Scenario: Eoin revising perimeter and area concepts
Country: Ireland
Grade (student age): 6th class (age 11-12 years)
Contributed by: Dolores Corcoran, St Patrick’s College, Drumcondra, Ireland
Context – national, curricular, professional, other
While espousing a reform approach with the mandated development of mathematics process skills, the Irish primary mathematics curriculum is relatively specific in that content to be taught is divided into five mathematical strands (number, algebra, measures, shape and space and data and chance). These are further divided into strand units with lists of content objectives (followed by very brief exemplars) specified for each strand for each class level over the eight years of Irish primary schooling. Mathematics textbook series from one of three different publishing companies are currently available in Irish schools and these are ubiquitous in Irish classrooms. Student teachers are encouraged to extend their mathematics lessons beyond the textbook and here a second year Bachelor of Education student on spring teaching placement interpreted this advice by supporting his textbook-based lesson with three posters.
Scenario
This scenario illustrates adherence to the textbook. Eoin’s lesson started with homemade posters designed to give instructions on how to find the area and perimeter of a rectangle, material which Eoin believed to be already known by the children but which he rehearsed in detail, demonstrating the procedures with further examples on the board.
Posters were A5 size, bearing the following legends:
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Poster 1 Poster 2 Poster 3
Eoin then gave the class the “question for today”:
Eoin: If we have two rectangles and when we add up the perimeter, if we get the perimeter, if they have the same perimeter, two different rectangles, do they have the same area? So we are going to have to prove that.
This ‘task’ was set as he pointed to the third poster, which echoed a highlighted text box on the page. Children were asked to open their textbooks at page 163 and start at question 1 (a), which Eoin read aloud and then drew on the board inviting children to chorus responses as he calculated the perimeter of a 5 x 1 and then a 4 x 2 rectangle, by writing and then adding 5+5+1+1 to get 12 and 4+4+2+2 to get 12 again. Children’s attention was drawn to these perimeters as being the same.
Then he invited the cooperation of the class to check the area of each rectangle. Following which, he concluded that as per the poster that “even though two rectangles may have the same perimeter they may not have the same area”.
Next, pupils moved to question 2(a), where Eoin demonstrated how to find the perimeter of the first (6x1) rectangle. Pupils were invited to do the second and third rectangles themselves and then ‘compare and contrast the rectangles”. This work continued through questions 2 and 3, for 18 minutes when Megan was invited to the board to demonstrate working 3(i) (a) and (b). The remaining parts were corrected on the board in chorus. While Eoin referred to his posters regularly, no reference was made to the two places on the textbook page where students are advised to “discuss your answers”. Twenty-five minutes into the lesson, Eoin drew the children’s attention to the last question on the page: draw five rectangles with a perimeter of 24cm each with a different length and width. Reference on the textbook page to using squared paper to draw the rectangles was ignored and Eoin started by giving an example of such a rectangle as 7 x 5. The 8 x 4 example soon followed but the 6x 6 option was discounted as a ‘square’ and therefore not a rectangle. When five different rectangles had been found and tested, Eoin reiterated the lesson objective:
Eoin: So from today’s lesson what have we learned? That although rectangles may have the same perimeter they may not have the same area. Everyone got that clear in their head? No questions at all on that?
Knowledge Quartet Coding Commentary
Contributed by: Dolores Corcoran, St Patrick’s College, Drumcondra, Ireland
Knowledge Quartet Dimension: Foundation
Knowledge Quartet Code: Adherence to textbook (ATB)
Scenario: Eoin revising perimeter and area concepts
This scenario illustrates adherence to textbook where overt subject knowledge of geometry as well as measurement would be needed to use the material fully or creatively. Eoin’s lesson was enacted from a single page of the class mathematics textbook although he changed the wording from “[R]ectangles that have the same perimeter need not have the same area” in the text (Edwards and McElwee, 2003, p. 163)to “[e]ven though two rectangles may have the same perimeter they may not have the same area’ on his poster”
While globally, textbooks constitute a major teaching resource, indications from Eoin’s lesson are that they must be used creatively to be helpful and their teaching impact is limited by the mathematics content knowledge of the teacher. The textbook page was designed to teach children that conservation of perimeter does not guarantee conservation of area. Eight examples were offered to illustrate the conjecture. These were graded with only a slight degree of variation (Marton and Morris, 2002) through 5 x 1 and 4 x 2, to 6 x 1, 5 x 2, 4 x 3, and 7 x 1, 6 x 2 and 5 x 3. Eoin appeared unaware of the visual and numerical patterns being shown here as he drew the first of these rectangles - to scale - on the board and invited children to draw the next three without offering them squared paper on which to work. Use of the l x b formula was the only option he accepted to find the area, though the first two examples in the textbook had the square centimetres drawn in. Eoin asked if his 60cm x 10 cm rectangle drawn using the mater stick resembled the one in their books (6cm x1cm), but did not mention his use of scale, which could have been a useful Connection to make for these 12 year olds and relevant to work they may have done previously in the same textbook. He invited children to “compare and contrast these rectangles” by which he actually meant check if perimeters and areas were the same. He demonstrated how to do this in a strictly, numerically procedural fashion.
