## Podcast: Conceptual vs procedural approaches to maths teaching (Part 1)

A podcast with Richard Andrew and Colin Kluepic

16th March 2017

A podcast with Richard Andrew and Colin Kluepic

16th March 2017

Here's another chat I had with Colin Kluepic for the 'More to the Point' podcast. This time we unpack the argument between conceptual and procedural approaches to maths teaching. Both the podcast and the transcript are provided below.

Effective mathematics teaching involves communicating what are often very complex ideas in simple ways. Educators need a strong understanding themselves but they also need their students to have the right level of buy-in or engagement. And what about procedures versus concepts. My guest today is Richard Andrew, a specialist in mathematics teaching and online learning. In this episode, we tease out the practical aspects of this tricky issue.

Colin: Richard, welcome back to the show.

Richard: Thank you, Colin, always good to be here.

Colin: In a recent article you wrote for LinkedIn you quoted a definition for conceptual teaching methods in a maths context, and it goes like this … “Conceptual understanding is knowing more than isolated facts and methods. The successful student (with conceptual understanding) understands mathematical ideas and can transfer their knowledge into new situations and apply it to new contexts.”

But your reaction to this is that definitions like this one are a poor representation of the ‘experience’ of actually making the conceptual connections. Why is that? Can you elaborate?

Richard: Well that particular definition was one of the better ones I could find … Definitions of conceptual understanding often sound more like this one … "Students demonstrate conceptual understanding in mathematics when they provide evidence that they can recognise, label, and generate examples of concepts; use and interrelate models, diagrams, manipulatives, and varied representations of concepts; identify and apply principles; know and apply facts and definitions; compare, contrast, and integrate related concepts and principles; recognise, interpret, and apply the signs, symbols, and terms used to represent concepts. Conceptual understanding reflects a student's ability to reason in settings involving the careful application of concept definitions, relations, or representations of either.

This type of definition is designed for teachers who may not have thought deeply about these ideas of conceptual understanding and to give ideas of what the teacher should be aiming for when teaching. And this is fine - I’m not saying we shouldn’t define conceptual understanding. But I do think we tend to over intellectualise things. The problem with definitions - and I agree they are important - but the problem with them is they tend to diminish the very thing we are defining. If I can paint a picture to explain this …

Colin: Sure, go ahead.

Richard: Let’s say I’m with my 6yo daughter in the botanic gardens and she sees her first gigantic tree. Let’s say it’s a magnificent American redwood tree with a trunk the girth as big as a small house which goes up into the sky like Jack’s beanstalk. My daughter’s jaw is on the ground. She is totally gobsmacked. Woooooow. But because I’m a redwood specialist (which I’m not btw!) and to further my belief that I’m a good father and good fathers need to deliver information I launch into: “This tree is of the Sequoioideae sub-family of Cupressaceae trees. And then continue with all sorts of facts about redwood trees - what makes a redwood a redwood … And then my daughter says “OK dad, can we go have an ice cream.’

Colin: Thanks for that dad, you have really bored that tree down for me!

Richard: Yes, I’ve really destroyed the moment. So I was doing some research recently and John Steinbeck wrote about the redwood, "The redwoods, once seen, leave a mark or create a vision that stays with you always. From them comes silence and awe. They are ambassadors from another time.”

Not that’s not a definition. But he’s using words to elicit the emotional and spiritual experience of being with a redwood tree. Now, of course, I didn’t do this with my daughter when we had this experience - we did really embrace the experience together. But I’d have been better off saying to my daughter “Let’s be still and take it in for a few minutes without saying anything. Or to suggest to her to imagine that the tree is from another world that we have just stumbled across. And NOT to jump to a standard definition or description. Mind you, with my daughter I don’t need to suggest we have stumbled across a different world … she’s there anyway!

So how does that relate to mathematics conceptual understanding?

Colin: Well I thought I might just jump in there and pick up on the words you used ‘Silence and awe’. Are you suggesting then that when you explain for example Pythagoras’ Theorem that you might get a class response that is ‘silence and awe’?

Richard: I think that aiming for silence and awe with Pythagoras’ Theorem at Year 8 might be a stretch! But there is a relationship here. My point is that when someone first gets’s something mathematical - a minor example is seeing a square number for what it is, for example seeing a 4 or a 9 or a 16 as a square. My wife recently mentioned this, she said: “I never realised before what a square number is - they are squares!” And I have to admit that I never realised this until later in my career when I finally saw the relationship with squares. I just hadn’t SEEN it before. Maybe I had been shown it before as a student but it had never stuck.

So my point is when we educators get around a table discussing conceptual understanding we talk definitions and … well … my brain starts hurting!! And I think we lose sight of the gravity of the experience and therefore - perhaps - of how important conceptual understanding is!!

