**Background**When teaching, do you generally adopt a 'traditional', direct, teacher-led approach, or do you go for a more 'progressive', discovery/inquiry-based, student-led approach? Perhaps you use a mixture of the two - but

**how do you decide**?

As a teacher you make this decision every time you plan a learning sequence. Do you make these decisions based on your

**beliefs**about teaching and learning? Or perhaps the

**content**and

**assessment**governs your decision? Or maybe you base your decisions primarily on your

**learners**?

There is a vast amount of research (and a vast range of opinions) on how we learn and, subsequently, how we should teach. This post contains some of the research, my thoughts and some conversations I've had with others regarding this question:

**How do you decide how to teach**

**?**

"...that which leads to improved student achievement using outcomes that matter to their future success. Defining effective teaching is not easy. The research keeps coming back to this critical point: student progress is the yardstick by which teacher quality should be assessed. Ultimately, for a judgement about whether teaching is effective, to be seen as trustworthy, it must be checked against the progress being made by students."

How do teacher beliefs and practice contribute to effective teaching? The report Effective Teachers of Numeracy found that (p28):

“Those teachers with a stronglyconnectionistorientation were more likely to have classes that made greater gains ... than those classes of teachers with stronglydiscoveryortransmissionorientations.”

How are each of these categories connected to beliefs about the level of teacher guidance? They describe each category as follows:

[In the discovery approach] Learning takes precedence over teaching and the pace of learning is determined by the pupils. Pupils' own strategies are the most important: understanding is based on working things out for themselves.The transmission orientation places more emphasis on teaching than learning. Thus teaching is believed to be most effective when it consists of clear verbal explanations of routines.The [connectionist] belief here is that teaching mathematics is based on dialogue between teacher and pupils, so that teachers better understand the pupils' thinking and pupils' can gain access to the teachers' mathematical knowledge.

Put

*very*simply, the discovery and transmission positions can be considered as opposite ends of a scale of teacher guidance, with the connectionists somewhere in the middle.

However, to make things complicated, the categories are far from exclusive; a teacher could practice connectionist principles (such as connecting different areas)

*and*transmission principles (such as clear verbal explanations). This is recognized in part by the authors:

These orientations are "ideal types": no one teacher did, or is ever likely to, fit exactly within the framework of beliefs of any one of the three orientations; many teachers combined several characteristics of two or more orientations.

However, we could consider the transmission and discovery positions as roughly opposites. The report describes the three positions further, such as this description of the connectionist position:

The connectionist orientation is a belief that most pupils are able to learn mathematics given appropriate teaching, which explicitly introduces the links between different aspects of mathematics

Given this description, it is hardly surprising that the report found that good teachers held the connectionist position - few teachers could argue against this. But what are the practical implications for the amount of guidance teachers should provide?

When I asked Mike Askew, one of the authors of the report, about his beliefs on the amount of guidance teachers should give, he replied:

This is of little practical use when deciding

*how*to teach. In a subsequent report, the same group of authors tried to identify how teacher beliefs translate to the classroom; it is much more difficult to determine the

*behaviour*of effective teachers:

We are left with the perhaps rather happy conclusion that the behaviour of effective teachers and less effective teachers are not easily characterised; much depends on the particular way that teachers and classes as people relate together.

There are signs that certain types of behaviour may often lead to higher gains, but there are always exceptions in both directions. Indeed we have several teachers who appear twice or more in our sample and there is sometimes a considerable difference in their effectiveness as measured by the gain scores with different classes.

Is there any

*conclusive*research evidence on which teaching method produces the best student outcomes? The answer is not really; there is evidence to support both approaches. It is probably impossible to say whether any single teaching method is

*inherently*effective or not, for a number of reasons, such as:

- Any learning depends on the learners, and that which is to be learned.
- It will depend on the quality of instruction, and whether the students are active or passive.
- Effective teachers use a range of methods.
- It is very difficult to measure effective teaching through observing teachers.
- There are a number of pedagogical, social and cultural factors that contribute to student outcomes, of which teaching methods are only one small part.
- It depends how you define effective - do you mean exam success, or something else?

