Mastery, rates of learning and closing the gap

In this blogpost I summarise some reading  on mastery learning and make some conjectures about the implications on how our teaching and curricula are planned.  This is not a manifesto for mastery learning and neither is it a rigorous piece of academic writing.  However, I felt that the papers I read had some interesting points worth contemplating.  Sharing on my blog often generates debate.  I value this interaction.  Especially with those who disagree. 

John B. Caroll’s “Model of School Learning” (1963)
In its simplest form, Carroll’s model proposed that if each student was allowed the time he needed to learn to some level then he could be expected to attain the level.  However, if the student was not allowed enough time, then the degree to which he could be expected to learn was a function of the ratio of time actually spent learning to the time needed. 

1.    Time allowed: The total time allowed for completion of the learning

2.    Perseverance:  The time the student is willing to spend actively involved in the learning.

3.    Aptitude:  The amount of time required to learn given optimal conditions.  Instead of being an index of the level a student is capable of learning to it is the amount of time required to learn to the required level.  Individuals have different aptitudes for different subjects, and even different topics within the one subject.

4.    Quality of instruction:  The degree to which the presentation, explanation and ordering of the learning task’s elements approached the optimum for each learner. 

5.    Ability to understand instruction:  The student’s ability to generally profit from the instruction.  Verbal intelligence is particularly important here.  EAL and ASN likely impact here too. 

The factors influencing time required for learning can be collectively called the ‘rate of learning’.  Individual students have different rates of learning.  These rates of learning are likely not static.  Academic factors such as previous learning are important, but one must remember that our learners are human beings and as such, are impacted by the weather, hunger, fatigue and emotional state etc. However, Suppes (1964) states that over the long term and over many numbers of learning tasks the rate of learning is constant.  J. P. Guildford (1967) suggest verbal ability, memory ability, spatial ability and so forth, form the aptitudes for each student in relation to specific learning tasks.

Carroll’s model is just that: a model.  It gives us a basis upon which to make some inferences, which may or may not be correct.

Bloom’s Proposal
The next few paragraphs assume some prior knowledge of the reader regarding the mastery learning cycle, and should by no means be considered an introduction to the idea.  I have previously written an introduction to mastery here.
Bejamin S. Bloom (1968) wrote that if students were normally distributed with respect to aptitude for a subject and if they were uniformly instructed then achievement would be normally distributed.

If students receive optimal quality of instruction and the required learning time then a majority would be expected to attain mastery.

Bloom states “since education is a purposeful activity in which we seek to have students learn what we teach, the achievement distribution should be very different from the normal curve if our instruction is effective.  In fact, our educational efforts may be said to be unsuccessful to the extent that student achievement is normally distributed.”
Bloom’s proposals for a mastery model are summarized:
·      For use in normal classroom situations, where time is fixed
·      Mastery should be defined as sets of content and cognitive behaviours
·      Curriculum content broken down into small learning units.  The absolutely essential objectives in each unit should be defined (objectives for which mastery is required for progression).
·      Lessons are taught via group based methods and supplemented with feedback and corrective procedures to ensure optimal instruction
·      Brief diagnostic assessments at the end of units
·      Supplementary instructional correction applied before the group continues, if required, at the end of units.
Interestingly Bloom talks not just of content but also of cognitive behaviours.  One argument against the idea of breaking a curriculum up into objectives is that it can be hard to capture the skills that permeate an entire subject.   Essentially, the curriculum becomes, merely, a list of “stuff” that needs to be taught.  In the case of mathematics examples of cognitive behaviours may include conjecturing, reasoning, specifying, generalising etc.  Many curriculum planners do not promote mathematical thinking explicitly, but opportunities for these cognitive behaviours can, and should, be planned in the curriculum.
Mastery can be seen from Bloom’s writing to have, at its core, what we now call formative assessment.  Scriven (1967) made the breakthrough in this field with what he called formative evaluation. Airasian (1969) explains that formative evaluation was designed to be an integral part of the teaching-learning process and to provide continuous feedback to both the teacher and the student regarding the process’ on-going effectiveness.   Bloom writes that his model required the constant flow of feedback information to the student and the teacher. 
The role of correctives for those who have not demonstrated mastery is very important.  This may be in the form of small group tuition, alternative explanations and models, alternative textbooks, concrete manipulatives etc.  Bloom explains “The important point is that the use of alternative methods of instruction and instructional materials is an attempt to improve the quality of instruction in relation to the ability of each student to understand that instruction.  A particular student may use whatever variety of methods and materials found most useful as he encounters difficulties in his learning… If a student can’t learn one way, he should be reassured that alternatives are available to him”
By 1971 there had already been around 40 studies in actual school conditions that supported the idea of mastery learning.  As well as increases in attainment it was noted that students exhibit greater interest and attitudes towards the subject when studying via a mastery curriculum.   Since then countless papers and pieces of research supporting mastery learning have been published.  Perhaps the most prominent contemporary advocate of mastery and Bloom’s legacy is Thomas Guskey.  A short summary of mastery by him is included here as are some links to research papers
Quality of Instruction
A variable in Carroll’s model, above, that we influence over is the quality of instruction.   Carroll states that quality of instruction is “The degree to which the presentation, explanation and ordering of the learning task’s elements approached the optimum for each learner.”  Students with low ability to understand are more affected by the quality of instruction  – it is vital that we get this right.  Carroll opines that attempts to measure teacher effectiveness by check lists and observational schedules can only be minimally successfully, mainly because the ability for the teacher to manage instruction is, in part, a matter of how he or she applies proper controls (feedback, evaluation, remediation) over a long period of time.  Observation of a lesson or even several separate occasions does not tell us this. 
I am of the opinion that rather than thinking only in terms of what teachers do on any given day, we need to look at how well the teacher observes and monitors the long-term progress of individual students and provides proper feedback and remediation.   The most effective teachers are planning on a day-by-day, medium and long-term basis.  I am of the opinion that this cannot be written into any curriculum plan.  There is no resource that can make a teacher more effective at this.  Diagnostic assessments, hinge questions, mini whiteboards etc can be provided in addition to detailed schemes of work and quality resources and pedagogical suggestions.  However, having the tools and knowing how to use them effectively– especially in terms of corrective interventions, is a fundamental skill we should expect of teachers.  
In our departments we can become too focused on debates around “which method do you teach kids for topic X”?  However, formative assessment and the subsequent interventions play just as important, if not more so, a part in student attainment.  When was the last time you heard colleagues passionately debating remediation strategies?  The teacher, who is an excellent explainer but who is less skilled at formative assessment cannot, in my view, be said to be delivering learning and teaching of a high standard.  We make judgments about each other’s teaching ability based upon the methods we use to teach topics.  We must also examine the macro level processes of assessment and remediation, which are of vital importance.
Influence on motivation
Perseverance is, rightly, a consideration in Carroll’s model, “the time the student is willing to spend actively involved in the learning.”  This correlates with, among many studies, the work of Gordon Carrier et al. (2015), which demonstrate that prior achievement and success in learning results in students being more motivated and, frequently, being able to demonstrate more perseverance.  (This should not be confused with Growth Mindset.)  Husen (1967) states that “if a student has found his past efforts rewarding, he is likely to spend more time on a particular task”.  Bloom makes the point that perseverance is not fixed; it can be increased by increasing the frequency of reward and learning success. 
It should also be noted that the need for perseverance can also be reduced by high quality instruction and by using suitable learning tasks. Mastery is about breaking the curriculum down into objectives – students should be able to feel success with each one.

