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.

**UNIVERSAL FEATURES**
**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 made. That is what makes lifelong learning so important and exciting. Clearly, our goal for all students is that they develop a relational understanding of mathematical ideas. This type of understanding is best developed by instruction that engages students in the processes of *doing* mathematics (using mathematical understandings in meaningful ways to develop deeper, more connected mathematical knowledge) in contrast to instruction that emphasizes the learning of isolated mathematical concepts and skills.

**Understanding Learning Characteristics of and Barriers for Struggling Learners**

The key to helping struggling learners achieve success with the K–12 curriculum is to understand *why* they have difficulty learning mathematics. This book focuses on two major areas of barriers for struggling learners in mathematics. One area involves a common core of learning characteristics, one or more of which struggling learners may possess. These learning characteristics, discussed in Chapter 5, include attention and memory difficulties, learned helplessness, passive approaches to learning, processing and metacognitive deficits, low levels of academic achievement, and math anxiety.

The second major area that can create barriers for struggling learners has to do with how the mathematical curriculum is delivered. These curriculum delivery– related issues, discussed in Chapter 6, include spiraling curriculum, mastery learning, algorithm– driven instruction, cyclical reforms, and use (or non–use) of effective teaching strategies.

Chapter 7 describes six stages of learning for struggling learners. These stages provide educators with a foundation for perceiving how assessment and instruction relate to *how* students learn. As educators better understand why struggling learners have difficulty learning mathematics, they become better equipped to understand how teaching methods have an impact on students–learning. In turn, this empowers teachers to make better instructional decisions about how they teach their students.

**Continuously Assessing Learning to Make Informed Instructional Decisions**

Continuous assessment of learning simply means that educators should evaluate what students know and can do before, during, and after instruction. Before teaching any mathematical concept, teachers should evaluate students' knowledge and experiences related to the concept. This includes evaluating their prerequisite knowledge and skills and their experiences and interests that might relate to the target concept. Chapter 8 describes how assessment can be done to gain a clear picture of students' knowledge about a target concept and prerequisite skills that are associated with the concept. This picture of learning includes students' level of competency in understanding mathematical concepts and performing mathematical skills; their level of understanding (concrete, representational, or abstract); whether they can choose an example of a concept or skill (i.e., receptive understanding) or show their understanding without being provided choices (i.e., expressive understanding); and whether they have procedural knowledge, conceptual knowledge, or both. If students are not proficient in the knowledge and skills that are necessary for understanding the target concept, then they will have difficulty being successful. If students do possess the prerequisite knowledge and skills that are needed to understand the target concept, then knowing their experiences and interests that relate to the concept can help teachers to develop meaningful learning contexts within which to teach the target concept. Chapter 8 describes the Mathematics Interest Inventory, a tool for learning about students' interests and experiences outside mathematics per se that is used to integrate students' interests and experiences in meaningful ways as teachers plan, assess, and teach their students.

During instruction, it is vital that teachers determine whether students understand the target concept and are able to use it proficiently, two skills integral to the success of struggling learners. Students may understand a target concept at different levels from one another and use it at different levels of proficiency. For example, some children, when adding, do not readily "count up" (e.g., when given two numbers, a student counts up from one of the numbers, adding the second number) but will "count all" (e.g., when given two numbers, the student begins counting up from zero until all numbers are added). Students in the latter example seem to understand adding and have some level of proficiency, but they lack sophistication in their thinking and approach. Evaluating student understanding *during* instruction allows educators to monitor the success of instruction so that changes can be made immediately. This prevents the loss of valuable instructional time and helps teachers to avoid surprises such as waiting until the class moves on to the next concept or unit before realizing—too late—that some students did not understand it.

After instruction, evaluating where students are in terms of their learning of the target concept provides teachers with a foundation for planning further instruction. In some cases, students might demonstrate sufficient understanding to move to the next target concept. In other cases, they may demonstrate the need for additional instruction or response opportunities to become proficient (i.e., they can demonstrate understanding of the concept or can perform the skill with a high level of accuracy and at a satisfactory rate). By determining this information after instruction occurs and before the next lesson is planned, teachers have a ormat for planning subsequent instruction so that it best meets the learning needs of the students. Chapter 8 describes a fluid process for continuously assessing students in practical yet informative ways to guide instructional decision making.

**Making Mathematics Accessible Through Responsive Teaching**

A primary purpose of this book is to help educators make the learning of mathematics accessible to struggling learners. *Access* is an important concept in successful mathematics instruction for these students. Because of the learning barriers that these students face, educators must be creative in their thinking about how to provide them with meaningful mathematics learning experiences. In the context of teaching struggling learners, access is defined as the methods, practices, or procedures that the teacher plans and implements that directly address the learning characteristics of these students, thereby increasing the likelihood that students have a clear understanding of the mathematics concept. When students can process the concept in ways that make sense to them or in ways that are accessible given their own learning abilities and needs, they are more likely to understand the concept.

**SUPPORT FOR THE FOUR UNIVERSAL FEATURES**

The four universal features for making mathematics meaningful for struggling learners, including an understanding of and instruction in both content and process big ideas, an understanding of learning characteristics of and barriers for struggling learners, continuous assessment of learning and instructional decision making, and an ability to make mathematics accessible, are not allencompassing. However, they provide educators with an informed framework for effectively teaching mathematics. Each universal feature represents an important element of effective instruction in mathematics for struggling learners and is supported by literature in both special education and mathematics education fields.