Eoin: Jenny can you tell me how to get the perimeter of that please?
Jenny: Eh, Add six and one
Eoin: (Writing) Six plus one …
Jenny: Equals seven.
Eoin: Equals seven ... Is she right? Can anyone write the whole thing? Jenny can you see … can you correct that yourself? Do you remember when we were doing perimeter?
Jenny: I said six and one and you add it
Eoin: Perimeter equals add … you’re right in that. Do you remember over here (pointing to poster B) when we were adding the four sides?
Jenny: Oh yeh, yeh.
Eoin: So would you correct yourself there? So add six plus one plus six plus one.
(Writing) 6 + 1 = 7
6 + 1 = 7
14
Seven plus seven equals fourteen. Does everyone agree with that?
It may be that Jenny’s apparent error was a Contingency opportunity where Eoin could have questioned the thinking, which led her to suggest, “I said 6 and 1 and you add it”. It may be that Jenny was seeing the duplicative aspect of the 6 + 1 + 6 + 1 formula of which Eoin appeared unaware, since he usually presented the dimensions in pairs of matching sides.
Eoin was insistent that ‘width’ of rectangle referred to the horizontal dimension and ‘length’ to the vertical dimension, only. His understanding of these terms as different is evident in his lesson objectives where he planned two separate objectives to determine the length and width of a rectangle given the other dimension and the area. This rigid use of terminology resulted in labelling the rectangles in the text such as the first one as L = 1cm W = 6cm.
When dimensions only were given this led to an interesting orientation of rectangles, different from those in the book being drawn in response to L = 7cm, W = 1cm. Again there was scope here for visual and spatial pattern comparisons which might help children to enjoy mathematics as an aesthetic experience and provide a sound basis for understanding area as “a measure of the amount of two dimensional space inside a boundary” (Haylock, 2006, p. 294) instead of the more usual, limited and limiting, explanation of area as “x centimetres squared”. Eoin’s pedagogic emphasis on area was on the use of the ‘ squared’ symbol (²), first mentioned when rehearing the example on poster A.
Eoin: To get the area of the rectangle, multiply 6 x 3 to get 18 cm squared. Very important the ‘squared’.
And later in the lesson, when correcting a child’s work,
Eoin: And you remembered your ‘squared’ this time.
The last problem, taken from the textbook could well have been a starting point for fruitful investigation of just how many, different rectangles there can be for any given perimeter, leading to generalisations about conservation of perimeter and transformation of area of rectangles and conservation of area and transformation of perimeter. There are for example, six possible rectangles of perimeter 24 and when a child suggested the 6 + 6 + 6 + 6 combination, Eoin discounted it as a square and so not eligible to be counted as a rectangle. The textbook rather ambiguously specified that “each of the rectangles must have a different length and width” so the inclusion of the six by six case should at least have been open to discussion. Earlier on the page such discussion had been invited “Discuss your results with your teacher and other pupils” (Edwards and McElwee, 2003, p. 163). The concluding sentence of Eoin’s lesson plan promised that “the results will be compared with the rectangle on the board and a discussion will take place on these results”. However, Eoin’s vision of mathematics as a difficult discipline of rules and procedures did not allow for discussion beyond constantly seeking reassurance “So everyone alright with that?” and frequent questioning, “Anyone confused so far?” Further, in our post-lesson discussion Eoin demonstrated his beliefs about mathematics as a difficult discipline.
I am afraid to leave … that I am leaving kids, because I know I am not covering a huge amount in the day … but I feel that I just have to hammer things home and do example after example.
Eoin appears to carry his own experience of learning mathematics at second-level into his teaching of primary mathematics and ‘does’ example after example to “hammer things home”. He followed all the numerical examples on the textbook page without deviation or discussion but appeared to miss the pedagogical implications inherent in their choice.
References
Edwards, M. and McElwee, S. (2003). Maths Matters 6 (Dublin, Educational Company).
Haylock, D. (2006). Mathematics Explained for Primary Teachers (London, Sage).
Marton, F. and Morris, P. (2002). What matters? Discovering critical differences in classroom learning. (Göteborg: Acta Universitatis Gothoburgensis).