There are many elements that make up effective teaching, many other contributing factors, and many different outcomes to consider, of which many are difficult to measure. However, this does not stop us debating and exploring effective methods of teaching.

To quote Deborah Ball's wonderfully reflective article With an eye on the Mathematical horizon: Dilemmas of teaching Elementary school Mathematics:

We shouldaim to create and explore practice that aims to be intellectually honest with both mathematics and the child.

__A conversation__Two maths teachers, D and T, were having a conversation about planning their next lesson...

D: I think we should let students discover this for themselves. Discovery is an important part of learning maths.

T: I don’t agree. We can't just let students decide willy-nilly what they are going learn. It's up to the teacher to guide the learning. Have you not read the article Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching? All of these 'discovery' approaches have been shown to be less effective that teacher-led approaches.

D: There are a number of refutations to this article here and here. Firstly, I'm not suggesting a

*pure*discovery approach, more of a

*guided-discovery*approach, where the student has chance to construct their own meaning under the guidance of the teacher. Secondly, these approaches are not the same. They are

*related*in that they are 'student-centred' and based on a constructivist theory of learning, and they often involve students working cooperatively to solve problems and guide their own learning (see this article)...

...but they are not the same thing. In discovery learning , we might want the students to investigate (say) right-angled triangles and perhaps arrive at a generalization or theorem of some kind. However, In inquiry learning, the theorem itself might form the prompt for further inquiry - discovery and inquiry are opposites in this sense. The inquiry is then conducted after negotiation between students and teacher, who acts as a 'more experienced knower' - they don't just work on whatever they feel like doing. You can find more information about inquiry-based learning here and here, and @inquirymaths has an excellent page on it here. And, in problem-based learning, students would work together in small groups to identify how to solve problems and direct their own learning - it's not just about solving problems. However, I would agree that the terms are not well defined.

T: Mmm, this is interesting. The approaches

*are*different but they are similar in that they

*de-emphasise teacher guidance*. I can see these methods might be suitable for older, more-developed students, in certain areas of study such as medicine, where perhaps the most important aspect of learning is the problem-solving element... But for our younger, less-developed students, I think it's better if we are very clear about what we want them to do, and guide them so that they know exactly how to do it. Surely it is better if we give them all the information rather than just some of it? (see this article)

D: It doesn't matter who we are teaching, we shouldn't do all their thinking for them:

Students should be

*participating*in their learning, developing their reasoning skills, not just

*acquiring*procedures and facts. We should let them find out things for themselves - we learn from making mistakes.

T: I agree, but they'll still be participating and making mistakes under my approach! I'll explain the concepts clearly, perhaps have a brief discussion and give them some examples, and then they can carry out some independent practice and solve some problems. They will form their own meaning through working on problems.

D: But they might just be repeating what you have told them, creating the illusion that they have learned; they need to construct their own meaning to really learn something. For this we need a discovery approach.

T: I agree that we learn by constructing our own meaning, but your fallacy is that this doesn't necessarily mean we have to

*teach*that way (see this article)...

We as teachers need to give clear guidance and give students lots of opportunity to practice basic skills. Teaching is all about memory - and this how we remember, through practice! Have you not read Practice Perfect?

D: Of course we need to practice; effective problem solving approaches enable practise to occur as part of the process...

...I think This is what Lampert means when she talks about

*reconnecting procedural and conceptual knowledge*in the excellent article Knowing, Doing and Teaching Multiplication...

T: My approach still allows the connection of procedural and conceptual knowledge; any teacher who knows anything about maths would agree that good teaching is about allowing students to make connections (from here).

D: But how will your students gain flexibility of knowledge of they don't learn in flexible contexts? How will they be able to transfer their knowledge? The excellent book How People Learn has lots on this (p62):

Research has indicated that transfer across contexts is especially difficult when a subject is taught only in a single context rather than in multiple contexts.