Colin: So what we are suggesting here is that the understanding the concept - rather than actually being able to (only) answer the question - was the thing that could bring real joy to the learning experience. Is that where you’re going with this?

Richard: Absolutely! 100% If I’m a student and your my maths teacher, the difference between me actually getting the concept underlying the maths your asking me to work with, and then answering questions based on that understanding - so I’m now using MY OWN LOGIC to answer the questions VS me ONLY remembering a routine you’ve just shown me for answering the next set of questions - is massive. Because if I’m only drawing on my memory of that routine then I’m operating as a machine. Number in … number out. The difference between the two is like chalk and cheese. And I suspect many maths teachers haven’t really thought about this. And I’m not criticising maths teachers. We don’t tend to operate in a culture which promotes this sort of discussion. For a long time, I certainly hadn’t considered these things. But participants of many of my Professional Learning (Learn Implement Share) courses certainly do … because to consider these issues deeply is a requirement of the course!! And it is fun to see teachers respond “OK, I get what you are showing us here”.

Colin: Actually just listening to you tell that story or explain that idea, it reminds me of when I was actually in high school learning physics. And we had a physics teacher who was doing his level best to explain to us concepts of acceleration and speed and motion. And we were talking in terms of formulas and we were using pro numerals like v, and a, and t (you know t for time, and v for velocity, and a for acceleration).

Richard: Yeah.

Colin: You know I was really trying to understand. And I thought "Yeah I kinda have got this. I see where he is going". But there were some questions where I just couldn't make the formulas work and I just couldn't get what was going on as well as other students.

And then I was also studying Maths and I was also part of the extension Maths class at that time and I remember one day our Maths teacher was explaining a particular concept in Calculus. I am sitting there thinking, "Yeah, I can see something unfolding here but I’m actually just not quite sure what it is but I will just sit here and watch." He was very very good at doing Theorem explanations and First Principles explanations on the board. And there’s this full chalkboard full of stuff. Okay, it was actually a chalkboard, not a whiteboard back then - and I am looking at this thinking “there’s something going on here”. And then he stops and turns around. And if you could picture this in your mind's eye - he still had the chalk touching the board but he turned his head and half of his body around and then he says, "Are there any physics students in the room?" And I look at him and say "Yeah, that's me." And then all of the sudden I thought, "Wow look at that. That thing that I was struggling with over in Physics class was actually masquerading as calculus. I totally get it1” And so what I then started to do was back in the Physics class instead of using the Physics formulas, I would actually start to use Calculus instead and things just become a lot easier. In fact, I used that working in my exam and had a lot more success with that.

Richard: Wow!

Colin: So before we actually get on to the actual methods for teaching conceptually, I just wanted to ask you - this is a bit of side question - because in that situation suddenly a whole other subject became much more useful to me. Extending that idea out a little bit, do you think that if you actually had conceptual understanding of most things in life then we would actually be just become more useful people?

Richard: Well, Colin I absolutely believe this. But I am biased because I’ve always wanted to know "why" I had to do something before doing it. So, I was thought the answer to the "why" question was vital. Although I have to say it took me a while to adopt this principle in my teaching because initially I couldn't even answer the "why" question. I would just say "No, no, no let me share you another example where this situation applies." In fact, I remember in my first year of teaching, this student was asking me the "why" question and I knew that I couldn't teach him what he wanted. All I could I do is giving him another example. And I walked out of the room thinking "I have no idea how to teach this subject." - and I was doing ‘well’ as a first-year teacher - “I have no idea how to teach this, but someone somewhere must know how to teach this. There has to be a way. There absolutely has to be.” It was as clear as a bell. And it is almost like that realisation propelled me into discovering ways to present mathematics that actually allowed students to understand. It gave me that yearning to find a better way.

Colin: Yeah.

Richard: So, that was a bit of a side-track there! Back to the original question - I always believe that "why" questions are important. But many adults I have encountered have the attitude "Just tell me what I need to do and don't bore me with all of the details.” I often hear that and I don't understand why people are like that. It is either because it's a personality thing - maybe some personalities are more interested in the "why" or maybe - and we have talked about this previously; all toddlers are into the ‘why’ - maybe interest in the ‘why’ get’s knocked out of some people between ages 4 and 6! I don't know. I have no proof, I’m just speculating.

Colin: I am just wondering whether a better conceptual understanding of more things in life might make it easier for us to for example cook a meal at home.

Richard: Absolutely. Well, I’m no cook but you know what it is like when you are with someone and they say “Well when you mix this with that the enzymes come out and that’s why the flavour is enhanced … “ and you go "WOW I didn't know about all that." And so the science of cooking makes so much more sense. So, yes, conceptual understanding equips you with a lot more skills - it makes you more useful.

