arrow_back Back to Research

Explicit Instruction

Explicit instruction may be thought of as teacher-led instruction. It is more interactive than simply lecturing, involving questioning and responsive teaching, but a key characteristic is that the teacher dictates the content and structure of the lesson, in contrast to more student-centered approaches. I often think of Explicit Instruction as comprising of four elements: explaining, modeling, scaffolding and practising. A great deal of research suggests this is the most effective way to help children learn, and despite having moved away from it over the course of my career, it is now the central structure upon which I build my lessons. These papers discuss the advantage and techniques involved in effective explicit instruction, which are then examined further in the section on Cognitive Load Theory.

Research Paper Title: Follow Through Evaluation (and for a lovely online summary of Project Follow Through and the work of Siegfried Engelmann click here)
Author(s): Siegfried Engelmann
My Takeaway:
When I interviewed Dylan Wiliam for my podcast, he described Project Follow Through as the most important piece of educational research ever conducted - and yet I had never heard of it. Project Follow Through was the most extensive educational experiment ever conducted. Beginning in 1968 under the sponsorship of the US government, it was charged with determining the best way of teaching at-risk children from kindergarten through to grade 3. Over 200,000 children in 178 communities were included in the study, and 22 different models of instruction were compared. The results were startling.  Siegfried Englemann's model of Direct Instruction outperformed all other models in basic skills, and indeed all but two of the other models returned negative numbers, which means they were soundly outperformed by children of the same demographic strata who did not go through Project Follow Through. But what is perhaps most interesting is that Direct Instruction also outperformed all other models on DI was not expected to outperform the other models on cognitive skills, which require higher-order thinking. Indeed, it was the only model to return positive scores across reading, maths concepts and and maths problem solving. And if that wasn't enough, students who followed the Direct Instruction Model also had higher self esteem and self-confidence. This image presents these findings clearly. The thing that struck me most was the effect Direct Instruction had on problem solving skills. I had always assumed that you use instruction to get the basics sorted, and then things like investigations, puzzles and rich tasks to develop students problem solving. But, as we shall see, it is a lot more complicated that than, and models of direct and explicit instruction are key to developing the problem solving skills that we all want our students to have.
My favourite quote:
The basic problem we face is that the most popular models in education today (those based on open classrooms, Piagetian ideas, language experience, and individualized instruction) failed in Follow Through. As a result there are many forces in the educational establishment seeking to hide the fact that Direct Instruction, developed by a guy who doesn’t even have a doctorate or a degree in education, actually did the job. To keep those promoting popular approaches from hiding very important outcomes to save their own preconceptions will take formidable help from persons like yourself. We hope it is not too late.

Research Paper Title:
Why Minimal Guidance During Instruction Does Not Work (there is also a shorter, easier to digest version Putting Students on the Path to Learning, along with an amazing sketchnote summary from Oliver Caviglioli)
Author(s): Paul A. Kirschner, John Sweller, Richard E. Clarke
My Takeaway:
This article, possibly more than any other on this page, changed my way of thinking. It provides evidence that for everyone apart from experts, partial guidance during instruction (the kind used in inquiry-based learning or project work) is less effective than full guidance (direct or explicit instruction). This suggests that the practice (often encouraged by previous Ofsted inspection guidelines and senior management alike) of encouraging students to take more ownership of their learning, "discover" key concepts, and for the teacher to be the "facilitator" of learning and not the expert imparting their knowledge, is detrimental to students' learning and long term development. Without facts and procedures stored in long term memory, students cannot become the problem solvers we all want them to be. Solving the kind of problems the students encounter during projects, inquiries or even multi-mark exam questions requires knowledge and procedures stored in long term memory. Without these - and crucially without guidance from the teacher - the student's fragile working memory is faced with trying to process too much information: What is the question asking? What knowledge do I need? What procedures do I need? Now how do I actually carry out out that procedure? This can lead to Cognitive Overload, which will be covered in detail later. I have four major takeaways from this:
1) Students can look like they are working hard on complex problems, and not actually be learning anything - this is the concept of problem-solving search via means-end analysis, and will be addressed in the Cognitive Load Theory section.
2) You cannot teach students to become problem solvers by simply giving them loads of different problems to solve. They need to be expertly supported by the teacher, and the problems selected carefully. These concepts will be covered in more detail in the Problem Solving section.
3) It is quite likely that throughout the course of minimal guided instruction, students will develop misconceptions, or incomplete and disorganised knowledge. This is because there is necessarily less control over the direction of their learning - which for many years I assumed to be a good thing!
4) I love an inquiry or a rich tasks, but now when considering one, I think about the opportunity cost. I ask myself: is there a more efficient way of my students acquiring the knowledge I want them to? When asking myself this these days, more often than not the answer is "yes". If I can tell students the knowledge - having clearly and carefully planned out my explanation - and then get them to practice applying that knowledge to begin the process of committing it to long term memory, as opposed to hoping they discover it, and then having to correct and reteach any areas that are missing, can I really justify not doing so?
Finally, this paper is also careful to point out that full guidance may not be the most suitable instructional technique for students once they reach mastery in a given domain (i.e. become experts) - but all too often I have assumed students are at that level before they actually are.
My favourite quote:
After a half-century of advocacy associated with instruction using minimal guidance, it appears that there is no body of research supporting the technique. In so far as there is any evidence from controlled studies, it almost uniformly supports direct, strong instructional guidance rather than constructivist-based minimal guidance during the instruction of novice to intermediate learners. Even for students with considerable prior knowledge, strong guidance while learning is most often found to be equally effective as unguided approaches. Not only is unguided instruction normally less effective; there is also evidence that it may have negative results when students acquire misconceptions or incomplete or disorganized knowledge.

