Scenario:

Chloe teaching strategies for subtracting 9, 11, 19 and 21

Country: UK

Grade (student age): Year 1/2 (age 5-7)

Contributed by: Fay Turner, University of Cambridge, UK

Context – national, curricular, professional, other

Chloe was a student teacher in the final term of a one year postgraduate teacher education programme.  The lesson took place during her final school placement.  In planning this lesson Chloe referred to curriculum guidance in the National Numeracy Strategy framework (DfEE, 1999) in place at that time.  A mental calculation strategy suggested for Year 2 in this guidance was ‘add/subtract 9 or 11: add subtract 10 and adjust by 1.  Begin to add/subtract 19 or 21: add/subtract 20 and adjust by 1’.  Chloe was following up a lesson in which her class had used these strategies for addition of 9, 11, 19 and 21. Chloe was a graduate pre-service teacher, and the lesson took place in a school-based placement towards the end of her one-year teacher preparation.

Scenario

Chloe was introducing strategies for subtracting 9, 11, 19, and 21 to a Year 1/2 (pupils age 5-7) class.  She used a large1-100 number grid and magnetic counters to represent these strategies by moving the counters on the grid.  After a starter activity in which children were asked to give the compliments of 10 and 20, Chloe revised strategies for adding 9, 11, 19 and 21.  She asked a child to demonstrate how this might be done using the 100 grid and counters.  The child successfully demonstrated this by, for example for ‘add 9’, moving a counter down one row and back one square.  After telling the class that they were going to learn strategies for subtracting these numbers, Chloe asked how they might subtract 19 from 70.   The first response was to subtract 20 and subtract one.  Chloe said this was incorrect and stated that to subtract 19 they should subtract 20 and add one.  She demonstrated this by moving a counter up two rows to 50 in order to subtract 20 and then around to 51 on the other side of the hundred grid in order to add one (Figure 1).  Chloe then pointed to each number in turn going backwards from 70 on the hundred grid (69, 68, 67 … 51) as she counted back 19 in order to check that this was correct.

1	2	3	4	5	6	7	8	9	10
11	12	13	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30
31	32	33	34	35	36	37	38	39	40
41	42	43	44	45	46	47	48	49	50
51	52	53	54	55	56	57	58	59	60
61	62	63	64	65	66	67	68	69	70
71	72	73	74	75	76	77	78	79	80
81	82	83	84	85	86	87	88	89	90
91	92	93	94	95	96	97	98	99	100

 

 

 

 

 

 

Chloe’s next question asked the children to say how they would ‘take away’ 9 from 70.  After some ‘incorrect’ responses Chloe reiterated that to subtract 9 they should subtract 10 and add 1.  She demonstrated this by moving a counter on the hundred grid up to the 60 and then across the board to 61.

Chloe then moved over to a white board where she used questioning in order to elicit and then write the “rules” for subtracting the four numbers. She questioned the class about the strategies needed for subtracting each number beginning with 19, followed by 9, 11 and finally 21.  Each time she recorded the strategy by writing an example of subtracting from 70 as shown below.

 

The children clearly found explaining the strategies difficult and offered several responses involving adjustment in the wrong direction, e.g. for subtract 19, it was suggested that they should “subtract 20 and subtract one”.  Chloe asked for strategies from different pupils until a correct response was given.

The children were told to refer to these ‘rules’ and not to use hundred grids when completing their worksheets of problems.   The lower achieving children were given worksheets requiring them to subtract 9 and 11 while the higher achieving children were given worksheets requiring them to subtract 19 and 21.  Many of the children encountered difficulties and Chloe worked with individuals to help them find the right strategy for the particular subtractions.

The lesson was concluded with a brief plenary in which Chloe asked children to give strategies for solving 30 – 9 and then 40 – 21.

