From Concrete to Abstract: Teaching for Transfer of Learning when Using Manipulatives

One of the most important uses of manipulatives in a classroom is to aid a learner to make connection from tangible concrete object to its abstraction. In this paper we discuss how teacher educators can foster deeper understanding of how manipulatives facilitate student learning of math concepts by emphasizing the connection between concrete objects and math symbolization with, preservice elementary teachers, the future implementers of knowledge. We provide an example and a model, with specific steps of how teacher educators can effectively demonstrate connections between concrete objects and abstract math concepts.

Nitrate leaching from Kentucky bluegrass soil columns predicted with anion exchange membranes

Ideal nitrogen (N) management for turfgrass supplies sufficient N for high-quality turf without increasing N leaching losses. A greenhouse study was conducted during two 27-week periods to determine if in situ anion exchange membranes (AEMs) could predict nitrate (NO3-N) leaching from a Kentucky bluegrass (Poa pratensis) turf grown on intact soil columns. Treatments consisted of 16 rates of N fertilizer application, from 0 to 98 kg N ha-1 mo-1. Percolate water was collected weekly and analysed for NO3-N. Mean flow-weighted NO3-N concentration and cumulative mass in percolate were exponentially related (pseudo-R2=0.995 and 0.994, respectively) to AEM desorbed soil NO3-N, with a percolate concentration below 10 mg NO3-N L-1 corresponding to an AEM soil NO3-N value of 2.9 micro g cm-2 d-1. Apparent N recovery by turf ranged from 28 to 40% of applied N, with a maximum corresponding to 4.7 micro g cm-2 d-1 AEM soil NO3-N. Turf colour, growth, and chlorophyll index increased with increasing AEM soil NO3-N, but these increases occurred at the expense of increases in NO3-N leaching losses. These results suggest that AEMs might serve as a tool for predicting NO3-N leaching losses from turf.