Brief Summary of the Academic Principles of Towermaths by Professor Koop Lubbinge


Brief Summary of the Academic Principles of Towermaths by Professor Koop Lubbinge


Brief Summary of the Academic Principles of Towermaths
  1. Operational model.
    • The Transformation/conversion model:

This model is based on Operational Management principles.

Inputs are Transformed/Converted by a process that changes the inputs into outputs. For the purpose of this discussion conversion applies to inanimate (dead) inputs while transformation applies to animate (live) inputs.

A transformation process is much more complicated than a conversion process. The reason is that an animate input actively interacts with the transformation process. Furthermore an animate input can have a memory of previous transformation processes.

There are two kinds of transformation processes:

  • Transformation of an input that is totally hardwired. This is an input that does not have consciousness. It always reacts to its environment in a predictable manner since its ‘behaviour’ is totally controlled by its genetic design. It also does not have a memory of previous transformations. An example is transforming seeds into a plant. Let us assume that ALL seeds have the same potential for germination. In this case the transformation process has to create (and control) the correct CONDITIONS for the seeds to germinate and grow into plants, from soil preparation, planting, growing to harvest.
  • Transformation of an input that is a combination of conscious and sub-conscious neuron networks. This input can be an individual, a group of individuals, an entire organisation or even a country. It is immediately evident that this is a much more complex input that demands a very sophisticated transformation process that should include (amongst other) measurement, feedback and LEARNING! This transformation process employs ACTION LEARNING!

1.2. Product versus Service: Transformation processes most often deal with services while conversion processes deal with products. A product is an object that can be seen, touched, stored and (sometimes) consumed. Services are invisible (cannot be seen, touched, stored etc.) and are ‘consumed’ as they are produced.

Learners are the input into a transformation process that changes (transforms/grows) their brains. The output of the transformation process is a learner with a changed (enlarged) brain in the sense that it (the brain) is now able (competent) to do things it could not do before being transformed. This also applied to entire organisations since they also have conscious (systems) and sub-conscious (culture) ‘brains’.

Teaching has all the attributes of being a SERVICE! Although is uses visual resources, its service is being consumed by the learners as it is supplied by the teachers. 

1.3. Process yield.

Process yield is the ratio of its output over its input. In the case of mathematics the output of grade R is the input of grade 1. The output of grade 1 is the input of grade 2 etc. etc. The total transformation process is a sequence of 13 sub-processes (grades R, 1 to 12). These sub-processes interact with (and are interdependent of) each other.

If each sub-process (grade) yields (successfully transforms) 90{59280e49db9ccb102d2acb137052d0edd228c7eb563e70164082eace3d812984} of its input, the total process will NOT yield (successfully transform) 90{59280e49db9ccb102d2acb137052d0edd228c7eb563e70164082eace3d812984} of its inputs but only:


This (unpleasant) reality is not generally appreciated by educational practitioners in mathematics.

1.4. Process variation:

Process variation is the variation in the output of the process. Not all learners will meet the minimum standards applied to the output. The design, operation and control of the transformation process will influence process variation. There are two fundamentally different causes of process variation:

  • Variations caused by the design of the process. These are called NATURAL This is best explained by an example: Imagine being a hunter. Live buffalo have to be ‘transformed’ into dead buffalo. (I know this is not a good example of a service to the buffalo). The transforming resource is a gun. Every gun will have natural variations due to its design. Even under ideal conditions its bullets will NEVER all hit the target in ONE SPOT (the bulls-eye). The bullet holes will be distributed around the eye. The amount of spread is called the NATURAL variation of the gun. It is measured by the guns process CAPABILITY. A gun with a process capability of 1 (100{59280e49db9ccb102d2acb137052d0edd228c7eb563e70164082eace3d812984}) will have ALL bullet holes INSIDE the defined TARGIT. A gun with a process capability of .5 (50{59280e49db9ccb102d2acb137052d0edd228c7eb563e70164082eace3d812984}) will only have half of its bullet holes inside the target area. This variation is NOT caused by the hunter or the environment. At best the hunter will only yield 50{59280e49db9ccb102d2acb137052d0edd228c7eb563e70164082eace3d812984} of the potential number of buffalo. Improvement demands a REDESIGN of the GUN!
  • Variations caused by the operation and environment. These are called ASSIGNABLE variations. These could be the ability of the hunter, availability of buffalo, wind etc.
  • Reducing process variations: Generally there seems to be an assumption that ALL process variables are ASSIGNABLE. In many countries teachers are evaluated on the YIELD of the transformation process, without regard for the NATURAL variations due to the DESIGN of the process. This is seriously INHIBITING any attempts at improvement.