Research Paper Title: Assisting Students Struggling with Mathematics
Author(s): Institute of Education Sciences
My Takeaway:
This is a fascinating paper on the most effective practices for assisting students who struggle with mathematics that would be of great value to any teacher who teaches lower achieving students or students with special educational needs. I am not afraid to admit that this is a particular area of weakness in my own teaching. The paper provides eight recommendations, some of which are aimed more at a senior leadership of over governmental level. However, one recommendation stood out to me in particular, and is directly relevant to our discussion in this section:
Recommendation 3: Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review. This recommendation is then followed by supporting evidence and practical strategies to carry it out.
In the past I have been guilty of changing my teaching approach for students who struggled with mathematics. I often tried to make it more about discovery, less teacher led, all with an aim of making mathematics less daunting for these students. However, the research suggests I should have been doing the opposite. As we have seen in the papers above, the only people likely to benefit from minimally guided instruction are experts, and hence the students most likely to benefit from a more teacher-led approach, together with careful use of examples, explanation, modelling and practice, are the students who find maths the most difficult. I will not be making that mistake again.
My favourite quote:
Our panel judged the level of evidence supporting this recommendation to be strong. This recommendation is based on six randomized controlled trials that met WWC standards or met standards with reservations and that examined the effectiveness of explicit and systematic instruction in mathematics interventions. These studies have shown that explicit and systematic instruction can significantly improve proficiency in word problem solving and operations across grade levels and diverse student populations.

Research Paper Title:
The Role of Deliberate Practice in the Acquisition of Expert Performance (for an easier to read introduction to Deliberate Practice that covers all the key principles, see The Making of an Expert)
Author(s): K. Anders Ericsson, Ralf Th. Krampe, and Clemens Tesch-Romer
My Takeaway:
To become an expert at something, you must practice in the right way. This paper (along with the excellent book, Peak), outlines a model of Deliberate Practice. A key feature of this model is that you breakdown a complex process, isolate an individual skill and work on it, receiving regular and specific feedback so you can improve you performance. Crucially (and this was a game-changer for me), the skill you are practising may not look like the final thing you are working towards. It is why young Spanish footballers start on tiny pitches, playing 5-aside, working on specific drills - you learn far more from that than you ever would in a big 11-aside game where you hardly touch the ball, and there are so many other factors out of your control. It is why professional musicians practice scales over and over again instead of constantly playing full pieces. The implications for teaching are huge. When we teach a topic, say adding fractions, perhaps we need to break the skills down into their most minute components. If we want students to have success in tricky exam questions (e.g. those carrying several marks, or those of a more problem-solving nature)  then it might not be the case that we should present these questions in their final forms. Instead we should break them up, isolate individual skills (such as basic numeracy, or identifying exactly what the question is asking), and practice these in isolation until students have mastered them before even going near an exam. The activities students do in practice may look nothing like what they will be expected to do in the final exam, but that is absolutely fine. Moreover - and for me this is just as important - students need immediate feedback on their practice. Think about a tennis player hitting hundreds of balls during a service drill. Each time they hit a serve, they immediately see the result and can adjust their technique to improve. How can we ensure students can do the same during their practice, so errors are identified and not compounded? It could be as simple as giving them immediate access to the answers. This is why two teachers I interviewed for my Mr Barton Maths Podcast always supply students with the answers - John Corbett during classwork, and Greg Ashman for homework. After all - and this is one of my alll-time favourite phrases - practice doesn't make perfect, practice makes permanent.
My favourite quote:
During a 3-hr baseball game, a batter may get only 5-15 pitches (perhaps one or two relevant to a particular weakness), whereas during optimal practice of the same duration, a batter working with a dedicated pitcher has several hundred batting opportunities, where this weakness can be systematically explored

