Church growth modelling attempts to use mathematics to bring understanding about the way churches grow and decline in numbers. As churches contain people there would appear no reason why the sort of mathematical models used in population modelling could not be applied, with appropriate modifications, to growing churches. However, such an approach could leave a Christian believer with feelings of unease as Biblical conversion is more than a social phenomenon but a work of the Holy Spirit. Surely the miraculous cannot be modelled!

The modelling work also encompasses sociology, a subject often distrusted by believing Christians as sociologists are perceived to be antagonistic to religion, especially anything supernatural. Further still, the work leans on ideas from the church growth community while at the same time discussing the hallowed subject of revival. Many Christians who are passionate about revival deeply distrust church growth methodology. They fear that it has compromised the supernatural and Biblical foundations of the church, replacing it with pragmatic and secular principles. Thus church growth modelling runs the risk of being seriously misunderstood by those who it seeks to help the most.

The following defence of church growth modelling is based on the explanatory notes for the first paper. It is inevitably long and must take the reader back to the meaning of both mathematical modelling and revival to be worthwhile. This defence does not extend to church growth methodologies or the use of sociology in religion. These subjects have little impact on the nature of the models or their results.

John Hayward

Mathematical Modelling

To help understand the limits of the church growth models, it might help if the reader can understand what mathematical modelling is and why it is used. This approach will take us into the realm of physics, where mathematics has its most successful applications. This section may require a bit of persistence!

Definition of Mathematical Modelling

A model of any real-world object or situation is an attempt to describe that object or situation by certain key features of interest while discarding those features which are not of interest. A mathematical model is such a model described using mathematics, usually with the purpose of explaining why something behaves the way it does, discovering some laws or patterns, and maybe making predictions. Thus, a model has a purpose, and mathematics is merely the language that enables the understanding and purpose to be expressed quantitatively and precisely. Purpose is essential for modelling.

This process of discarding uninteresting features is called abstraction, and assumes one has a context for what is interesting or what is not! Context is also essential for model construction. In the case of mathematics, anything that cannot be expressed quantitatively is jettisoned since it is outside the bounds of the model.

An example of a non-mathematical model would be a scale model of a car. Its purpose is to achieve a miniature replica. In doing so many features of the real-world original are abandoned such as a working engine!

Much of science involves modelling. Every model inevitably has some compromise built-in; mathematical models are no exception. What is compromised depends on the situation being modelled.

Fundamental Physics

The best known mathematical models are those of fundamental physics. Typical of these are Newton’s laws of motion. They can be expressed mathematically and are perhaps the most extensively tested of all laws of science. So much so no mechanical engineer would ever doubt them and together with Newton’s law of gravitation were used to send Apollo 11 to the moon and back without the slightest doubt that the model would break down. To call such theories “models” is almost underselling something which seems part of the way God made the universe. In fact, they are usually referred to as “theories” – models with a high degree of confidence.

In practice, Newton’s theories break down in three areas:

  • high speed – solved by special relativity;
  • small distances – solved by quantum theory and special relativity;
  • high acceleration – solved by general relativity.

As yet, there is no theory that covers all scenarios, however, a search for such a model (sometimes called a unified field theory) continues with the assumption that one must be found.

However, not all modelling concerns such fundamental laws of the universe. Even given such laws, further modelling needs to be done to handle solvability and complexity.

Modelling to Handle Solvability

Given the fundamental laws of the universe, simple situations can be described and predicted with very high accuracy; for example, the motion of one planet orbiting the sun. However, extend this to many planets orbiting the sun then, although it can be described precisely with mathematics, it can’t be solved, the equations are too hard. Thus, further modelling needs to take place in order to achieve any predictability out of the model. This further modelling is achieved by making assumptions. Thus there is now a less than perfect model of the real-world situation in order that some answers can be found. It started with a fundamental law, but assumptions are needed to simplify the mathematics and obtain a result.

Such modelling is familiar in engineering. Although based on Newton’s laws, complicated arrangements for machinery are governed by equations that need further assumptions to be made in order to solve them. The arrangement of simple things can be made so complicated that mathematics (which is essentially simple) does not have the power to produce perfect results. However, with time and effort, the results can usually be made accurate enough.

