NWSC November Math Cohort Meeting WELCOME! Nancy Berkas Cyntha Pattison

Intervention from a Concrete Perspective Nancy Berkas Cyntha Pattison

The 9 Columns for the NCTM Bulletin 1. What Is Intervention and Why Is Everyone Talking About It? 2. Intervention Lenses Learning Significant Mathematics Knowing the Mathematics Assessment and Data Gathering Quality Planning and Delivery Alignment

The 9 Columns for the NCTM Bulletin - Continued 3. Technology: The Unused Possibilities 4. Manipulatives: More Than A Special Education Intervention 5. What Can We Learn From Colleagues in Other Content Areas? Part 1 6. What Can We Learn From Colleagues in Other Content Areas? Part 2

The 9 Columns for the NCTM Bulletin - Continued 7. Creating or Selecting Intervention Programs 8. Differentiated Instruction and Universal Design for Learning 9. Accepting Responsibility for Every Student

What Is An Intervention? About teaching and learning and the opportunity to learn Significant mathematics Need established by close scrutiny AND valid and reliable data Need can only be “real” if it is preceded by quality teaching with multiple opportunities to learn

What Makes Mathematics Different from Other Content Areas? Mathematics, more than any other content area, calls for multiple entry points throughout our educational lives since at any point the progression through mathematics may bring in an entirely new concept. Those new concepts may or may not be dependent on and/or linked to previous mathematics.

What Makes Mathematics Different from Other Content Areas? At the same time, a majority of people, both in and outside of education still believe mathematics is linear in its design and one’s progression through math stops when you do not internalize one specific concept in the progression. This inattention to the workings of the human brain continues to impede student success in other areas of math due to lack of exposure.

What Makes Mathematics Different from Other Content Areas? In reality, there are indeed developmental progressions in mathematics just as there are in reading, science, social studies, physical education and etc. However, in most cases, student progress through mathematics is most limited by the lack of multiple perspectives and contexts used in the teaching of mathematics.

What Does All of this Mean? Actually, much has been learned about the teaching and learning of mathematics since the publication of the Curriculum and Evaluation of School Mathematics by the National Council of Teachers of Mathematics in Much of that learning has focused on the use of manipulatives to provide concrete experiences and understanding of difficult concepts. Researchers often delegate those concrete encounters to those with learning difficulties. However, what research on the teaching and learning is actually beginning to show is that ALL students can profit from a concrete to representational to abstract sequence of learning experiences.

Concrete-Representational-Abstract It is especially important to note that CRA does not in fact define a model for planning lessons since each stage is a level of understanding in itself and thus necessitates a check for student understanding before proceeding to the next stage or level. In fact, in some cases students can use manipulatives to demonstrate their Representational and/or Abstract understandings. Of significance in understanding this sequence is the fact that in the literature, skills and concepts require a slightly different set of definitions when using the CRA sequence. This difference is most obvious in the Representational and Abstract portions of the CRA sequence.

Concrete In the Concrete portions of the sequence, students use concrete objects to explore the number, idea, concept, or skill. Students need ample time to form an understanding at the concrete level before proceeding to the representational stage. This may require several different types of concrete objects or manipulatives and several different explorations with those tools. In addition, manipulatives that are used for measuring and graphing require more sophistication than those that are used for counting and modeling numbers and equations.

Representational In the Representational stage, students are asked to draw pictures of and/or record the manipulative work they have done. They also might be given pictures in books or workbooks and asked to define the representation or articulate their understanding of the picture. Oftentimes, authors propose that the picture is the concrete and the number or concept or model that the student articulates is the representation.

Representational This issue is further confounded by the NCTM Representation standard in the Principles and Standards for School Mathematics published in Representation Standard Instructional programs from prekindergarten through grade12 should enable all students to— – create and use representations to organize, record, and communicate mathematical ideas; – select, apply, and translate among mathematical representations to solve problems; – use representations to model and interpret physical, social, and mathematical phenomena. (NCTM 2000 p. 67)

Abstract In the Abstract stage, students are able to demonstrate understanding of processes and concepts. It is at this stage of CRA that the difference in skills and concepts is most marked. Abstract when dealing with procedures and skills is using symbols to perform operations. Oftentimes, this is oversimplified and equated with “mental math.” In actuality, students at the Abstract stage of skill development can use algorithmic procedures to solve problems using a variety of modalities

Abstract From a conceptual perspective, the Abstract stage of development is closely linked to the application of knowledge and uses portions of the Representation standard articulated by NCTM. Most state and national standards are stated at the Abstract level of the sequence of understanding. Depending on the concept and/or strand of mathematics in question, mathematics at an Abstract level can take a number of forms and most of those forms cannot be assessed in multiple choice or short answer formats. Only with skills is this possible.

Manipulatives: More Than A Special Education Intervention Concrete-Representational-Abstract Research on the use of manipulatives to teach mathematics show generally positive impacts when manipulatives are combined with: 1) virtual manipulative software, 2) reflective practices, 3) cooperative learning, or 4) learning activities that are exploratory and deductive in their approach.

Manipulatives: More Than A Special Education Intervention The Conundrum What constitutes good instruction and what constitutes intervention?

Contact Information Cyntha Pattison Nancy Berkas

Manipulatives: More Than A Special Education Intervention Learning Significant Mathematics – If every student should learn significant mathematics, do we have an obligation to extend our investigation of the situations and combinations that enhance concrete understanding of concepts through manipulative-based activities? Knowing the Mathematics – Can we identify concepts that are best developed with a concrete, manipulative base? Is our mathematical understanding sophisticated enough to know how to use manipulatives most effectively? Assessment and Data Gathering – Can we assess concrete understanding? Are our assessments and definitions of success limited by specific pathways/mindsets?

Manipulatives: More Than A Special Education Intervention Quality Planning and Delivery – Do we have the ability to plan and deliver quality manipulative-based lessons? Alignment – Can or should we use standards and benchmarks to determine which areas are most appropriate for the use of manipulatives? Are there certain grade levels where it would not be appropriate to teach some concepts or skills by using manipulatives?