Research Paper Title: Deliberate Practice and Performance in Music, Games, Sports, Education, and Professions: A Meta-Analysis (and for a critical view of deliberate practice, I would recommend this article from the New Yorker)
Author(s): Brooke N. Macnamara, David Z. Hambrick, and Frederick L. Oswald
My Takeaway:
Just when I was beginning to thin that Deliberate Practice was the answer to all my problems, I came across this paper. It is a meta-analysis covering all major domains in which deliberate practice has been investigated. The authors found that deliberate practice explained 26% of the variance in performance for games, 21% for music, 18% for sports, and less than 1% for professions. What was the figure for education?... wait for it... 4%! The authors conclude that deliberate practice is important, but not as important as has been argued. So, what are we to make of this? Firstly - and perhaps unsurprisingly - deliberate practice cannot explain all difference in performance between students. Factors such as their class teacher, school culture, measures of ability, and others are bound to play a role. But why is the effect size of deliberate practice so low? Well, the authors themselves offer a plausible suggestion - deliberate practice is not well defined in education. As far as I could tell from following up the papers mentioned in this meta-analysis, no-one has studied deliberate practice properly in education. By that I mean comparing two cohorts of students, with one tackling a topic such a simultaneous equations by working through questions as a whole, whilst the other works at individual components (identifying whether you add or subtract, manipulating algebraic expressions, solving linear equations, etc), practicing each component many times, getting immediate feedback, nailing each one before moving on to the next component, before finally putting it all together. For me, that is effective deliberate practice. That is more like the deliberate practice that is seen in sports, where the effect size is far greater. Until I see research related to that form of deliberate practice, I am not giving up on it just yet!
My favourite quote:
Why were the effect sizes for education and professions so much smaller? One possibility is that deliberate practice
is less well defined in these domains. It could also be that in some of the studies, participants differed in amount of prestudy expertise (e.g., amount of domain knowledge before taking an academic course or accepting a job) and thus in the amount of deliberate practice they needed to achieve a given level of performance.

