There is a
common thread with many students I have come in contact with over the last few
years, when teaching and discussing mathematics. Many students are: 1. not remembering math from year to year; 2. not
able to easily transfer math skills to other subjects, and 3. not able to
problem solve. As a teacher of mathematics and math coach, this is
troubling.
In this
microwave era of, “I want it now,” and “what’s taking so long,” many teachers of
mathematics have resorted to equipping students with a series of algorithms and
procedures to commit to memory. However,
as we are preparing our students to be college and career ready, long gone are
the days where mathematics equates to numbers only. To equip our students to compete locally and
globally, our focus must shift to helping students develop insight, versus only
procedural skills. Thus, the much
debatable topic of teaching for conceptual understanding and learning is worth
delving into. It is not surprising that when polled, many K-12 teachers are not
familiar with what teaching conceptually looks like, and how it is carried out
in the classroom. Many of us sat in classrooms where we watched a teacher go
through solving a problem, wrote down the “steps,” and then proceeded to practice
fluency by working another 25 to 50 problems. No wonder many parents have
labeled their child’s math homework as “new math!” This is understandable because
of the early focus in education to produce industry workers (i.e. assembly line
workers, welders, automobile workers). But what this teaching lacked was
equipping students to: make sense of problems and persevere in solving them, reason
abstractly as well as quantitatively, construct viable arguments and critique
the reasoning of others, model with mathematics, use appropriate tools
strategically, attend to precision, look for and make use of structure, and look
for and express regularity in repeated reasoning. Sound familiar? It should, because these are the Standards
for Mathematical Practices, the way in which we should deliver math content. Now,
our task as teachers is to prepare students for jobs and careers that may not
exist yet, and teaching through the lens of the Standards for Mathematical
Practice will ensure conceptual understanding. This knowledge takes them far
beyond procedures and fluency, arming them with the critical thinking and
problem solving skills needed to be successful today and in the days to come.
The Learning
Principle from the NCTM Principles and Standards for School Mathematics (2000)
is a good resource to gain an understanding of conceptual knowledge. This principle states: “Students must learn
mathematics with understanding, actively building new knowledge from experience
and prior knowledge.” This supports the fact that rote memorization is not the
key to high achievement in mathematics, especially if our students don’t understand
the math. However, I do not discount procedural
fluency in any way. But, there must be a
balance in order to improve student achievement.
So, how do
we correct this lack of conceptual understanding in the mathematics
classroom? Allowing students to model
concepts, use manipulatives in class, and express their findings in words will
help students gain an understanding of complex ideas. Give students opportunities to express and defend
their thinking, as well as receive constructive feedback from peers and the
teacher. This should not be limited to learning at the elementary level. Take time to incorporate the practice of estimating.
Also, giving students an opportunity to
express math concepts in multiple ways leads to conceptual understanding, in
essence, understanding the why before
practicing procedural fluency (the how).
Dan Meyer says it best in his TED talk, Math Class Needs a Makeover, and offers
examples on how to begin changing our delivery to best meet the needs of
students of mathematics.
After
sharing these ideas with my middle level colleagues, I was pleased to find that several
teachers have changed their practice to include the modeling of concepts with
an understanding of why/how it works, before delivery of procedural knowledge.
Also to my delight, while conferencing with a reflective teacher, she shared a
new practice she will incorporate to begin assessing conceptual understanding. By simply inserting a “think check” component into
her daily lesson, she will learn if students are taking away an understanding of why, and not only how. This will also remind her to plan for
conceptual understanding in her delivery, so students will be able to answer
such questions each day.
Here is an
image of an example:
Related Resources to aid in teaching mathematics for conceptual understanding:
3-Acts Math Tasks Inspired by Dan Meyer - Storytelling to provide a framework for certain
mathematical tasks that is both prescriptive enough to be useful and flexible enough to be usable.
One minute of video or one photo to tell the start of a mathematical story that
will engage learners in asking a question.
Mathalicious
-Real-world lessons help middle and high school teachers address Standards
while challenging their students to think critically about the world.
Conceptual Math.org - Promotes math as a tool for understanding yourself and the world around you.
Statistics
Education Web - Improve statistics
education at all levels, with relevant,
useful, and meaningful applications.
Works Sited:
National
Council of Teachers of Mathematics (2000). Principles
and Standards for School Mathematics. Reston, VA: NCTM.
