Blooms 2 Sigma | Pedleysmiths Blog

Roger Federer is widely regarded as one of the greatest tennis players of all time and often referred to as a master of tennis. His extraordinary talent, remarkable achievements, consistency, and longevity in the sport have solidified his status as a tennis legend. He also seems to be a very nice bloke!

Relevance to learning?
To improve your position in the tennis rankings, you must first prove yourself at the lower levels before you move to the higher ones, which seems like a pretty solid idea, and yet that’s not how it works in education. If you think back to your school days, although there were different recognised levels of ability within a year, everyone progressed to the next stage of learning based on age. This could mean that you were studying something at a higher level having not mastered the basics at the lower one.

What is Mastery
Mastery learning is an educational approach where learners are expected to achieve a high level of proficiency or mastery in a particular subject or skill before progressing to more advanced material or a higher level. In other words, you must demonstrate mastery of the current material before moving on. For example, in a math’s lesson learners may be required to demonstrate proficiency in solving algebraic equations before moving on to more advanced topics such as calculus. Similarly, in English, learners must demonstrate their proficiency in grammar and punctuation before progressing to writing essays.

Benjamin Bloom, remember him, is often credited with pioneering the concept of mastery learning. In the 1960s. Bloom proposed the idea as a response to the limitations of traditional instructional methods, which often resulted in some learners falling behind while others moved ahead without mastering the material. He emphasised the importance of ensuring that all learners achieve a solid understanding of core concepts before progressing to more advanced topics. This approach required personalised instruction, continuous assessment, and opportunities for remediation to support every learner in reaching mastery. An example of this type of mastery teaching can be found in the personal tutor market, where parents pay an expert to coach and mentor their children so that they will be able to ace high stake tests.

There is also substantial evidence supporting the effectiveness of mastery learning. Several studies have highlighted the advantages, indicating that learners taught using mastery typically exhibit superior academic performance, greater retention of information, and deeper comprehension compared to those instructed using traditional methods. For instance, a meta-analysis published in the “Review of Educational Research” in 1984, analysed 108 studies on mastery learning and concluded that learners consistently outperformed their counterparts on standardised tests and other metrics of academic attainment.

Mastery grade – Although the emphasis in mastery is on ensuring that learners understand and can apply the material, rather than achieving a specific grade, if the assessment method includes questions, then there has to be a “pass mark”. Although the exact percentage may vary, I saw 80% – 85% mentioned, the consensus seems to be 90% or higher.

“Ah, mastery… what a profoundly satisfying feeling when one finally gets on top of a new set of skills… and then sees the light under the new door those skills can open, even as another door is closing.” Gail Sheehy

The impact of technology
While we have seen that the effectiveness of mastery learning has been proven it is not without its challenges. One issue is the increased time needed, the result of personalised tuition, and additional resources e.g. more questions and course materials. This can put a significant strain on any organisation that might result in them cutting corners in practice. However, this is where technology can really help, we are now in the age of AI and adaptive learning which has the potential to offer the high levels of personalisation required to deliver at scale on the mastery promise. Which could mean that mastery and all its benefits becomes well within the reach of everyone rather than a privileged few.

But, but, but the devils in the detail
In practice there are of course other problems, what role if any will teachers play, will there need to be significant retraining? Who will pay for all of this, and what of the social stigma that may result for those held back because they are not “bright enough”?

Three bigger questions:

Firstly, there is the argument that by focusing on mastering the component parts of a subject, the wider learning is lost, for example the cultivation of critical thinking and problem-solving skills. In addition, attaining genuine mastery for all learners within a given timeframe poses challenges where the demographic is more diversified in terms of learning styles, backgrounds, and abilities.
Secondly, how do we really know if a subject has been learned, let alone mastered. Although someone might have moved to the next topic having scored 92% 3 months earlier, what if the test itself wasn’t sufficiently robust for the level of understanding required at the higher level?
And lastly, we know that as we progress our tendency to forget increases, unless of course that original information is revisited, think here about the forgetting curve. While the speed at which we forget varies greatly, the consistent observation from decades of research is that, with time, we inevitably lose access to much of the information once retained.

Now there are answers for most of these, however there is the potential for well-meaning organisations to promote so called “mastery courses” when in reality they are fundamentally flawed.

What does this all mean
For learners – Mastery is compelling and should be kept in mind when studying. Going back over something to make sure you understand it will not only reinforce what you already know but builds a solid base from which to move forward. However, there will be times when you don’t fully grasp something and time runs out, leaving you no alternative but to move on. The secret here is not to worry, it might be that this particular piece of knowledge or skill is not required in the future and if it is, you can always go back and learn it, again!

For educators – Mastery is certainly something worth pursuing but be careful, it’s very easy to get caught up in the ideal and create something that looks like it’s working but it’s not because of a lack of attention to the detail required to make this work in practice.