Research Paper Title:
Educating the Evolved Mind: Conceptual Foundations for an Evolutionary Educational Psychology
Author(s): David C. Geary
My Takeaway:
This paper is brilliant. The author argues that there are two types of knowledge and ability: those that are biologically primary and emerge instinctively by virtue of our evolved cognitive structures, and those that are biologically secondary and exclusively cultural, acquired through formal or informal instruction or training. Evolution over millions of years has led to us having brains that eagerly and rapidly acquire those things that are biologically primary, whereas the brain has simply not has enough time to adapt to make biologically secondary knowledge and ability as easy to acquire. Much of what students learn in school is biologically secondary. For me, this has three major implications for teaching. The first is it goes someway to explain why students seem to enjoy speaking to their friends more than engaging in my lovely discussion of how to solve a quadratic equation. The first is biologically primary (folk psychology - interest in people), whereas the second is not. Secondly, it explains why subject such as mathematics should be explicitly taught and not "discovered" through minimal guided instruction - learning maths is simply not natural for humans and hence is too difficult on your own without the help of an expert. Thirdly, it does open up a potential limitation to the model of explicit instruction - we assume students will believe what they are told. When that involves something like how to calculate the percentage of an amount, or how to use trigonometry to find missing sides in right-angled triangles, this is probably fine as students are unlikely to hold any intuitive (i.e. biologically primary), contradictory beliefs. However, I am not convinced that is true for all topics. Take something like probability. Probability is filled with so many counter-intuitive results. The birthday paradox, and the Monty Hall problem are two classics that catch out everyone, but we've all seen students hold erroneous beliefs about the probability of getting a Head and a Tail when tossing two coins, or rolling a "lucky" 6 on a dice. Such results directly contradict the natural intuition held by students. Geary tells us that such biologically primary knowledge is not easy to shake off, so that simply being told you are wrong and what the correct answer is may not be enough. For topics such as these, students may well need to be convinced of the answer - for it is only when they are convinced that their deeply geld intuitions can be changed. Is this also true of misconceptions? Take a skill such as adding two fractions together. By instinct, students are likely to simply add the numerators and denominators. Now, this is perhaps not representative of a biologically primary piece of knowledge, but such misconceptions are so prevalent among students (how many Year 11s have you seen making that mistake despite 6 years of being taught it?), that once again they may need convincing as opposed to just being instructed. How do we convince them? It is not easy, but in short, we need to create cognitive conflict - a concept that will be discussed further in the next paper. Students need to see and believe the real results for themselves. We have traditional tools such as demonstration and mathematical proof, as well as more modern approaches such as Dan Meyer's 3 Act Math structure.
My favourite quote:
From an evolutionary perspective there are several key points: First, secondary learning is predicted to be heavily dependent on teacher- and curriculum-driven selection of content, given that this content may change across and often within lifetimes. Second, for biologically primary domains, there are evolved brain and perceptual systems that automatically focus children’s attention on relevant features (e.g., eyes) and result in a sequence of attentional shifts (e.g., face scanning) that provide goalrelated information, as needed, for example, to recognize other people. Secondary abilities do not have these advantages and thus a much heavier dependence on the explicit, conscious psychological mechanisms of the motivation-to-control model— Ackerman’s (1988) cognitive stage of learning—is predicted to be needed for the associated learning. Third, children’s inherent motivational biases and conative preferences are linked to biologically primary folk domains and function to guide children’s fleshing out of the corresponding primary abilities. In many cases, these biases and preferences are likely to conflict with the activities needed for secondary learning.

Research Paper Title: Numerical Cognition: Age-Related Differences in the Speed of Executing Biologically Primary and Biologically Secondary Processes
Author(s): David C. Geary and Jennifer Lin
My Takeaway:
Since reading the Geary paper above, the distinction between what is biologically primary and secondary with regard to mathematical skills has fascinated me. As well as providing an interesting study on the effect of age on the speed of mathematical computations, this paper addresses maths specific biologically primary and secondary skills.
1) Enumeration: Human infants and animals from many other species are able to enumerate or quantify the number of objects in sets of three to four items. This process that is termed subitizing and is defined as the ability to quickly and automatically quantify small sets of items without counting.  For the enumeration of larger set sizes, adults typically count the items or guess This likely reflects a combination of primary and secondary competencies. Primary features include an implicit understanding of counting, and secondary features include learning the quantities associated with numbers beyond the subitizing range. Thus, subitizing appears to represent a more pure primary enumeration process than does counting.
2) Magnitude Comparison: The speed of determining which of two numbers is smaller or larger becomes slower as the magnitude of the numbers increases, but becomes faster and less error prone as the distance between the two numbers increases. The most conservative approach is to assume that the limit of these innate representations is quantities associated with 1 to 3, inclusive.
3) Subtraction: It appears that human infants, pre-verbal children, and even the common chimpanzee (Pan troglodytes) are able to add and subtract items from sets of up to three (sometimes four) items. Although this primary knowledge almost certainly provides the initial framework for the school-based learning of simple addition and subtraction, most of the formal arithmetic skills learned in school appear to be biologically secondary. For example, effective borrowing is dependent on a conceptual understanding of the base-10 structure of the Arabic number system and on school-taught procedures.
Finally, it is fascinating to note the findings from this paper: Componential analyses of solution times suggested that younger adults are faster than older adults in the execution of biologically primary processes. For biologically secondary competencies, a pattern of no age-related differences or an advantage for older adults in speed of processing was found. In other words, the younger you are the quicker you are at biologically primary knowledge, but that relationship is not seen (or may even reverse) when it comes to biologically secondary. One conclusion from this might be to highlight the importance of how students are taught the biologically secondary skills.
My favourite quote:
The primary–secondary distinction also provides a means of considering the extent to which experiences in childhood, such as schooling, might influence cognitive performance in old age. The result of this and other studies (e.g., Bahrick & Hall, 1991) suggest that the development of strong academic competencies, that is, secondary abilities, in childhood can have important mitigating effects on any more general declines in cognitive performance with adult aging, perhaps by facilitating their use throughout the life span. This latter implication leads us to wonder about the arithmetical competencies of the current generation of younger American adults as they age.

