A Guide to Success with Math: An Interactive Approach to Understanding and Teaching Orton Gillingham Math

By Marilyn Wardrop

An Engaging- Interactive- Successful - Math Guide
for Teachers, Parents and Literacy Practitioners
Based on the Orton-Gillingham Approach to Literacy Instruction
Understanding and Teaching OG Academic Math:
For some students, mathematics is a road travelled in small steps. A successful multi-sensory approach leads the student through small increments of understanding toward unifying themes in mathematics. Student’s strengths and needs must be recognized and addressed with built-in “checks for error,” as well as a built-in system for building confidence and competence.
The OG interactive, cognitive approach for learning math enables students to develop more effective ways of thinking about math and applying new skills.
OG Academic Math Handbooks, Training Courses and Workshops are designed to provide teachers, instructional support staff, Orton-Gillingham practitioners and parents with hands-on, interactive math training.

• Language: English
• Publisher: BookBaby
• Release date: Dec 1, 2014
• ISBN: 9780993724619

Book preview

Introduction

The inspiration for writing this book comes from my firm belief that all children want to learn and be successful in school. Too often when it comes to math, many students struggle with the concepts and the way math is taught. In 2005, Gallup conducted a poll that asked students to name the school subject that they considered to be the most difficult. Not unexpectedly, mathematics came out on top of the difficulty chart. So what is it about math that makes it difficult? For many students, math is not something that comes intuitively or automatically.  For these students, math is a road traveled in small steps requiring appropriate instruction, effective interventions and support from parents and teachers who possess the persistence to do what it takes for their children and students to succeed. The OG Academic Math Approach outlines one path to developing success with math. .

I have been an Orton-Gillingham literacy instructor for thirty years. During that time I have worked in different school settings with students in a wide range of grade levels from kindergarten to 8th grade. Within these classes were students both with and without special learning needs.  When my students were struggling with math, I was motivated to find an approach that combined the effectiveness of Orton-Gillingham with a cumulative, structured, multisensory approach to mathematics instruction. My research and training resulted in the development of the OG Academic Math Approach for teaching mathematics.

I have had the privilege to train educators and parents in the OG Academic Math Methodology in Canada, the United States, the Middle East and the Philippines. These educators include teachers, Orton-Gillingham practitioners, para-educators, school psychologists and administrators. It has proven to be a useful math intervention for all of these supporters, educators and professionals. 

My goal is that this book becomes a valuable reference, tool, guide and resource for helping students to learn and be successful with math.

I wish your children and students great success as you meet the math challenges ahead.

Marilyn Wardrop

Chapter One

Why and How We Learn Mathematics

A quadratic equation belonged to the world of Alice in Wonderland and the Differential Calculus was a dragon.

- Winston Churchill, Siegel, 2013

In commencing this book, we must first answer the question that almost any student in any mathematics classroom has asked: Why do we have to learn math? The fact is that math is all around us. That may seem to be an overused adage, but it is the truth. Math is simply a science of patterns and relationships—two aspects of our everyday lives. In this world, we must understand how to read a clock, balance a chequebook, and modify a recipe. We even need to know math to comprehend the significance of ten percent battery life remaining on whatever electronic device we happen to be using at the moment!  Without this comprehension, we would be perpetually late and helplessly broke. More importantly, without math, houses and buildings would be askew and unstable; timetables for ferries, planes, trains, and other modes of transportation would be incorrect, possibly resulting in collisions; and misplaced decimals could result in over or under-paying employees, bills, and more.

Part of a well-rounded math education requires students to solve word problems. As much as students tend to detest these problems, these problems assist students to develop more than just the basic math skills. These questions ask students to go beyond the simple numbers and use reasoning skills to determine the correct formulas and identify the variables. These are skills that can translate into many contexts in life: calculating how many eggs you need to double a recipe or determining how many children will be on each team for an intramural tournament. Furthermore, these reasoning skills give students practice in following a process to reason out an answer regardless of whether it is a math question or another topic. Simply put: this world would not function without math skills, including the most basic skill of counting to the most difficult skills learned in calculus.

How We Learn Math

If we were to ask any adult today, he probably cannot give an exact time when he first learned to count to ten. Some adults will remember the success they felt when they were finally able to count to one hundred. Others may remember the confusion they felt when they first encountered fractions. Regardless of what memories an individual may have, one’s memories of learning math should follow a sequential pattern of growing difficulty as one progressed in school.

Math instruction builds from the basic skills of number sense to simple operations (addition and subtraction) to more complicated operations (multiplication and division) and so on. The premise with this type of educational sequence is that students will master the skills at each level, allowing them to progress to the next level. If we had not learned the definition of a fraction in primary school, we would be lost when we began to add, subtract, multiply, and divide fractions in the intermediate grades. Unfortunately, the sequence and style of math instruction provides an exceedingly narrow understanding of the concepts taught (Woodward & Stroh, 2004), leaving many students without a full grasp of the concept and its application. Often, students who did not master a skill in a previous year will struggle the following year in math, and they will have a sense that they are always playing catch up with their classmates. These students are frequently left behind to relearn the concepts on their own to prevent delaying the rest of the class from staying on pace with the curriculum.

