For teachers and parents who seek a theoretical framework for their teaching.

Math Stages

“The Five Stages of Math Achievement” offers an in-depth perspective of how Math is accommodated and assimilated by most learners in the K-12 range. It breaks the process down into a set of distinct interlocking stages that suggests how a learner may be helped.

In the Epilogue, Shad proposes “visual mediation” (sight) as the “phonics” (sound) equivalent for teaching Math. He demonstrates how this makes Math both readable and understandable like any formal language.

A brief section is devoted to a commentary on Differentiated Learning and the current “Math Wars”.

Shad’s book on the 5 Math Stages will be available soon.
You can download Shad's preface to "The Five Stages Of Math Achievement" also as PDF:
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Math Stages - The Never Ending Story

Author's Preface

"In the construction of the Five Stages, I confess my bias against rootless learning. I believe that whatever is learned is tied to the learner’s inexhaustible desire for meaning-making..."
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"Stages 1 and 2 begin with the slow transformation of Math concepts into functioning knowledge. At the beginning, mathematical concepts and their symbolic representations are assimilated..."
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"Knowing the learner helps parents and teachers correctly diagnose difficulties, especially in the higher problem-solving Stages 4 and 5. The risks of attributing incorrect reasons for why..."
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"Some of the most pressing questions around Math education are addressed in Stage 5. These are provoked by the observation that Stage 4 learners, despite attaining a high level of mastery in problem-solving, continue to perform below expectations..."
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AUTHOR’S PREFACE

The Stages proposed in this book have grown out of my own clinical experience of teaching Math to K-12 students or their equivalent in different parts of the world for over 35 years. A lot of it has been gleaned from clinical observations, teaching sessions, notes from clinical assessments, parent, student interviews, and reflections following my training workshops. Some aspects of the Stages have been distilled during my 15-year phase of developing the Karismath program.

The perspectives that I have drawn of the Five Stages are entirely mine, and open to debate.

The Five Stages of Math Achievement are being offered with certain qualifications and within a certain context:

  1. The Five Stages cover the K-12 school range and suggest increasing stages of complexity in the understanding and applying of Math concepts for that range.

  2. The Stages do not necessarily begin at any particular age. Stage 1 applies to students of all ages between K1-12, since everyone, at some point or another, is introduced to new mathematical concepts.

  3. These Five Stages are unique and distinct in and by themselves, and are distributed across the progression of increased mathematical understanding. They can be clearly demarcated for assessment purposes.

  4. More than one stage may co-exist during a particular period of math learning. Multiple stages may also be triggered off like a cascade during which different stages may get activated with varying gaps of separation between them.

  5. The Stages also suggest to an extent, levels of cognitive readiness for assimilating increasing levels of mathematical complexity within the K-12 range.


In the construction of the Five Stages, I confess my bias against rootless learning. I believe that whatever is learned is anchored to the learner’s inexhaustible desire for meaning-making. The Stages have been presented from the Math teaching point of view where meaning-making for students is often a function of how the teacher communicates the math concepts. This may be very unlike the way mathematicians make meaning out of Math or how they see Mathematics being taught in the light of all its useful applications.

My own experiences confirm that Math teaching principles and approaches bear fruit when they are in sync with the Five Stages. For one thing, they offer a useful framework for developing evaluation approaches by teachers. For another, they encourage students with their own self-evaluation. Students get their bearings on where they stand vis-a-vs their efforts with Math.

These Stages are being proposed because there is an urgent need for Math teachers to ground their practice in some kind of a basic theory amidst a storm of conflicting arguments about the best way forward. The theory can be extended with more research and contribution from peers. The Five Stages, like the Karismath program, serve as the first small step in a journey of a thousand miles.

They also come at a time when concerned parents and teachers in North America are asking for change in systemic and curricular aspects of Math education. Much of the debates emerge in highly polarized forms around the so-called “Math Wars” and discussions on Differentiated Learning.


Differentiated Learning
First, the continuing debate on differentiated instruction.

Differentiated learning accommodates the learning habits, attitudes, interests, prior knowledge, personal preferences of learners. Current thinking suggests that its success depends upon the effectiveness of futuristic, possibly utopian educational approaches that can gratify multiple learning styles simultaneously. The expectations that drive such thinking may impact on how the effectiveness of new courseware will be evaluated.

