Miller's book is for those who struggle with Maths

Five stars

When I was at Primary School in the early 1950s, and already acquiring a reputation for mathematical incompetence, our Headteacher came into the classroom one day carrying a small cube - 10x10cm I guess - made of pieces of wood, some black, some white, some cubes with 1cm edges, some rods 10cm long, others of various sizes. 'That looks as if it might make sense', I thought. I suspect it was part of a new scheme for teaching maths. After a while, the interesting cube was taken away and we went back to grumpily learning our tables. I suppose that was my first awareness that there might be another way of doing mathematics.

The years went by and I narrowly passed 'O' level Elementary Maths. I would not say that I was numerically incompetent, however. I had simply devised for myself methods of everyday computation that seemed easier than the procedures we had been taught. Many of us do that. An interesting feature of Miller's book is that he offers several ways of doing a sum, leaving the reader to choose which suits him best. This alone helps one to feel that one owns one's mathematical tools and thus helps to dispel the cloud of gloom that envelops the subject for so many learners.

After 'O' level, because of the exigencies of the school timetable, I found myself in a class of my contemporaries who were preparing to take the Additional Maths exam, which involved an introduction to Calculus. Our teacher held me spellbound. Here at last was a branch of mathematics that I could relate to the real world. I did not take the Calculus exam, but I was left with an awareness that maths could fascinate even me.

This developed as I read G.H.Hardy's Mathematician's Apology, and in occasional conversations with real mathematicians. Then along came Marcus du Sautoy. I never thought I would come across a maths book that I couldn't put down, but that's how it was with his The Music of the Primes.

Miller's book is for those who struggle with Maths, as I still often do. But its value for me has been that from the very beginning he enables me to glimpse the fascination of maths, even when dealing with those dreaded multiplication tables. Having stripped away plenty that we were unnecessarily obliged to learn, he brings the reader face to face with the fact that some of them just have to be learnt - there is no other way. Du Sautoy has taught me that these non-negotiable brick walls - or Mount Everests - are part of the fascination of maths. Miller, in several places, enables us to see even elementary maths in a similar way.

Passing from integers to rational numbers, Miller explains that this is a passage from counting to measuring - precisely what had fascinated me when introduced to Calculus. I'm sure I could have discovered the charm of maths earlier, had I been more diligent and receptive. But Miller has enabled me to revisit that road of discovery, and to appreciate more of what I meet on the way.

As the book progresses, allusions become more frequent to a second volume that will deal with more advanced topics, including the dreaded algebra. It is as if one is being given glimpses of the mountains ahead. With such a guide, I feel I will be able to make the ascent, and so I look forward to Part Two.

Bruce Harbert, Wednesbury


Maths is now exciting and stimulating. I now want to learn more from the 2nd part.

Five stars

My experience of using the book:

While reading the book in response to the request to review it, I found myself diverted into doing the exercises.  I experienced real pleasure in being able to complete the sums in the early sections before seeing the answers.

Consequently, I was drawn into using the book as it is intended.

I found the book very easy to read, clear & logical.

At the end of the introduction, the sentence ‘It is never too late to learn’ cheered me enormously. I found it encouraging that there were sections where I already knew things.  I recognised the blocks to learning referred to, such as the emotional impact of school teaching & my lack of the basic sequential knowledge.

The instruction to take my time if I went blank was so helpful.  Being told to go back again if I did not understand something & then being able to ‘get it’ when I did return to the section was wonderful.

The formula of BODMAS suddenly gave a logical way of approaching every calculation in arithmetic involving brackets. I could grasp the sequences.  I could enjoy doing the process of sums.  This achievement added to my pleasure & gave me the motivation to explore new things in the book.

I never knew that there was a framework & rules that govern arithmetic.  The book sets them out clearly & in simple language.  The new vocabulary & sense of order was stunning.

I was struck by the way the key elements are highlighted on the page; by the sequencing of calculations; by the ‘rules’ of each process; by the little tricks for mental arithmetic; by the methods & the alternatives that I could follow; & the conundrum of rational numbers.