Colin: Well let’s get down to some detail. When I was learning mathematics back in the late 80s - and I just gave that description of that ‘aha’ moment for me - but essentially it was still drill and practise. We were always shown the proofs and theorems first. But essentially it was a textbook and classroom exercise. In your article, you refer to compartmentalising. What do you mean by that?

Richard: So in the vein of what you were just saying let me use this analogy from my PE teaching days. Let’s say you are a kid who has never learnt to throw a baseball properly - you’ve simply never mastered the skill, and I’m going to teach you. So I’ll show you the whole action first but I won’t expect you to replicate the whole skill first. I’ll break it into parts. I’ll teach you the wrist action first. It’s really easy - anyone can do it. So we’ll throw 20 times back and forth just with the fingers and a wrist action and from close range. Then we will keep throwing but incorporate the forearm as well. And the student is thinking “Wow, I can do this. Then you step apart a bit and add a full arm action. Then you add the weight transfer from back foot to front foot. Then a full throw with a few steps run up. So I teach in parts and add parts to make up the whole. It’s not an exact analogy to teaching mathematics traditionally but it is close because the teaching is somewhat compartmentalised. Now when it comes to mathematics it seems logical to use a similar approach. The example I use in the article is basic trigonometry. Basic trigonometry or the trigonometry of right angled triangles allows us to calculate - within right-angled triangles - the length of a side given one side and an angle OR the size of an angle given 2 sides. So at the end of a unit, we want students to look at real life situations and solve the problem which will involve calculations of angles and sides within right angled triangles. And the traditional approach - which I assume we’ve used for hundreds of years - is to break the skills into parts. We teach the 3 trig scenarios - sine, cosine and tangent - but we teach them in isolation. So what happens is we first teach the Sine situation, show an example and then have the students do ‘ten questions’ based on that. So they are only thinking about the procedure to get the answers for the sine situation. And then we do the same for the cosine situation. Now students are only thinking about the procedure to get the answers for the cosine situation. And then we repeat for the tangent situation. All the time we have only been finding lengths of sides. So next, we show them how to find angles using sine, then cosine, then tangent. Then we progress to ‘real world’ situations and finally onto bearings. Now the problem with this is that at each of those parts students are thinking only about the routine for that one situation.

Colin: Yeah, they are thinking about those three parts rather than the entire concept of what’s going on.

Richard: Exactly! But it seems logical to teach this way. And the other thing that seems logical about this compartmentalised approach is that it gives teachers the (false) impression that we can herd these students through a unit and think they are all progressing somewhat uniformly. You hoodwink yourself into thinking this works. Although I couldn’t! Because I knew damn well it wasn’t working. It was depressing. I had my top students bored - I could see it. I was boring them senseless. And I’d lost the bottom end using this method. And those in the middle weren’t exactly inspired either. And I had to find another way.

Colin: So what is the alternative?

Richard: The alternative is what I call a conceptual approach. Now it is tough to explain to people who have not taught this way because it involves ideas and labels which will mean ten different things to ten different people. So it is very difficult to explain what I’m referring to as a conceptual approach in a podcast or an article. In a 4-month long course - i.e. the course ‘Engagement - Winning over your mathematics class’ - these ideas are developed slowly and teachers trial various ideas in their classroom as they progress through the course. So everyone knows exactly what the conceptual approach looks like by the end of the 4-month journey. But again, trying to explain this within a 25-minute podcast or long article it is very difficult, but let’s give it a shot anyway - I’ll try to explain the conceptual approach to teaching right-angled trig.

Colin: Sure

Richard: With the trigonometry example, rather than compartmentalising into the sine situation and the cosine situation and so on, the introduction is a conceptual intro into what trigonometry is. And I discovered this when I sat down one day thinking “This is driving me nuts (my lack of success with the conceptual approach) there has to be another way. What is trigonometry?” And I just looked at it differently and thought "Oh my God it’s just similar triangles.” No one had shown me this before. And this is not earth-shattering - most mathematics teachers probably realise this but I had never seen it. I thought, "Okay. So I’ll simply demonstrate similar triangles and then I’ll ask leading question to allow students to see the principles at work. Now there is software that makes this so easy. If you go to the article https://www.linkedin.com/pulse/conceptual-understanding-mathematics-what-why-do-we-need-andrew?trk=prof-post , there is a video in the middle of the article showing what this introduction actually looks like with the software GeoGebra. So, this is the introduction to the Conceptual Approach to trigonometry.

And then what happens is the students begin to tackle a series of questions. So these are questions on paper, but it is a worksheet where rather than the questions being compartmentalised, they are using the trigonometry principles to work out whether each triangle is a sine situation or cosine or tangent situation.

