and inferential statistics. The book yields 6 sections of Mathematics each diverging into its further branches which divide further into its constituent topics. It delves into the epidemy of each division and ushers a multiple count of theorems or relations(formulas) and its constituent proof along with abstract examples of its types and visual representation and reasoning of geometrical types. For example, in Chapter 2, section 2.1, part 2.1.2

The liner Diophantine equation at+by=c Suppose that a, b, c are integers and suppose that we wish to find all possible solutions of the linear Diophantine equation ax + by = c. First of all we need a condition on a, b, c in order to guarantee the existence of a solution.

THEOREM: The linear Diophantine equation ax + by = c has a solution if and only gcd (a, b) 1 c.

PROOF. Set d-gcd (a, b) and assume that c = kd for some integer k. Apply the Euclidean trick to find integers s and t with sa + tbd. multiply through by k and get a(sk) +b(tk) = kd = c.  A solution is therefore x-sk and y tk. Conversely, assume that ax + by = c. Then since dla and dlb, we see that d| (ax + by), hence, proving the theorem.

The book aims to allocate and furnish an understanding of a diversity of different genre of general mathematics up to pre-university standards with topics any one might have come across during their years in IGCESE or IAL, along with unfamiliarly new segments of the same topics that we are already familiar with. For example – In IAL Statistics –1, everyone yielded rudimentary learning about the topic Normal distribution from the chapter continuous random variables, this book takes it a notch further to exponential distribution and delves further into Discrete Random Variables where further geometric and binomial distribution is introduced.

The main notion of the book to create a tangible understanding of mathematics like no other, not on how to solve and learn on basis of formulas. It helps one to develop an elemental style to understand and answer mathematical problems, not solve it.

Reveiw 3: Concluding Review

ADVANCED HIGH-SCHOOL MATHEMATICS

The title- “Advanced High School Mathematics” may be a bit deceiving as the content of this book at first glance may seem better suited for freshman or sophomores attending University but any high school student capable of handling this stellar will certainly be ready for the top-notch mathematical course universities have to offer.

The book covers clearly in detailed presentations, elements of the advanced Euclidian geometry (classical construction of triangles, circles and their intersections), basic combinatory links between graph theory, inequalities and extreme points, basic abstract algebra, infinite series and differential equations and the elements of probability theory and infernal statistics.

While going through the book, I came across topics like the division algorithm. This algorithm was taught in school but we were never made familiar to the proof but this book covers proofs like this for almost all the theorems, thereby providing us with a better idea of the math. It also covers several computations and tricks- talks about Taylor and Maclaurin series and provides us with interesting short cuts to tackle the sums. It further teaches us the concepts of the Hamiltonian cycles (a graph or closed loop that visits each node exactly once). This concept is used in computer graphics and is vital for any student studying computer science. It isn’t part of the high school curriculum but I feel as though these topics should be introduced as they will help strengthen the basics and prepare us for more serious mathematical courses, it will also ensure that we have a well rounded preparation for the SAT-II math level two exams. All the topics will not be common with the SAT syllabus but the ones which are, namely, inequalities, sets, functions, prime numbers etc will provide us with a wider knowledge of the topic.