In the section on fractions, there was vocabulary I needed to learn.  I needed to re-read these pages several times to acquire the unfamiliar terms.

As I progressed through the book it took me more time to read & absorb the information.  The book becomes increasingly demanding of attention, checking of understanding & retention of details.

I found that if I go back & work slowly forward, I do begin to understand & can progress. If I sat too long trying to work things out, it was counterproductive.  Short sessions were more beneficial.

I want to go through the book again & download the exercises to confirm my understanding & refresh my ageing memory!

My recommendations to other potential users:

I shared my enthusiasm for the book with my son.  When he looked at it, he felt that he could also benefit from reading it to complement his string of medical qualifications.

As an adult learner who missed out on numbers for almost a lifetime, I would recommend this book as a wonderful experience.

It establishes all the basic building blocks of numbers in a logical sequence.

The presentation is clear & visually accessible.

The dialogue with the reader is amusing & empathetic.

I wanted to read it & read it as fast as I could to get to the end of the book & know that I could ‘do it’!

I look forward to grappling with Part 2.

Can’t wait!

My background:

I left school at 13 ½ years of age with no qualifications.

My experience of being taught arithmetic at secondary school was at the hands of a strict disciplinarian.  He made liberal use of the cane.

The effect of this treatment created an emotional block to learning the processes of arithmetic. This contributed to my failure to comprehend. The use of the cane was mentally & emotionally disabling to any ability I had to understand & apply the rules of numbers.

Consequently, I escaped from this experience by truanting & leaving school as soon as possible.

Nevertheless, I had learnt enough at primary school & through the necessity to work at an early age as the 2nd of 14 children, in order to manage financial transactions.

At 8 years old, I was paid by the bag for picking potatoes. I kept a close tally of my earnings.  I gave ¾ of my wages to my mother & ¼ was my pocket money.  I could work out this use of numbers because it was my source of wealth to buy icecreams & go to the cinema.

In the laundry van, I worked out the mileage & use of petrol, as well as collecting money, giving change & totalling the amount at the end of the day.

During my 5 year apprenticeship as a loom tuner in a woollen mill, I learnt to follow design instructions from a card system that set the pattern for weaving on the looms.  Spacers of different numbers created the design for weaving.  This was achieved by eye & simple counting. A knowledge of basic arithmetic was all that was required.

In the rest of my career, I chose work that did not require mathematics.  This limited my choice of profession, but did not bar my access to University twice.

However, it mattered to me that one day I would know & understand the concepts & processes of higher learning.  I found a maths teacher who was the room mate of my wife-to- be at university.  She gave me a series of lessons & I was happy to re start my learning.  Unfortunately the lessons did not continue long enough because she moved away & I became absorbed in work & family life.

Then I watched each of my 3 children learning arithmetic & mathematics & succeeding with it. Two of them used mathematics to study A level physics & progress to careers in science. To my regret, I could not follow what they were doing. I felt very proud of them.  I wanted to be able to learn like them, but I did not know how.

Since I retired some years ago, I have attended a Cafe Scientifique monthly meeting which my elder daughter helped to set up while she was at university. It was at a meeting on teaching & learning mathematics that I asked the Emeritus Professor of Mathematical Thinking at Warwick University if it was possible to learn maths at 75 years of age.

Stephen Miller was also at that meeting.  He put his new book in my hands & asked me to review it for him.

John Gordon, Sutton Coldfield


A valuable resource, taking a highly structured approach to Maths difficulties.

Five stars

In recent years, several maths text books have appeared on the market written specifically for adults who find maths difficult or who missed out on learning the basics at school.

These books have usually followed a strongly practical curriculum, based on giving people the numeracy skills they use most in everyday life. Many of them were written with a focus on the multiple choice National Tests in Numeracy for adults.

However, in recent years the written Functional Skills tests that replaced the National Tests have moved towards examining understanding of the process skills of mathematics. There has been a greater emphasis on learners needing to show they can work towards a correct answer systematically, not just arrive at the correct answer by whatever means possible.

