The graphs and figures that illustrate the derivation of a hierarchical cluster profile based on the Nine Nemp A Sample Tasks follow. [4]

The initial cluster solution is illustrated below in Figure 6 shaded at the 7 cluster solution.

[4] Note that in the following analysis – the collections of math facts and algorithm tasks were removed as independent validation aids (and because they threatened to produce a over-weighting of this particular “school mathematics” in the analysis).

The heirarchical cluster solution was based upon the flexible cluster procedure with beta set to .25. No cases were removed from the analysis but they appeared in the binary variable set for the analysis with zero scores for all responses. The scree slope of the pseudo t statistic which led to the decision to utilise the seven cluster solution is illustrated in Figure 7 below. The flattening out of the pseudo-t line past the 7 cluster point may be because the amalgamations between clusters past that point do not represent joinings between profoundly different clusters.

(Click to enlarge)

Some support for the seven cluster solution is found in the visual depiction of cluster solutions in Figure 8 below. While the colour density indicates quite a dispersed set of objects (students), it can be seen that in general there tends to be a slightly higher colour density within clusters than between clusters.

The structure evident in the truncated tree in Figure 9 emphasizes the difference between this cluster and the other four.

In terms of validation of the non-randomness of the cluster solution at 7 clusters, CLUSTAN GRAPHICS allowed the comparison of the series of fusion coefficients (distances) of the actual sample compared to a series of Bootstrap-derived samples. The results of this comparison is detailed in Figure 10 below. The largest deviations of the Bootstrap derived samples from the other samples between 3 and 7 clusters. . Based upon these results, the decision to utilise the 7 cluster solution in further analysis has justification.

Cluster group 1 showed a good grip on all tasks:

• High competency in everyday money ratio task. (Based for many on computation strategy)
• Very high competence on digit task testing knowledge of place value. On whole numbers it was virtually maximal. There was a slight drop off in dealing with decimals and for about 20% some difficulty in obtaining the lowest valued decimal digit to two significant figures.
• Estimation of the lengths of objects was the best of all the groups (around 50% on whole). Their competency at measurement was good (best of all groups).
• The group was the only group in the moptorway task to show a preference for estimation over algorithm.
• This group was competent at marking the 0-1 number line for all items (only group that this was so for).
• Showed greatest statistical knowledge of all groups (both at the relative probabilities and the computational level).
• It was the only group to have a sizeable portion of responses correct in estimation of Wallies (suggesting a working estimation strategy).
• Was the greatest identifier of a conventional algorithm or first tens then units approach on strategies for two digit addend addition.
• Was the most likely to use non-algorithmic sophisticated compensations in subtraction.

• High competency in everyday money ratio task. Used estimation as preferred strategy in deciding change and so was correct on 25% of time. Half were correct in ascertaining the cost per kilo.
• High competence on digit task testing knowledge of place value. There was a slight drop off in dealing with decimals. Only about 50% could give a place-valued read-back when decimals were involved.
• Estimation of the lengths of objects was moderately accurate. Their competency at measurement was good (except for height).
• Low level of accuracy at the motorway task favouring the use of algorithms in solution attempts.
  This group had difficulty in marking decimals and fractions on the 0-1 number line except for 1/10.
• On the statistical task showed competence at relative probabilities only.
• On the Wallies task showed the pattern typical of most groups of a slight underestimation (but some in the group were also inclined to a large over-estimation).
• Was the group most likely to not identify a way of addition.
• Was the least likely to identify a strategy in the subtraction. However when using concrete objects were least likely to use a mixed strategies of counting and removing a partition of the collection. (More likely than other groups to use the strategy of count to the initial amount and then counting out the second amount.)

• High competency in everyday money ratio task. Used estimation as preferred strategy in deciding change and so was correct on 25% of time. Half were correct in ascertaining the cost per kilo.
• High competence on digit task testing knowledge of place value. There was a slight drop off in dealing with decimals. There was a considerable drop off compared with Cluster 1 and Cluster 2 in dealing with the problem of obtaining the lowest valued decimal digit to two significant figures. Only about 50% could give a place-valued read-back when decimals were involved
• Estimation of the lengths of objects was less accurate than that for Clusters 1, 2,4. Their competency at measurement was good (especially for height)
• Low level of accuracy at the motorway task favouring the use of algorithms in solution attempts.
• This group had profound difficulty difficulty in marking the number line except for the 50% and 100% questions.
• On the statistical task showed second highest competence, They succeeded at the level of relative probabilities about a quarter of this group could calculate chance of drawing particular classes.
• On the Wallies task showed the pattern typical of most groups of a slight underestimation (but some in the group were also inclined to a large over-estimation).
• Was most likely to use a conventional algorithm or first tens then units approach as strategies for two digit addend addition.
• Was the least inclined to work through their identified strategy in subtraction. However when using concrete objects were likely to use a mixed strategy of removing a partition of the initial amount, and then counting out the amount being subtracted.

