It is not uncommon to hear both children and adults proudly proclaim their lack of skill in math. “I am not very good at math,” seems to be a socially acceptable statement. Students say it, parents say it, even teachers say it, and then we all laugh about it. Studies have shown that attitudes and social forces do have an impact in academic performance, particularly in mathematics, whether that influence is coming from parents (Soni & Kumari, 2017) or teachers (Beilock, Gunderson, Ramirez, Levine, & Smith, 2010). As special educators, we have to work to combat this societal acceptance of math phobia and employ specific strategies to help students with learning disabilities in math (MD) become proficient in a topic that may be a source of anxiety.
Students with MD often display poor attitudes toward mathematics. Studies have determined that these struggles often begin as early as first grade, with a negative relationship between attitude and achievement observed across all grade levels (Ma, 1999). Given the link between attitudes and performance in math, it is imperative for special educators to recognize the role teachers play in creating positive and negative learning experiences in math, and how powerful and memorable those experience are for students.
Teacher skill and attitudes
Numerous studies show that teacher content knowledge holds more power than curriculum or methods, and directly affects student outcomes (Hill, Rowan, & Ball, 2005, National Mathematics Advisory Panel, 2008; Good, 1979). Teachers need a high level of mathematical understanding in order to recognize student misunderstanding and analyze student errors (Shields, 2005) then respond quickly with corrective feedback and appropriate practice opportunities.
Content preparation in special education is unique when compared to general education. Many states license special education teachers K-12, without emphasis or focus on a specific content area (i.e. math, reading, science). This means that special education teachers could support learners in a range of mathematical content areas, from early numeracy skills to high school algebra, and many do so after taking only one or two math courses, required as part of a teacher preparation program. It is a challenge for special educators o become highly skilled in all areas of mathematics, but it is also a matter of professional obligation to reach a level of minimal mathematical competency and to work toward deeper content knowledge across grade levels (Humphrey & Hourcade, 2010).
Personal Math Stories
When working with both preservice and in-service special education teachers, to improve their math instruction, I often start with an activity where participants write their own math stories regarding experiences they remember most in math. Participants report that their negative feelings regarding math are vividly tied to specific educational interactions. This activity is used to demonstrate how much a teacher can influence a student’s attitude toward math well into adulthood. Here are a few quotes from those math stories.
“She made you feel so dumb for asking a question.”
“I failed math, my only F ever. So I had to take a credit recovery class, which was all hands on, and I passed with flying colors.”
“Math made me cry, because once I got home, I couldn’t remember how to do the “tricks” the teacher taught us.”
“I kept asking the same question, because I did not understand, and the teacher just kept giving the same explanation, so I just gave up.”
Some common themes are revealed throughout these teacher math stories. First, many of the memories are linked to a specific teacher. Second, positive experiences usually involve instruction that incorporated conceptual instruction, teachers who could modify instruction to meet the needs of all learners, and teachers who were excited about teaching math. Third, negative experiences typically involved teachers who focused on math “tricks”, teachers who seemed to struggle with providing explanations, and timed math tests.
McCulloch, Marshall, DeCuir-Gunby, and Caldwell (2013) also used math autobiographies with K-2 teachers to analyze their own experiences around the people and the activities that helped to form their math identity. It was found that even though teachers were asked to recall both people and activities from their own childhood, they generally focused on the people. Positive experiences were generally attributed to teacher practice, such as high expectations, high levels of support, and effective teaching strategies, while negative experiences focused on frustration and lack of support. These findings suggest that teachers may have a strong influence on student attitudes towards mathematics and the potential to shift these beliefs. Therefore, teachers who experience negative feelings towards mathematics need to examine ways to adjust their attitude through pedagogical and content development.
Knowing that many of our teachers seem to have had similar negative experiences as our struggling learners, it is imperative that preservice and in-service teachers have the opportunity to increase their own confidence in teaching mathematics through emphasizing the appropriate teaching strategies and methods that we know positively influence student learning.
Strategies to reset math attitudes
Certain common instructional techniques have been shown to negatively influence math attitudes. Inflexible teaching methods, such as insisting on one method to solve a problem, can cause anxiety in learners (Oberlin, 1982). Strict adherence to a prescribed algorithm over the examination of multiple methods to solve a problem can be a source of frustration, especially for student who may be able to find the correct answer through a different method. Focusing on quickly obtaining the one correct answer can also be a source of frustration (Shields, 2005) and emphasizes the algorithm over conceptual understanding, often through the use of math “tricks” which have no basis in mathematical reasoning (Cardone, 2015).
Understanding the cause of anxiety, stress, and lack of confidence in mathematics instruction can help us to identify techniques for improving math attitudes. In working with preservice and in-service special education teachers who work with students with MD, I emphasize three key instructional techniques that can be implemented regardless of grade level taught: teach for conceptual knowledge before procedural knowledge, teach to flexibility within declarative knowledge, and avoid using math tricks.