Colin: Yeah.

Richard: That's a much different question and, by the way, students do not need to be highly skilled in answering these. And so once they decide which situation it is they write the appropriate ‘trig sentence’ and solve the equation and get an answer. The next question might be a cosine situation and so they have to work that out, then follow the procedure to find that answer. But all the questions are deliberately mixed. So students are required to draw on their conceptual understanding of the trig principles to calculate the answers rather than rote drilling one type of situation.

And then, all of the sudden - after, perhaps 20 questions - they encounter a question asking them to find an angle. We haven’t talked about how to find an angle. So the first students to encounter this ask “Mr A … how do you do this?” Well, we’ve just created a need to learn! Previously I would have explained this up front: “OK, everyone stop. I am going to teach you how to find angles because that’s what is coming next."

Colin: Yeah, that's a nice language that you are using. You have created a need to learn.

Richard: Created a need to learn. In the course, I call it a Brick Wall. i.e. the Brick Wall strategy - I had to give it a name because it is a simple yet powerful strategy and some teachers use it without knowing what they are actually doing.

Colin: Sounds good.

Richard: So, of course, the faster students reach this question first and I’d say, ‘tongue in cheek’ "That's easy Joe. You’ll work it out." And I say that to stall for time waiting for the next few students to reach this point AND because it gives Joe some time to think about the situation. And then I’ll give a short mini lesson but this time it is literally a 1-2 minute demonstration - not a 5-minute lecture - of how to find the angle to the few students who have that need. Because it is still the same principles being applied, but finding the angle requires a slight twist of the same conceptual approach they have been using. So we have saved 5-8 minutes because before we used to do this whole 5-10 minute lecture on how to find an angle before students needed it. And so with the mini lesson approach they go, "Oh, is that all it is … OK." And then you keep repeating this mini lesson a number of times and until everyone has passed this point.

Colin: So is that why it save's time?

Richard: Yes, I taught using both methods a lot. Once a student can answer sine, cos, and tan question and find angles then they are ready for the ‘real world questions’ (usually just a diagram in a book) is. With a traditional, compartmentalised approach it would often take me between 5-7 lessons for students to reach the real world questions. In the conceptual approach, some students reach the real world questions in 2 lessons, and all of them by 4 lessons. This is based on my experience with many classes. So do the maths - there’s a time saving of 2-4 lessons in a unit, and that’s just to the halfway point.

Colin: Let's just give some room to the sceptics for a minute; because some people might argue that we have been achieving all this great stuff in the world without new ideas in Maths teaching. The example I like to give is that decades ago we put people on the moon. And the people developing the space program presumably came out of classes taught with drill and practice. Or alternatively, you have the teacher who says, "I have been getting great results for the last 10 or 20 years so why do I need to change?” What would you say to them?

Richard: Yeah. It is a really good question. I would say there is no doubt that the traditional approach has worked. There is no doubt that it worked. But I’d also say it has only worked for a small percentage of students. And perhaps those students in which the traditional approach did not work possibly didn't go into the space program.

Colin: Well, I mean it is hard to quantify really, isn't it?

Richard: It is.

Colin: Because we really are talking about something that is I guess in one essence quite subjective and there is no way that you could ever really test for that result but you are still gonna get sceptics, aren't you?

Richard: Well, the first thing you say to a sceptic is “What information would you need to hear to change your mind?” And most of them will say, “None!”. So there is no point talking to them anyway. You know it is true. Just take climate change sceptics for example. Most of them have no intention of changing their minds. They have an opinion that helps fuel their self-image or their pockets.

Colin: So, perhaps we shouldn't be talking to the sceptical teachers, we should be talking to their bored students!

Richard: Perhaps! But I am convinced that a well scaffolded, well run conceptual approach - and this is why it is difficult to define because someone will say “Well I tried that once and it never worked!” But what exactly is it that they tried? Was it just throwing a bunch of equipment to students and saying “See if you can learn trigonometry.” I mean, what I am advocating is a very targeted approach but it is specifically not just teaching skill and drill. And I am convinced that this type of Conceptual Approach which allows students to run into problems and brainstorm and think things through collaboratively definitely equips potential spaceship builders much better than an approach premised on only drilling procedures. I am convinced of that. Now, I don't have any proof and I am not particularly interested in the proof, to be honest. And other people who teach this way would agree. Once you experience it becomes obvious. Perhaps it is bit like that the flat earthers and the round earth’s a few centuries ago. The round Earthers say, "Well it is obvious it is round, you know? How can you say it is flat?" But the flat earthers are going, "No, no, no, it is supposed to be flat, don't tell me it’s round!”

Colin: So, if our listeners would like to ask you a question or would like to engage with you a little bit more how do they connect with you?