**Support in Content and Understanding**

The concept of teaching the big ideas in mathematics for struggling learners has been a topic of discussion in the literature (e.g., Cawley et al., 1998; Carnine, Dixon, & Silbert, 1998; NCTM, 2000; Parmar & Cawley, 1991; Educational Resources Information Center/Office of Special Education Programs (ERIC/OSEP) Special Project, 2002). For example, Cawley and colleagues (1998) emphasized the importance of moving beyond basic skills instruction for struggling learners. They advocated concentrating on how students *reason* about the mathematics that they do and on helping students build connections between and among mathematical concepts. Carnine and colleagues (1998) advocated for teaching big ideas that cut across the mathematics curriculum (e.g., use of arrays or area models for multiplication of whole numbers) as a method for helping struggling learners apply the same idea to other mathematical ideas (e.g., multiplication of fractions, decimals and polynomials).

Mathematics educators long have espoused the need to teach the big ideas, not only the big ideas that compose the *what* of K–12 mathematics but also the big ideas for doing mathematics. The NCTM clearly identifies both content standards (the *what* of K–12 mathematics) and process standards (the *doing* of mathematics; NCTM, 2000). Several mathematics educators have advocated the integration of these standards with the need to apply them on the basis of the individual strengths and weaknesses of struggling learners (e.g., Baroody, 1987; Van de Walle, 2005). For example, Baroody (1987) contended that traditional ways of teaching mathematics (i.e., skills-only approach) does not meet these students’ developmental or psychological needs, resulting in a lack of understanding and significant gaps in the students’ mathematical knowledge. In essence, focusing also on the big ideas instead of solely on individual skills and concepts provides opportunities for students to construct connections between various skills and concepts.

**Support for Understanding Barriers**

Special education has long held as major tenets both the importance of understanding the learning characteristics of struggling learners and the importance of applying that understanding to teaching these students (Minskoff, 1998). Similarly, mathematics education long has advocated for developmentally appropriate instruction whereby teachers use their knowledge of child development to provide mathematics instruction that is consistent with students' developing cognitive abilities (Carpenter, Fennema, Franke, Levi, & Empson, 1999; Kami, 2000). Without the knowledge of how children learn mathematics, educators tend to teach in ways that can be detrimental to students' learning of mathematics in meaningful ways (e.g., Kami, 2000). Related to struggling learners specifically, much has been discovered regarding why these students have difficulty comprehending mathematics. As a result of characteristics that derive from disability, language differences, and cultural diversity, barriers occur for these students and make it difficult for them to make meaning of mathematics. Two primary types of barriers are reported in the literature: barriers that result from the learning characteristics of struggling learners and barriers that result from the interaction of these learning characteristics and how the mathematics curriculum is taught (e.g., Allsopp, Lovin, Green, & Savage–Davis, 2003; Baroody, 1987; Cawley et al., 1998; Cawley, Parmar, Foley, Solomon, & Roy, 2001; Gagnan & Maccini, 2001; Mercer, Harris, & Miller, 1993; Mercer, Jordan, & Miller, 1996; Miller & Mercer, 1997).

**Support for Continuous Assessment**

In addition, the use of continuous assessment of students' understanding to make informed instructional decisions in mathematics long has been advocated in the literature. Assessment procedures such as curriculum-based assessment and curriculum-based measurement result in teachers' greater awareness of both their students' learning needs and their students' day–to–day progress in meeting learning goals and objectives (e.g., Allinder, Bolling, Oats, & Gagnon, 2000; Miller & Mercer, 1993; Shafer, 1998; Woodward & Howard, 1994). The NCTM maintains that assessment should be an integral part of instruction, providing not only the teacher but also the student with information about the student’s learning. With this information, teachers will be better able to modify their instruction and students will be better able to modify their activity, all with the ultimate goal of students coming to a deeper understanding of the concepts being studied. A review by Black and Williams (1998) of more than 250 studies strongly supports the notion that students’ learning is improved considerably when teachers consistently use formative assessment to guide their instruction.

**Support for Providing Access**

The literature that supports the final universal feature of providing access to successful mathematical learning experiences for struggling learners through effective teaching practices builds on the literature support for the first three universal features. As the literature evolves, there is greater awareness that mathematics instruction should be more than direct teaching of basic skills and that struggling learners can learn mathematics at much deeper levels of understanding than previously believed. This is especially true when instructional practices that are characterized by different levels of teacher support are implemented (Mercer, Lane, Jordan, Allsopp, & Eisele, 1996). This continuum of instructional choices for struggling learners is implemented most successfully when conscious thought is applied to *what* students are ready to learn, how they will be asked to *do* it, how it relates to their *knowledge*, and the *barriers* that might make learning difficult for them. When such responsive teaching is applied to teaching struggling learners, success is more likely to occur. Fortunately, a growing body of literature documents mathematics instructional practices that respond to the needs of struggling learners (e.g., Baxter, Woodward, & Olson, 2005; Bottge, Heinrichs, Metha, & Hung, 2002; Cawley et al., 1998; Lock, 1996; Kroesbergen & van Luit, 2002, 2003; Maccini & Gagnon, 2000; Mercer, Jordan, et al., 1996; Miller, Butler, & Lee, 1998; Owen & Fuchs, 2002; Vaughn, Gersten, & Chard, 2000).

The model for effectively teaching mathematics for struggling learners described in this chapter can be used to help educators conceptualize, plan, and evaluate their own mathematics instruction. As the reader learns more about how each universal feature can be implemented, it will be helpful to refer back to the universal features model periodically and reflect on how the particular methods described "fit" with the model and his or her own teaching. This model can also serve as a structure for evaluating the extent to which an adopted curriculum addresses the mathematics learning needs of struggling learners.