An important distinction to understand is the one between conceptual and procedural knowledge. In math, we are normally taught procedural knowledge—"how to do something. This knowledge is surface knowledge. It allows us to look at a question, pick a formula, and solve the problem. But what if the question does not follow the exact outline of questions we have seen before? This situation calls for conceptual knowledge—an understanding of what features in the task mean . . . [that] allows one to understand why the procedure is appropriate for that task" (Booth, 2011). There must be a balance between the drill and practice method of instruction and problem-based instruction so that students develop both of these types of knowledge and are able to master both the comprehension and application aspects of the topic (Woodward & Stroh, 2004).

Steve Chinn and Richard Ashcroft (2007) discuss in detail in Mathematics for Dyslexics: Including Dyscalculia the two mathematical styles of thinking. The two styles are intuitive and sequential, and both styles are almost equally prevalent in learners. Intuitive thinkers are said to be right brain thinkers and think more conceptually. Metaphorically, an intuitive thinker can be represented by a grasshopper because he does not follow a specific path; instead, he hops around to achieve his end result. These thinkers tend to break large numbers and problems into small numbers and equations that they are more familiar with so that they are able to solve the problem. For instance, if asked to divide 96 by 6, intuitive thinkers might do the following:

96 ÷ 6 = ?

Step 1: 6 x 10 = 60

96 – 60 = 36

36 ÷ 6 = ?

Step 2: 6 x 5 = 30

36 – 30 = 6

Step 3: 6 ÷ 6 = 1

Step 4: 10 + 5 + 1 = 16

96 ÷ 6 = 16

To a sequential thinker, this process seems extremely confusing. For an intuitive thinker, his understanding of the concepts of multiplication, subtraction, and addition supports him to break down the division process into steps and successfully find the solution.

Sequential thinkers follow a thought pattern that most closely aligns with the way students are taught math in most schools. They tend to be left brain thinkers and can be compared to inchworms (Chinn & Ashcroft, 2007). Inchworms move along bit-by-bit, never deviating from the straight line they follow. These thinkers use known procedures to solve math problems, which can lead to difficulties if the basic math skills have not been mastered. This is not to say that sequential thinking is not effective. In fact, many students who use this method are quite successful in math courses, but it is uncertain how much they retain once the course ends. The main difference between a sequential thinker and an intuitive thinker is that the sequential thinker bases his thinking on procedural knowledge that he has acquired; on the other hand, the intuitive thinker uses his conceptual knowledge of numbers and operations to solve the problem. This conceptual knowledge is more likely to be retained and used much longer after the course ends.

The same question, 96 divided by 6, looks much different when solved by a sequential thinker:

96 ÷ 6 = ?

16

6 ) 96

-6

36

-36

0

96 ÷ 6 = 16

We can see that the sequential learner would revert to long division—probably due to the memorization of the division process—to discover that 96 divided by 6 equals 16. But what if the student’s skills in long division were subpar? Or what if the student was unable to memorize his times tables and did not know that 36 divided by 6 was 6? It is easy to see how a student could quickly err following the sequential method if the basic math skills have not been mastered. Thus, the sequential way that math is taught in schools provides the same pitfalls for students who have not yet mastered these skills or who struggle with these aspects of math.

In considering these two thinking styles, Chinn (2005), in his article Maths & Dyslexia, argues that most learners are between the two and this is the best position to take. Math requires a balance of these two styles such that students have the flexibility of acting intuitively to solve problems but thinking formulaically to apply the concepts and skills they have learned. Dyslexic students, and probably other students with learning disabilities, tend to stick to the formulaic way of thinking because this is what was taught in class and it seems safe to them. Adhering themselves to only one thinking style may be successful for learning basic concepts, but these students will struggle once word problems and other higher order math is introduced because they will be unable to link the concepts or rearrange the formulas to serve the purpose the students require.

Math in schools is taught almost clinically by following the prescribed curriculum and moving through the lessons and units at a predetermined pace. This pace continues from year to year as teachers at each grade level pick up where the previous teacher finished. At the beginning of the year, teachers usually begin with a miniature unit to refresh students on the concepts they learned the year before that will be useful for the new math they will be learning in the coming year. However, after a two-month summer break, many students have already forgotten what they have learned the year before. In some cases, if students are on a semester system, it may be an entire year before they enter another math classroom! With these large time gaps between instructional periods, teachers are left frustrated as they are caught in the middle, deliberating whether to stick to the curriculum or bring their students up to speed. Usually, the curriculum wins.