Richard: Through learnimplementshare.com or richard@learnimplementshare.com

Colin: And if someone writes you an email will you respond to them?

Richard: Absolutely. And in under 24 hours. Definitely!

Colin: Very great to speak with you, Richard. Thanks very much for your time.

Richard: Thank you.

Colin: I am Colin Klupiec. Until next time, bye for now.

Richard: Thank you, Colin, always good to be here.

Colin: In a recent article you wrote for LinkedIn you quoted a definition for conceptual teaching methods in a maths context, and it goes like this … “Conceptual understanding is knowing more than isolated facts and methods. The successful student (with conceptual understanding) understands mathematical ideas and can transfer their knowledge into new situations and apply it to new contexts.”

But your reaction to this is that definitions like this one are a poor representation of the ‘experience’ of actually making the conceptual connections. Why is that? Can you elaborate?

Richard: Well that particular definition was one of the better ones I could find … Definitions of conceptual understanding often sound more like this one … "Students demonstrate conceptual understanding in mathematics when they provide evidence that they can recognise, label, and generate examples of concepts; use and interrelate models, diagrams, manipulatives, and varied representations of concepts; identify and apply principles; know and apply facts and definitions; compare, contrast, and integrate related concepts and principles; recognise, interpret, and apply the signs, symbols, and terms used to represent concepts. Conceptual understanding reflects a student's ability to reason in settings involving the careful application of concept definitions, relations, or representations of either.

This type of definition is designed for teachers who may not have thought deeply about these ideas of conceptual understanding and to give ideas of what the teacher should be aiming for when teaching. And this is fine - I’m not saying we shouldn’t define conceptual understanding. But I do think we tend to over intellectualise things. The problem with definitions - and I agree they are important - but the problem with them is they tend to diminish the very thing we are defining. If I can paint a picture to explain this …

Colin: Sure, go ahead.

Richard: Let’s say I’m with my 6yo daughter in the botanic gardens and she sees her first gigantic tree. Let’s say it’s a magnificent American redwood tree with a trunk the girth as big as a small house which goes up into the sky like Jack’s beanstalk. My daughter’s jaw is on the ground. She is totally gobsmacked. Woooooow. But because I’m a redwood specialist (which I’m not btw!) and to further my belief that I’m a good father and good fathers need to deliver information I launch into: “This tree is of the Sequoioideae sub-family of Cupressaceae trees. And then continue with all sorts of facts about redwood trees - what makes a redwood a redwood … And then my daughter says “OK dad, can we go have an ice cream.’

Colin: Thanks for that dad, you have really bored that tree down for me!

Richard: Yes, I’ve really destroyed the moment. So I was doing some research recently and John Steinbeck wrote about the redwood, "The redwoods, once seen, leave a mark or create a vision that stays with you always. From them comes silence and awe. They are ambassadors from another time.”

Not that’s not a definition. But he’s using words to elicit the emotional and spiritual experience of being with a redwood tree. Now, of course, I didn’t do this with my daughter when we had this experience - we did really embrace the experience together. But I’d have been better off saying to my daughter “Let’s be still and take it in for a few minutes without saying anything. Or to suggest to her to imagine that the tree is from another world that we have just stumbled across. And NOT to jump to a standard definition or description. Mind you, with my daughter I don’t need to suggest we have stumbled across a different world … she’s there anyway!

So how does that relate to mathematics conceptual understanding?

Colin: Well I thought I might just jump in there and pick up on the words you used ‘Silence and awe’. Are you suggesting then that when you explain for example Pythagoras’ Theorem that you might get a class response that is ‘silence and awe’?

Richard: I think that aiming for silence and awe with Pythagoras’ Theorem at Year 8 might be a stretch! But there is a relationship here. My point is that when someone first gets’s something mathematical - a minor example is seeing a square number for what it is, for example seeing a 4 or a 9 or a 16 as a square. My wife recently mentioned this, she said: “I never realised before what a square number is - they are squares!” And I have to admit that I never realised this until later in my career when I finally saw the relationship with squares. I just hadn’t SEEN it before. Maybe I had been shown it before as a student but it had never stuck.

So my point is when we educators get around a table discussing conceptual understanding we talk definitions and … well … my brain starts hurting!! And I think we lose sight of the gravity of the experience and therefore - perhaps - of how important conceptual understanding is!!

Colin: So what we are suggesting here is that the understanding the concept - rather than actually being able to (only) answer the question - was the thing that could bring real joy to the learning experience. Is that where you’re going with this?