Many assume that an educational program should satisfy the needs of a widening spectrum of learning styles. Such differentiated learning delivered from a single central source e.g. a selection of courseware, a program or a teacher, etc. is a wonderful ideal though still an invisible speck on the horizon.

When it comes to Math learning, there is a flip-side to this whole debate: differentiated learning in Math education may, in fact, reflect the consequences of differentiated teaching.

Consider the astonishing diversity of adults teaching Math to our children in elementary school: they vary from creative, innovative super-teachers at one end, to those who are less able and/or willing at the other. There are teachers who know their Math and who can anticipate its extension into secondary and post-secondary education and also teachers who struggle with concepts beyond elementary basics; language-arts teachers have taught Math; even gym-teachers, student-counselors, school-principals, art-teachers, teachers who excel primarily in teaching accounts and economics, social-sciences teachers, etc. All have taught elementary Math at school. The existing diversity in Math teaching methods and approaches has evolved over generations with confusing inputs from different pedagogical schools.

Consequently, generations have grown up learning early and middle Math in inexhaustible ways. Our children have been exposed to differentiated teaching in Math: it is taught according to the different needs, styles, habits, tastes, prior knowledge, biases, interests and experiences, even gender and ages, of those teaching it. The pervasive belief that Math is understood in different ways is more likely a reflection of how it was taught in the first place. Filtering out the purely cognitive aspects of math learning from such earlier experiences is not easy and requires more extended research and study.

Is “differentiated learning” a polite way to request for “customized” learning, as in customized cars, customized jewelry, customized golf-clubs and customized banking? If so, this may be more a matter of privilege than rights. The prosperous professional who orders customized products and services today probably got there not via differentiated learning but by either (a) working hard against an inequitable educational system or (b) with the gift of superior intelligence or (c) with supplementary assistance from passionately committed helpers.

To make Math learning more comfortable and appropriate for the needs of different learners does not necessarily call for differentiated learning in the sense of “customization”. Math education can be offered in ways that appeal to a majority of today’s under-achieving population of learners. It can carry an intrinsic universal appeal. It can have colours, music, sound and dynamic imagery. Concepts can be communicated and practiced to mastery, using intelligent design. To do this well means integrating current research on how children learn numeracy and numerical reasoning, into the design.

Today, technology offers opportunities for innovative educational designs. Together with expert knowledge, practice and experiences of math educators, the Math-learning experiences of a variety of learners can be radically ...and swiftly...transformed.

All the more so amidst rising concerns around the current and projected economic conditions. The current and future projections of the global economy portends a different set of delivery needs. Products and services need to be cheap, effective, optimal and capable of producing the desired impact in less time.

In other words they should have the power to shape learning habits without compromising on quality, by displacing the good with the better, and without mangling the long-term goals with distracting objectives. This revolution has already happened in the world of communication and computers and in the telecom industry. We can get more for less. We can do more for less. Its impact is being felt in the health services and increasingly now in education services.

We are able to learn more for less. This is not utopia. It is real. In such a world, differentiated learning in Math may have other implications.

What new-generation educational programs in Math should offer is pedagogy-embedding i.e. teaching and learning tools embedded within the courseware that actually do the Math teaching with minimal teacher-parent intervention. Such courseware frees the most important ally (teacher or parent) to switch gears: i.e. personalize and customize Math education for differentiated learning. Learners recognize and experience personalization as a form of customization of the learning environment in which they feel better understood by their closest allies. Fostering such teacher-learner congruency probably serves the most important goal of differentiated learning. New-generation Math courseware would do well to reconfigure the traditional teacher’s role: while the teaching tools within the courseware do the teaching, the teacher is liberated to do all the differentiating in learning.

To achieve this, teachers need teaching implements just as dentists need their dental tools. These have to be designed for maximum effectiveness while addressing different needs as they arise. Teaching implements can free teachers and parents from playing the conventional teacher role where they have to manufacture their own unique teaching materials (charts, manipulatives, worksheets, visual stimuli) after dinner, for the next morning’s lesson! Imagine a dentist doing that every night.