 

 

 

 

 

 

 

 

 

 

 

Knowledge Quartet Coding Commentary

Contributed by: Fay Turner, University of Cambridge, UK

Knowledge Quartet Dimension: Connection

Knowledge Quartet Code: Anticipation of complexity (AC)

Scenario: Chloe teaching strategies for subtracting 9, 11, 19 and 21

 

The way in which Chloe differentiated the work that children were asked to complete suggested that she did not anticipate the relative complexity of strategies for the four subtrahends.  The higher achieving pupils were given problems involving subtracting 19 and 21 and the lower achieving pupils, problems involving subtracting 9 and 11.  Anticipation of complexity might have led Chloe to give the higher achieving pupils problems involving 9 and 19 and the lower achievers, 11 and 21.  Addition and subtraction of 11/21 entail a sequence of actions in the same direction whereas 9/19 require a change of direction for the final unit i.e. compensation.  Therefore, subtracting 11 or 21 requires less complex strategies than subtracting 9 or 19, i.e. subtract 10 or 20 then subtract one more is less complex than subtract 10 or 20 and then reverse the direction and add one (Heirdsfield, 2001). Chloe differentiated by the size of the numbers rather than the complexity of the strategy involved. The wording of the curriculum guidance (DfE, 1999) which suggested that children in Year 2 should - ‘add/subtract 9 or 11: add subtract 10 and adjust by 1.  Begin to add/subtract 19 or 21: add/subtract 20 and adjust by 1’, may have encouraged Chloe to differentiate in this way.

The first subtrahend Chloe chose to use in her demonstration was 19.  Of the four numbers addressed in this lesson, it might be argued that the strategy for subtracting 19 is the most complex.  Subtracting 19 may be considered to be more complex than subtracting 9 as it seems necessary to understand how to subtract ten before understanding subtraction of multiples of ten.  Therefore a subtraction which involves subtracting ten and then adjusting would seem to be a logical forerunner of a subtraction involving subtracting twenty and then adjusting.  Chloe followed her demonstration of subtracting 19 with a demonstration of subtracting 9.  Strategies for the subtraction of 11 and 21 were not addressed using this representation. Chloe’s choice of 19 and then 9 as the subtrahends for her demonstration examples suggests that she did not anticipate the complexity of the strategies needed to carry out these subtractions relative to those needed for the other subtrahends.

Following her demonstration, Chloe used questioning to elicit strategies for each of the subtrahends which she wrote in symbolic form on the white board.  She had pre-prepared the board with headings for each of the subtrahends suggesting she would address them in the order 9, 19, 11, 21.  However, Chloe began by writing ‘70 – 19 = ’ under the heading ‘Subtract 9’ which she later changed to ‘Subtract 19’.  She then addressed strategies for the other three subtrahends in the order 9, 11, 21.   This order suggested that Chloe recognized that strategies for 19 and 9 were similar to each other and different to those needed for subtracting 11 and 21.  However, that she chose to address subtract 9 and 19 before 11 and 21 suggested she had not anticipated that the former would be more complex.

While Chloe’s sequencing of examples could also be seen as an example of DS, it is here used as an example of AC for the following reasons. Chloe gave examples involving subtraction of 9 and 11 to the lower achieving children and 19 and 21 to the higher achieving children suggesting she had not anticipated that subtraction of 9 and 19 would be more difficult than subtraction of 11 and 21 when she planned the lesson.  For her demonstration, she chose the harder examples involving subtraction of 9 and 19.  This may have been because she realised they were more difficult and thought that it would be important to demonstrate these.  However, starting with 19 rather than 9 suggests her choice of these examples was probably random.  Finally, the order in which she addressed the subtractions in the symbolic form seemed to confirm that she did not anticipate that subtracting 9 and 19 would be more difficult than subtracting 11 and 21.

 

References

DfEE  (1999)  The National Numeracy Strategy: Framework for Teaching Mathematics.  Sudbury:  Department for Education and Employment Publications

Heirdsfield, A.  (2001)   ‘Integration, compensation and memory in mental addition and subtraction’, in M van del Heuvel-Panhuizen (ed.),  Proceedings of the 25^th Annual Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 129-136) Utrecht: Freudenthal Institute, Utrecht University.