Modelling to Handle Complexity

This complexity can be taken many stages further when living things are examined. Consider population modelling. Populations are composed of people, who are made up of basic chemicals, which in turn are composed of particles that obey fundamental laws. However, the situation of a growing population is so far removed from the underlying fundamental model that any attempt to even construct models, let alone derive behaviour, from this has to be abandoned as the situation is so complex. Instead, a model is started from scratch based on observation and some sweeping assumptions. These assumptions are made to reduce complexity in the model description. They are often called “empirical” models.

For example, the simplest model of a growing population is the “exponential law”, which can be expressed as “a population will double in number in a fixed interval of time”. The USA population 1790-1850 doubled in number about every 15 years. This “law” is easily derived from the assumption that, on average, family size is constant.

Most mathematical modelling comes down to this type of empirical modelling, whether they are models of traffic flow, the economy or industrial processes. Clearly, such models cannot be thought of in the same light as the fundamental laws of the universe since they are so dependent on the assumptions made. For example, the assumption of constant family size is dependent on people’s behaviour (very unpredictable) as well as outside factors such as famine, disease etc. Thus the exponential law will only ever have approximate validity for short periods.

If such empirical models have such a limited validity, why are they constructed? The main reason is that they increase understanding of the situation modelled, either numerically or in the form of a principle. Thus, in the case of the exponential law, it becomes clear that if family size remains fixed, and the population is growing, then it will ultimately get ridiculously large. The model will also give some indication of the time scale on which this increase occurs. If it takes thousands of years, it is no problem; if it is a hundred years, then policies are needed to slow the growth. Thus, these models can make limited numerical predictions.

However, there is also a principle. The only way to stop a population from growing exponentially is to reduce family size! Even if no numbers can be predicted, some strategy can now be tried that effectively reduces family size. Perhaps different strategies can be built into the model and results compared.

The church growth models are this type of empirical model. They make no claim to being fundamental laws of the universe. Neither do they make any attempt to predict human behaviour, let alone God’s influence on the growth of his church. All they say is that given certain assumptions, then a certain type of growth follows. These assumptions nearly always reduce down to saying “given God continues to act in the same way”, or “people continue to act in the same way”. No attempt is made to model how God’s ways might change, such as when he might pour out his Spirit on the church. Such a model would make God subject to cause and effect and dependent on his own creation rather than the sovereign and totally self-sufficient God he has revealed himself to be.

Empirical models have for many years had a certain amount of success in modelling complex social situations, provided one realises they are not fundamental laws and that the models are very limited in their area of application. The advantage of such models are:

  • the ability to make predictions;
  • discover principles;
  • examine the effects of changing parameters and strategies.

The disadvantage is that people think the models are saying something more profound than they really are! Mathematics is merely an attempt to describe, in quantitative terms, the processes we see. It does not explain how the process is caused. Assumptions have to be made to explain the process; the mathematics works out the consequences of these assumptions.

Understanding Revival

Definitions of Revival

Some thought needs to be given to the relationship between the church growth models and the phenomena of revival. This relationship is a complex issue as there are many definitions of revival. Some definitions are theological, centring on what God does, while some definitions give more weight to the effects produced in revival on people or the community. Consider the following definitions:

