DI Math – Unit 2
Differentiating Mathematics Instruction for All Students
Unit 21: Essential Elements of Mathematics – What’s It All About?
Unit Objectives
Welcome to Unit 2 of the Differentiating Mathematics Instruction for All Students module.
This unit will explore the processes of developing math competence and proficiency while also investigating the latest research on mathematics pedagogy and the importance of understanding the Concrete-Representational-Abstract (CRA) methodologies for instruction. Additionally, Florida’s Next Generation Sunshine State Standards and Access Points for mathematics will be reviewed as they relate to the National Council of Teachers of Mathematics (NCTM) recent publication Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics (2006). Finally, this unit will explore the barriers and misconceptions about students developing proficiency in mathematics.
As you read the sections of this unit, continue to reflect on the Responsive Decision-Making Framework below. As you reflect on the various methodologies, research, and Florida’s standards in mathematics, think about the students you teach, their background knowledge in mathematics, their interests and diverse cultures they bring into your classroom as well as the barriers that may exist in developing proficiency. Where do you, the teacher, currently fit in this model? Is your classroom setting about teaching all students at the same rate of progression? Or do you utilize various methods of instruction and assessment to pace students based on their readiness levels? While you further investigate the research behind effective mathematics instruction and the links to Florida’s Next Generation Sunshine State Standards and Access Points and Response to Intervention, refer to this framework as your resource for creating a differentiated classroom.
Click on the link below to hear how Ms. Matheson, whom you first met in Unit 1, is utilizing the information she has been learning about through professional learning opportunities, to better create and implement a differentiated classroom environment for her 6th grade mathematics students.
Ms. Matheson Reflection:
“With the recent research and articles I’ve been reading and the professional learning opportunities in which I’ve participated, I’ve made some self-discoveries about myself and my classroom. I feel that when really taking into consideration my students opinions and feelings about mathematics, it has given me a better understanding of where to begin with my planning.
Of course, I need to know my students’ ability levels, but also knowing what interests them, excites them and even how they like to show me their talents has been such an eye-opening experience for me. This change in my attitude and how it’s reflected in the way I plan, has helped me to create not just “interesting and fun” activities, but lessons that really want to make my students engage in the learning – lessons that are meaningful because the information is presented with more clarity and is used through real-world experiences to which my students can relate.
I want to make sure my lessons are covering the foundations that are necessary for them to achieve success with the standards, but through discovering my students’ interests and talents, I’m hoping to create an environment where they can trust me not only as their math teacher, but their mentor, their guide, and their support system. I’ve even noticed some of the more advanced students taking on roles to help those who are struggling. They know what the goals of the lessons are and can then work in groups to help one another achieve those learning goals, those “Do’s” from my K-U-D chart.
What’s really been helpful as well is reflecting on how my students’ conceptual understanding of the benchmarks is driving how I further plan. The types of assessments I use, not just data from the FCAT, has really been an “a-ha” for me. You’ll see a little later on in Unit 3 how I implement various pre-assessments and analyze the items and errors. This method of item-error analysis puts a whole new spin on CRA instruction.
I’ll catch up with you again in Unit 3. This unit has some really important information about the foundations for teaching mathematics. I know you’ll enjoy the research and activities as much as I did.”
End of screen page
Below is the KUD (your learning objectives) for this unit of the module.
To begin your learning activities for this unit, please click on “Learning Activities” or go to the “Learning Activities” menu item on the left.
(FCIM: end of screen page & Unit 2 Introduction)
Differentiating Mathematics Instruction for All Students
Unit 2: Section 1
Essential Elements of Mathematics –What’s It All About?
Developing Mathematics Competence
What is math? To say that mathematics is a study of numbers and patterns would not fully give the subject credit. It’s actually more complicated than that and there have been several definitions over the years as to what truly defines mathematics. Many would argue that math is a language, a way of expressing relationships between numbers, patterns, and situations.
So what is math competence? Several educators in the field have defined what students need to do to be successful or competent in mathematics. Some of these abilities include:
· Ability to recognize patterns
· Ability to sequence
· Estimate quantities with reasonable guesses
· Visualize and manipulate patterns
· Demonstrate mental mathematics
· Demonstrate fluency
· Demonstrate spatial awareness and organization
· Demonstrate deductive reasoning
· Demonstrate inductive reasoning
Students demonstrating difficulties in one or more of these areas may have difficulty mastering the necessary skills to perform higher-level thinking required to demonstrate proficiency in algebra, which is a required content strand for high school mathematics. Mathematics development can be compared to that of reading development regarding the foundational skills necessary for proficiency.