Professionals do not overtly serve their professions: they serve the needy. Dentists do not treat dentistry: they treat clients. Most teachers understand the value of teaching the learner rather than teaching Math. It helps them draw useful insights about different learners’ personalities: their interests, learning attributes, levels of understanding, attitudes, likes and dislikes, relationships with peers and so on. Active student-observers, with a well-grounded understanding of Math and its pedagogy, and teaching tools, are better able to monitor student progress and understand how they learn. Five different learners can make the same mistake in Math for five different reasons. Often the reasons have to do with the way each of the five think of and respond to a given problem. And how they respond has much to do with who they intrinsically are as individual learners. Pain, they say, has a wider spectrum of responses than pleasure.

Knowing the learner is key to correct diagnosis. The risks of attributing incorrect reasons for why a student performed poorly in a Math “quest” are otherwise great. Often, a false premise on the part of the teacher detracts students from learning from their own mistakes. Math teachers aren’t known for setting good examples in this area. Many trust their personal diagnosis because they feel strongly about incorrect mathematical processes and the mistakes that follow. An error in Math is quickly attributed to an error residing in the thinking process of the learner. By contrast, correct and accurate diagnosis comes from matching the pattern of mathematical errors with the pattern of specific impediments encountered during a student’s learning process.

The consequences of an incorrect diagnosis of errors and difficulties also contaminate teacher-learner relationships. When teachers rigidly pursue incorrect remedies and prescriptions for remediation it gets harder to manage the growing alienation between them and learners. In Math, teachers are not always right about when and why a student is wrong. But in their diagnosis of the cause of errors, students always know when a teacher is wrong. And so they switch off.

Not surprisingly, teachers and parents are often wrong about things that matter more to the students than Math. Which is why knowing the learner brings valuable benefits to teachers and parents than just “knowing their Math”. Of course, the opposite comes with equally precarious risks: knowing your student but not knowing your Math!

Differentiated learning, in order to be effective, requires
(a) a powerful set of effective teaching tools that are highly organized and structured, and which cater to different learning needs . Thus, the teaching tools should contain some mechanism for calibrating difficulty-levels in times of need. The difficulty levels should cover (i) comprehension (ii) recall (iii) speed and (iv) accuracy in numerical processing and reasoning. Such a mechanism allows the teachers and parents to focus more upon the learner than on the Math to be taught.

(b) a solid understanding of Math as a subject: its concepts, its applications, its skills development, and its continuing trajectory onto higher levels. New-generation Math programs (like Karismath) are and will be, self-learning. Teachers will be able to learn-as-they-teach, and teach-as-they-learn. Nevertheless, the “don’t-teach-Math, teach-Johnny” aspect is best addressed by undergoing some formal theoretical training.


The Five Stages
The Five Stages of Math Achievement is being presented as an attempt to introduce a theoretical foundation for Math teachers aspiring to be reflective practitioners. It acknowledges the power of mathematical concepts as much as of the mastery of standard algorithms.

In the Stages, the conceptual development of mathematical ideas forms the bedrock of all future conceptualization and abstractions needed for the higher task of mathematical problem-solving. There is a Stage for mastery of concepts, and there is another for mastery of skills. And there is another where the two are integrated to become a powerful intellectual approach for problem-solving. There is a stage where Algorithms are induced, drawing upon the natural intuitive heuristics of learners. These are skillfully drawn out and developed with training, into a formal set of algorithms. Most of such developmental processes lead to what we know as “Standard Algorithms” developed centuries ago! There is a Stage where an understanding of processes and procedural aspects guide problem-solving. It is the point where the answers are less important than the processes that lead up to them. And there is a Stage where the answer becomes important: its extraction with speed, fluent thinking and accuracy becomes a highly desirable goal.

Stages 1 and 2 begin with the slow transformation of Math concepts into functioning knowledge. At the beginning, mathematical concepts and their symbolic representations are assimilated in small, discrete and incremental doses. Between the first two stages they are recycled into bits and bytes of “mathematical knowledge”.

From the teaching point of view, everything that is understood and learned during the Stages 1 and 2 should be experienced as part of a continuing Math Story, a long saga. Each step of the way learners receive bits and pieces that are not scattered fragments but complete units in themselves, like coherent chapters in a continuing epic. Like chapters in any story, they eventually end but the Math story continues. If teachers and parents, with the help of a Math program, succeed in generating their learners’ interest and curiosity in the Math Story, then something of great value has been achieved.

