Assessment of Early Number / Arithmetic Learning
Children develop and use a range of methods of solving problems. The strategies they use increase in sophistication as children gain experience and develop better ways of solving problems. The Early Learning Framework classifies the various strategies used by children into five stages as children progress from 'Early Learning Strategies' to 'Early Arithmetical Procedure'. Starting at the emergent stage of counting, children develop more sophisticated counting techniques, recognise forward and backward number sequences, perceive composite units and move on to early multiplication and division concepts and base ten strategies.
The following excerpts show a case study of Julia, aged 4 years and 9 months being interviewed by Joanne Mulligan of Mcquarie University. Assessment tasks are used to find out the child's understanding and strategies for multiplication and division concepts. It also exemplifies how the teacher needs to observe the child very closely during these assessment sessions. Some of the seemingly unimportant gestures of the child are actually indications of the way she approaches the task. This is brought out by probing further, sometimes requiring specifically structured questions to ascertain the level of strategy use.
Julia is presented with counting, multiplication and division tasks of varying difficulty levels which determine her early number knowledge and strategy use. It is obvious from the tasks that Julia can count correctly to numbers beyond 30. She is able to count concealed items, but uses her fingers and generally uses the count all strategy, showing perceptual counting even in cases where she seemingly worked out the answers in her head.
Some sample tasks
Here are some examples of the tasks presented to Julia and her responses:
Task 1: Composite unit of 3 with perceptual counting
Julia is shown a row of 3 yellow counters. She is told that two more identical rows are placed under a card and asked how many altogether?
Julia gives the answer 9.
"How did you work that out Julia?"
Click here to see Julia's response:
It is seen that Julia uses her fingers, goes 3, 3, 3 then looks at all the fingers used and says 9. It makes us wonder what strategy Julia has used here. Julia probably recognises the pattern of 9 fingers knowing that she has ten fingers in all. Does she count by ones to 9? Does she use skip counting in threes?
Task 2: Unable to skip count
Julia is shown a long row of counters and asked to count them in 2's.
She is unable to skip count.
Julia estimates that there are 12 counters and then correctly counts in ones to find there are 14. She is again asked to try counting in 2's but fails.
Click here to see Julia's response:
It is clear from this task that Julia is able to count confidently in ones, her estimate is quite good and she can see the row in pairs. But she cannot count as 2, 4, 6, Š. So Julia has not achieved skip counting yet. Probably because this is usually a skill taught in school and she has not yet started school.
Task 3: Perceptual counting composite units
Julia is presented with a card consisting of 5 bands of different colours. When asked "how many colours?", Julia responds "5". Then she is shown another identical card and asked "how many now if this card is placed as an extension of the first card?". Julia answers almost instantly "10" and explains that she already knows that 5 and 5 make 10. Now the two cards are placed side by side and a third identical card is placed alongside but turned upside down so that the five colours are not visible.
Again Julia is asked "how many colours?"
Click here to see Julia's response:
Julia counts 1, 2, 3, ŠŠ , 15. Note that she aligns the third card with the others and counts from 11 to 15 by actually touching the places she imagines the five colours to be. Here Julia displays perceptual counting and is able to count concealed items, but starts from 1.
Now let us summarise our conclusions about Julia's level of achievement on the early learning framework.
Julia can:
- Count accurately and knows her numbers
- Count composite units provided she can use her fingers or other items for perceptual counting
- Count concealed items but uses the count all strategy
Julia does not:
- Skip count in a pattern of 2, 4, 6
- Count concealed items as a composite unit rather, she uses perceptual counting
- Seem to use five or ten as a base in the tasks shown above
Exercise
Here are some more tasks that were presented to Julia.
Based on what we have observed in Julia's responses to the earlier tasks, can you predict Julia's responses to these tasks?
In Task 2, Julia fails to count in 2's but counts in ones to find there are 14 counters in the row. The row is then split into two parts with a break after 6 counters. Julia is asked to count in 2's to find out how many in the first part and how many in the second part.
What do you think Julia's response will be and how do you expect her to arrive at that answer?
Click here to see Julia's response:
Click here to see comments on Julia's response
Julia is shown 9 yellow counters placed in a 3x3 array, and asked "how many counters ?"
What do you think Julia's response will be and how do you expect her to arrive at that answer?