Unfortunately I couldn’t find anything about mastery from Roger Federer, but I’m sure he would wholeheartedly endorse this by a master with a different skill.

“If people knew how hard I worked to get my mastery, it wouldn’t seem so wonderful at all.” Michelangelo

When people see the year 1984 most think of George Orwell’s book about a dystopian future, but a few other things happened that year. Dynasty and Dallas were the most popular TV programs and one of my favorite movies, Amadeus won best picture at the Oscars. You can be excused for missing the publication of what has become known as the two Sigma problem by Benjamin Bloom, of Blooms taxonomy fame. He provided the answer to a question that both teachers and students have been asking for some time – how can you significantly improve student performance?

One of the reasons this is still being talked about nearly 40 years later is because Bloom demonstrated that most students have the potential to achieve mastery of a given topic. The implication is that it’s the teaching at fault rather than the students inherent lack of ability. It’s worth adding that this might equally apply to the method of learning, it’s not you but the way you’re studying.

The two-sigma problem
Two of Bloom’s doctoral students (J. Anania and A.J. Burke) compared how people learned in three different situations:

A conventional lecture with 30 students and one teacher. The students listened to the lectures and were periodically tested on the material.
Mastery learning – this was the conventional lecture with the same testing however students were given formative style feedback and guidance, effectively correcting misunderstandings before re-testing to find out the extent of the mastery.
Tutoring – this was the same as for mastery learning but with one teacher per student.

The results were significant and showed that mastery learning increased student performance by approximately one standard deviation/sigma, the equivalent of an increase in grading from a B to an A. However, if this was combined with one-to-one teaching, the performance improved by two standard deviations, the equivalent of moving from a C to an A. Interestingly the need to correct students work was relatively small.

Bloom then set up the challenge that became known as the two-sigma problem.

“Can researchers and teachers devise teaching/learning conditions that will enable the majority of students under group instruction to attain levels of achievement that can at present be reached only under good tutoring conditions?”

In other words, how can you do this in the “real world” at scale where it’s not possible to provide this type of formative feedback and one to one tuition because it would be too expensive.

Mastery learning – To answer this question you probably need to understand a little more about mastery learning. Firstly, content has to be broken down into small chunks, each with a specific learning outcome. The process is very similar to direct instruction that I have written about before. The next stage is important, learners have to demonstrate mastery of each chunk of content, normally by passing a test scoring around 80% before moving onto new material. If not, the student is given extra support, perhaps in the form of additional teaching or homework. Learners then continue the cycle of studying and testing until the mastery criteria are met.

Why does it work?
Bloom was of the opinion that the results were so strong because of the corrective feedback which was targeted at the very area the student didn’t understand. The one to one also helped because the teacher had time to explain in a different way and encourage the student to participate in their own learning which in turn helped with motivation. As you might imagine mastery is particularly effective in situations where one subject builds on another, for example, introduction to economics is followed by economics in business.

Of course, there are always problems, students may have mastered something to the desired level but forget what they have learned due to lack of use. It’s easy to set a test but relatively difficult to assess mastery, for example do you have sufficient coverage at the right level, is 80% the right cut score? And finally, how long should you allow someone to study in order to reach the mastery level and what happens in practice when time runs out and they don’t?

The Artificial Intelligence (AI) solution
When Bloom set the challenge, he was right, it was far too expensive to offer personalised tuition, however it is almost as if AI was invented to solve the problem. AI can offer an adaptive pathway tracking the student’s progression and harnessing what it gleans to serve up a learning experience designed specifically for the individual. Add to this instructionally designed online content that can be watched by the student at their own pace until mastery is achieved and you are getting close to what Bloom envisaged. However, although much of this is technically possible, questions remain. For example, was the improvement in performance the result of the ‘personal relationship’ between the teacher and student and the advise given or the clarity in explaining the topic. Can this really be replicated by a machine?

In the meantime, how does this help?
What Bloom identified was that in most situations it’s not the learner who is at fault but the method of learning or instruction. Be careful however, this cannot be used as an excuse for lack of effort, “its not my fault, it’s because the teacher isn’t doing it right”.

How to use Blooms principles.

Change the instruction/content – if you are finding a particular topic difficult to understand, ask questions such as, do I need to look at this differently, maybe watching a video or studying from another book. Providing yourself with an alternative way of exploring the problem.
Mastery of questions – at the end of most text books there are a number of different questions, don’t ignore them, test yourself and even if you get them wrong spend some time understanding why before moving on. You might also use the 80% rule, the point being you don’t need to get everything right

In conclusion – It’s interesting that in 1985 Bloom came up with a solution to a problem we are still struggling to implement. What we can say is that personalisation is now high on the agenda for many organisations because they recognise that one size does not fit all. Although AI provides a glimmer of hope, for now at least Blooms 2 Sigma problem remains unsolved.

Listen to Sal Khan on TED – Let’s teach for mastery, not test scores