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**V.8 #1 Mathematics - Probing Dyscalculia: Strategies to Strengthen Elementary Students' Understanding of Number Relationships and Estimation**

Brought to you by Learning Disabilities Worldwide (LDW®) through the generosity of Saint Joseph's University.

"What is dyscalculia?" is a question that I pose to my preservice elementary teachers in my mathematics methods course at the start of every semester. This question is usually met with blank stares and silence. Most have not heard the term and for those who have, they have no idea how to define it. However, when I ask "what is dyslexia?" hands go up and typical responses include that it is a learning disability which affects one's ability to deal with, acquire, and process language. Dyscalculia, a learning disability that affects one's functioning in mathematics, is not a term most preservice elementary teachers have heard yet it is important for both special and general educators to understand some of its characteristics as it is estimated that roughly 4-6% of the population is affected, equating to approximately one child in any average classroom (Bird, 2013). This article explores developmental dyscalculia, indicating a developmental problem as opposed to one that has been acquired through accident, illness, or other adverse circumstances (Hannell, 2013), and how elementary teachers can structure early number sense experiences to help those affected gain greater foundational knowledge.

Characteristics of Dyscalculia

According to the National Center for Learning Disabilities (2014), dyscalculia refers to a "wide range of lifelong learning disabilities involving math. There is no single type of math disability. Dyscalculia can vary from person to person. And, it can affect people differently at different stages of life." Even within this broad definition are characteristics which people with dyscalculia may possess. Synthesized from Bird (2013), Hannell (2013), Sousa (2008), and Stowe (2005), some characteristics of people with dyscalculia include:

Difficulty understanding what numbers mean

Puzzled by the relative values of numbers

Memory weaknesses such as difficulty memorizing math facts

Challenged by the reasoning needed for math such as not knowing which mathematics processes to use and the appropriate sequence of steps required to solve a problem

Spatial issues such as the ability to place numbers in correct places on a number line

Difficulty with mental manipulation of graphic images such as symmetry and tessellations

Poor comprehension of math symbols

Difficulty with everyday math such as telling time on an analog clock, making change with money, directions, or estimating quantities

Difficulty with the language of mathematics such as position, relationships, and size

Generally, people with dyscalculia lack an intuitive grasp of numbers (Bird, 2013). In other words, they have no feel for numbers. They lack the ability to estimate even small quantities and have no idea whether an answer to a problem is reasonable or not. The neurological basis of developmental dyscalculia appears to be an abnormality in the person's innate ability to subitize, to recognize the number of items in a small set without counting (Sousa, 2008). Without these fundamental skills, learning advanced concepts or processes can become very challenging.

Fostering Early Number Sense

Since students with dyscalculia lack strong number sense, it is important for teachers to work with them early on to develop these fundamental skills. Without them, more abstract and advanced concepts will become increasingly difficult. Therefore, greater attention must be paid to number sense in the early elementary grades because it forms the basis for a strong foundation.

It is essential for teachers to devote considerable time to develop core number sense in their students. Students with dyscalculia benefit from direct instruction and lots of opportunities for practice and mastery. They best learn new skills beginning with concrete examples and later moving to more abstract applications such as with the Concrete-Pictorial-Abstract (CPA) approach based on the work of Jerome Bruner (Bruner, 1960)[see SSL 2013, 6(2) Gujarati, for more detail on the CPA approach]. Teachers should provide enough time for practice and consolidation of skills at each stage and then revisit previously mastered skills often. Throughout the different stages, it is important for the teacher to routinely ask probing questions to have a greater sense of where students are at in their mathematical thinking.

Students with dyscalculia need increased activities to help develop their intuitive number sense, more intense and explicit teaching about the number system, more practice in using the number system, and concrete experiences with both large and small quantities (Hannell, 2013). Fundamental challenges that students with dyscalculia have are their difficulties in understanding the number line and how numbers are positioned on it, a poor sense of order of magnitude of numbers, and understanding relationships that numbers have with each other. Another challenge is the concept of estimating. To address the aforementioned challenges, the next sections showcase strategies to help students explore and strengthen their understanding and skills relative to number relationships and estimation. These strategies/activities were met with success when I taught elementary school.