Research Paper Title: An Evolutionary Upgrade of Cognitive Load Theory
Author(s): Fred Paas & John Sweller
My Takeaway:
We still have the delights of Cognitive Load Theory awaiting us in the next section, but I include this paper here as it adds an extra dimension to Geary's interpretation of biologically primary and secondary knowledge and the implications for teaching. Here the authors argue that working memory limitations may be critical only when acquiring novel information based on culturally important knowledge that we have not specifically evolved to acquire. Cultural knowledge is known as biologically secondary information, and as we have seen includes most of the maths students are taught at school. Working memory limitations may have reduced significance when acquiring novel information that the human brain specifically has evolved to process, known as biologically primary information. If biologically primary information is less affected by working memory limitations than biologically secondary information, it may be advantageous to use primary information to assist in the acquisition of secondary information. One interesting application to maths is “mindful movement”, defined as the use of body movements, for instance children forming a circle, for the purpose of learning about the properties of a circle. The use of mindful movement was expected to be particularly effective for children who are able to cooperate but are not yet capable of high-level verbal interaction. Research quoted found that, compared with the conventionally taught control group, the experimental group using mindful movement in cooperative learning obtained better results. Also, an explanation of the modality effect (see next section) is proposed: we may have evolved to listen to someone discussing an object while looking at it, hence why the process of describing an image or an animation is effective. We certainly have not evolved to read about an object while looking at it because reading itself requires biologically secondary knowledge.
My favourite quote:
The major purpose of this paper has been to indicate that biologically primary knowledge that makes minimal demands on working memory resources can be used to assist in the acquisition of the biologically secondary knowledge that provides the content of most instruction and that imposes a high working memory load. Evidence for this suggestion can be found in the collective working memory effect, the human movement effect and in embodied cognition through the use of gestures and object manipulation. The collective working memory effect indicates that our primary skill in communicating with others can be used to reduce individual cognitive load when acquiring secondary knowledge. The human movement effect demonstrates that we are able to overcome transience and the resultant cognitive load of animations if those animations deal with human motor movement because we may have evolved to readily acquire motor movement knowledge as a primary skill. The use of gestures and object manipulation are primary skills that do not need to be explicitly taught but can be used to acquire the secondary skills associated with instructional content.

Research Paper Title:
How Do I Get My Students Over Their Alternative Conceptions (Misconceptions) for Learning?
Author(s): Joan Lucariello and David Naff
My Takeaway:
In the papers above we were introduced to the potential difficulties of relying on explicit instruction to convince students to go against their biologically primary knowledge and intuition. This paper takes this further by looking at practical ways we may help students change the erroneous misconceptions they hold. The author provides a handy list of effective strategies for changing students' misconceptions, including asking them to write down their existing preconceptions of a topic before teaching it, and using diverse instruction to present a few examples that challenge multiple assumptions, rather than a larger number of examples that challenge just one assumption. These are all covered in detail, with practical strategies and supporting research. However, my main takeaway is related to concept of creating cognitive conflict. Cognitive conflict can lead to conceptual change or the accommodation of current cognitive concepts. The authors suggest two strategies for bringing about this cognitive conflict, both of which have direct relevance to mathematics:
1) Present students with anomalous data. This is data that does not accord with their misconception. An example from maths might be to get students to type in 1/3 + 1/4 on their calculator and let them see for themselves that the answer is not 2/7. If they have built up sufficient trust in their calculator over the years, this should be enough to induce cognitive conflict which will make them more amenable to hearing alternative approaches.
2) Present students with refutational texts. A refutational text introduces a common misconception, refutes it, and offers a new (alternative) theory that proves to be more satisfactory. An example might be: "some students think you can just add the numerators and the denominators when adding fractions together. You cannot. If you could, then 1/4 + 1/4 would be 2/8, which is just 1/4. We know 1/4 + 1/4 = 2/4, which shows us that the denominators do not get added together". What I like about this is it confronts the misconception head on, dismisses it with a reason, and then offers a more viable solution. This had really influenced the notes I give my students - no longer just explaining the answer, but confronting wrong answers as well. I feel this is important, because can you fully understand a concept unless you also know the possible misconceptions associated with it? The matter of misconceptions and how to deal with them is discussed further in the Formative Assessment section.
My favourite quote:
Alternative conceptions (misconceptions) interfere with learning for several reasons. Students use these erroneous understandings to interpret new experiences, thereby interfering with correctly grasping the new experiences. Moreover, misconceptions can be entrenched and tend to be very resistant to instruction. Hence, for concepts or theories in the curriculum where students typically have misconceptions, learning is more challenging. It is a matter of accommodation. Instead of simply adding to student knowledge, learning is a matter of radically reorganizing or replacing student knowledge. Conceptual change or accommodation has to occur for learning to happen.Teachers will need to bring about this conceptual change.