Richard: Absolutely! 100% If I’m a student and your my maths teacher, the difference between me actually getting the concept underlying the maths your asking me to work with, and then answering questions based on that understanding - so I’m now using MY OWN LOGIC to answer the questions VS me ONLY remembering a routine you’ve just shown me for answering the next set of questions - is massive. Because if I’m only drawing on my memory of that routine then I’m operating as a machine. Number in … number out. The difference between the two is like chalk and cheese. And I suspect many maths teachers haven’t really thought about this. And I’m not criticising maths teachers. We don’t tend to operate in a culture which promotes this sort of discussion. For a long time, I certainly hadn’t considered these things. But participants of many of my Professional Learning (Learn Implement Share) courses certainly do … because to consider these issues deeply is a requirement of the course!! And it is fun to see teachers respond “OK, I get what you are showing us here”.

Colin: Actually just listening to you tell that story or explain that idea, it reminds me of when I was actually in high school learning physics. And we had a physics teacher who was doing his level best to explain to us concepts of acceleration and speed and motion. And we were talking in terms of formulas and we were using pro numerals like v, and a, and t (you know t for time, and v for velocity, and a for acceleration).

Richard: Yeah.

Colin: You know I was really trying to understand. And I thought "Yeah I kinda have got this. I see where he is going". But there were some questions where I just couldn't make the formulas work and I just couldn't get what was going on as well as other students.

And then I was also studying Maths and I was also part of the extension Maths class at that time and I remember one day our Maths teacher was explaining a particular concept in Calculus. I am sitting there thinking, "Yeah, I can see something unfolding here but I’m actually just not quite sure what it is but I will just sit here and watch." He was very very good at doing Theorem explanations and First Principles explanations on the board. And there’s this full chalkboard full of stuff. Okay, it was actually a chalkboard, not a whiteboard back then - and I am looking at this thinking “there’s something going on here”. And then he stops and turns around. And if you could picture this in your mind's eye - he still had the chalk touching the board but he turned his head and half of his body around and then he says, "Are there any physics students in the room?" And I look at him and say "Yeah, that's me." And then all of the sudden I thought, "Wow look at that. That thing that I was struggling with over in Physics class was actually masquerading as calculus. I totally get it1” And so what I then started to do was back in the Physics class instead of using the Physics formulas, I would actually start to use Calculus instead and things just become a lot easier. In fact, I used that working in my exam and had a lot more success with that.

Richard: Wow!

Colin: So before we actually get on to the actual methods for teaching conceptually, I just wanted to ask you - this is a bit of side question - because in that situation suddenly a whole other subject became much more useful to me. Extending that idea out a little bit, do you think that if you actually had conceptual understanding of most things in life then we would actually be just become more useful people?

Richard: Well, Colin I absolutely believe this. But I am biased because I’ve always wanted to know "why" I had to do something before doing it. So, I was thought the answer to the "why" question was vital. Although I have to say it took me a while to adopt this principle in my teaching because initially I couldn't even answer the "why" question. I would just say "No, no, no let me share you another example where this situation applies." In fact, I remember in my first year of teaching, this student was asking me the "why" question and I knew that I couldn't teach him what he wanted. All I could I do is giving him another example. And I walked out of the room thinking "I have no idea how to teach this subject." - and I was doing ‘well’ as a first-year teacher - “I have no idea how to teach this, but someone somewhere must know how to teach this. There has to be a way. There absolutely has to be.” It was as clear as a bell. And it is almost like that realisation propelled me into discovering ways to present mathematics that actually allowed students to understand. It gave me that yearning to find a better way.

Colin: Yeah.

Richard: So, that was a bit of a side-track there! Back to the original question - I always believe that "why" questions are important. But many adults I have encountered have the attitude "Just tell me what I need to do and don't bore me with all of the details.” I often hear that and I don't understand why people are like that. It is either because it's a personality thing - maybe some personalities are more interested in the "why" or maybe - and we have talked about this previously; all toddlers are into the ‘why’ - maybe interest in the ‘why’ get’s knocked out of some people between ages 4 and 6! I don't know. I have no proof, I’m just speculating.

Colin: I am just wondering whether a better conceptual understanding of more things in life might make it easier for us to for example cook a meal at home.

Richard: Absolutely. Well, I’m no cook but you know what it is like when you are with someone and they say “Well when you mix this with that the enzymes come out and that’s why the flavour is enhanced … “ and you go "WOW I didn't know about all that." And so the science of cooking makes so much more sense. So, yes, conceptual understanding equips you with a lot more skills - it makes you more useful.

Colin: Well let’s get down to some detail. When I was learning mathematics back in the late 80s - and I just gave that description of that ‘aha’ moment for me - but essentially it was still drill and practise. We were always shown the proofs and theorems first. But essentially it was a textbook and classroom exercise. In your article, you refer to compartmentalising. What do you mean by that?

