| Design Principles
Our goal is to design lessons that support the learning of students
with a wide range of understanding. In heterogeneous LMR classrooms,
a teacher’s role is to foster mathematical thinking and facilitate
the ’travel of ideas’ through whole class and small group
discussions. There are 5 principles that guide our lesson design.
- Lessons should target core mathematical
ideas. Core mathematical ideas for integers on the number line, for
example, include:
- Integers can be represented on the number
line as magnitudes. On the number line, magnitude is the measure of
the distance between two points.
- Positive integers are magnitudes from
0 to the right of 0 on the number line. Negative integers are magnitudes
from 0 to the left of 0 on the number line.
- Integers can be represented as vectors
on the number line, with the properties of magnitude and direction.
- Lesson should be organized as a mathematically
coherent lesson series. Prior lessons provide a foothold into the
big ideas introduced in subsequent lesson, and subsequent lessons
provide new insights into the ideas that were the focus of prior lessons.
- Lessons should engage all students in
reflection on core mathematical ideas. One way to encourage reflection
is to ask students to choose the best of several solutions to a challenging
problem; this strategy will be most effective if students choose from
typical solutions offered by students. As teachers ask students to
consider the thinking of their peers, conceptual challenges emerge
that students need to confront and resolve. Teachers can facilitate
the discussion in a way that encourages students to apply knowledge
gained from prior lessons.
- Lesson design should build on student
thinking within and across lessons. Because it can be challenging
for teachers to interpret student thinking while managing a complex
lesson, we provide teachers a framework for interpreting student thinking
in each LMR lesson based on the findings from our research.5. Lessons
should be feasible to implement, and one way to ensure feasibility
is a predictable lesson structure. As we describe below, LMR lessons
often begin with a Problem of the Day that focuses on core mathematical
ideas and elicits variations in student thinking; what follows is
a structured sequence of whole class and small group discussions that
support reflection, listening, and building mathematical consensus.
Each LMR lesson also builds on prior lessons in explicit ways, so
no lesson feels like a unique event.
As shown in the figure above, LMR lessons begin with a ’Problem
of the Day’ (PoD). This problem features a non-routine treatment
of the number line featuring unequal intervals. First students work
on the PoD, followed by teacher-guided whole class and small group
discussions focused on students’ differing solutions. In the
figure, the PoD is to identify the number indicated on a number line
partitioned into unequal intervals. Each lesson has a 5-phase structure,
and at each phase, the PoD has a central and organizing role.
Phase 1. Students work independently to
choose one of five solutions to the problem of the day and justify
their choice in writing. This phase provides time for students to
reflect as they construct their solution, and the student work provides
the teacher an initial assessment of students’ mathematical
thinking.
Phase 2. Several students present their
solutions and justifications. This phase allows the teacher to assess
the variation in student thinking in the class.
Phase 3. The teacher organizes students
in small work groups. Students present their thinking to one another
and try to reach consensus on a single solution and justification.
Phase 4. The class re-assembles, and the
teacher orchestrates discussion towards the correct solution with
an extended explanation.
Phase 5. The lesson concludes with a final
assessment consisting of extension problems. The teacher uses students’
responses to evaluate the lesson’s effectiveness.
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