Preparing
for PARCC
Aligning
Mathematics Instructional Practices
This post is part of our blog
series on PARCC. In this series, we offer tips and strategies you can use to
ensure that your students perform at their very best on the PARCC tests.
Regardless of how you feel about PARCC, or any standardized test, I think that in PARCC states we can all agree, at this moment it is necessary to prepare our students for the experience. And to be honest, I don’t think that preparing for PARCC is a waste of instruction time. PARCC is a test that evaluates students’ progress toward college and career readiness. It is a test of our students’ competence regarding the Common Core State Standards. Therefore, when we are preparing students for PARCC we are applying and practicing the Common Core. That is what we are supposed to do.
But what does a fully aligned mathematics
classroom look like?
“The PARCC assessments
are aligned to the Common Core State Standards (CCSS) and were created to
measure students' ability to apply their knowledge of concepts rather than
memorizing facts.” (NJDOE)
The mathematics
PARCC assessments require students to:
- Solve problems using mathematical reasoning
- Be able to model mathematical principles
Mathematical Reasoning
What Is Mathematical Reasoning?
According to G.W. Martin, et al., “Reasoning can be thought
of as the process of drawing conclusions on the basis of evidence or stated
assumptions…Sense making can be defined as developing an understanding of a
situation, context, or concept by connecting it with existing knowledge.” (Martin,
G.W. and Kasmer, L. “Reasoning and Sense” Mathematics
Teacher Dec. 2009/Jan. 2010).
The
ability to reason
is essential to understanding mathematics. Teachers should use effective
questioning techniques to promote their students’ reasoning abilities. Students
need opportunities to respond to effective questions that require critical
thinking, and to share ideas and clarify their understanding. When students are
able to connect mathematical ideas, they develop a deeper and lasting
understanding of mathematics. The process of reasoning has three stages: conjecture,
generalization, and justification.
The Process of Reasoning
- Conjecturing: developing statements that are tentatively thought to be true but are not known to be true
- Generalizing: extending the reasoning beyond the scope of the original problem
- Justification: a logical argument based on already-understood ideas
Types of Reasoning Tasks
- Proof and Justification Tasks: Students are asked to use reasoning to provide an argument for why a proposition is true or is not true.
- Example: the student draws a comparison between two fractions and provides proof that the comparison is true, using a mathematical model.
- Critiquing Tasks: Flawed reasoning is presented and students are asked to correct and improve it. Example: the student reviews an answer created by a fictitious student and must identify and explain possible flaw(s) in the reasoning, correct the answer, and provide an explanation supporting the correct reasoning and answer.
- Mathematical Investigations: Students are presented with a problem and invited to formulate conjectures and prove one of their conjectures.
- Example: the student tests an idea, such as, “Is it always true that when two fractions are multiplied, the product is less than the two fractions?”
Modeling in Mathematics
Concrete
models and pictorial models can be used to demonstrate the meaning of a
mathematical idea and/or communicate the application of mathematics to solve a
real-world problem.
“Students
can develop a conceptual understanding of mathematics through modeling,
following a progression of representations: concrete, pictorial, and abstract.”
(Strategies for Successful Learning, Vol. 6, No. 2, January 2013)
Concrete
representation is often demonstrated with manipulatives. Pictorial
representation can be various drawings, such as graphs, number lines, object
drawings, Ten Frames, and visual fraction models. Abstract representation is
the use of numbers, letters and symbols to represent the mathematics.
Consider
these examples of the three types of representation:
“There
are three times as many cats as dogs; there are 15 dogs. How many cats are
there?”
In the
Common Core State Standards, each grade level addresses distinct operations and
number relationships.
Here is a list of the distinct operations and/or number
relationships for grades 2 through 6:
- Grade 2: addition and subtraction
- Grade 3: multiplication and division
- Grade 4–6: fractions and ratios
The operations and number relationships are
developed sequentially, to allow students to visualize and solve increasingly
complex problems. Solving for an unknown quantity at the concrete
and pictorial stages aids in the transition to the abstract.
Mathematical
Methods and Representations within the Standards
Many of
the Common Core Standards for Mathematics are very specific about which methods
and representations need to be used to develop understanding of the
mathematical concept(s).
To
demonstrate this, let’s examine a grade 4 Standard:
4.NBT.5
Multiply a whole number of up to four digits by a one-digit whole number, and
multiply two two-digit numbers, using strategies based on place value and the
properties of operations. Illustrate and explain the calculation by using
equations, rectangular arrays, and/or area models.
The main concept is multi-digit multiplication; the specific digits are
provided. The methods are place
value strategies and properties of operations (commutative, associative, distributive).
The representations are equations,
rectangular arrays, and area models. The standard states the specificity
clearly; the expectation is that classroom instruction would include the specificity
as stated. What could this look like?
An example of an equation that
demonstrates place value and properties of operations:
3 x 27 = 3 (20 + 7)
A rectangular array can be
demonstrated using a manipulative, such as tiles or base-ten blocks, with a
place-value mat.
An example of an array model:
Conclusion:
The PARCC assessment is closely
aligned to the Common Core State Standards. When considering classroom
instruction and the students’ demonstration of understanding, the specificity
of the Standards cannot be ignored. Since the students are expected to reason
mathematically and use modeling to represent mathematics on the PARCC
assessment, they need opportunities to communicate reasoning and provide
modeling in classroom tasks.
It is our
sincerest wish that you find value in these ideas and resources and begin to
integrate the concepts that students will experience on PARCC. Please let us
know if we can help you make your classroom or school more fully aligned with
the Common Core and PARCC.
Standards
Solution and Inspired Instruction offers 540 PARCC lesson plans, online
PARCC-like assessments with technology-enhanced items, PARCC workshops, and
PARCC demonstration lessons. Please contact Judy Cataldi for more information:
Judy.cataldi@standardssolution.com or call 908-223-7202