Richard: So in the vein of what you were just saying let me use this analogy from my PE teaching days. Let’s say you are a kid who has never learnt to throw a baseball properly - you’ve simply never mastered the skill, and I’m going to teach you. So I’ll show you the whole action first but I won’t expect you to replicate the whole skill first. I’ll break it into parts. I’ll teach you the wrist action first. It’s really easy - anyone can do it. So we’ll throw 20 times back and forth just with the fingers and a wrist action and from close range. Then we will keep throwing but incorporate the forearm as well. And the student is thinking “Wow, I can do this. Then you step apart a bit and add a full arm action. Then you add the weight transfer from back foot to front foot. Then a full throw with a few steps run up. So I teach in parts and add parts to make up the whole. It’s not an exact analogy to teaching mathematics traditionally but it is close because the teaching is somewhat compartmentalised. Now when it comes to mathematics it seems logical to use a similar approach. The example I use in the article is basic trigonometry. Basic trigonometry or the trigonometry of right angled triangles allows us to calculate - within right-angled triangles - the length of a side given one side and an angle OR the size of an angle given 2 sides. So at the end of a unit, we want students to look at real life situations and solve the problem which will involve calculations of angles and sides within right angled triangles. And the traditional approach - which I assume we’ve used for hundreds of years - is to break the skills into parts. We teach the 3 trig scenarios - sine, cosine and tangent - but we teach them in isolation. So what happens is we first teach the Sine situation, show an example and then have the students do ‘ten questions’ based on that. So they are only thinking about the procedure to get the answers for the sine situation. And then we do the same for the cosine situation. Now students are only thinking about the procedure to get the answers for the cosine situation. And then we repeat for the tangent situation. All the time we have only been finding lengths of sides. So next, we show them how to find angles using sine, then cosine, then tangent. Then we progress to ‘real world’ situations and finally onto bearings. Now the problem with this is that at each of those parts students are thinking only about the routine for that one situation.

Colin: Yeah, they are thinking about those three parts rather than the entire concept of what’s going on.

Richard: Exactly! But it seems logical to teach this way. And the other thing that seems logical about this compartmentalised approach is that it gives teachers the (false) impression that we can herd these students through a unit and think they are all progressing somewhat uniformly. You hoodwink yourself into thinking this works. Although I couldn’t! Because I knew damn well it wasn’t working. It was depressing. I had my top students bored - I could see it. I was boring them senseless. And I’d lost the bottom end using this method. And those in the middle weren’t exactly inspired either. And I had to find another way.

Colin: So what is the alternative?

Richard: The alternative is what I call a conceptual approach. Now it is tough to explain to people who have not taught this way because it involves ideas and labels which will mean ten different things to ten different people. So it is very difficult to explain what I’m referring to as a conceptual approach in a podcast or an article. In a 4-month long course - i.e. the course ‘Engagement - Winning over your mathematics class’ - these ideas are developed slowly and teachers trial various ideas in their classroom as they progress through the course. So everyone knows exactly what the conceptual approach looks like by the end of the 4-month journey. But again, trying to explain this within a 25-minute podcast or long article it is very difficult, but let’s give it a shot anyway - I’ll try to explain the conceptual approach to teaching right-angled trig.

Colin: Sure

Richard: With the trigonometry example, rather than compartmentalising into the sine situation and the cosine situation and so on, the introduction is a conceptual intro into what trigonometry is. And I discovered this when I sat down one day thinking “This is driving me nuts (my lack of success with the conceptual approach) there has to be another way. What is trigonometry?” And I just looked at it differently and thought "Oh my God it’s just similar triangles.” No one had shown me this before. And this is not earth-shattering - most mathematics teachers probably realise this but I had never seen it. I thought, "Okay. So I’ll simply demonstrate similar triangles and then I’ll ask leading question to allow students to see the principles at work. Now there is software that makes this so easy. If you go to the article https://www.linkedin.com/pulse/conceptual-understanding-mathematics-what-why-do-we-need-andrew?trk=prof-post , there is a video in the middle of the article showing what this introduction actually looks like with the software GeoGebra. So, this is the introduction to the Conceptual Approach to trigonometry.

And then what happens is the students begin to tackle a series of questions. So these are questions on paper, but it is a worksheet where rather than the questions being compartmentalised, they are using the trigonometry principles to work out whether each triangle is a sine situation or cosine or tangent situation.

Colin: Yeah.

Richard: That's a much different question and, by the way, students do not need to be highly skilled in answering these. And so once they decide which situation it is they write the appropriate ‘trig sentence’ and solve the equation and get an answer. The next question might be a cosine situation and so they have to work that out, then follow the procedure to find that answer. But all the questions are deliberately mixed. So students are required to draw on their conceptual understanding of the trig principles to calculate the answers rather than rote drilling one type of situation.

