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.

Now You See it, Now You Don't: Why Do Real Estate Agents Withhold Available Houses from Black Customers?

Potential home buyers may initiate contact with a real estate agent by asking to see a particular advertised house. This paper asks whether an agent's response to such a request depends on the race of the potential buyer or on whether the house is located in an integrated neighborhood. We build on previous research about the causes of discrimination in housing by using data from fair housing audits, a matched-pair technique for comparing the treatment of equllay qualified black and white home buyers. However, we shift the focus from differences in the treatment of paired buyers to agent decisions concerning an individual housing unit using a sample of all houses seen during he 1989 Housing Discrimination study. We estimate a random effect, multinomial logit model to explain a real estate agent's joint decisions concerning whether to show each unit to a black auditor and to a white auditor. We find evidence that agents withhold houses in suburban, integrated neighborhoods from all customers (redlining), that agents' decisions to show houses in integrated neighborhoods are not the same for black and white customers (steering), and that the houses agents show are more likely to deviate from the initial request when the customeris black than when the customer is white. These deviations are consistent with the possibility that agents act upon the belief that some types of transactions are relatively unlikely for black customers (statistical discrimination).