When a subject is taught in multiple contexts, however, and includes examples that demonstrate wide application of what is being taught, people are more likely to abstract the relevant features of concepts and to develop a flexible representation of knowledge.

T: I can still present multiple contexts under my approach. Anyway, I would argue that the road to flexible maths is paved with inflexibility. There is a time and place for drill and practice. I'm not convinced your approach will develop the necessary level of fluency required by the National Curriculum. Students need to focus on gaining mastery of the basics before we can start solving problems... that’s what they do in Asian classrooms!

D: We can gain fluency through solving problems. However, I would agree that mastery is incompatible with, say, the inquiry approach, as it is too teacher-led. I find it hard to believe that

*every*Asian classroom follows a teacher-led approach?

T: Well, that’s what I hear. I mean, I've never been...

D: And surely there's more to the success of Asian students that just adopting this approach? Having read the book The Teaching Gap, I would suggest our stereotypical view of Asian classrooms probably isn't true. In the 1995 TIMSS study, they found that many Japanese lessons started off with students solving problems, and then the teacher used these solutions to analyse the best methods for solving problems - a structured problem solving approach:

T: This doesn't sound much like the mastery approach, more like your inquiry approach?

D: Well yes, initially student-led with increasing teacher guidance. But then not

*all*Japanese lessons were like this - there were some strongly teacher-led lessons too. There was not one common method, but rather a common theme of

*coherence*between (parts of) lessons.

T: The connectionist approach...

D: Indeed. In the follow-up study in 1999 they analysed lessons from a wider range of 'successful' countries and found again that there was no common method across all successful countries - although they

*did*say that perhaps a sequence of learning similar to the Japanese example would be worth further examination...

But most importantly, they concluded that teaching is a

*cultural activity*, and that any reform must involve the whole system of teaching, rather than individual teaching methods, and it will take time Any individual changes will just be swallowed up, or may have negative effects. This probably explains why teaching in the UK hasn't changed much.

T: Are you saying that we shouldn't just jump in and adopt the Mastery approach?

D: I'm saying that there's not much point just adopting the latest fad like we (or should I say, the government) always seem to do. We're pretty sure there isn't a magic bullet to better teaching - it's probably the case that a mixture of our approaches is probably best. All we can really do is base our decisions on who and what we are teaching, and make sure we have thought about what the research might tell us... then plan together and analyse carefully what works and what doesn't...

T: I agree with this, but I still have concerns over your approaches. For example, don't students sometimes go off on a tangent? They might not learn what I want them to learn, or, worse still, they might not get to where I want them to get to - and I'll end up telling them the answers anyway!

D: Mmm, there is a chance this might happen - it takes a lot of skill to teach using inquiry. But, skilfully handled, these 'tangents' as you call them allow us to make valuable connections to other parts of the curriculum...

T: I’m not sure students

D: If I design the curriculum and activities properly, they will cover the curriculum, there will be structure. They don’t necessarily have to learn exactly the same thing at exactly the same time, do they? As long as they have all covered the curriculum by the end…?

T: Mmm, but how will you assess and monitor it all? If they are all solving different problems, how can you make sure they are all learning what they need to learn? Isn’t it preferable that all students learn the same thing every lesson... you know, the

D: As I say, it does place a great burden on the teacher, but it

T: Perhaps, but I'm not sure many teachers can pull that off, or students for that matter! It would only work with certain teachers in certain classrooms I think.

D: I think this point of view is much of the problem. All teachers should be aiming to teach in the way that is best for their students - what Lampert would call intellectually ambitious teaching. There's a danger that we rely on traditional approaches because they are easy, for us and the students; all learning must satisfy our students' intellectual need.

T: But we just have too much content to cover in too little time.