And then, all of the sudden - after, perhaps 20 questions - they encounter a question asking them to find an angle. We haven’t talked about how to find an angle. So the first students to encounter this ask “Mr A … how do you do this?” Well, we’ve just created a need to learn! Previously I would have explained this up front: “OK, everyone stop. I am going to teach you how to find angles because that’s what is coming next."

Colin: Yeah, that's a nice language that you are using. You have created a need to learn.

Richard: Created a need to learn. In the course, I call it a Brick Wall. i.e. the Brick Wall strategy - I had to give it a name because it is a simple yet powerful strategy and some teachers use it without knowing what they are actually doing.

Colin: Sounds good.

Richard: So, of course, the faster students reach this question first and I’d say, ‘tongue in cheek’ "That's easy Joe. You’ll work it out." And I say that to stall for time waiting for the next few students to reach this point AND because it gives Joe some time to think about the situation. And then I’ll give a short mini lesson but this time it is literally a 1-2 minute demonstration - not a 5-minute lecture - of how to find the angle to the few students who have that need. Because it is still the same principles being applied, but finding the angle requires a slight twist of the same conceptual approach they have been using. So we have saved 5-8 minutes because before we used to do this whole 5-10 minute lecture on how to find an angle before students needed it. And so with the mini lesson approach they go, "Oh, is that all it is … OK." And then you keep repeating this mini lesson a number of times and until everyone has passed this point.

Colin: So is that why it save's time?

Richard: Yes, I taught using both methods a lot. Once a student can answer sine, cos, and tan question and find angles then they are ready for the ‘real world questions’ (usually just a diagram in a book) is. With a traditional, compartmentalised approach it would often take me between 5-7 lessons for students to reach the real world questions. In the conceptual approach, some students reach the real world questions in 2 lessons, and all of them by 4 lessons. This is based on my experience with many classes. So do the maths - there’s a time saving of 2-4 lessons in a unit, and that’s just to the halfway point.

Colin: Let's just give some room to the sceptics for a minute; because some people might argue that we have been achieving all this great stuff in the world without new ideas in Maths teaching. The example I like to give is that decades ago we put people on the moon. And the people developing the space program presumably came out of classes taught with drill and practice. Or alternatively, you have the teacher who says, "I have been getting great results for the last 10 or 20 years so why do I need to change?” What would you say to them?

Richard: Yeah. It is a really good question. I would say there is no doubt that the traditional approach has worked. There is no doubt that it worked. But I’d also say it has only worked for a small percentage of students. And perhaps those students in which the traditional approach did not work possibly didn't go into the space program.

Colin: Well, I mean it is hard to quantify really, isn't it?

Richard: It is.

Colin: Because we really are talking about something that is I guess in one essence quite subjective and there is no way that you could ever really test for that result but you are still gonna get sceptics, aren't you?

Richard: Well, the first thing you say to a sceptic is “What information would you need to hear to change your mind?” And most of them will say, “None!”. So there is no point talking to them anyway. You know it is true. Just take climate change sceptics for example. Most of them have no intention of changing their minds. They have an opinion that helps fuel their self-image or their pockets.

Colin: So, perhaps we shouldn't be talking to the sceptical teachers, we should be talking to their bored students!

Richard: Perhaps! But I am convinced that a well scaffolded, well run conceptual approach - and this is why it is difficult to define because someone will say “Well I tried that once and it never worked!” But what exactly is it that they tried? Was it just throwing a bunch of equipment to students and saying “See if you can learn trigonometry.” I mean, what I am advocating is a very targeted approach but it is specifically not just teaching skill and drill. And I am convinced that this type of Conceptual Approach which allows students to run into problems and brainstorm and think things through collaboratively definitely equips potential spaceship builders much better than an approach premised on only drilling procedures. I am convinced of that. Now, I don't have any proof and I am not particularly interested in the proof, to be honest. And other people who teach this way would agree. Once you experience it becomes obvious. Perhaps it is bit like that the flat earthers and the round earth’s a few centuries ago. The round Earthers say, "Well it is obvious it is round, you know? How can you say it is flat?" But the flat earthers are going, "No, no, no, it is supposed to be flat, don't tell me it’s round!”

Colin: So, if our listeners would like to ask you a question or would like to engage with you a little bit more how do they connect with you?

Richard: Through learnimplementshare.com or richard@learnimplementshare.com

Colin: And if someone writes you an email will you respond to them?

Richard: Absolutely. And in under 24 hours. Definitely!

Colin: Very great to speak with you, Richard. Thanks very much for your time.

Richard: Thank you.

Colin: I am Colin Klupiec. Until next time, bye for now.

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