Math is Difficult for Some Students

The 2009 Nation’s Report Card warned that, amongst American eighth graders, 27 percent had not mastered basic math and a total of 66 percent were not proficient in math (Mazzocco, 2011). The way math is taught in our schools leads students to believe that math is a black and white subject—there is no gray area in which they can achieve a couple points for a good idea. Instead, they familiarize themselves with the sight of red x’s scattered across their page and learn to cope with getting by instead of succeeding. These students know that they must complete math to graduate; yet, they see no real purpose for their learning outside the walls of the classroom. These students may detest math and prove unsuccessful at it for various reasons. One of the reasons for the lack of success may be that the student possesses a Mathematical Difficulty (MD).

The education community has identified the umbrella term MD to describe students who struggle with math. Most MDs are a result of other learning difficulties and/or environmental influences. Students with a Mathematic Learning Disability (MLD) fall under this group, but they are a highly specific category, accounting for only 10 percent of all students in 2007 (Mazzocco, 2011). We can differentiate between MLDs and MDs based on test scores. Students with MLDs score low on basic non-symbolic (i.e. judging amounts without counting) and symbolic (i.e. understanding that five represents a specific amount) numbers tests, indicating a specific deficiency of skills and conceptual knowledge relating to math. On the other hand, students with MDs score around the median or higher on these same tests. Based on this data, we can conclude that students with MLDs struggle with numeracy and math itself, but students with MDs struggle as a result of an outside influence or other learning disability. Some of the main causes of MDs are language processing problems, visual-spatial confusion, memory and sequencing problems, and unusually high anxiety. Each disability or outside influence has an effect on students’ processing abilities, including input, cognition, and output.

Input

Input can be auditory, visual, tactile, or kinaesthetic. In most math classes, teachers rely on auditory and visual instruction. A teacher will write a math problem on the white board and show how to complete the problem while explaining aloud. In the first grade, students are given worksheets with problems resembling the one below:

4

+ 3

Teachers have demonstrated and explained what students must do, and they may or may not have used pictures to show what the equation represents, but most often, it is the teacher who has written the work or drawn the pictures. Students are not given the ability to practice the tactile aspect of the lesson. Many of today’s teachers still rely heavily on students’ auditory modality—the ability to listen to verbal directions and learn from them. If the child does not understand after the initial instruction, the teacher will re-explain the concept aloud, unwavering in her approach. For some students, it can be very difficult to process this information because only auditory and visual inputs are used.

Cognition

Cognition refers to one’s memory, understanding, and organization of inputted information. Cognition is a domain that causes many students to have difficulty with math due to the memorization and the organization of information. If input is effective, students may still struggle to process it because they fail to remember it or remember it incorrectly due to problems with short-term, working, and/or long-term memory. (Howard, 2011; Worling, 2005) Additionally, some students struggle to process information because their brains do not store the information logically, making it difficult for students to retrieve the information when necessary.

Output

Output refers to the information that an individual contributes through reading aloud, talking, drawing, and more. It is the final stage in the processing cycle and depends on the other stages for success. Students with motor disabilities, speech impediments, or other disabilities with output will struggle to give correct answers in a specified medium. In most cases, students are only given the ability to demonstrate their knowledge in one fashion, which creates difficulties for those students who cannot quickly or simply give the desired output in that manner.

Remember the discussion of how students are sometimes left behind to relearn concepts on their own time so that the class can continue moving forward? These students might have Math Difficulties (MDs). Maybe they were not exposed to numeracy until they entered kindergarten or perhaps they have never grasped a specific concept in a previous math course. Students can have learning disabilities (LDs) that are non-math related but that affect the students’ abilities to process information, resulting in an MD. There are many factors that could cause a student to develop an MD. MDs can generally result from directional confusion, sequencing problems, visual difficulties, spatial awareness difficulties, short-term and working memory problems, long-term memory problems, speed of working difficulties, language of mathematics, cognitive style or thinking style, conceptual ability, anxiety, stress, and/or low self-image (Chinn & Ashcroft, 2007). The following sections describe several types of disabilities that may contribute to the development of an MD.

Mathematic Learning Disability

Students who struggle with basic math despite thorough instruction can be identified as having an MLD, which is a domain-specific deficit in understanding or processing numerical information (Mazzocco, 2011). These students tend to score low on basic non-symbolic and symbolic numbers tests. Their problems stem from neurological differences in their development that change the way they learn and process information, specifically in regards to numerical information. Their problems lie in basic number sense and basic math skills with which other students who fall under the MD umbrella do not struggle. These students usually have good problem-solving skills but lack the computational skill to be successful (Siegel, 2013). Dyscalculia falls under this specialized category because it directly affects an individual’s math abilities, preventing the individual from achieving success in math. In fact, MLD and dyscalculia are often used interchangeably (Geary cited in Chinn & Ashcroft, 2007).

Dyscalculia

Dyscalculia is a math-based learning difference that interferes with students’ abilities to learn math regardless of the instruction they receive. Whereas dyslexia affects math because of difficulties with language and memory, dyscalculia is a condition in which a student has difficulty learning math because of something other than language. In fact, some of these students are actually above average in their language acquisition. There are only roughly 3 to 6.6 percent of students who are classified as having