Not Your Parents' Math

Stephanie Huie

The Windward School Math Program—Research-based and Comprehensive

As the world becomes more complex and technologically driven, the need to understand mathematical concepts and to apply them to everyday activities increases. From planning vacations, to shopping at the grocery store, to making schedules, and many other daily tasks, we apply mathematics more often than we realize.

Math is not just in daily life activities, either. According to the Skills, Technology, and Management Practices 2016 study by M. J. Handel, 94% of workers perform some form of math application in their jobs. While most people end up using division, fractions, and multiplication more so than Algebra 2, Handel and others agree that completing advanced math courses has clear benefits in helping students succeed in college and get good jobs.

The way you might have learned math is most likely not the way your children are learning it, especially if they are attending The Windward School. By integrating direct instruction, a research-based teaching method with specific multisensory teaching strategies, Windward’s math program ensures that students learn in an engaging and optimally challenging classroom environment that enables them to succeed in mainstream classrooms. 

The Windward School Math Program— Research-Based & Comprehensive

Although The Windward School operates on three campuses, the math department functions as a cohesive unit providing a strong academic curriculum that adheres to both New York State standards and the specific needs of Windward students.

At Windward, math lessons are created to teach computation and application skills, enhance conceptual understanding, and develop each child’s ability to communicate using the language of math. Using a multisensory curriculum that enhances students’ understanding of mathematical concepts, the math program is designed to give each student the skills necessary to be successful in a mainstream math class. Classes are grouped homogenously with approximately 6-12 students per group. Grouping this way allows teachers to introduce skills and tailor instruction so that students can learn at an appropriate pace.

Through direct instruction, homogenous grouping, and deliberate planning of the language component of each lesson, teachers build a solid foundation of basic concepts, computation, and number sense. The language of math is taught using techniques from the Windward Language Arts Program. Similar to a language arts class, math teachers intentionally plan questions and comments to check for understanding, to foster discussion, and to connect concepts to real-world applications. Since linguistic struggles are common for Windward students, direct instruction in solving word problems is part of every unit. Teachers emphasize vocabulary in each topic, and oral explanations are components of daily class discussions. Math units reinforce foundational skills to build number sense, procedural fluency, and math problem-solving skills.

Multisensory is Appropriate for All, Essential for Windward Students

Similar to the methodology employed in the language arts program, all math groups receive multisensory, direct instruction, which allows students to learn concepts using more than one sense at a time in a clear, explicit manner.

Multisensory instruction uses specific senses including visual (language we see), auditory (language we hear), and kinesthetic- tactile pathways (language symbols we feel). Manipulatives, kinesthetic activities, mnemonics, literature, and technology are integrated into daily instruction to promote student engagement and retention of math concepts. Using multiple senses simultaneously enhances retention of written language and concepts. “Multisensory learning is particularly helpful for students with language-based learning disabilities. Since many of our students have trouble with visual or auditory processing, it makes it hard for them to learn information through only reading or listening. By incorporating more senses, students are given more ways to connect with what they are learning,” explains Lori Squillante, Westchester Lower School Math Coordinator. Windward teachers use lessons that encourage real-world applications of mathematics while developing skills in a sequential, structured manner.

The CPA Approach

One of the specific multisensory approaches to teaching math at The Windward School is through the Concrete-Pictorial-Abstract (CPA) approach. CPA is a three- step instructional approach that is highly effective in teaching math concepts and is based on the work of psychologist Jerome Bruner. This approach to teaching mathematics provides students with a deeper understanding of math concepts so that they gain greater conceptual knowledge, rather than just procedural knowledge.

Windward teachers employ concrete representations to introduce new concepts to students using manipulatives that can be moved, grouped, and rearranged to illustrate a given problem. Manipulatives are widely used in grades 1-5 to introduce major mathematical concepts; however, certain types of manipulatives continue to be incorporated in higher grades, too.

The pictorial stage, or the seeing stage, involves using images to represent objects to solve a math problem. These images include, but are not limited to, diagrams, bar models, charts, and other types of graphical representations. Middle school teachers usually begin lessons at the pictorial stage but may use concrete examples depending on their math group’s previous knowledge.

In the abstract stage of the CPA approach, also known as the symbolic stage, numbers and symbols are used to solve math problems.

Since CPA is a gradual, systematic approach, each stage builds upon the previous one and is usually taught in sequence. If at any point a student struggles with the way a math concept is presented, the teacher will go back to the previous stage and reinforce its foundation. Throughout CPA, teachers provide an appropriate number of examples to ensure students’ understanding.

Mathematics teachers need to organize content into concepts and provide instruction that allows students to process the new learning in meaningful and efficient ways” (Jordan, Miller, & Mercer, 1998). By using the CPA approach, students learn math concepts in a structured way and are able to build better connections when moving through the stages of the approach.