  • An outpouring of the Holy Spirit e.g. Acts 2:17. (Many authors, e.g. Edwards 1965, Jonathan Edwards on Revival, p17)
  • A manifestation of God’s glory, with reference to e.g. 2 Chronicles 5:13-14. (Many authors.)
  • A repetition of Pentecost ( Lloyd-Jones 1984, Joy Unspeakable,”the Way To Revival”, p278 in the first edition; Lloyd-Jones 1886, Revival, ch.8, p100).
  • The Holy Spirit comes down upon a number of people together (Lloyd-Jones 1886, Revival, ch.8, p100]). Lloyd-Jones uses many expressions p.100 gives “The Spirit of God has descended in their midst. God has come down and is among them. A baptism, an outpouring a visitation.”
  • An extraordinary movement of the Holy Spirit producing extraordinary results. (Roberts 1991, Revival, p16)
  • A community saturated with God (Duncan Campbell, quoted in Edwards 1990, Revival: A People Saturated With God, p. 26).
  • A remarkable increase in the spiritual life of God’s people, accompanied by an awesome awareness of the presence of God, intensity of prayer and praise, a deep conviction of sin with a passionate longing for holiness and unusual effectiveness in evangelism, leading to the salvation of many unbelievers (Edwards 1990, Revival: A People Saturated With God, p28).
  • A sovereign work of God in which God pours out his Spirit on all flesh (Riss & Riss 1997, Images of Revival, p.1)
  • An extraordinary work of a sovereign God that revitalises the church and transforms society (Greig 1998, Awakening Cry, p.xv).
  • An outpouring of the Holy Spirit, brought about by the intercession of Christ, resulting in a new degree of life in the churches and a widespread movement of grace among the unconverted (Murray 1998, Pentecost Today, p23-4).

Clearly, some of these statements do not mean the same thing. A community saturated with God could adequately describe some situations in the Old Testament Israel when God made his glory known. However, these could not be described as a “repetition of Pentecost”. Pentecost hadn’t taken place! They are not even an “outpouring of the Spirit”, as the Spirit was only given, in the sense of his anointing, to selected individuals before Pentecost. However, all definitions apply to the situation in the Christian church and can be summed up by saying that God has done something to people.

Revival and Church Growth

The popular view of revival is that it adds many people to the church. Strictly speaking, this is a result of revival, not revival itself. Riss & Riss 1997 has strongly emphasised this fact in the wake of the Toronto Blessing. Revival is something that happens in the lives of people, both believers and unbelievers; however, in the latter case, it is often referred to as an awakening. If such revival happens in a large number of unbelievers, the growth of the church results.

By contrast, if revival is largely confined to believers, then the growth, initially, will be in the numbers of “revived” Christians within the church. This pattern has characterised the charismatic renewal of the 1960s to the present, as well as the recent Toronto Blessing. Both movements have led to substantial growth in the number of “charismatic” influenced Christians, who have come from within the church, rather than converts from outside.

In fact, churches can grow numerically without any divine intervention at all! Churches can attract “social converts” – those who have adopted Christianity as their religion but for whom there is no supernatural or spiritual change within. It shouldn’t happen, but it does, and it is often difficult to tell the difference between real converts, who have been changed by God, and social ones. In times of revival, this effect can get worse, particularly if churches fail to test converts.

Some Christians struggle to recognise the existence of social conversions; however, without them, there is no other way of explaining the growth of other religions. I doubt if any of us who are Christians would attribute the rapid growth of Mormons, Jehovah’s Witnesses or Islam to an outpouring of the Holy Spirit! The dynamics of growth of these groups is similar to the Christian church, but the source of the growth is very different as far as a Christian is concerned.

The Church Growth Model and Revival

The models of church growth describe “revival-type” growth, whether it comes from spiritual conversion or social conversion. The key feature is that belief is spread by contact between believers and unbelievers. Since God normally changes people through the preaching of the Gospel, and via the witness of Christians, then “spread by contact” is true, even for spiritual conversion. In the case where God changes people directly without the normal means (e.g. the apostle Paul?) then the model would not apply.

Growth patterns vary through a change in the parameters of the model. One parameter depends on the effective contact between the believer and the unbeliever. This mechanism is an averaging effect over a large number of people but must ultimately be determined by how much actual contact takes place; the nature of the contact (what is said by, or seen in, the believer); and how receptive the unbeliever is to change. In the case of social conversion, the latter would be some psychological change in the unbeliever. The other changes in the believer would have to be interpreted in terms of their enthusiasm. Still, it would be difficult to find a model for the cause of their changed behaviour, particularly if such changes occurred suddenly.

