Teaching Mathematics Meaningfully: Solutions for Reaching Struggling Learners - Softcover

Dr. Allsopp is Assistant Dean for Education and Partnerships in addition to being the David C. Anchin Center Endowed Chair and Director of the David C. Anchin Center at the College of Education at the University of South Florida. He is also Professor in the Department of Teaching and Learning–Special Education Programs. Dr. Allsopp holds degrees from Furman University (B.A., Psychology) and the University of Florida (M.Ed., Learning Disabilities; Ph.D., Special Education). Dr. Allsopp teaches at both the undergraduate and doctoral Levels, and his scholarship revolves around effective instructional practices, with an emphasis on mathematics, for students with high-incidence disabilities (e.g., specific learning disabilities, attention-deficit/hyperactivity disorder, Social-emotional/behavior disorders) and other struggling learners who have not been identified with disabilities. Dr. Allsopp also engages in teacher education research related to how teacher educators can most effectively prepare teachers to address the needs of students with disabilities and other struggling learners. Dr. Allsopp began his career in education as a middle school teacher for students with learning disabilities and emotional/behavioral difficulties in Ocala, Florida. After completing his doctoral studies at the University of Florida, Dr. Allsopp served on the faculty at James Madison University for 6 years. He has been a member of the faculty at University of South Florida since 2001.

Dr. Lovin began her career teaching mathematics to middle and high school students before making the transition to Pre–K through Grade 8. For over 20 years, she has worked in elementary and middle school classrooms. Then and now, Dr. Lovin engages with teachers in professional development as they implement a student-centered approach to teaching mathematics. At the time of this publication, she focused her research concerning teachers' mathematical knowledge for teaching on the developmental nature of prospective teachers' fraction knowledge. She has published articles in Teaching Children Mathematics, Mathematics Teaching in the Middle School, Teaching Exceptional Children, and the Journal of Mathematics Teacher Education. She coauthored the Teaching Student-Centered Mathematics Professional Development Series with John A. van de Walle, Karen Karp, and Jenny Bay-Williams (Pearson, 2013). Dr. Lovin is an active member of the National Council of Teachers of Mathematics, the Association of Mathematics Teacher Education, and the Virginia Council of Teachers of Mathematics.

Excerpted from Chapter 2 of Teaching Mathematics Meaningfully: Solutions for Reaching Struggling Learners by David H. Allsopp, Ph.D., Maggie M. Kyger, Ph.D., & LouAnn H. Lovin, Ph.D.

Copyright © 2007 by Paul H. Brookes Publishing Co. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

Struggling learners must have three important needs met in order to achieve success in the K–12 mathematics curriculum: 1) teachers who are committed to ensuring that they learn mathematics; 2) teachers who have a conceptual framework for why struggling learners have difficulties learning mathematics; and 3) instruction that addresses their learning needs, thereby allowing them to understand mathematics.

In Chapter 1, four universal features of meaningful mathematics instruction for struggling learners were introduced: 1) understanding and teaching big ideas in mathematics and the big ideas for doing mathematics, 2) understanding learning characteristics of and barriers for struggling learners, 3) continuously assessing learning and making informed instructional decisions, and 4) making mathematics accessible through responsive teaching. As Figure 2.1 (which is repeated from Figure 1.1 in Chapter 1) suggests, each of these four anchors for meaningful and effective mathematics instruction for struggling learners is connected integrally to each other. The double-ended arrow between the responsive teaching anchor and the learning characteristics/barriers anchor shows the reciprocal relationship between these two universal features. For example, for teachers to respond to students' learning needs, they need to understand how to implement effective instructional practices (responsive teaching). Conversely, teachers must understand how barriers to learning have an impact on what and how students learn (learning barriers) so that they can select instructional practices that are responsive to students’ needs. However, teaching mathematics does not happen in a vacuum. Teachers must understand the big ideas of mathematics, including content big ideas (the what of mathematics) and process big ideas (how mathematics is done). Such understanding supports responsive teaching by informing teachers about what they need to teach (content big ideas) and the multiple ways in which their students can do mathematics (process big ideas). By understanding the mathematics curriculum, teachers are equipped to make good decisions that are based on the learning needs of their students. The arrows that flow from the big ideas anchor to the responsive teaching anchor and the learning characteristics/barriers anchor show these relationships. Similar to the big ideas anchor, the continuous assessment anchor informs the responsive teaching anchor and the learning needs/barriers anchor. Responsive instruction is informed by student performance data collected continually that show whether students are learning so that timely instructional decisions can be made. In turn, this informs teachers about the extent to which particular instructional practices meet the learning needs of their students. The arrows that flow from the continuous assessment anchor to the responsive teaching anchor and the learning characteristics/barriers anchor depict these relationships.