Stages 1 and 2 assume that learners will be left with questions. As in an epic saga, getting bits and pieces of a larger story, can be frustrating. But frustration often feeds on unsatisfied curiosity. Teachers should not hesitate to lead their students to wonder: “What happens next?”. Even when learners may not like the Math they are doing they at least feel secure and satisfied about understanding what they learn, every step of the way. One can therefore rely on the exasperating curiosity that follows. “OK” they might say, “ I GET this......but so what?”

Interpret this to mean: “What is all this leading to?”. That’s a good enough place to be. The Thousand and One Nights’ tales began with the first story.

Stages 3 and 4 both relate to problem-solving. There is a lot of emphasis on examining, understanding and interpreting word-problems rather than on “finding the right answers”. This approach enables learners to become more process-oriented and less results-obsessed. It is an important digression from conventional approaches. Today’s knowledge-economy demands a new skills-set: innovative ideas that lead to the creation and design of habit-transforming products in everyday lives. Often the solution already exists in terms of a “vision” for a new product. The skills-set needed for tomorrow will draw on the ability of knowledge-workers to examine how the procedural aspects of conception, creation, design and production can be managed in the making of desirable products (or known solutions). These must suit the needs of a rapidly changing world. The swift global spread of technology promises diverse delivery-platforms for goods and services. Many are now designed and produced in virtual environments. Already the trajectory for academic life and a successful career path is getting established in the screen-oriented habits of children and school goers as young as 5 years old. These are here to stay. Consequently Math education needs to adapt to a new social and global environmental and prepare to redefine learning outcomes for this century.

Some of the most pressing questions around Math education are addressed in Stage 5. Stage 4 learners, despite attaining a high level of mastery in problem-solving, continue to perform below expectations in major examinations. One reason for this is that final examinations, or state wide, provincial or national examinations cover multiple Math Topics. They demand a wider range of recall and application of math concepts and algorithms, from students. Nowhere is failure due to insufficient examination-readiness more evident than in Stage 5. And yet, everywhere in the world, bottom-line student proficiency in high school Math is evaluated via such broad-spectrum examinations covering multiple Math Topics. A small minority performs well. Most succumb to the psychological pressures of time, anxiety and an attack of nerves. Some of it can be attributed, of course, to lack of preparedness. But other more serious reasons exist beneath the surface.

Viewed from the macro perspective many questions arise as to why a majority of the world’s student population fails Math. What is it that most nations share in common with each other when it comes to the systemic delivery of Math education? What is so unique about the history of Math education that anticipates the failure of an overwhelming majority as perfectly normal and understandable? Why does public opinion unflappably support the belief that Math is tough and only smart kids can excel in it?

When viewed from the micro perspective, other issues come to surface. Most are related to the design and content of Math curriculums everywhere. One could argue that there is something in the design of the curriculum that makes exam preparedness at the level demanded, a difficult, if not impossible proposition for most students. For one thing, the school system does not provide enough time or resources within the math curriculum. How much review and rehearsal work is required for attaining 100% examination preparedness across multiple Topics covered over, say, the last 2 years of school ? A lot. Compared to the time and resources currently allocated to schools, the amount actually needed would be dismissed as being outrageously phenomenal. How then, does one explain the continuing success of a minority group functioning in Stage 5 despite these limitations? What can we learn from them? More importantly, what in their practices, are not replicable or reliable enough to help the vast majorities of failing students ? Where then do we turn for solutions?

Admittedly, in the book such concerns are raised within the context of the Five Stages of Math Achievement. They give rise to many more questions that fall outside the purview of my discourse. Some suggestions are offered from the curricular perspective, while challenging the urban myth: “Math is for Smart Kids Only”.

Indeed there exist other urban myths around Math. Some aren’t myths but just misconceived notions that have simply gone unchallenged. For instance, a continuing but stubbornly ignored question regarding Math curriculums the world over is : “Who needs all the Math taught in schools, in real life?” Successful professionals from all walks of life (including Science!) have repeatedly protested that of all the Math they learned at school, only 1-3% of it is retained for actual use in their everyday working lives. The rest is doomed to pure redundancy. Hence, why the need to impose a wider Math curriculum upon the entire school-going population? Perhaps much could be gained from the time and resources saved by minimizing the width and depth of the Math curriculum across K-12. The savings could, perhaps, be diverted to delivering a high quality, though minimalist Math curriculum that attracts far greater responsiveness from students. Arguably, such a curriculum could inspire and motivate students to attain much higher levels of success in Math than is presently possible, or even conceivable.

