Australasian Journal of Early Childhood

■■a 40-minute, one-on-one interview■■extensive professional development.The impetus for the ENRP was a desire to improvechildren’s mathematics learning. To quantify suchimprovement a framework of pivotal growth points inmathematics learning, also described as key steppingstones along the path to mathematical understanding,was developed in nine domains including the threemeasurement domains of Time, Length, and Mass.The one-on-one interview consisted of 61 assessmenttasks created to match the growth points. Within theENRP, teachers interviewed each child in their class atthe beginning and end of the school year; children wereposed questions in each domain up to the point wherethey encountered difficulty. The teacher would thenmove onto the next domain, or sometimes to a detour.Relevant elements of length-learning andassociated interview itemsFour growth points from the ENRP form the basis ofthe data presented below. For each growth point therewas an interview item, each using particular equipmentsuch as string, paperclips and a ruler. The followingdiscussion presents the respective growth points, theinterview item, and consideration of associated issuesthat are inferred from the students’ responses to theitems. The interview items are shown as they werepresented to the teachers, with italics indicating whatthe teachers were to do, and the normal text indicatingwhat they were to say.The comparing lengths growth pointThe first of the ENRP growth points relevant for thediscussion is:■■Compares, orders, and matches objects by length.The interview item was as follows:The string and the stickDrop the string and the skewer ontothe table (See diagram at right).a) By just looking (withouttouching), which is longer: thestring or the stick?b) How could you check? (touchingis fine now)The string and the stickc) So … which is longer?Following the work of Piaget and his colleagues(Piaget et al., 1960), this item explicitly addressesmeasurement by direct comparison. We took asuccessful response to suggest that the student wasconserving length, even though we did not specificallyassess understanding of conservation nor developa growth point for this. Further, we did not includean item assessing transitivity, although clearly this isrelated to indirect comparison and forms part of thestudent’s learning in the early years.The quantifying lengths growth pointThe second of the ENRP growth points relevant for thediscussion is:■■Uses uniform units appropriately, assigningnumber and unit to the measure.The interview item was as follows:The straw and the paperclipsGet the straw and show the child the long paperclips.Here are some paperclips. Here is a straw.a) Measure how long the straw is with thepaperclips. (If child hesitates) Use the paperclipsto measure the straw.b) What did you find? (No prompting)If correct number is given (e.g. 4), but no units, ask‘4 what?’This item infers the use of non-standard but consistentunits to quantify a length measurement. This includesthe idea of iteration where the unit is seen as part ofthe whole and the unit is translated successively. Asindicated in the interview item detailed above, weprovided the children with multiple copies of a unit (8paperclips) and did not assume the stance of Kamii(2006), that unit iteration is demonstrated throughthe use of one unit, such as one block, being usedrepeatedly. However, our approach did allow us to seethe children who could use informal units and attend tomeasurement principles of iteration, tiling, identificationof unit and number assignment, as compared to thosewho had not moved beyond comparing. For example,those children who laid out all the paperclips and saidthey were longer than the straw, or who laid just onepaperclip next to the straw and said the straw waslonger, were judged as not yet quantifying length.We see iteration using a single unit repeatedly as anassociated idea, and one which can form part of thelearning experience for the students at this level.The using standard units growth pointThe third of the ENRP growth points relevant for thediscussion is:■■Uses standard units for estimating and measuringlength, with accuracy.The interview item was as follows:30A u s t r a l a s i a n J o u r n a l o f E a r l y C h i l d h o o d