The Mathematics of Students
By David Wees, posted July 6,
There is a tremendous drive in the United States, and in many other
countries, to support and develop teachers’ use of formative assessment
practices in mathematics education. I work for a project called Accessing
Algebra Through Inquiry, in which one of our objectives is to support teachers’
formative assessment practices across 31 New York City high schools.
But what knowledge does one need to be able to implement regular,
responsive formative assessment in the classroom every day? Clearly, the better
one understands the mathematics one is teaching, the easier it is to teach. For
formative assessment practices, however, one also needs to understand the
typical ways that students understand mathematical ideas.
Science educators have known this about teaching science for a long
time. Check out this amazing and well-organized database of the
different ways students understand scientific concepts!
Unfortunately, there is no such database for mathematics teachers.
Until such a collection is created, each mathematics educator who wishes to
incorporate formative assessment practices has to begin to develop his or her own
understanding of the common ways that students conceive of and use mathematical
What does this formative assessment cycle look like in practice?
First, anticipate what you expect students to do in response to a chosen task.
Next, design a set of responses, perhaps by selecting a different task,
choosing what feedback you will give students, or deciding how you will
structure your classroom discussions. Once you implement your response in the
classroom, reflect on its effectiveness and then revise your anticipation and
response for the next formative assessment cycle.
Observing student work and looking for trends in how students represent
their thinking are incredibly valuable. Consequently, the strategy I suggested
in my second blog post should, over time, lead to a better understanding of students’
mathematics acumen. It should also lead to stronger formative assessment
Here’s an example from Elizabeth Green’s book, Building a Better Teacher, from the
knowledge and experience of Magdalene Lampert:
Take a moment and hypothesize why a student might think this is
If you have no idea, based on this written sample, don’t despair.
One excellent way to make sense of students’ ideas is to interview the actual
students. If you had asked this student, he or she might tell you that 12 divided
by 7 is 1, with a remainder of 5, so he or she wrote the answer as 1.5. In this
case, the student has interpreted the fraction in an unconventional way and
written the answer differently than perhaps expected. Knowing this information
makes it easier to plan your response.
The big idea from this series of blog posts is that making sense of
student work is a valuable process that can help inform your teaching and
support your developing understanding of student thinking. These qualitative
approaches shouldn’t stop you from finding quantitative ways to describe
student growth, but they can inform and expand your available information with
which to make decisions.
David Wees, firstname.lastname@example.org, is a Formative
Assessment Specialist for New Visions for Public Schools in New York. He tweets
at @davidwees and blogs at http://davidwees.com.