Aligning Mathematics Instructional Practices
This post is part of our blog series on PARCC/NJSLA. In this series, we offer tips and strategies you can use to ensure that your students perform at their very best on the PARCC/NJSLA tests.
Regardless of how you feel about PARCC, NJSLA, or any standardized test, I think we can all agree that 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/NJSLA is a waste of instruction time. PARCC/NJSLA 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/NJSLA 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/NJSLA 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/NJSLA assessments require students to:
- Solve problems using mathematical reasoning
- Be able to model mathematical principles
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:
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:
The PARCC/NJSLA 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/NJSLA 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/NJSLA. Please let us know if we can help you make your classroom or school more fully aligned with the Common Core and PARCC/NJSLA.
Inspired Instruction offers hundreds of PARCC/NJSLA lesson plans, online PARCC/NJSLA-like assessments with technology-enhanced items, PARCC/NJSLA workshops, and PARCC/NJSLA demonstration lessons.
Please contact Michele Regan for more information: Michele.Regan@inspiredinstruction.com or 908-223-7202
Inspired Instruction and Standards Solution Holding, LLC are not owned by or affiliated in any fashion with PARCC, Inc.