D: I know there is a lot to cover, especially at A-level, but we must to find a way to include mathematical reasoning and problem solving in our teaching. Focusing solely on content is one of Wiggins and McTighe's twin sins of instructional design:

*should*direct their own learning - isn’t this the job of the teacher? If students are in charge of their learning, they might not learn what is needed by the curriculum. It is our job to make sure students are learning the right things, and give them*structure*to their learning; this structure will help them learn.D: If I design the curriculum and activities properly, they will cover the curriculum, there will be structure. They don’t necessarily have to learn exactly the same thing at exactly the same time, do they? As long as they have all covered the curriculum by the end…?

T: Mmm, but how will you assess and monitor it all? If they are all solving different problems, how can you make sure they are all learning what they need to learn? Isn’t it preferable that all students learn the same thing every lesson... you know, the

*objective*? Doesn’t it just get too complicated and confusing for everyone involved?D: As I say, it does place a great burden on the teacher, but it

*is*possible - you have to plan very carefully and keep a careful record of what the students have learned so that you ensure the curriculum is covered. But when it is done well, it is fantastic.T: Perhaps, but I'm not sure many teachers can pull that off, or students for that matter! It would only work with certain teachers in certain classrooms I think.

D: I think this point of view is much of the problem. All teachers should be aiming to teach in the way that is best for their students - what Lampert would call intellectually ambitious teaching. There's a danger that we rely on traditional approaches because they are easy, for us and the students; all learning must satisfy our students' intellectual need.

T: But we just have too much content to cover in too little time.

D: I know there is a lot to cover, especially at A-level, but we must to find a way to include mathematical reasoning and problem solving in our teaching. Focusing solely on content is one of Wiggins and McTighe's twin sins of instructional design:

T: I agree with this, though I still think that we can teach students all of this more effectively through my approach, if we do it skilfully... While we're quoting Wiggins and McTighe - how does your approach fit with their other 'sin' - that of activity-oriented design?

D: Well, you do have to be very careful about this. You must have a clear focus on objectives when designing these discovery based activities, and guide students in a meaningful direction.

T: That worries me slightly, but OK... My main argument is that my method is

*more*effective that yours for teaching problem solving. I

*would*argue that any method that is posited on minimal guidance from the teacher will actually make it

*harder*for students to learn how to solve problems (see this article and also this one)...

Cognitive Load Theory says that we have a limited working memory, the implication being that the process of solving problems uses up loads of our working memory, and that learning this way doesn't leave much space for remembering the content and so on...

Using worked examples instead of a problem solving approach have been shown to be a much better way of instructing students how to solve problems:

D: Well, Moreno says that worked examples don't always work - their success depends on their quality, whether they are intrinsic to the object of learning, the students, and so on...

T: Yes, of course the examples (as with any instruction) have to be of a high quality. But use of examples is a valid approach for teaching maths - see the article 'Same/Different - A 'Natural' Way of Learning Mathematics' (p2-113 in this document) and Engelmann's Theory of Instruction. Then there's the research into interleaving - and finally let's not forget the use of Learner Generated Examples, as advocated by one of my heroes, John Mason.

D: Do I sense a bit of constructivism coming out here?

T: As I say, my approach is based in constructivism too, just in a different way.

D: Mmm, maybe... But I come back to this: How do you go about teaching students problem solving strategies if you are going to show them what to do all the time?

T: Well, CLT says that we don’t solve problems using separate 'knowledge' and ‘strategies’ as such. When solving a problem, we are really recalling information from a bank of problem solving approaches in our long-term memory, stored in structures called schemas (this all comes from this article)...

Studies have confirmed that the major factor distinguishing novice from expert problem solvers was not knowledge of sophisticated, general problem-solving strategies but, rather, knowledge of an enormous number of problem states and their associated moves.

*Acquiring expertise is the acquisition and automation of schemas:*

D: Interesting... so how do you build these schemas in the first place?

T: That’s a good question... Erm, we just learn them, like we learn anything I suppose... There's a wonderful article on how experts and novices solve problems here, and the schemas they use.