Research Paper Title: Cognitive conflict, direct teaching and student's academic level
Author(s): Anat Zohar and Simcha- Aharon Kravets
My Takeaway:
The suitability of cognitive conflict to overcome misconceptions is far from universally accepted, and research findings into its effectiveness are mixed. This piece of research suggests a reason for that, and with it an important - and logical - condition on when cognitive conflict is likely to be a useful way of helping students overcome their misconceptions. The study sought to compare the effectiveness of two teaching methods - Inducing a Cognitive Conflict (ICC) versus Direct Teaching (DT) for students of two academic levels (low versus high). 121 students who learned in a heterogeneous school were divided into four experimental groups in a 2X2 design. The key finding was that the ICC teaching method was more effective for high level students following a test of retention 5 months later, while the DT method was more effective for low level students. Specifically, high level students benefited from the ICC teaching method while the DT method delayed their progress. In contrast, low -level students benefited from the DT method while the ICC teaching method delayed their progress. Why would this be the case? Well two common reasons given for the apparent failure to demonstrate the effectiveness of cognitive conflict in the classroom are:
1) students often fail to reach a stage of meaningful conflict that requires a certain degree of both prior knowledge and reasoning abilities
2) students may not have an appropriate degree of motivation (goals, values and self - efficacy) that are potential mediators in the process of conceptual change
The authors speculate that students with low academic aptitudes and achievements tend to have a lower degree of prior knowledge, less advanced reasoning abilities and a lower degree of motivation than students with high academic aptitudes and achievements. These explanations thus suggest that as a group,students with low academic aptitudes and achievements will tend to benefit from instruction using cognitive conflict less then students with high academic aptitudes and achievements. Perhaps such instruction even obstructs the learning of low-achieving students compared to other teaching methods such as direct teaching. If this is correct, then it implies that academic ability/achievement is key determinant of the success of cognitive conflict. This makes logical sense if you consider the point that students cannot experience cognitive conflict without first understanding the related domain specific concepts involved in the method being demonstrated.
My favourite quote:
The findings confirm our hypothesis that inconclusive findings regarding the effectiveness of teaching with cognitive conflict may be caused by an interaction effect between students' academic level and teaching method.

Research Paper Title:
The Privileged Status of Story
Author(s): Daniel Willingham
My Takeaway:
This lovely paper is directly related to Geary's argument about biologically primary and secondary knowledge and skills above. I include it as a potential model for explicit instruction, as well as to show the explicit instruction is not simply lecturing to students in a dull, uninspiring way.  Here, Willingham argues that planning a lesson around the structure of a story is a good way not just to engage students but to help them learn and retain the content. He argues that stories are both easier to comprehend and easier to remember than if that information was presented another way. Why should this work? Well, to return to Geary, story-telling has existed in human societies since the evolution of language, whereas communicating in written form is a relatively new development, as is sitting in the whole school environment for that matter. Learning from stories may be viewed as more biologically primary than other forms of instruction. With this in mind, Willingham suggests the following key components of a story: causality, conflict, complications and character. How easy is it to structure a maths lesson around these components without it feeling forced, false and downright unbelievable to the students? Well, not as difficult as you might think. Causality is as simple as showing one concepts follows directly from another. Conflict can be presented as a problem we need to solve. Complications exist when our current mathematical tools let us down, or we encounter a surprising result. Character is perhaps a little more difficult, but searching the rich history of maths to see where the great ideas started from can breathe humanity into otherwise abstract ideas. Once again I return to Dan Meyer's 3 Act Math structure, and his amazing aspirin-headache approach, as possible tools to incorporate these key components, but I am convinced that it doesn't even need to be that complicated. Simply considering the structure of a story when planning lessons, possibly incorporating one or two of these components, may just lead to an experience that is easier to understand and more memorable for your students - and that, after all, is the goal of teaching.
My favourite quote:
Screenwriters know that the most important of the four Cs is the conflict. If the audience is not compelled by the problem that the main characters face, they will never be interested in the story. Movies seldom begin with the main conflict that will drive the plot. That conflict is typically introduced about 20 minutes into the movie. For example, the main conflict in Star Wars is whether Luke will succeed in destroying the death star, but the movie begins with the empire's attack on a rebel ship and the escape of the two droids. All James Bond movies begin with an action sequence, but it is always related to some other case. Agent 007's main mission for the movie is introduced about 20 minutes into the film. Screenwriters use the first 20 minutes—about 20 percent of the running time—to pique the audience's interest in the characters and their situation. Teachers might consider using 10 or 15 minutes of class time to generate interest in a problem (i.e., conflict), the solution of which is the material to be learned.