Time Allowed & setting

The following are my conjectures, which are based upon a tentative acceptance of all of the above.  I am hesitant about including the following few paragraphs, the reasons for which may be obvious upon reading.

Whatever time is allowed for learning of mathematics on your school’s timetable, it is likely to be too little for some students and more than enough for others.  Assuming the idea of the rate of learning, from above, then Bloom concludes that “most students can achieve mastery, if they are allowed and do spend the necessary amount of time on a learning task.” He asserts that “an effective mastery strategy must find ways of altering the time individuals need for learning as well as providing the time necessary for each student.”  We can impact upon the quality of instruction. Optimal instruction will improve the rate of learning for all.  We can only ensure that learners gain mastery by having the required time allowed.  However, altering the time allowed is complicated in the context of a normal school environment. This leads me to my conclusion that closing the gap is a fallacy (in normal conditions). 

We know that pupils arrive at secondary school with different levels of existing knowledge and that each pupil will have a different rate of learning.  Likely, in the majority of cases, those with the most coverage and mastery of maths in primary school will be those who have the fastest rates of learning.  Similarly, those with the least coverage and mastery of maths in primary school will be those who have the slowest rates of learning. 

While the following will be based around an assumption of setting, I do think the argument would hold equally for mixed attainment.  I am open minded on this, however, and would welcome critique of my thinking.

As part of a well-implemented mastery model, pupils would be set into homogeneous groups – primarily by rate of learning.  There are obvious challenges with this, but provided that the setting is flexible pupils can be moved to classes working at a more appropriate pace.  
At the outset the faster classes have significant advantages.  They will need to spend much less time going over previous primary work, and will have covered more at primary to a better level.  They have a good basis upon which to build. In contrast, the slowest group will have covered less and to a poorer level of mastery.  This means that while the fastest groups accelerate ahead, the middle classes may consolidate more at the start of topics before moving ahead.  The slowest group will likely need significant intervention before progressing onto new material with the majority of topics.
The class with the fastest pupils will progress quickly with high levels of achievement.  The other classes will, in turn, each progress slower to ensure adequate time for mastery, allowing high levels of achievement for these pupils too.  However, we end up with a scenario that the faster classes are significantly ahead of even the medium paced classes.  In essence, the gap increases, even when pupils are achieving mastery.
We can decrease the rate at which the gap continues to grow by differentiating the curriculum somewhat.  In each topic, if we take Bloom’s advice, we can plan so that the absolutely essential objectives are defined.  Mastery is required of these objectives for progression.  Other material in the unit may provide more depth/width, which although valuable, would not be required of all classes.   Essentially, we need to trim the curriculum for those classes going slower.  We need to keep the focus as the focus. 
So, putting all of this together, and we can expect high levels of achievementfrom the majority of learners but the gap will continue to widen as the most able will continue to progress faster than the slowest.  Even by trying to decrease the rate at which the gap opens, we’d be doing less content with the slower classes.  This in itself means there is a gap of knowledge and experience – although in this case less of an issue as it is non-essential material.
In Scotland, come S5 exams this can mean many things.  In my own context, it might mean two or three Higher classes as usual, but with the slowest Higher group taught perhaps a more focused version of the course to ensure that everyone passes.  Perhaps the entire course is covered, but with the less frequently appearing material not laboured.  The fastest class, on the other hand, may spend much more time labouring over the minutia as mastering the essential skills will be a quicker process for them, allowing this luxury.  Similarly, the National 5 classes may have different experiences – each aimed at ensuring that the needs of the majority of that class are met by providing instruction that permits adequate time on the key topics for exam success. 
So, even in raising attainment, and improving examination results for all learners it is, in my view, likely that there will be a gap of some form between students – unless we find the required additional time for those pupils who need it.  

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