In the case of spiritual conversion, which is the main interest of the paper, the changes in the believer and the unbeliever come about by God’s activity. Note that the more involved definitions of revival (especially the last one above) pick up on this dual change, both in the believer, which makes them an enthusiast, and the unbeliever which brings about their salvation. This view depends on a theological model of conversion which views the cause of a person’s conversion lying with God, but the means of their conversion is the interaction between a believer who witnesses and the unbeliever whose heart is opened by God.

The mathematical model merely describes the growth that takes place given a combined average value for these effects, but cannot say how that average value is achieved, or whether it stays constant over time or location. As stated before, the model only works given that God continues to act in the same way. The cause of the changes in believers and unbelievers lies firmly with God.

The Church Growth Model and Conversion

This view of conversion leads to a significant controversy in connection with revival governing the extent of God’s involvement and human involvement in conversion. In the early years of the 19th century in the USA, a different theological model for conversion arose which placed a greater emphasis on a person’s ability to change their mind regarding Christian things and become spiritually changed by an act of their will. This controversy, often associated with Charles Finney and called the “new measures” controversy, led to a change in the conduct of Christian meetings where more human “pressure” was exerted to secure a conversion. Such meetings came to be called revival meetings.

As such, the word “revival” was used in the USA to mean any type of evangelistic meeting. The UK churches have not generally adopted this use of the word revival (even if they agree with the “new measures” theology) and only use revival if some significant spiritual change takes place. All the papers and pages on the Church Growth Modelling website use the word “revival” in the older sense of the word, although the model for growth could be used for any non-revival situation that involves evangelism.

The “new measures” beliefs still hold considerable sway in the Christian church, although most people would probably place themselves on a point somewhere between the old and the new views. Proponents of new measures will be able to interpret the church growth model in terms of their beliefs, but they may have a harder time interpreting the makeup of the parameters in the model, just as they have a harder time distinguishing social converts from spiritual ones. An important assumption in these models is that the changes in both believers and unbelievers are brought about by God even if he uses other people as instruments. The theological backdrop of the models generally follows Murray 1998 as this seems to fit the Biblical evidence better.

The Church Growth Model and the Reviving Work of the Spirit

A further controversy concerns the change that takes place in the believer. Is it a fixed effect, moving from an “ordinary” Christian to a “Spirit-filled” one, or is it something that occurs in varying degrees? Undoubtedly the enthusiasm and ability in witnessing can occur in different degrees, even in the same person. In the model, this is reflected in the parameter called the “conversion potential” being able to take a variety of values.

Murray 1998 argues that the underlying work of the Spirit that causes these revival effects also occurs in different degrees. Even putting aside any Biblical arguments, this view makes sense. It would be hard to see how a fixed spiritual change could make different degrees of changes in the same person unless the person themselves contributes to those changes. That would undermine the work of the Spirit as a sovereign work of God. Thus, God himself must baptise with the Holy Spirit in different degrees.

Revival, in Murray’s view, is not taking the church back to where it should be but giving more than a normal influence of the Holy Spirit leading to an increase in spiritual fervour, which is ultimately used by God in more conversions. Whether a “normal” Christian can ever be interpreted in any absolute sense or is only an average of recent experience is something the author has never seen adequately tackled. In the model of church growth, there is no definition of normal values for the parameters. Rapid growth comes from positive changes in those parameters which lie in positive changes in believers and unbelievers. Thus no definition of a “normal” Christian is needed to interpret the model.

The model does use two categories, “Inactive Believers” and “Active Believers”. These categories could be treated as averages of believers with a “normal ” work of the Holy Spirit and those with the “exceptional” work. However, there is no need to view it this way. The use of two categories is purely to simplify the model and ultimately the mathematics and in no way implies two classes of Christians. The only attribute of these categories modelled is their effective witness, how many conversions the people are used in. There may well be people in the “inactive” group for which this differs little from some in the “active”. But on average there is a difference, the average is taken as representative of the whole.

Likewise, there is also a sliding scale of revival, and there is bound to be a grey area where it is difficult to classify an event as a revival. “Were there enough revived Christians and conversions to justify the word?” can be an impossible question to answer sometimes.