D: But surely the best way to learn these problem solving schemas is to solve problems...?

T: No, CLT tells us that we learn them more effectively if we do it gradually, through carefully structured instruction... What I find really interesting here is the idea that there's no real distinction between

*knowledge*and

*understanding*, it’s all just remembering, and can be automated in the same way. Understanding (the ability to solve problems) can be considered as just remembering more complex information - see the section on 'understanding' in this article.

D: Even if I did believe all this, won't we miss building some of the most important parts of our problem solving schemas if we don't attempt to solve 'ill-structured' problems. Surely working on problems will lead to better understanding in the long run, even if it takes time?

T: Well, CLT, and this article in particular, suggests not!

D: But this is all just theory, isn't it? Although there is evidence to suggest it may be true, it is really still a conjecture about how we learn; we do not know exactly how our memory works, or exactly how we solve problems. And secondly, how do we measure cognitive load exactly? It will be different for different people, with different levels of prior knowledge, rendering the theory difficult to use in practice. Thirdly, there seems to be some kind of paradox here; how do we build a ‘

*knowledge of an enormous number of problem states and their associated moves’*unless we solve lots of problems? There are more arguments against CLT here.

T: These are all good points, but I think we should consider limitations to working memory, and the idea that

*no learning has taken place unless there is a change in long-term memory*, when designing our lessons!

D: You're probably right! I suppose my biggest issue with all this is that students just won't be engaged using your approach. They just sit there being bored, and then practice some questions and get even more bored. Hattie said that students "

*may only have up to 10 minutes before attention fades*" when listening to the teacher talking (Visible Learning and the Science of How We Learn, p50).

T: That’s just not true - the research he cites says differently:

Students’ attention does vary during lectures, but the literature does not support the perpetuation of the 10- to 15-min attention estimate. Perhaps the only valid use of this parameter is as a rhetorical device to encourage teachers to develop ways to maintain student interest in the classroom. If psychologists and other educators continue to promote such a parameter as an empirically based estimate, they need to support it with more controlled research. Beyond that, teachers must do as much as possible to increase students’ motivation to “pay attention” as well as try to understand what students are really thinking about during class."It would make sense that the amount of time learners can actively listen depends on a number of factors, such as their stage of development anyway.

D: Ah, well, your approach is still boring. I want students to have fun in my lessons.

T: But learning isn’t always fun, sometimes it’s hard and a little bit boring. Have you read the book Why Student's Don't Like School?, or read any of Daniel Willingham's articles Ask the Cognitive Scientist?

D: I have, but there's just such a danger that if we teach this way all the time, learning will become boring and students will become passive - and then they definitely won’t learn anything! Whatever happened to the joy of maths?

T: Mmm, I see what you're saying, this approach does seem to lose a bit of the joy of maths. It all depends on how you do it - I’m not just going to talk at them for a whole hour! I'm going to plan to use a mixture of different styles of talk, for as short a time as possible, with as clear examples as possible, to ensure my explanation is as clear as possible - and then the students can work on problems, either on their own or together. And maybe chuck in a bit of interesting stuff here and there. Here are some excerpts from the article The concept of learning demand as a tool for designing teacher sequences (Leach, Scott 2000) that describe my process quite nicely:

The authoritative voice of the teacher is heard as new ideas are introduced and developed on the social plane of the classroom. At the same time there must be opportunities for the teacher to check students’ developing understandings, through dialogic exchanges with the whole class, small groups, individuals and also through short written tasks. There must also be opportunities for the students themselves to begin to try out these new ideas, through discussion both with the teacher and with other students. The teaching sequence thus consists not only of the [authoritative element], but crucially also the talk and various forms of interaction through which the model is staged. These are planned in advance and form an integral part of the teaching sequence.

The aim is for the teacher to gradually ‘handover’ responsibility for applying the model to the students as they work through a number of different contexts. With this approach, the scaffolding function of the teacher is achieved both through dialogic interactions with the students and through careful selection of learning activities.