Research Paper Title: Cognitive Supports for Analogies in the Mathematics Classroom
Author(s): Lindsey E. Richland, Osnat Zur and Keith J. Holyoak
My Takeaway:
Related to the importance of a story structure to lessons discussed above comes the finding described in this paper that students respond particularly well to the use of analogies during instruction. The work by Daniel Willingham in the Cognitive Science section outlines how students connect new knowledge to existing knowledge in the formation of schema and how this is crucial to long-term learning, and hence it would appear to be good practice to use anaoliges wherever possible during instruction. The authors identified Six strategies involving analogies after studying maths lessons in the US, Hong Kong and Japan. Teachers:
(A) used a familiar source analog to compare to the target analog being taught;
(B) presented the source analog visually
(C) kept the source visible to learners during comparison with the target
(D) used spatial cues to highlight the alignment between corresponding elements of the source and target (e.g., diagramming a scale below the equals sign of an equation)
(E) used hand or arm gestures that signaled an intended comparison (e.g., pointing back and forth between a scale and an equation);
(F) used mental imagery or visualizations (e.g., “picture a scale when you balance an equation”)
Fascinatingly, each of these strategies were used less by teachers in the US than by teachers in the other two (higher performing) regions. The strategies seem obvious, and I guess I have always been doing parts of them, but knowing these strategies has made me focus on them more explicitly in my delivery of concepts. They all provide that extra support that might be crucial in the early stages of concept development. The researchers offer an explanation (see quote) as to why analogies are successful in maths instruction, and this supports the findings from Cognitive Load Theory. This may go some way to explaining the recent popularity of visual approaches such as Bar Modelling, which could potentially tap into more than one of the strategies outlined above. A word of caution though: unsuccessful analogies may produce misunderstanding that can even lead to nasty, lingering misconceptions. Choose your analogies wisely!
My favourite quote:
If the source analog is not familiar and not visible, then students may struggle with processing. First, students will need to perform a taxing memory search to understand the source. Then, assuming that memory retrieval is successful, lack of visual availability will place further burdens on working memory during production of the relational comparison. Finally, lack of supporting cues to guide the comparison itself may result in the student learning much less than, or something quite different from, the new relational concept the teacher means to convey.