Practice Makes Permanent

Research shows that combining the direct instruction teaching model with multisensory methods results in many benefits to student learning. Among the strengths of direct instruction are additional time for students to practice their skills and teaching that is thoughtful and organized. “The (direct instruction) program works across various sites and types of children…It produces positive achievement benefits in all subject areas – reading, language, math, and spelling. It produces superior results for basic skills and for higher-order cognitive skills in reading and math. It produces the strongest positive self- esteem…” (Adams, 1996).

Multisensory Teaching Strategies

Visual and symbolic representations are powerful tools for developing students’ number sense and for communicating mathematical concepts. By incorporating the CPA approach and direct instruction, students are able to make connections and develop concrete understanding.

The National Council of Teachers of Mathematics (NCTM) recommends the use of manipulatives in teaching mathematical concepts at all grade levels. “At Windward, we use manipulatives to give students a concrete example of the concept before moving on to the other stages. Also, we have a variety of manipulatives that can be used throughout all the grades,” says Victoria Scarinci, The Windward School Manhattan Campus Math Coordinator. The multisensory teaching strategies used at Windward include base ten blocks, which are used in the lower schools to help build number sense and an understanding of place value. As students continue to build their conceptal foundation, teachers introduce fraction strips to help students explore fraction concepts and numerical operations.

To learn about properties of two-dimensional figures and how to calculate area and perimeter, students will use geoboards to help visualize these concepts. Geoboards continue to be used in middle school for geometry and for calculation of slope. To learn about three-dimensional figures, geometric nets are used. Interactive whiteboards help teachers of all grade levels to actively engage students in problem solving through kinesthetic and visual means. Virtual manipulatives are introduced in conjunction with interactive whiteboard activities, so students can see mathematical relationships by manipulating different objects using a computer mouse. Bulletin boards are another visual tool created to aid in students’ understanding. These are only a few of the hands-on, multisensory tools used to help students make connections and develop understandings of mathematical concepts.

A Visual Solution to Word Problems

One of the most important multisensory teaching strategies used throughout all grades at Windward is model drawing. Since word problems can be challenging for all students, but particularly for students with language-based learning disabilities, it is important to incorporate a variety of strategies to help students learn to solve multi-step word problems while building their confidence in this area. Although The Windward School teaches several different problem-solving strategies to decipher and solve word problems, the model drawing has proven to be particularly effective for our students.

Model drawing, often called “bar modeling,” is a systematic method of representing word problems and number relationships that is explicitly taught beginning in second grade and extending all the way to secondary algebra. In the CPA approach, it falls under “pictorial” and is a step-by-step approach to problem solving where students draw rectangular “bars” to represent the relationship between the known and the unknown numerical quantities in word problems. Model drawing appeals to learners who benefit from creating a picture to understand written words. It also includes very clear and specific steps to help keep students organized and consistent in their approach to problem solving. “By using model drawing, students are able to visualize the problem and know what operations to perform by using the rectangular bars to identify the unknown quantity. This means students are viewing problems from an algebraic perspective, even in the early elementary years,” says Laura Bottari, Westchester Middle School Math Coordinator. To see an example of model drawing, watch this video as Laura Bottari explains bar modeling to her eighth-grade class: 

Moving the Needle & Building a Mathematical Toolbox

Upon entrance to Windward, students’ skills in language arts and mathematics are assessed using achievement tests. In the past, the Stanford Achievement Test was administered to students in second through eighth grade. More recently, it has been replaced by the Iowa Assessments. Analysis of results (see Figure 1) on standardized tests administered upon entrance reveals that over the period of 2005 to 2017, 36% of students new to Windward scored below average, 53% scored average, and only 11% of students scored above average in math computation. Achievement tests administered at outplacement, however, produced starkly different results: 9% of students scored below average, 57% scored average, and 34% of students scored above average in math computation. This means that 91% of Windward students were performing in the average to above average range in math computation prior to leaving Windward.

A similar result occurred in students’ abilities to learn strategies for math problem solving. Students’ scores upon entrance were 48% below average, 43% average, and 9% above average. Just prior to being mainstreamed, the results showed only 12% of students below average, 49% average, and 39% above average in their abilities to solve word problems. Overall, 88% of Windward students were performing in the average to above average range in math problem solving.

The Windward School’s math program is intentionally grounded in research. The structured curriculum, multisensory instructional model, and diagnostic teaching methods create a strong foundation of mathematical concepts reinforcing and building upon new skills as students progress in their mathematics education. Ultimately, the program ensures students build a toolbox of mathematical strategies to be engaged and successful learners in both Windward classrooms and the mainstreams schools they attend once leaving Windward.