Teach for conceptual understanding before procedural knowledge
The need for teaching to conceptual understanding before procedural knowledge is a key piece in improving the success of students with MD. Conceptual knowledge is about acquiring an understanding of mathematical concepts and why mathematical procedures work. Conceptual understanding also includes linking information and patterns to see how concepts work together, for example, understanding that addition and subtraction are inverse operations, or explicit understanding of how fractions and decimals are related (Miller & Hudson, 2007). Procedural knowledge is knowing how to use a set of steps to solve a mathematical problem. This is the “how” of solving mathematical tasks. Both types of knowledge are equally important in becoming proficient mathematicians, but it is important that as special educators, focus is on ensuring conceptual understanding prior to moving on the procedural knowledge. When students possess the conceptual background, mathematical procedures are not just random steps to follow, but instead hold meaning. This understanding leads to student success, which in turn leads to improved math attitudes.
To develop conceptual understanding before procedural knowledge, one effective evidence-based practice is the concrete-representational-abstract (CRA) framework. The CRA framework begins with teaching concepts in a concrete manner. This means that students have manipulatives in hand, performing mathematical tasks without the focus on numerals and operational symbols. Once students show mastery at the concrete level, instruction moves to the representational level. In the representational level, students use drawings and pictures instead of manipulatives. It is important to be explicit in the transfer from concrete to representational by initially using both manipulatives and drawings at the same time to show the relation between the two. As mastery is shown in the representational phase, it is appropriate to then move to the abstract. Again, it is necessary to be explicit in the transfer from representational to abstract, showing both together to reinforce the relation between the conceptual to the abstract. The abstract phase focuses on solving problems without the use of manipulatives or drawings, just with numerals and operational symbols. This is where students strengthen procedural knowledge. Special education teachers need to understand how to move fluently through the CRA framework. It is not necessarily a process where one moves from concrete to representation to abstract and is then finished. As more difficult problems and concepts are introduced, it may be necessary to move back and forth through the framework, as students need help in understanding and making connections from conceptual knowledge to procedural knowledge.
Declarative knowledge flexibility
Declarative knowledge in mathematics is also a key piece to student success. Declarative knowledge involves information that needs to be recalled immediately, whether that is recognizing quickly that 5 + 7 = 12 or that a certain figure is a triangle. This quick retrieval of information aids students in moving through more complicated math skills (Miller & Hudson, 2007). Many students with MD struggle with working memory, making declarative knowledge and the accompanying speed testing a stressful activity. When teaching and assessing declarative knowledge, it is necessary to assess using some sort of time-bound procedure such as Constant Time Delay or timed probes, as the key is that students recall information automatically.
An effective way to approach teaching basic math facts is through interactive activities. Games on tablets or computers can be one way for students to practice basic facts in a way that brings down anxiety levels. Williams (2000), studied basic fact declarative knowledge practiced through either traditional paper and pencil timed tests or a computer multiplication puzzle game. Results showed that participants using the computer game for practice significantly increased performance on a timed multiplication fact test, where those in using the paper and pencil practice did not. In another study of using computer-based practice, Kanive et al (2014) found that students in the computer-based practice group outperformed peers.
Another way effectively teach math facts, while also decreasing the stress connected to memorizing these facts, is to emphasize the patterns and relations between known and unknown facts. If we were to memorize all of the basic multiplication facts zero to twelve, there are 169 math facts. That number can be cut in half when students understand and apply the commutative property of multiplication. Then focusing on some key facts, such as multiplying by 2s, 5s, and 10s, will give students a foundation. Instructional strategies that focus on number sense are helpful in expanding past those base facts. For example, if a student immediately knows 6 x 5 = 30, then quickly calculating 6 x 6 = 36 by knowing to add on another 6, should not be as difficult.
Math tricks or sayings often have no basis in mathematical reasoning, and students have no idea why they work (Cardone, 2015). For example, many students are taught to add and subtract fractions using the “butterfly method” (see Figure 1). While this trick may help students by quickly creating a common denominator by which to add or subtract, it often leads to irregular fractions that need to be simplified thus adding extra steps. In addition, using this method does not require students to conceptually understand how to find a common denominator. The introduction of tricks to help a student get through the lesson and solve an algorithm correctly, but the tricks quickly become meaningless without strategies that lead to understanding the true concept. When students can apply concepts, they can check to see if their answers are appropriate instead of relying on the meaningless magic of a trick.
With older students, it may be difficult to undo some of the tricks that have been taught. One resource to support in reversing the overreliance on tricks is the website NixTheTricks.com. Cardone (2015) worked with math colleagues to collect common tricks and provide fixes that lead to conceptual understanding.
Research has shown that “math stories” or personal experiences in math have a profound effect on attitudes and confidence in mathematics. We know when math attitudes improve, performance improves (Ashcraft, 2002). In order to improve attitudes toward math, special educators need to focus on developing their skills in using teaching strategies that promote conceptual development. Whether this be using CRA, focusing on flexibility in declarative knowledge, or removing tricks from math instruction, it is clear that an important element in improving math instruction is teacher attitudes and skills needed to differentiate mathematics in meaningful ways that lead to conceptual understanding. Research needs to continue to examine the relationships between teaching attitudes, pedagogical skills, and student performance in mathematics. Examining these connections will likely reduce the accepted and expected math phobias and fears in both children and their teachers, and lead to increase confidence in math.
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