Introductionxiiiindirect. For example, when teachers ask students to explain their answer, unless
the expectations have been established that an explanation includes elements
such as a sequence of steps and an accurate use of a representation, students, espe-
cially those without a solid mathematical foundation, cannot fulfill the request.
Being clear about expectations and goals helps all students, but explicit teaching
is particularly important for fragile learners.
Linking Assessment and Teaching
The essays in this section illustrate that assessment must be ongoing and must
inform planning, as opposed to being used only to measure learning at the end of
a unit of study. Assessing students who struggle involves finding out about their
strengths as well as their weaknesses, and planning accommodations accordingly.
Throughout this section, you will see evidence of teachers’ deep knowledge of ele-
mentary school mathematics content and how mathematical ideas develop. This
knowledge forms the basis for their teaching and assessment decisions. Although
finding time for ongoing assessment is difficult, because these teachers had spe-
cific goals in mind, they were able to do assessment in a manageable period of
time: taking notes as they observe children working in small groups, remember-
ing children’s comments during whole-group discussions, or meeting with stu-
dents for targeted one-on-one interviews.
Building Understanding Through Talk
In recent years, there has been an acknowledgment of the importance of talk in
elementary mathematics classrooms. According to the NCTM standards, mathe-
matics instruction should allow students to:
- organize and consolidate their mathematical thinking through communi-
cation - communicate their mathematical thinking coherently and clearly to peers,
teachers, and others - analyze and evaluate the mathematical thinking and strategies of others
- use the language of mathematics to express mathematical ideas precisely
(NCTM 2000)
Yet teachers find that including all their students in discussion is challenging. We
were often asked, “I want to include all students in class discussions, but some of
my students who struggle tune out during meetings. What can I do to make them
feel included?”