LEARNING DISABILITIES IMPACTING MATHEMATICS Ann Morrison, Ph.D.

Your Own Experience  Think back to when you were in elementary, middle, and high school. What was your experience learning mathematics? Was it easy for you? Was it difficult? Is remembering your experiences learning math making you happy or anxious at all?

Table Discussion  What is the difference between people who are just “bad” at math and people with dyscalculia?  Share anything you want about your own experience learning math

Dyscalculia  Recognizing numbers  Fluidity and flexibility  Visualizing  Counting  Estimating  Measurement  Manipulation of numbers  Patterns  Spatial relations  Rules

CO Prepared Graduate Competencies in Mathematics, 1 st half  The prepared graduate competencies are the preschool through twelfth-grade concepts and skills that all students who complete the Colorado education system must master to ensure their success in a postsecondary and workforce setting.  Prepared graduates in mathematics:  Understand the structure and properties of our number system. At their most basic level numbers are abstract symbols that represent real-world quantities  Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error  Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency  Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning  Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

CO Prepared Graduate Competencies in Mathematics, 2 nd half  Solve problems and make decisions that depend on understanding, explaining, and quantifying the variability in data  Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations  Make sound predictions and generalizations based on patterns and relationships that arise from numbers, shapes, symbols, and data  Apply transformation to numbers, shapes, functional representations, and data  Make claims about relationships among numbers, shapes, symbols, and data and defend those claims by relying on the properties that are the structure of mathematics  Communicate effective logical arguments using mathematical justification and proof. Mathematical argumentation involves making and testing conjectures, drawing valid conclusions, and justifying thinking  Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

Factors Complicating Mathematics Education  Sometimes parents can help, sometimes they can’t  Bottom-up versus top-down approaches oftentimes don’t match with students’ learning  Skills for algebra are different than for geometry  “Minute math” – timed computation  Inflexible teachers  Teachers for whom math came very easily

Learning Disabilities Including Dyscalculia  Typically stem from challenges in one of the following cognitive processes:  attention  language  visualization  metacognition  memory, storage and retrieval

Language for Mathematics  Difficulties with language can be caused by challenges with:  phonological awareness which slows decoding, particularly of multisyllabic words.  connecting language with imagery, which impacts language comprehension.  memory storage and retrieval, which slows the speed at which words can be read and understood.

Language for Mathematics  Functions presented as expressions can model many important phenomena. Two important families of functions characterized by laws of growth are linear functions, which grow at a constant rate, and exponential functions, which grow at a constant percent rate. Linear functions with a constant term of zero describe proportional relationships. Colorado Common State Standards for Mathematics, 2009

Visualization and Mathematics  Mathematics often requires students to create a mental model based on partial visual models that are provided or entirely on language.  Solve the problem on the next page. While you do, stay aware of how you use visualization to solve it.

Visualization and Mathematics

 How did you use visualization strategies to solve the previous problem?

Metacognition and Mathematics  Metacognition is a skill that includes:  Before completing a task: Planning an approach or strategy for how to complete the task Deciding how you will know whether you completed the task successfully  While completing the task: Evaluating how well the approach or strategy is working Changing to another approach or strategy if the first one isn’t working  After completing the task: Deciding whether you completed the task successfully If the task wasn’t completed successfully, deciding how to approach it differently on the next attempt

Metacognition and Mathematics  The next slide has a math problem on it.  Use metacognitive skills in solving it.

Metacognition and Mathematics

 How did you use metacognitive skills in solving the previous problem?  Before solving it?  While solving it?  After solving it?

More concrete More Abstract Symbolic Iconic Enactive

 The image provided on the previous slide is iconic in the symbolic, iconic, and enactive model  Create an enactive model  Create a symbolic model  There are many ways to do both, no one answer How Can You Represent the Dots?