Steve Miller's Numbers Explained is an ideal guide for those learners who want to give themselves structured and robust methods of working out maths problems. This is not a 'pick and choose what you need to know' book, but a course within a book, taking learners right back to the absolute basics.

For example, it reminds us of the rules of counting, which include starting at one, not counting the same number twice and progressing in an ordered sequence. In reminding us that there ARE rules, Steve provides a valuable service - mistakes and misunderstandings can be quickly identified.

Although written for adult learners, I felt that the early chapters on the basic rules of Maths would be of great value to early years educators in enabling them to teach children methods that eliminate misunderstandings at source.

I would wholeheartedly recommend this book to all who teach primary mathematics in fact, so that they can both check their own understanding and equip themselves with some solid techniques for getting the basics right, right from the start.

I would hesitate in saying this is a book for all those who struggle with Maths as adults.

Its formulaic approach and relatively minimal use of graphics may put off those who struggled with conventional maths textbooks at school. Learners with dyslexia may find the text a challenge.

I calculated a readability level of 14, which roughly equates to being able to read and understand the editorial of the Daily Express. Chambers Adult Learners' Guide to Numeracy has been approved by the British Dyslexia Association and may well be a better choice where reading texts is challenging in itself.

Where Numbers Explained does score over other books is in having a wide range of resources available online as free downloads, particularly solid exercises to test newly gained skills, with worked examples made available, so that understanding can be fully checked. The large A4 format of this softback also makes it an ideal size to safely tuck these printable resources away for future reference.

 The book also has an extremely useful Glossary of mathematical terms, along with a very handy summary of the key points made in the book. Whilst not designed as a 'dip-in' book, a useful index makes it easy to find a particular topic quickly. Well-spaced pages with wide margins make it particularly easy to make helpful notes of one's own.

This is the first book of a two - part series. It includes most of the basics such as the rules of number, fractions and decimals, but does not include percentages, ratios and standard form - these will appear in Book 2, along with other more 'advanced topics'.

Despite this, I would definitely recommend this book as a useful resource for the majority of adults who want to improve their mathematical understanding, and also for teachers and tutors in giving them some fresh ideas in how to tackle the stumbling blocks faced by many who 'can't do maths'. 

Andrea McCulloch, online Maths tutor, Newton Aycliffe

Good for Numeracy Teachers

Five stars

This thorough, detailed and patient book provides a foundational understanding of basic numeracy concepts and methods. But it appears to be written without a clear idea as to the intended audience. It is ideal, I feel, for those teaching numeracy (with some caveats), but I suspect that it would overload those fearful of maths. So the rating of 5 stars would drop to 3 or 4 for those who fall into that category.

After devouring the book in a weekend, my sister who teaches secondary GCSE mathematics was impressed by the book, not least because it provided some 'nuggets' that were new to her. And this touches on one of the key merits of the book - its thoroughness. It adopts the linear number line as the basis for addition and subtraction and the repeated versions of these - multiplication and division. It covers multiple methods of calculations in splendid stepwise fashion, leaving no stone unturned, and providing a historical context for the terminology.

For a maths struggler, however, it provides too much terminology, and too many side-boxes that sometimes go into great detail on diversionary matters. This is enlightening for a teacher but potentially frightening for someone fearful of maths. The point here being that the approach one should adopt for a teacher is quite different from that for a learner. For the latter, greater use of real-world similes such as the excellent, but all too brief use of cabbages for representing groups of numbers would have been appropriate.

BODMAS (the order of brackets, division, addition etc in calculations) is taught in schools and the author explains it at length here. But this also might confuse a struggler - this ugly acronym adds complexity without teaching numeracy. We are rarely presented with basic calculations in the real world that have the ambiguity caused by bracket omission that BODMAS was introduced for. A book for strugglers should help them become fluid with the basics rather than sidetrack them with something that they will rarely use. If it is really that important, it should appear in book two for those wanting to progress beyond the basics.

So an excellent book for teachers but potentially too foreboding for those revisiting a subject they formerly struggled with.

Neil Moffat, Cardiff

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