The 104 teachers in this study came from urban and suburban school districts in
the greater Los Angeles area. The average number of years teaching was 7.2 years
(SD = 6.1) ranging from one to 38 years. While 87 teachers had a general teaching
credential, only 15 were credentialed in mathematics. During an online survey of
their pedagogy, teachers were asked to explain the distributive property (Task 1),
use of the distributive property to add fractions (Task 2), and why denominators
stay the same when adding fractions with the same denominators (Task 3).
47.5.2 Associations Between Quality of Mathematics
and Language Features
There were significant associations between the quality of mathematics demon-
strated by the teachers and the language and discourse features of the explanations
they gave in their written responses to Task 1. Mathematics quality required
teachers to demonstrate accuracy, precision and appropriateness of the mathematics
content in their explanations (dichotomously coded present or not), as well as the
depth and type of knowledge being conveyed (scaled 1–3). Teachers with accuracy
and depth of knowledge of mathematical concepts always elaborated their expla-
nations by including two or more of the language and discourse features (di-
chotomously coded elaborated-unelaborated). These features were explicitly
providing a definition of one or more instances of the key terminology; sequencing
with discourse connectors (e.g.,then, as,finally); using real-life or analogous
examples that clarified the mathematical concepts; using mathematical examples
that explained how the problem would be worked out; and referencing multiple
modes of communication, such as how the teacher would use a manipulative,
graphic, etc. Those teachers showing conceptual knowledge (but not integrated with
procedural knowledge) used fewer mathematical examples in their explanations.
For Task 2, teachers with accuracy and depth of knowledge of mathematical
concepts included elaborated mathematics examples in their explanations. Task 3
had the largest number of significant associations with the inclusion of appropriate
examples or analogies related to the use of elaboration in mathematics examples.
There were also positive associations between the integration of procedural and
conceptual understanding and the provision of elaborated definitions, the use of
real-life examples, and the use of elaborated mathematics examples.
47.5.3 Intersection of the Quality of Mathematics
and Teacher Explanations
Table47.1shows the contingencies between high language/discourse quality (i.e.,
two or more elaborated features) and high mathematics quality (i.e., highest possible
ratings).
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