It is worth remembering that “revival” is our word to explain what we see. At all times, God is working in different degrees. If he executes such significant changes that people are left in no doubt that something different is happening, we are more likely to call it a revival. Thus, the models only talk about “revival-type” growth, rather than revival. Revival-type growth is strictly defined by an increase in the net rate of infection, i.e. converts to the active category, This is the threshold of the epidemic. There are probably many situations where this occurs that we would not call a revival. In conclusion, the models only model changes in numbers, not the spiritual experience of Christians.

Example of a Church Growth Model (Words Only!)

If all the talk of modelling has seemed a bit abstract consider the following simple model of church growth.

A church starts with 50 young people all in their twenties. Over the next few years, no one leaves or dies, and no one joins or is born. What does the church look like in 20 years?

No mathematics or systems dynamics is needed to give the answer! The church is composed of 50 middle-aged people. This is a “model” of church growth. In this case, a mental model is sufficient to draw a conclusion.

Note the assumptions: all in their twenties; none leave; none die; no births; no deaths. A highly contrived situation but deliberately constructed to reduce the complexity of the problem. If we cannot understand this situation, there is little point in making it more complicated.

Now relax the assumptions about death, assuming the church members are subject to the normal death rate of the whole population. Assume that they now marry (only within the congregation) and have children, again at the national average. Assume that no one leaves or dies. What happens over the next 20 years?

Again common sense says there will be a reasonably healthy Sunday school for several years and a growing youth group towards the end of that period. To get the exact numbers, some maths would now be needed. However, arithmetic would be enough to get a reasonable estimate. One thing is clear. The church will grow as deaths will not become important for at least 30 years. If everyone marries, the church will more than double (2.4 children per family!).

Now a harder question. What will the church look like in 100 or 200 years time?

As things average out, the congregation eventually grows at the same rate as the whole population. However, it takes a fair bit of maths to show a more surprising result. The age groups of the church eventually reflect that of the whole population. It has children, young people, middle-aged and elderly in the expected proportions. It is this type of result that mathematics is good at showing.

This yields a principle that can be used even in less contrived situations: if a youth church is started, it cannot remain a youth church. It will eventually become like any other church. To some, this may seem obvious, but mathematics could calculate how much the conversions have to favour young people to keep it a youth church – and almost certainly show that it is impossible unless older people are forced to leave!

The above example is the essence of modelling: make assumptions, draw conclusions, establish some principles.

Now assume that the church also loses people. In the UK, the leaving rate is about 6% per year. Additionally, about 50% of children of church members fail to join the church. The church has a few people who are spiritually alive, good at sharing the gospel and inviting people to services and have still retained a few non-Christian friends. How many contacts does each person need to see come to faith, and how many of these new converts need to become such evangelisers themselves?

Dealing with these issues is the essence of the limited enthusiasm model! It takes a lot of mathematics and simulation to unravel a problem like this. Again, it is all about assumptions. It doesn’t say “this is how the church is” at any given time. But it does represent the sort of situations the church does face and can establish principles that help it, such as “more effective evangelisers are better than more of the same sort of evangelisers”.

This principle has a direct spiritual consequence. If God does make some Christians more effective – baptising them with the Holy Spirit as on the day of Pentecost – then if that work continues the increase in the growth of the church is dramatic. Indeed, it doesn’t take a large change in individual Christians to produce that growth. However, if the church is small, we will have to be patient to see that growth – it starts quite slow! This is an outline of the results of the limited growth model.

At no time is the model undermining God’s sovereignty in growing his church. The results are all conditional on what God does, both in conversion and in reviving his people.


Mathematical models of church growth say nothing about what God does in his church, or with any individual. They do not predict growth, revival or decline. They only explore the consequence of certain assumptions and help understand the situations the church is facing or may face in the future.

To close, here is one conclusion of a model as applied to the UK. If the pattern of losses and conversion of the last 20 years continues, then the most optimistic scenario is that most of the older denominations in the UK will be extinct by the middle of the 21st century. All that will be left will be the Pentecostal, New Paradigm churches and some similar congregations from the older denominations. If that result drives Christians to pray for an outpouring of the Spirit on the church (and themselves), then church growth modelling will have been worthwhile.