With reading, students achieving success with phonemic awareness, phonics, and decoding will have greater success with fluency, thus leading to comprehension success. Similarly, students who demonstrate number sense and computational skills have greater foundations for developing fluency and problem solving skills required for abstract learning seen with and necessary for algebraic thinking.
All students must have foundational competence across various mathematical principles in order to acquire functional skills, organize numeric problems, understand relationships between quantities, or participate with every day life activities, just to name a few.
Suggested reading:
Brown, C. (2009). A road map for mathematics achievement for all students: Findings from the national mathematics panel [Electronic version]. Learning Point Associates.
http://www.centeroninstruction.org/files/A%20Road%20Map%20for%20Math%20Achievement.pdf
National Council of Teachers of Mathematics (2006). Agenda for action: Measuring success [Electronic version]. Curriculum focal points for prekindergarten through grade 8 mathematics. Retrieved August 23, 2009 from
http://www.nctm.org/standards/content.aspx?id=17284
Differentiating Mathematics Instruction for All Students
Unit 2: Section 2
Essential Elements of Mathematics - What’s It All About
Concrete-Representational-Abstract Understanding
The concrete-representational-abstract (CRA) approach to mathematics, originally coined concrete-pictorial-abstract and based on the work of Jerome Bruner in the 1960s, is an instructional method where the teacher utilizes three stages for instructing mathematics. In addition it is commonly referred to as the process for which students learn mathematics.
Concrete Stage:
In the concrete “doing” stage, the teacher utilizes objects or manipulatives (base-ten blocks, tangrams, counters, coins, etc.) to move, manipulate and demonstrate the math. Students working with objects develop a better understanding of the skills, such as moving pattern blocks to create shapes or using real coins to count change. Students functioning at the “concrete level” may demonstrate skills such as needing to touch and maneuver things in order to “get it”, count on their fingers for even the simplest calculation tasks, or rely on teacher models using manipulatives to see the relationships within the activity. Using manipulatives is essential for building and developing conceptual understanding of mathematics.
Click on the links below from MathVIDS! to view teacher lessons utilizing manipulatives within a whole group lesson and within cooperative learning groups:
http://www.fcit.usf.edu/mathvids/videos/strategies/em/concrete/etmc_clip3_med.mov
David: How do I attach your slide show with audio that sets up the above video clip?
http://www.fcit.usf.edu/mathvids/videos/strategies/em/concrete/etmc_clip3_elab_med.mov
http://www.fcit.usf.edu/mathvids/videos/strategies/pda/pda_clip1_med.mov
Differentiating Mathematics Instruction for All Students
Unit 2: Section 2
Essential Elements of Mathematics – What’s It All About?
Concrete-Representational-Abstract Understanding
Representational Stage:
In the representational “seeing” stage, the teacher incorporates visuals, drawings, or pictures to “show” the math concepts. It’s especially beneficial for struggling math students to continue using manipulatives during this stage to develop a stronger grasp of concepts. Drawing the visual representation of colored Cuisenaire Rods to make fractional parts allows a student to see how the shorter pieces, when put together, can equal one longer piece. Students at the “representation stage” may benefit from always using a graphic organizer, visual model or drawing when lessons are presented. Incorporating picture drawing as a strategy during this stage helps students later in the “abstract” stage to become better problem solvers. Computer games representing manipulatives, such as those on the National Library of Virtual Manipulatives website, may also assist students at this stage of learning. Students with more significant math difficulties may benefit from the actual picture or photograph of the manipulative that is being referred. For instance, if a lesson requires the use of base-ten blocks, the student may still utilize base-ten blocks but also need a picture of the actual blocks being used for a graphic representation. This can also assist with the transference of skills from a hands-on-to-textbook lesson.
Click on the link below from MathVIDS! to view a teacher’s lesson utilizing visual representations within a whole group lesson:
http://www.fcit.usf.edu/mathvids/videos/plans/rnth/rounding3b_med.mov
Differentiating Mathematics Instruction for All Students
Unit 2: Section 2
Essential Elements of Mathematics – What’s It All About?