1.5. Process effectiveness.

This is defined as the success rate of the process in meeting its PURPOSE! The PURPOSE of the grade R transformation process is (should be) to:

(a) Ensure that ALL learners ENJOY the transformation process.

(b) Ensure that a DEFINED percentage succeeds to perform to (meets) DEFINED STANDARDS. This takes into account variations in the INPUT.

1.6. Process efficiency.

This is defined as the COST per successfully transformed learner. It is influenced by the cost of the INPUTS as well as the process YIELD. Process yields of 45{59280e49db9ccb102d2acb137052d0edd228c7eb563e70164082eace3d812984} will DOUBLE the cost per learner compared to a yield of 90{59280e49db9ccb102d2acb137052d0edd228c7eb563e70164082eace3d812984}. In many instances improvement focuses on EFFICIENCY. Many technology applications are but one example.

1.7. Process measurement.

Measurement is done for three distinctly different purposes:

  • Measurement for score keeping. A score is neutral information. It merely states facts numerically. Examples are: Information on the schools, teachers, trainers and learners involved in a project.
  • Measurement for evaluation of results. A result is a comparison of the measured variable with a standard or an intended outcome. Progress report of a project against a plan. Assessments of results per learner, grade, school and project against defined standards are examples.
  • Measurement for learning. This concentrates on identifying cause and effect relationships. It is an integral part of ACTION LEARNING. It is invariably used in experimentation. It attempts to answer the questions: What happens if I do that? What do I do if it happens? What do I do if it does not happen?
  1. Design of the human brain:
    • Neuron plasticity: Up to late 1980 it was generally believed that the mature human brain was of a fixed size and that neural circuits (mental programmes) are formed by means of repeated actions. With sufficient repetition these programmes become ‘hard wired’ in a memory.

Recent research has shown that the adult brain is ‘plastic’ in the sense that, through a process of focussed thinking/learning, it can grow. The human brain can continue to grow even at an advanced age IF it is subjected to regular thinking (learning).

Brain growth depends on LEARNING! Millions of stem cells are continuously created (secreted) in the brain. These stem cells are reabsorbed after 4 days if NOT stimulated. Stimulation is by means of a transformation process which transforms these stem cells into infant-neurons. Only a transformation process of LEARNING succeeds in initially saving these stem cells from annihilation by re-absorption.

Infant neurons have to be regularly (daily) stimulated to grow into neurons that are incorporated into the neuron-networks of the brain. This process continues throughout a person’s life. There is considerable medical evidence that a damaged brain can ‘repair’ itself if sufficiently stimulated by inputs:

Terry Willis was 19 years old when he was involved in an accident, after which he was in a coma depending on life support systems. For 19 years his mother kept him informed of all the normal events DAILY as though he was alive and well. During all this time Terry showed no evidence of any normal brain function. One day Terry regained consciousness and within one year his brain was functioning normally, although he remained paralysed from the waist down.

Using brain scans neurologists were able to examine how Wallis’ brain had repaired itself. During his 19 years in a coma his brain, being daily stimulated by his mother, effectively ‘grew’ new neuron-networks and essentially ‘rewired’ itself. Once sufficient neuron-networks had been grown his brain ‘rebooted’ itself.

His ‘miraculous’ healing has made him a world-wide celebrity and a scientific legend.