Research Paper Title: Inflexible Knowledge: The First Step to Expertise
Author(s): Daniel T. Willingham
My Takeaway:
We have seen in the Cognitive Science section that experts and novices think in fundamentally different ways due to the more developed set of schema that the former possess. This allows experts to see the deeper structure of problems, and to make the sort of connections and provide the sort of the solutions that are out of the novices' grasp. This lovely paper by Daniel Willingham argues that in order to transform novices to experts we must first accept that they need to first develop inflexible knowledge. Willingham distinguishes between flexible,inflexible and rote knowledge. Knowledge is flexible when it can be accessed out of the context in which it was learned and applied in new contexts. Inflexible knowledge is meaningful, but narrow; it's narrow in that it is tied to the concept's surface structure, and the deep structure of the concept is not easily accessed. Rote knowledge is devoid of meaning. Willingahm's point is that we as teachers often think rote knowledge is bad (which it is), but most of what we consider as rote knowledge may in fact be inflexible knowledge, and the development of inflexible knowledge is a necessary step long the path to expertise. Take something like "angles on a straight line add to 180 degrees". The first time students encounter this, they are unlikely to understand why, and will probably only be able to answer questions that are presented in a straight-forward manner (focusing on the surface structure). Their knowledge still has meaning, albeit limited, hence their knowledge is inflexible not rote. However, the more questions they encounter, and the more situations they see this concept appear in (different shapes, algebraic, etc), then the more they will begin to appreciate the deeper structure and their knowledge becomes more flexible. For me, there are a few interesting takeaways from this:
1) It is fine - and indeed necessary - for students to develop inflexible knowledge of concepts
2) We aid the path to expertise by ensuring this knowledge is secure and then showing it in lots of different scenarios via carefully chosen examples
3) Sometimes it may be sensible to teach the "how" before the "why". This is somewhat controversial, and I first thought about this in my conversation with Dani Quinn on my podcast, but I think it makes sense. Go back to my angles example. Is there any point trying to convince students why the angles on a straight line add to 180 degrees when they first encounter the topic? Is it not better to get them fluent at the calculations, competent at answering questions with the same surface structure, confident in their abilities, and then later on return to the why when they have a greater appreciation of the concept? Relating this to Cognitive Loads Theory, my concern introducing the why and the how at the same time is that it is too much for the students to take in and they end up learning nothing.
My favourite quote:
There is a broad middle-ground of understanding between rote knowledge and expertise. It is this middle-ground that most students will initially reach and they will reach it in ever larger domains of knowledge (from knowing how to use area formulas fluently to mastering increasingly difficult aspects of geometry). These increasingly large stores of facts and examples are an important stepping stone to mastery. For example, your knowledge of calculating the area of rectangles may have once been relatively inflexible; you knew a limited number of situations in which the formula was applicable, and your understanding of why the formula worked was not all that clear. But with increasing experience, you were able to apply this knowledge more flexibly and you better understood what lay behind it. Similarly, it is probably expecting too much to think that students should immediately grasp the deep structure beneath what we teach them. As students work with the knowledge we teach, their store of knowledge will become larger and increasingly flexible, although not immediately.

Research Paper Title: Antagonism Between Achievement and Enjoyment in ATI Studies
Author(s): Richard Clark
My Takeaway:
This fascinating paper reaches an incredibly important conclusion: students often report enjoying the method from which they learn the least.  It appears that students make inaccurate judgments about the amount of effort they will have to expend to achieve maximum learning outcomes. Specifically, low ability students typically report liking more permissive instructional methods (such as inquiry based learning), apparently because they allow them to maintain a "low profile" so that their failures are not as visible. However, as we have seen in this section and will see again in the section concerned with Cognitive Load Theory, in order to experience maximum achievement low ability students actually require less permissive methods (such as explicit instruction) which lower the information processing load on them. Conversely, high ability students like more structured methods which they believe will make their efforts more efficient, and yet these lower load methods seem often to interfere with their learning, as we will see when we consider the expertise-reversal effect. However, high ability students actually seem to learn more from more permissive approaches (such as problem solving independently) which allow them to bring their own considerable skills to bear on learning tasks. As well as highlighting the differing needs of experts and novices that is a recurring theme throughout this page for me, this is another argument against letting students choose the type of activity that they life - their preferences are likely to be influenced by their levels of enjoyment, which are poor indicators of how much they are learning.
My favourite quote:
The reason for this antagonism between achievement and enjoyment may stem from a situation where students seem to enjoy investing less effort to achieve and inaccurately assess the effect of investing less effort on their subsequent achievement. They appear to make judgments based on their perceived efficiency. They will report enjoying methods which appear to them
to bring maximum achievement with less investment of time and work. The decision that one method is superior in efficiency may come from a mistaken judgment that they are familiar enough with a method to profit from it. It is the methods which students perceive to be more familiar, however, for which Berlyne (1964) would predict the greatest enjoyment scores. But again it is not dear whether students accurately assess the extent to which they actually are familiar with a method. It is possible that the manifest and nominal characteristics of a method may only seem familiar to students. Berlyne (1964) offers evidence that objectively complex or demanding stimuli are often perceived as subjectively simple by subjects who locate familiar features in a complex display and categorize the whole display as "familiar".