Next Julia is asked how many counters would be left if one row of 3 counters is removed, and she promptly answers "6".
Julia is then asked how many counters will there be if one more row of 3 counters is added on to the 9 counters.
What do you think Julia's response will be and how do you expect her to work out the answer?
Click here to see Julia's response:
Click here to see comments on Julia's response
In task 3, Julia counts 15 colours in three cards. After the exercise is repeated with four cards, Julia is asked to imagine a fifth identical card with 5 colours beside the four cards and asked "how many colours will be there altogether?"
How do you expect Julia to tackle the problem?
What do you think her response will be?
Click here to see Julia's response:
Click here to see comments on Julia's response
Julia is given 12 counters and 3 plates. She is asked to place the same number of counters on each plate.
She first places 5 on the first plate and 2 on the second. Goes to the third plate realises something and goes back and takes 2 off the first plate and places on the third. Then places one each on each of the plates to get 4 in each plate.
Then she is asked how many altogether?
What do you think Julia's response will be and how do you expect her to arrive at that answer?
Click here to see Julia's response:
Click here to see comments on Julia's response
Julia is shown one blue block with 4 lego people on it. She is asked to imagine two more blue blocks with 4 lego people on each. "How many lego people altogether?"
What do you think Julia's response will be and how do you expect her to arrive at that answer?
Click here to see Julia's response:
Click here to see comments on Julia's response
Julia can do more than we have seen
Teachers should observe children as they perform various tasks and keep in mind that such observations may not always be uniform as children respond differently in different situations. Predictions made on such observations may consequently be wrong at times. For example Julia has consistently shown a certain level of achievement and we have come to expect certain responses from Julia in a given situation. However here are two tasks in which Julia demonstrates that she can actually do more than we have seen.
Task 4: Counting-on from larger
This is the initial assessment task of counting on from the larger number.
Julia is shown 3 yellow counters on the table and then 9 more counters concealed under a card. "When asked how many counter altogether?", she promptly replies "12".
Click here to see the task:
Note that on further questioning on how she worked that out, Julia's response is:
"9" (pointing to the concealing card), "then worked out 10, 11, 12".
In this task Julia seems to count on from the larger, which is a strategy she has not shown in the other tasks where she counts all.
Task 5: Ten as a unit (Intermediate Concept)
This is the initial assessment task for base ten strategies which tells the teacher that she is ready for equal grouping.
Julia is shown 4 dots on a green card and asked "how many dots?"
4 is the immediate response.
Another green card with 10 dots is added on and Julia promptly goes "14"
"How did you work that out?"
"I count 10 and then counted on to make 14".
One more card with 10 dots is placed beside the other two. When asked how many, Julia goes:" 10 and then 10 and then 4 make 24"
She is then asked to imagine another green card beside these three cards and asked how many dots there will be. Julia's response is:
"10, 20, 30, 34".
She is then asked to imagine one more green card with 10 dots and asked how many dots there will be.
And Julia goes " 10,10,pause, 20, 30, 40, 44"
Click here to see the task:
In this task again Julia counts in tens first and then counts on to add 4 rather than counting from 14 to 24 etc. Note the way she uses ten as a counting unit .
What other strategy do you think would have been more sophisticated?
What can we conclude?
Julia does not seem to count on from the larger number in most instances because the tasks are all groups and not addition, but makes an exception in Task 4.
She generally goes 2, 2, 2,Š.. rather than 2, 4, 6, Š..
When she does go 3 and 3 make 6 it could be that Julia recognises a pattern and knows the doubling and halving strategy. She clearly knows her addition doubles.
She displays knowledge of 5 and 5 make 10 and is able to count in 10's in Task 5. However in Task 3 she makes no attempt to count in 5's or even 10's, but counts in ones all the way to 25. Why is this so?
Julia is using different strategies because of the tasks and she doesn't have the skill of skip counting. Or she might be noticing a pattern in some tasks, which she fails to notice in others. Julia is really in transition between levels. Most of the time it is hard to say which level she is at. What is important is what she can do and how we can promote her to more efficient strategies.
In any case this case study does show us that close observation of children's strategies does give us insight into their thinking and the ability to reasonably predict their approach to other tasks. However it also shows us that such predictions will not always be accurate as children respond differently in varying situations and may develop benchmarks which makes certain tasks unexpectedly easy for them.