Epigenetics is the study of how changes in GENE EXPRESSION is influenced by GENES that are switched ON or OFF. What does this mean? Why is this important to teaching of mathematics? Popular ‘science’ has conditioned us (programmed our sub-conscious mind) that we are mere products of our genes. We are to believe that there is a ‘gene’ for just about every human trait. Furthermore we are under the control of these genes. In short: If a learner does NOT have the ‘mathematics gene’ it is told: Mathematics is not for you! This is like a life sentence.

What if there is indeed an ‘inherited mathematics gene’ that prevents us from succeeding in mathematics? Epigenetics suggests that this gene can be prevented from expressing itself by switching it OFF! Furthermore: There are strong indications that this can be done by an act of WILL!

BELIEVING that this gene determines your ability to ‘do’ mathematics (mathematics is NOT for you) will ACTIVATE (switch on) this gene! This is a physical demonstration of the destructive impact of a SELF FULFILLING prophecy!

BELIEVING that mathematics is for you will DEACTIVATE (switch off) this gene. Again, this is a physical demonstration of the constructive impact of a self fulfilling prophecy!

  • The human brain is an analogue (analog) system and NOT a digital system (wetware)

An analogue system receives and processes information from many sources continuously and simultaneously. This information is not exact but a reasonable estimate. A good example is an analogue clock of which the hands move smoothly in a circle. The position of the hours/minutes hands is an ESTIMATE of the actual time.

Information from sources A, B, C, D, E and F is activating relevant brain functions SIMULTANEOUSLY. These functions are ALL interacting with each other AT THE SAME TIME as shown:fig.1An example is driving a car: The brain is continuously receiving inputs (distances, images, sounds, smells, movements, changes etc) simultaneously from the environment. These inputs are NOT exact in digital units: Distances are NOT measured in meters. Light is not measured in lumen. Sound is not in measured in decibels. Movement is not measured in miles/hour etc. etc. These inputs are estimates that only have to be accurate enough to allow us to drive the car safely to its intended destination.

A digital system receives information from many sources by means of numbers which are processed sequentially as shown:fig.2Counting is a digital activity that does not come to the brain naturally. Primitive tribes only count 1, 2, 3 and many. Neuroscience is providing new discoveries in the field of numerical cognition. It is now possible to see that a whole group of neurons is lighting up when a number is observed. Some neurons have a ‘preferred’ number that makes them most active. For example several thousand neurons shine brightly for 1, less brightly for 2 and still less brightly for 3 and so on. Another group of neurons prefers the number 2, etc. etc. Thus explains why our brain naturally favours an approximate understanding of numbers.

If our brains can naturally only represent numbers approximately, how did we ‘invent’ numbers in the first place? Exact ‘number sense’ seems to be a uniquely human property which became essential when we became involved in trading. This demanded that each number is given a unique NAME which later was represented by a unique SYMBOL.

Numbers are constructed by MAN and are NOT something that we acquire innately.

  • The human brain functions on a quantum level.

Quantum physics (mechanics) is the study of the behaviour of matter and energy on the subatomic level. It deals with fundamental (elementary) particles smaller then atoms. Why is it essential to know about quantum physics in a discussion about learning mathematics? What is its relevance to teaching? Synapses are the spaces between the neurons of the brain that conduct signals using parts of ions (charged atoms). These ions function according to the rules (laws) of quantum physics.

Two laws are of particular importance:

  • The quantum Zeno effect: Elementary particles can be unstable and change from one form to another. This is called ‘decay’. Physicists have found that if they observe (measure) an elementary particle continuously it never decays, even though it would decay when not observed. The physicist is in fact holding (capturing) the unstable particle in a given state by the ACT of observing (measuring) it. Because your brain is a quantum system, IF you FOCUS on a given IDEA (THINKING!) you hold its PATTERN of connecting NEURONS in place. The IDEA will DECAY if not OBSERVED by FOCUSSED THINKING! This requires a DECISION of WILL! The ACTION of HOLDING an IDEA in place is a DECISION to continue to OBSERVE it. This is the REASON that only FOCUSSED THINKING (LEARNING) is able to prevent stem cells from decaying. It is only THINKING that can rescue them and GROW them into neurons.