The Five Stages does not address such issues or others related to them. But it may make questions such as the one above, more understandable, possibly even more compelling and urgent. More so, as long as:

(i)
Mathematics continues to be touted in narrow medieval terms as the “Language of Science” and little else.

(ii)
Mathematics is perceived as something “inherently complex, difficult and accessible only to superior minds.”

(iii)
The K-12 Math curriculums across the world remains unchanged in its rickety design and content.

(iv)
Math pedagogy is not synchronized with current research on how evolving minds understand quantity, numeracy, numerical reasoning and how they process numerical information.

(v)
Math teaching programs do not create pedagogy-embedded worksheets whose design guides the learning process


Very few cogent reasons are offered to parents, teachers and the general public re: the rationale behind existing Math curriculums. Let us accept for gospel truth that mediocre or inferior Math achievement is the established scientific norm for a majority. The bell-curve confirms that some will do poorly in Math, a majority will receive average grades in Math, and a few will excel in math. That goes for running or tying shoe-laces. So what is this fuss all about? It orbits around the following fact:

“The United States has one of the worst high school dropout rates in the industrialized world, and its students regularly rank far below those in other Western and Asian countries in reading and math scores. Slightly more than half of the population has only a high school diploma. One out of every two American university students drops out before completing their post-secondary studies.”

Secondly, “average” grades (determined decades ago) in Traditional Math curriculums for a 21st century population of Math learners does not necessarily translate into anything meaningful or revealing. It could merely suggest that the probability of the curriculum failing our students may be the same as for our students failing the curriculum.

Thirdly, the US has, to this day, displayed global leadership in higher education. It has the highest number of Ph.D’s, high-tech engineers and technicians, researcher faculties and members, think tanks and universities in relation to its population. It has also the highest number of patent ownerships and pending patents in the world (recent competition, and a serious one in patent ownership is now coming from China and India). This is the outstanding record of a nation that has understandably led the world in seeding the knowledge-economy that is harvested by many nations. How can one then accept the dismal record of mediocrity and failure in US schools? It is not just a matter of national pride. True, many in the US are distressed by school failure in Math because it is a sad betrayal of the standards that the US has set for the world in most academic disciplines. Many more worry about what continuing failure portends for the US national economy in the coming decades.

The world at large, including North America, may justify the existence of the age-old “national Math curriculums” on moral grounds. This may surprise many but here is how the argument goes:

“We have to offer Math, with its broad curricular sweep from K-12, to all students because it is their inherent right. We cannot offer it to brighter students because that would be discriminatory. By following such an equitable approach, we cannot be blamed for not having offered each and every student an equal opportunity to succeed and become an aeronautical engineer, or an astronaut, pilot, engineer, scientist, physicist, doctor, and so on. The burden of failure and its consequences fall upon their shoulders, not ours”

The most expensive Hotel in London takes mischievous pride in its claim that: “The revolving doors of our Hotel are open to all without any discrimination whatsoever”.

The analogy holds for the offer of Math education for all. There remain systemic pitfalls, gender-bias hurdles, curricular limitations, inadequate design, social and pedagogical inequities plaguing the delivery of “Math education for all”. When all of these are set into motion, only the privileged few get to pass through its many revolving doors. It is odd that all those doors are powered by the notion of democratic equality.

The Five Stages of Math, as proposed in this book, are not set in stone. They do not answer many questions, but do raise others that readers are likely to nod at. The Stages will certainly undergo some transformations in their form and content. It is hoped that valuable contributions from my peers will hone the Stages further with sharper edges. I expect that its evolution into a more elegant and comprehensive form will continue with extensive support from formal research.

In the meantime I hope that readers accept the Stages in the spirit in which it is offered: to provoke intelligent debate and discourse on all those thoughts, ideas and feelings that we hold very dear when it comes to Math education for our children.


Shad Moarif,
December, 2009.



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