Scenario: John teaching properties of 3-D shapes
Grade (student age): Year 9 (age 13-14)
Contributed by: Anne Thwaites, University of Cambridge, UK
Context – national, curricular, professional, other
The National Curriculum for mathematics in England includes properties of 3D shapes in Key Stage 3 (years 7-9, pupil age 11-14) including surface area and volume of 3D shapes based on prisms. John had introduced the idea of surface area and volume of 3D shapes in a previous lesson and was moving on to consider the properties of a cylinder in this lesson. John was a graduate pre-service teacher, and the lesson took place in a school-based placement towards the end of his one-year teacher preparation.
John was teaching in an open entry secondary school (pupil age 11-18) in a small town in the UK. The school divides the year group into maths sets (by ability) and John was teaching one of the two parallel second groups in Year 9 (pupil age 13-14). This was the final lesson in a short sequence.
For the initial activity of the lesson, John asked the pupils to match formulae, written on a whiteboard, (bh, πr², ½bhl, wlh, πr²l, ½bh, 4/3πr³, πd) with shapes (rectangle, sphere, circle, triangle, cuboid, cylinder). In his lesson plan John had noted that he would “Look at dimensionality of formulas”. Once the pupils had had time to discuss the pairings, John asked for answers and recorded these on the board. Then he highlighted two of the formulae and asked, “What is the difference between this one [4/3πr³] and that one [ πr²]?” and immediately got a response “It’s 3D”. John continued by discussing the dimensionality of area and volume in terms of length, height and depth.
John: What is the difference between this one [4/3πr³] and that one [ πr²]?
Student 1: It’s 3D
John: What does 3D mean?
Student 2: Three dimensional.
John: So it’s got... So length, it’s got length, and area has got length and base or height. So volume has got to have what?
Student 2: Em, length, height and depth?
John: I like that – depth. So we’ve got, em, a height, a length, and then also a depth, or a third sort of, em, a third sort of dimension. That’s exactly what it has. [Draws three axes on the board to represent the three dimensions] So let’s match up a couple more. …
Student 3: wlh is volume of a cuboid.
John: Yeah. Volume of the cuboid, yeah, and you know it was a volume because?
Student 3: It’s got three lengths.
John: It’s got... yeah, it’s got three, what do you call these letters? …
Student 4: Dimensions
John: Yeah, so they are, they are three dimensions, aren’t they?
Whilst considering the expression wlh he encouraged the pupils to suggest the term ‘variable’ rather than ‘lengths’ or ‘dimensions’. He went on to use ‘variable’ in his subsequent discussion.
In this cycle of our research, we were interviewing the student-teacher after the lesson. During this meeting we viewed episodes from the video of the lesson and one member of the team led a discussion with the student-teacher in the spirit of stimulated-recall. This discussion was audio recorded and in the first part we asked about John’s planning for this lesson.
Researcher: … You, in your plan, specifically mentioned dimensionality. … Was that something that you felt was important or is that something that is in the scheme of work?
John: It’s just something that I think, I think is important. It seems like a fairly fundamental thing in understanding shapes and … where these formulas come from. It’s important that you know that you are going to need three variables to calculate a volume. So it was just something that I thought I should add in because it’s something that they don’t usually meet. It’s not in the scheme … it’s not deemed to be such a big deal. But I think, it’s also interesting as well. It’s just something that they’ll find interesting and it’s crucial to shapes and things, so I’ve added it in.
Researcher: … So this is an area that you … really want to highlight?
John: Yep, yep. Because … I’ve … introduced volume to lower school classes and they find it really hard to get this idea of volume being inside a shape. And it’s really tricky and you can help that by focussing on that, ... So that’s why dimensionality came in because it’s a way of thinking about, well it’s that extra dimension. … we’ve got three variables to worry about is something totally different to what we were looking at before. So that’s why it’s in there now.
Later in the interview we asked him about using the term ‘variable’:
Researcher: ... you also quite deliberately brought in the term variable. You needn’t have done. …
John: … I’ve got a thing … about using mathematical language. … it’s … a bit of a philosophy, … that if you can teach kids to talk like mathematicians they will be better mathematicians. So, that’s why it came in, … it would be a good point to think about these three variables and link that with the three dimensions. So it was just a chance to get a bit of extra language in there really. …
Researcher: … I think it’s that sort of abstract concept of what a variable is, that here, … they can visualise what these shapes are, I think.
John: … because those variables actually have something that they can imagine rather than thinking. … you can’t imagine a variable unless it’s attached to something like this.
Knowledge Quartet Coding Commentary
Contributed by: Anne Thwaites, University of Cambridge, UK
Knowledge Quartet Dimension: Foundation
Knowledge Quartet Code: Theoretical underpinning of pedagogy (TUP)
Scenario: John teaching surface area and volume
In this scenario, John is introducing the properties of 3D shapes and has deliberately planned to discuss dimensionality with the pupils. This is an idea that is normally reserved for sixth form teaching (post-compulsory, pupil age 16-18) but he has included it and uses the volume of specific 3D shapes to introduce this idea. The fact that he has gone beyond the statutory curriculum is interesting and made us wonder why he was keen to introduce this idea. The post lesson interview enabled us to ask John about this. His explanation reveals his own understanding of key ideas within mathematics and what he feels should be introduced when opportunities arise.
“I think [it] is important. It seems like a fairly fundamental thing in understanding shapes and … where these formulas come from. … So it was just something that I thought I should add in because it’s something that they don’t usually meet. … But I think, it’s also interesting as well.”
John had introduced the term ‘variable’ to the discussion with the pupils and wove this into his subsequent work with them. Again he had a clear rationale for this during the interview:
“… I’ve got a thing … about using mathematical language. … it’s … a bit of a philosophy, … So it was just a chance to get a bit of extra language in there really.”
The interview has revealed something of John’s beliefs about teaching mathematics - matters about which he has strong feelings. John considers these two ideas (dimensionality and variable) as underpinning several areas of mathematics and therefore ones which can, perhaps, connect different parts of mathematics. He wants the pupils to use appropriate mathematical language to express some of these ideas, to learn to “talk like mathematicians” (Pimm, 1987; Johnston-Wilder and Lee, 2008), and so is reinforcing some of the common threads that exist between different parts of the mathematics curriculum.
The manner in which John deliberately built these opportunities into this lesson make us consider this as evidence of in-depth theoretical subject knowledge underpinning his planning and subsequent teaching. He holds a conviction that dimensionality is important, even though the curriculum does not specify it, which directs an approach to the formulae which transcends mere memorisation. Both of these aspects of theoretical underpinning relate to his beliefs about how students best learn mathematics (reference to dimension, learning and using language).
Focusing on his content knowledge, this scenario could also be an example of Overt Subject Knowledge (OSK). However we consider that it is also a good example of TUP because it has directly affected his planning and teaching of this lesson. In the interview, the way in which he highlights the importance of fundamental ideas that cut across the subject – the “Big Ideas” (Kuntzte et al., Cerme 7) - illustrates this.
Pimm, D. (1987). Speaking mathematically. London: Routledge and Kegan Paul.
Johnston-Wilder, S. and Lee, C. (2008). Does articulation matter when learning mathematics? Proceedings of the British Society for Research into Learning Mathematics 23(8): 54-9
Kuntzte, et al Cerme 7