In its relaxed (neutral) state the brain meanders from one thought to another to another. In any moment in time this keeps all possible options for future action open UNTILL a DECISION of WILL is taken for a specific action. This decision of WILL captures ONE option (IDEA) which becomes the FOCUS of the brain. Human WILL prevents the decay of the IDEA into mindless meandering! It is this FOCUSSED LEARNING (DEEP THINKING) that GROWS the brain! It is the human MIND that takes this decision. It is the MIND that controls the brain! Have you ever heard anyone say: I have made up my brain?

  • Elementary particles can be entangled (in the same state) despite not being in contact with each other (at a distance). This effect has also been observed for more complex material systems such as molecules. Brains also do not act in isolation but can be (and are) influenced (at a distance) by other brains. Since the mind is in control of the brain we can talk about ENTANGLED minds. This means that two independent minds can act as ONE although they are physically apart. This is called the Psi effect. There is convincing (odds against chance of 1-millions) that Psi does exist. The INSPIRATIONS and FEARS of one MIND can create similar INSPIRATIONS and FEARS in ANOTHER distant mind. This implies that the FEARS of a person (the teacher) involved in TRANSFORMING the MIND of another person (the learner) can be transmitted to the learner, not only physically but also through entanglement. Similarly the INSPIRATIONS (neuron patterns) of the TRANSFORMER (teacher) are creating similar INSPIRATIONS (neuron patterns) in the TRANSFORMED (learners). We are all familiar with the lasting influence of an INSPIRED TEACHER on learners. Sadly we are also familiar with the Mathematics teacher who instils a FEAR of mathematics in learners.
    • The human brain functions on a conscious and a sub-conscious level.

The sub-conscious brain continually monitors all physical variables and controls all bodily functions. The functioning of the sub-conscious brain is NOT a function of WILL! The neuron-networks are hardwired.

The conscious brain is under the control of our FREE WILL. Its actions are undetermined until a DECISION is taken. Repeated repetition of conscious actions results in neuron-networks that operate at a sub-conscious level. “Practice makes permanent”. This is when the conscious brain is on ‘auto pilot’. These learned ‘hard wired’ neuron-networks have many advantages since it frees brain capacity for more important decisions. A major disadvantage is that outdated/undesirable learned “hard wired’ neuron-networks are VERY difficult to unlearn. Furthermore there is convincing evidence that the sub-conscious neuron networks of a teacher entangle with sub-conscious neuron networks of learners.

  • Optimum brain growing conditions

A transformation process does not physically grow its LIVE input’s brains but creates the OPTIMUM conditions for growth. Living inputs will react naturally to favourable conditions for growth. For the brain these are:

  • Play: This is the oldest and most common human addiction. A transformation process that is designed as a game creates optimum conditions for learning. A game concentrates the mind on the outcomes and provides instant feedback based on the rules of the game. Outcomes can be neutral (although pleasant) or indicate winning and losing the game.
  • Patterns: The brain demands patterns. Without a PATTERN the brain cannot consistently grow meaningful neuron-networks since its absence prevents the brain from being able to FOCUS! Without a consistent pattern there is no connection between the past, the present and the future. Yesterdays IDEAS (learning/outcomes) will conflict (contradict) with those of today. Today’s ideas learning/outcomes) might be totally unrelated to those of tomorrow resulting in confusion and an absence of neuron growth.
  • Pleasure: The brain demands pleasure (craves dopamine) after action. The results of learning have to be pleasurable. One way is to give recognition for the RESULTS of learning. Another (much neglected) way is to design pleasure into the learning process. Still another is to design pleasurable OUTCOMES into the learning process.
  • Senses: A maximum number of senses (sight, hearing, touch, smell etc.) have to be activated in the learning process. These simultaneous inputs enhance the transformation process.
  • Challenges: The learning process should pose a challenge to the learner. Watching TV is not a learning process. Listening to a story, that demands that the learners ‘live’ the story and create their own images, is.
  • Individual and groups: The learning process should include individual as well as group learning. Constructive interaction with other learners has to be LEARNED! Learning mathematics involves the development of IQ (Intellectual intelligence quotient) as well as EQ (Emotional intelligence quotient).
  • Previous learning: Learning should be based on (connected to) previous learning since this modifies (enhances) neuron-networks that already exist, instead of having to build entirely new neuron networks.