Understanding and Teaching the Big Ideas in Mathematics and the Big Ideas for Doing Mathematics

The K–12 mathematics curriculum can be organized according to five content strands. These can be thought of as the what of the mathematics curriculum: 1) number and operations, 2) algebra, 3) geometry, 4) measurement, and 5) data analysis and probability (National Council of Teachers of Mathematics [NCTM], 2000). Each of these content strands has unique features and ideas that allow for their categorization into separate areas of mathematics. Unfortunately, this separation frequently is carried to an extreme: Some people think of mathematics as separate bodies of ideas that consist of isolated rules and procedures that often simply must be memorized. This perspective of mathematics offers a very narrow and false view of what mathematics really is about. It is true that mathematics is composed of different areas, such as algebra and geometry, but these areas actually are connected to each other in numerous ways. When seeking ways in which the areas of mathematics are related, one can see how mathematics really is a connected body of ideas that does make sense. Teachers must understand the connections among concepts within each strand as well as those between strands, especially as they relate to the particular K–12 curriculum that they teach. For example, in measurement, the notion of unit is paramount, whether you are talking about inches or square feet. The notion of the unit or the whole also is important when working with number and operations in part–whole fractions and proportions. Students' understanding of the notion of unit and its role in mathematics can make generalization to other areas easier. Moreover, teachers must prioritize the task of helping their students make these connections so that students can achieve success with the K–12 mathematics curriculum.

In the development of a more comprehensive view of mathematics, these five processes of doing mathematics also should be considered: 1) problem solving, 2) reasoning and proof, 3) connections, 4) communication, and 5) representation (NCTM, 2000). The NCTM maintains that the processes of doing mathematics "highlight ways of acquiring and using content knowledge" (NCTM, 2000, p. 29). All students, including struggling learners, should and can engage in mathematics using these processes. Although some students with significant cognitive disabilities may be stronger in certain ways of doing mathematics than in others, generally all struggling learners can do mathematics in ways that are consistent with each of these process standards. All too often, struggling learners are provided with instruction that emphasizes singular ways of doing mathematics (e.g., computations) because the thought is that they cannot do mathematics in ways that require greater levels of higher order thinking (e.g., reasoning, problem solving). Such practice is based on false assumptions about what struggling learners can do. It also is a practice that contradicts the current literature base (e.g., Cawley, 2002; Cawley, Parmer, Yan, & Miller, 1998; Chard & Gersten, 1999).

This book places an emphasis on the big ideas that are related to how students do mathematics because how students learn mathematics is tied intricately to what they ultimately learn and understand. Understanding something means more than "you have it or you don't." Understanding exists on a continuum. At one end of the continuum exists an understanding in which the ideas are not very well connected to other ideas. This type of understanding, called instrumental understanding (Skemp, 1978), is often the result of ideas that have been memorized and "learned" without meaning. Ideas understood in this manner have a higher likelihood of being forgotten. If a person is provided with opportunities to actively seek and make connections between ideas, he or she can develop a deeper, more connected network of ideas that results in a type of understanding called relational understanding (Skemp, 1978). This type of understanding consists of a meaningful network of concepts and procedures that, over an active lifetime of learning, only becomes more densely connected and useable. One is very likely to never have a complete, full understanding of a particular idea because as one learns about new ideas, new connections to existing ideas can be...