• High competency in everyday money ratio task. Used estimation as preferred strategy in deciding change and so was correct on 25% of time. Half were correct in ascertaining the cost per kilo.
• High competence on digit task testing knowledge of place value. There was a slight drop off in dealing with two place decimals.
• Estimation of the lengths of objects was moderately accurate. Their competency at measurement was good (except for height).
• Low level of accuracy at the motorway task favouring the use of algorithms in solution attempts.
• This group had difficulty in marking the 0-1 number line for all stipulated points.
• On statistical task showed competence at relative probabilities and about 20% of this group could estimate chances.
• On the Wallies task showed the pattern typical of most groups of a slight underestimation (but some in the group were also inclined to a n over-estimation).
• Was likely to not identify a way of addition.
• Was not likely to identify a strategy in the subtraction. They were the group who mentioned concrete materials most. However when using concrete objects were least likely to use the objects. Their preference was for removing a partition of the initial amount, and then counting out the amount being subtracted.

• Low competency in everyday money task.
• Reasonably competent on ordering whole number digits. The performance on all tasks was poor especially interpreting decimal place values..
• Estimation of the lengths of objects was poor. Was the only group showing a profound lack of competency at measurement of actual lengths.
• Very low level of accuracy at the motorway task favouring the use of algorithms in solution attempts.
• This group had difficulty in marking the 0-1 number line for all stipulated points.
• On statistical task showed some competence at relative probabilities only.
• On the Wallies task showed the pattern typical of most groups of a slight underestimation (but some in the group were also inclined to an over-estimation).
• Wasnot likely to not identify a strategy for addition.
• Was not likely to identify a strategy in the subtraction. When a strategy was mentioned the conventional algorithm or counting up was preferred.

• High competency in everyday money ratio task. Used estimation as preferred strategy in deciding change and so was correct on 20% of time. Less than a third were correct in ascertaining the cost per kilo.
• Reasonably competent on ordering number digits to obtain largest or smallest quantities. The performance on all tasks involving read back multidigit numbers was poor including interpreting decimal place values..
• Estimation of the lengths of objects was moderate. Their competency at measurement was good (but less so for height).
• Very low level of accuracy at the motorway task favouring the use of algorithms in solution attempts (if a strategy afforded).
• This group had difficulty in marking the 0-1 number line for all stipulated points except for 100%.
• On statistical task showed some competence at relative probabilities only.
• On the Wallies task showed the pattern typical of most groups of a slight underestimation (but some in the group were also inclined to an over-estimation).
• Was most likely to not identify a strategy the conventional algorithm or first units then tens when adding.
• Was not likely to identify a strategy in the subtraction. When a strategy was mentioned the conventional algorithm or counting up was preferred. Using concrete objects like Group 2 they had a lower likelihood of using a mixed strategy of counting and removing a partition of the collection. They would also use the strategy of counting to the initial amount and then counting out the second amount.

• Students from low SES decile schools were proportionately less likely than students from high SES decile schools to be found in the best performing Cluster 1 and more likely to be in the less well performing Clusters 5 and 6.
• Mäori students were proportionately less likely than European students to be found in the best performing Cluster 1 and more likely to be in the less well performing Clusters 5 and 6.
• Students from larger schools (schools in larger centers) were proportionately more likely than students from smaller schools to be located in the best performing Cluster 1 and less likely to be in the less well performing Clusters 5 and 6.
• Students from intermediate schools were proportionately more likely than students from full primary schools to be located in the best performing Cluster 1 and less likely to be in the less well performing Clusters 5 and 6.

• What maths activities do you like at school? Compared to other groups, Clusters 5 and 6 having difficulty with mathematics were far more inclined to say that they enjoyed working in their textbook or doing worksheets than the other groups.
• What maths activities do you like at school? Compared to other groups, Clusters 5 and 6 were likely to place less importance on knowing mathematics facts and more importance on developing good classroom behaviours.
• What are the interesting maths you do in own time? Compared to other groups, Clusters 5 and 6 were likely to mention basic facts and tables and they were less likely to mention solving problems or life skills mathematics.
• What do you do with really hard things? Compared to other groups, Clusters 5 and 6 were far more likely to ask a teacher and less likely to persevere. Those students in the best performing Cluster 1 were most likely to indicate that they would persevere.