2.7. Conditions that INHIBIT brain growth.

The most prevalent condition is FEAR of mathematics. Fear for mathematics can be caused by parents, teachers or the general environment. One of the reasons could be that present teaching methodology is not BRAIN FRIENDLY. Many people on public and social media confess to a fear for mathematics. Fear for mathematics is often sub-conscious. This makes it very difficult to treat. There are three options open for a brain that fears mathematics:

  • Fight: This option is not normally open to foundation phase learners. I did come across a grade 4 learner who told the teacher: “I hate sums! I hate you! I will burn down this school!” He was asked to leave the school because he became totally unmanageable.
  • Freeze: This is a frequent option. This fear is most often expressed when writing a test or an examination. Some parents try to solve this problem by administering sedatives before tests.
  • Flight: This is the most prevalent option. Numerous learners simply abandon mathematics with disastrous consequences for their future employment opportunities.
    • Geometry.

Geometry (Geo-earth, metry-measurement) basically deals with MEASUREMENT of space, shapes, position, size, time and actions. It represents its results by means of PICTURES using lines, points, coordinates (in up to three dimensions), angles, position, sequence, order and movement etc. Basic geometry dates back to earliest human civilisations. Early (primitive) measurement of geometric variables was based on estimation and patterns.

Geometry is part of our daily life from birth to death. Dealing with geometry (measurement of our environment) starts at birth and continues as a daily (at times subconscious) activity.

Teaching of mathematics in the foundation phase should START with GEOMETRY since this is the FOUNDATION that is part of a pre-school learner’s first 5 years.

Algebra is a language using symbols to describe geometric (and other) phenomena. Its results are expressed in NUMBERS. Numbers originated about 10 000 years ago. Algebra is NOT part of our daily life. It is a recent innovation that has allowed the development of modern science and technology.

  • Number systems.

A number is a VALUE expressed by a word, symbol or figure representing the quantity of a geometric variable. A number (numeral) system is a WRITTEN system for expressing numbers. It is a mathematical notation for representing numbers using digits in a consistent manner. Many number systems have developed over time. The DECIMAL number system is generally used while the Indo-Arabic notation is the international standard.

  • Decimal system.

All number systems use the 1 as a unit. The symbols 2, 3, 4, 5, 6, 7, 8, 9 represent multiples of 1. The number 0 is a recent invention that allows us to write any number using only the above symbols. The first 10 digits of the decimal system are NOT: 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 but: O, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Contrary to popular belief the digit 0 is NOT nothing! Every journey starts at 0. Every set of coordinates has a 0 position. Every reconciliation of financial statements has to result on 0. When you are stationary your speed is 0 km/h etc. etc. Independent of its practical value it also serves as a ‘place holder’. This is based on the principle that different POSITIONS of a number represent different ORDERS of numbers. For instance the numbers five does NOT have ONE value (5). It could have any other values like 50, 500, 5000 etc. depending on WHERE it is POSITIONED in the number.

  • Mathematics is a language.

Brain scans reveal that doing mathematics (sums and calculations) activates the same neural networks that are associated with linguistic processing. This is important since a word sum that is stated in English has to be translated into Mathematics before it can be solved. This is not recognised in present mathematics curricula.

  1. Design of the transformation process for foundation phase learners.

4.1. Process capability.

Process capability is WHAT the process can do. Starting with grade R the input into this transformation process is a typical pre-school learner. The following question has to be answered:

What is the required PROCESS CAPABILITY for a grade R transformation process? To answer this we have to define the REQUIRED OUTCOMES for grade R.

4.2. Process capacity.

Process capacity is HOW MUCH the process can do. The following question has to be answered: What is the required PROCESS CAPACITY to transform a group of 36 grade R learners over a typical academic school year of 40 weeks.