Concrete-Representational-Abstract Understanding
Abstract Stage:
Finally, in the abstract “symbolic” stage, the teacher uses numbers, symbols, or notations to teach and create the mathematical concept. For older students or those with better developed readiness skills, numbers are replaced with letters and signs and are more intangible (think: < >, =/=). Students at the “abstract stage” can decipher and explore equations using symbols or variables. Students do not need to rely on manipulatives or graphics for counting or producing results. Students with math difficulties most definitely benefit from continued use of manipulatives and/or visual representations during this stage in order to be able to connect the processes of mathematics (see Section 3). Some educators may argue that some students may never completely become “abstract thinkers” in mathematics, therefore, the use of manipulatives and visual representations should continue to be implemented during tasks requiring higher-order problem solving. This does not mean these students cannot have exposure to abstract problems or methodologies, but it does require the teacher to differentiate the lesson based on the student’s readiness.
Not all students move through the CRA stages at the same rate, however representation of numbers can be an essential approach for building foundational skills and conceptual understanding even for older students. The graphic below demonstrates how teachers may utilize the CRA approach or how students may move through the stages, where fluctuation may exist from lesson to lesson or benchmark to benchmark.
Click on the link below from MathVIDS! to view a teacher lesson utilizing the abstract approach (mental math) within a whole group lesson:
http://www.fcit.usf.edu/mathvids/videos/plans/rnth/rounding4c_med.mov
Differentiating Mathematics Instruction for All Students
Unit 2: Section 2
Essential Elements of Mathematics – What’s It All About?
Concrete-Representational-Abstract Understanding
More than Just Manipulatives:
The National Council of Teachers of Mathematics (NCTM) emphasizes the importance of using manipulatives for students at all ages and grade levels, but it is a recommendation that middle and high school teachers may not take as seriously. Manipulatives may be looked at as immature or childish and only for students in elementary grades or for students with special needs.
Check it out!
Required reading:
Berkas, N. & Pattison, C. (November, 2007). Manipulatives: More Than a Special Education Intervention. NCTM News Bulletin. [On-line]. Available:
http://www.nctm.org/news/content.aspx?id=12698
Hartshorn, R. & Horen. S. (1990). Experiential learning of mathematics: Using manipulatives. ERIC Digest. [On-line]. Available:
http://www.eric.ed.gov:80/ERICDocs/data/ericdocs2sql/content_storage_01/0000019b/80/20/a4/c5.pdf
Differentiating Mathematics Instruction for All Students
Unit 2: Section 2
Essential Elements of Mathematics – What’s It All About?
Concrete-Representational-Abstract Understanding
Suggestions for Using Manipulatives:
As all students benefit from the use of the CRA instructional approach, below are suggestions for implementing manipulatives with students:
· Select manipulatives that are age-appropriate (teddy bear counters probably won’t go over well in a middle school classroom.)
· Choose manipulatives that truly reflect the intent of the lesson or concept.
· Allow exploration of newly introduced manipulatives before introducing the lesson. These experiences can also lend to rich language and student-discovery of concepts.
· Model use of the manipulatives for proactive purposes such as reduced misbehavior and more on-task behavior.
· Provide explicit instructions with the lessons. Allow for errors or questions and use those as teachable moments.
· Allow students to explore through cooperative learning groups so “talk” is encouraged. This is beneficial for students with language difficulties as well as those who don’t have foundational or readiness skills for the targeted task.
· Do not take away manipulatives from students who still don’t “get it.” Allow for learning time, which for some students, may require multiple opportunities. Remember, no one student learns at the same rate or pace.
· Allow the use of manipulatives throughout the CRA process to provide a more fluid transition from the concrete to abstract stage.
For more information about the CRA instructional approach, click on the link below:
Concrete-Representational-Abstract Instructional Approach. (October, 2004). The Access Center. Retrieved August 11, 2009 from
http://www.k8accesscenter.org/training_resources/documents/CRAApplicationFinal_000.pdf
Concrete-to-Representational-to-Abstract (C-R-A) Instruction. Special Connections. Retrieved October 4, 2009 from
http://www.specialconnections.ku.edu/cgi-bin/cgiwrap/specconn/main.php?cat=instruction&subsection=math/cra
Differentiating Mathematics Instruction for All Students
Unit 2: Section 2
Essential Elements of Mathematics – What’s It All About?
Concrete-Representational-Abstract Understanding
Test Your Knowledge:
Directions: Click and drag the descriptive activities from the right-hand side box and match them under the appropriate instructional stage.