“Already
the place of the Practice Standards in the classroom is being undermined by
superficial approaches that boil down to ‘we are doing all the Practices all the
time.’ If these Practices are happening ‘all the time,’ the result will be that
none of them are happening with any attention or depth. If they are only listed
on the wall, they will soon be treated like wallpaper and ignored.” (Russell,
2012, p. 52).
It has been my experience talking
about SMP's at EdCamp, most people come to the session and say,
"What are the SMP's?" Excitedly, I always try to explain using
concrete examples, because, let's face it, there isn't a whole lot out
there.
Here is each SMP, a quick
definition, and a specific example of an instructional shift around that
standard. I define an instructional shift as something happening in classrooms
now that wasn't happening (to my knowledge) ten years ago when we were
accountable to the 1997 standards. (* = favorite standard. It's good to have
favorites.)
MP.1. Make sense of problems and persevere in solving them.
Definition- students
find ways to solve problems by making a plan, rather than just giving an
answer. They are able to use the knowledge and skills from similar
problems to persevere. Part of the perseverance piece is for
students to have the vision to “change course” in solving the problem if
necessary. Students are able to check their answers to make sure it
is an appropriate answer. They are also able to understand varying
approaches to the problem by other students as well. Instructional
Shift- The implication for teaching is that this standard cannot
be demonstrated by a multiple-choice test. The shift is in assessment- make a
new one.
MP.2. Reason abstractly and quantitatively.
Definition- students
demonstrate an understanding of the numerical data in the problems and the
relationship of that data to the situation posed by the problem. Students
should be able to represent the problem with appropriate symbols. This
includes students understanding what the question is asking in its
context.
Instructional
Shift- students think
about problems in ways to make sure the answers are reasonable. For
example, if there were seven children going on a field trip and each car holds
a maximum of two children, students should answer that four cars would be
needed to take the students. The contextualization of the problem takes place
in understanding the remainder of dividing seven children into cars. In
the above-mentioned problem, it is unreasonable for the answer to be 3.5 cars. A
student would have to reason that it is impossible in that situation to have
half of a car.
*MP.3. Construct viable arguments and critique the reasoning of others.
Definition- students prove their mathematical
statements are true. Students support the truth of their statements with
examples, counterexamples, and non-examples. They communicate their
findings to others and respond to the arguments of others. They are able
to choose the correct reasoning that supports the correct answer and explain why
other methods are incorrect. Students are able to ask useful questions to
understand the arguments of others.
Instructional Shift- Gallery walks via a re-engagement
lesson. Students represent their thinking and arguments on posters that
are displayed around the room. The gallery walk takes place when students
walk around the classroom and look at each other’s thinking as it is displayed
on the posters. Students have post-its and they leave questions and
comments about the mathematical thinking on the posters. Students then
have the opportunity to respond to the questions and comments on the
post-its. This gives students the chance to explain, defend and even
change their thinking.
MP.4. Model with mathematics.
Definition- students are able to solve
real world math problems that are age appropriate. Additionally, students
make assumptions and approximations, while realizing they may need to revise
their work later. Students represent their thinking, as well as the data
from the problem with diagrams, tables, graphs, formulas and flowcharts.
Instructional Shift- curriculum.
Past practice has given students the opportunity, regardless of grade level, to
work with numbers without any real world application or connection. Much
of mathematics as a subject has been focused on the memorization of algorithms
and getting one right answer. For the most part, elementary math has not
consisted of problems related to students’ lives where they make their own
surveys, interpret and analyze the results to their surveys, and display the
results in an organized way. In evaluating elementary math curricula,
almost all of the problems are low in cognitive demand. If students are to
begin age appropriate modeling with mathematics, the tasks in which students
engage in can only be high cognitive demand tasks. The responsibility of
having students actively engage in real world problems rests with
administrators and teachers.
MP.5. Use appropriate tools strategically.
Definition- students are able to decide what
mathematical tool is appropriate for solving a given problem. The
following is a list of some tools that a student might consider: pencil and
paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a
computer algebra system, a statistical package, or dynamic geometry software.
Instructional Shift- deciding
when to allow students to choose to work with a calculator. Before CCCS-M,
I did not let sixth grade students use a calculator until the third
trimester. After the CCCS-M were adopted, I allowed calculators in
the room the entire school year. My practice shifted because of
MP.5. If students were not given access to appropriate tools for the
entire year, how would they become proficient at selecting them? I didn’t
think they would, so I changed my practice around calculators in the classroom.
*MP.6. Attend to precision.
Definition- students communicate precisely to
others. Students are able to explain why they chose specific labels
and symbols, as well as using the equal sign consistently and appropriately. Students
calculate answers accurately and precisely within the given context of the
given problem.
Instructional Shift- text messaging. Attend to
precision always makes me think about text messaging. If I am not precise
in my text message to you, problems can arise. For example, if I was
sending a group text to my faculty members about a faculty meeting and sent out
“BYOB” instead of “BYOD,” the implications and ramifications of the mistake
because of one letter would be highly embarrassing. Since my text lacked
precision, my staff would think I was saying to ‘bring your own beer’
to the meeting, instead of the intended message of ‘bring your own device.’ It
is similar in math. For example, when dealing with percents, decimals and
fractions. Let's say if a basketball player makes one third of her
free throws, then she makes thirty-three percent of them. However, in a
different mathematical situation dealing with the fraction one third, the
repeating three tenths may not be able to be overlooked.
MP.7. Look for and make use of structure.
Definition- students look closely to discern a
pattern or structure within the context of a problem. This is where the skills
to think about numbers mentally and decompose them become crucial.
Instructional Shift- number talks. An example of a
number talk would be when students compute mentally the following problem:
thirty-seven plus forty-eight. Students can decompose the number in many
ways to find the sum. Students use the structure of place value to break
apart or decompose the numbers. Thirty-seven may become three tens and
seven ones. Forty-eight may become four tens and eight ones. Three
tens and four tens would be seven tens. Seven ones and eight ones would be
one ten and five ones. Seven tens plus one ten plus five ones would be the
sum of the digits totaling eighty-five.
MP.8. Look for and express regularity in repeated reasoning.
Definition- students notice if calculations
are repeated, and look for general methods and shortcuts. Mathematically
proficient students see repetition in division when working with repeating
decimals. Students not only notice a pattern, but also are able to
understand what meaning that pattern has within the context of the
problem.
Instructional
Shift- a different lesson around long division and what it means
when one divided by three equals 0.333333...333 and why it keeps occurring
forever. “Another way of looking at it: 0.3333.... means .3+ .03+ .003+
.0003+...= 3∑
∞n=1.1nand an ‘infinite sum’ is defined as the limit of the partial sums, NOT the
partial sums themselves, which is what you are doing in talking about
continuing to ‘add 3s.’” http://www.physicsforums.com