 4.3. Process outcomes.

Present grade R outcomes are based on present ‘truths’ about the mathematical limitations of the pre-school learner’s brain. These limitations should NOT be accepted as given. A typical grade R learner has considerable language skills. Foundation phase mathematics is endlessly LESS complicated than any spoken language. It seems very likely that a typical grade R learner’s mathematics potential is much higher than previously assumed. The design of the grade R transformation process should NOT be based on present grade R outcomes.

4.3. Process inputs.

Any normal five year old can identify hundreds of individual objects and name most of them. They can do many tasks. Most of these are based on the GEOMETRY of their ENVIRONMENT! The design should build on the conscious skills of:

  • Visual identification of objects by shape (pictures)
  • Auditory identification of objects by sound (names)
  • Visual identification (estimation) of position of objects in space.
  • Auditory identification of position objects in space.
  • Visual identification of actions by changes in position (movement)
  • Execution of verbal instructions.
  • Relating actions with outcomes.
  • Evaluating actual outcomes with intended outcomes.
  • Remembering previous events.
  • Projecting future events.
  1. Design of transformation process for grade R and grade 1 teachers.

                 5.1. Change management.

Foundation phase teachers bring their own conscious and sub-conscious networks that have to be changed. Furthermore, they are bound by curriculum and organisational constraints. Ideally they should participate in the transformation process.

For change (solving a problem) to succeed there have to be three conditions that have to be satisfied.

  • The problem has to be perceived. A problem that exists but is not perceived is NOT ready for solving. Education experts do perceive that there is a problem with the teaching of mathematics in the foundation phase. Foundation phase teachers do perceive that there is a problem with teaching of mathematics
  • The need for a solution has to be expressed. This is normally only the case if the price for NOT solving the problem is high. There is agreement that solving the problem is urgent. Initiatives to date do not seem to have made a noticeable impact on results.
  • A solution has to exist. Most initiatives seem to focus on doing present things better (efficiency). All ‘new’ designs are still based on past/present (outdated) education theories about the limitations of grade R learners. There seems to be a growing realisation that the process has to be redesigned. SINGAPORE MATHS is seen as an option as are some Oriental methods like Kumon.

It is NOT always appreciated that only a PUZZLE has a (unique) solution. A PROBLEM does NOT have a solution but only OPTIONS. Each option has advantages and disadvantages.

Excellence in mathematics should not be confused with being able to do sums correctly and quickly. Solutions to do sums quickly and accurately have existed for many centuries: The Japanese and Chinese abacus, (widely used by the Romans for trade transactions) Russian schoty, Chinese suan-pan and Mayan nepohualtzintzin, are all excellent examples of visual methods of calculating quickly. Neural imaging scans show that the parts of the brain activated by the abacus are DIFFERENT from the parts activated by mathematics. The abacus neural networks are associated with VISUOSPATIAL information. This is closely associated with sports like tennis.

There is no practical evidence that excellence in doing sums quickly is related to excellence in mathematics as there is no evidence that winners in spelling competitions are likely to become authors or poets.

The following example demonstrates this: A SOROBAN champion (in quick calculations) and a grade 1 (Towermaths trained) learner where asked the following mathematics question:

The sum of two consecutive (adjacent) numbers is 8422. What are the numbers?

  • The SOROBAN champion is still ‘calculating’ the answer.
  • The grade 1 learner (Annabella Wolfaardt) answered: This cannot be correct since the sum of two adjacent numbers is ALWAYS UNEVEN!

A more recent development takes into account that there is more to improving teaching than software and hardware. A new terminology is developing around WETWARE! It seems the teaching profession is finally waking up to the reality that teaching is a process that TRANSFORMS the BRAIN of the LEARNER!

Any ‘solution’ to the teaching of mathematics in the foundation phase has to be BRAIN BASED!

CONCLUSION: There are convincing experimental indications that Towermaths goes a long way in realising the full potential of foundation phase learners. This is discussed in a following communication called: A brief summary of the development of the Towermaths methodology.

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