D: This sounds a bit like my interactions with students. But how do you get around the fact that

*you*are the only one who gets to decide what the students will learn - how do you ensure it has

*relevance*for students? In

*Learning and Awareness*(1997), Marton and Booth state one of the two conditions for learning is relevance for the learner.

T: What, do you mean like making Harry Potter worksheets? What more relevance could there be than passing your exams?

D: Oh dear. That's such a narrow view - there’s more to maths lessons than passing exams! Learning needs to be relevant. Have you watched this video from @ddmeyer?

T: Ah, that was fun. I still think that my approach will better helping students to pass their exams.

D: There’s no conclusive evidence...

T: Well, that Sutton Trust document seems to be advocating a reasonably teacher-led approach, such as Rosehshine's Principles of Instruction. And what about the rest of the research highlighting the success of teacher-led methods, such as Engelmann's Direct Instruction in Project Follow-Through in the US?

D: From what I hear, Direct Instruction was not very popular...

T: I suppose some teachers don't like to be told what to do, they seem to value their autonomy, maybe to the detriment of their students?

D: Ouch...! Well, Direct Instruction isn't very well defined, is it?

T: You're right, there are lots of uses of the term, as described here - one of which is the negative connotation of teachers talking and students sitting passively. Here is a good definition of direct instruction and learning, from this article:

Direct instructional guidance is defined as providing information that fully explains the concepts and procedures that students are required to learn as well as learning strategy support that is compatible with human cognitive architecture. Learning, in turn, is defined as a change in long-term memory.

So you see, direct instruction is not just about teacher talk, it's more of a process where the teacher (1) reduces the initial difficulty of the task (2) provides scaffolds and support through modelling, examples, etc (3) provides feedback, corrections and so on, and then (4) provides extensive independent practice.

D: Isn't this what teachers have been doing for over 100 years?

T: I mean, yes. In his book A Conception of Teaching

*,*Gage describes this model as "

*embodying something fundamental in the nature of teaching.*" (p75) - it has been used across the world for decades and decades. There must be a reason for this.

D: This doesn't mean it's the best way. As I say, this approach is prone to abuse. You get lots of teachers just rocking up to a classroom and talking for ages without really planning what they are going to say. Discovery methods came about because the same old tried and tested ways were not considered to be working:

T: You're right, I've seen quite a few lessons like this in my time . There's room for both approaches - the report What to do about Canada's declining maths scores is pretty scathing about discovery approaches, but still advocates their use 20% of the time, a bit like the CAME project - an 80-20 approach.

D: I'm not sure I'm happy with 80-20, but some mixture of the two approaches sounds like a good idea. We keep coming back to the same ideas. It all depends on the quality of instruction, the content and the students - but you probably shouldn't teach one way all the time, and your preferences as a teacher are less of a priority!

T: I totally agree... So can we make

*practical*use of all this theory and research?

D: Well, the finding from the international studies about the

*order*in which types of learning and instruction occur was interesting, wasn't it? It seems to be a good way to give learners a chance to construct their own meaning first, create some

*relevance*, a need to integrate their ideas - and develop reasoning and problem solving skills...

T: ...before receiving some direct instruction. I like it. It reminds me of something similar I read in an article called A Time for Telling. The authors carried out some research where students were given a task that required them to analyse a concept for homework, a week or so before they were taught it.

D: This sounds good - shall we work together to design a couple of pre-teaching problems for our next lesson? Then we can watch each other teach them, and see if it works.

T: Great. We want the questions to be accessible, but will make them think - we want them to be able to start constructing their own meaning . And if we design a few, then they could choose the one that interests them the most.

D: Now you're getting it! Then we could build on this at the start of the lesson with a discussion about their understanding - it will make the exposition more meaningful! And it should mean that students are coming to the lesson with a less variability in prior knowledge!

T: That's assuming they all do the homework....

__Pre-teaching problems__

Here are a couple of pre-teaching prompts D&T came up with...