Strategies to Strengthen Understanding Relationships of Numbers

Since retention and comprehension of more complex concepts lay on students' fundamental understanding of number sense, it is critical for students to have a strong grasp early on, notably of the number line. Having an intuitive grasp of the number line is essential to comprehending the relationship numbers have with each other. Students need to understand that the interval between each whole number is the exact same. Without an internal representation and understanding of the number line, students lack a reliable reference point for mathematical reasoning. Below are some strategies teachers can employ at the early elementary grades to aid with comprehension of the relationships among numbers. All the activities can be tailored to target specific student skills and grade levels.

Human Number Line. Either in a small group or with the whole class, give some students an index card with a number written on it. Have those students line up in order from smallest to largest. For those who did not receive a number card, ask probing questions such as: "what number is larger 3 or 7?"; "what number is one more than 8?; "what number is two less than 6"; "which numbers are between 15 and 20?; or "is 27 closer to 20 or 30?" Students can verbalize their answers or respond non-verbally by having to point to the person holding the correct number card(s). Students can also look for patterns. You can color code benchmark numbers such as multiples of 5 or 10. The numbers used as well as the probing questions depend on the skill level of your students and where you wish for them to advance. The beauty of this activity is that it is interactive and allows students to explore number relationships appropriate for their developmental levels.

Linear Board Games. Have students make their own linear board games. On grid paper, have students write the numbers 1-50 (or another range of numbers). They can decorate their game boards, even selecting a particular theme. Have students play in pairs. Give students dice and they have to roll and move that many spaces on their board. You can even specify that if they roll a certain number, they have to move backwards that many spaces. So, you can designate the forward and backward numbers. Premade 100s charts or commercial board games (or even computerized board games) could serve the same purpose, but it is often more motivating if students can create their own and take ownership of it. Then, they can bring it home to play with their families. Working with dice also aids in subitizing as students can gain greater pattern recognition with the dice dot patterns.

Tracking School Days. Track the days students have been in school on a number line. Many elementary classrooms count school days for a special celebration when the 100th day is reached. Ask probing questions such as: "if today is day 13, how many more days until the 20th day of school?"; "today is day 32 so five days ago we were in school for how many days?"; or "are we closer to the 50th day of school or the 100th?" Make sure to have a number line hung up around the room which is large enough to at least reach 100 and that the correct school day is visibly marked somehow. This activity has a real-world application which is appealing to students because it pertains to the number of days they have actually been in school in a particular grade level.

Strategies to Strengthen Estimation Skills

Estimation is a higher-level skill that requires students to be able to conceptualize and mentally manipulate numbers (Van de Walle, Karp, & Bay-Williams, 2013). Estimating is an important skill because it enables students to determine the reasonableness of their answer. The activities listed below can be tailored to specific developmental levels.

Estimation Jar. Each week, one student can be assigned to fill the estimation jar and bring it to class at the start of the week. This jar should be small/medium sized so as to not overwhelm students with the estimation task. Make sure that the jar is clear and unbreakable. Often, students like to fill the jar with edible treats but the jar can be filled with any objects such as pennies, paper clips, blocks, etc. Throughout the week, students have to estimate how many objects are inside the jar. They can record their estimates privately on individual slips of paper or students can go around and share their estimates publically while the teacher writes them on the board. Students, as a class, can then count the items in the jar. They can count by 1s, 2s, 10s, etc. They can even group the objects by benchmark numbers. After determining the actual number of objects in the jar, ask students to explain how they arrived at their estimates and talk about which strategies for estimating were the most successful.

Set Estimation. These activities involve comparing an unknown to a known quantity. Show students a stack of playing cards. Tell them that you are holding 25 cards in your hand. Then, show them another stack of cards (which could be larger or smaller than what you are holding) and have students estimate how many cards you have based on the information already given to them. This can be done with any set of objects such as unifix cubes, coins, or candy. Students have to think about the reasonableness of their answers since they have a reference point.

In sum, it is important for teachers to know the characteristics of dyscalculia to be able to assist those who need greater mathematics support. This article showcases a few strategies teachers can utilize which are interactive, can be used in small and large groups, and can be tailored for individual and class needs. Once students are more secure with number relationships and concepts of estimation, they have a greater chance of comprehending more abstract or complex concepts since they will have more reliable reference points for mathematical reasoning. However, a solid foundation has to be there first and that takes time and effort to develop.

References

Bird, R. (2013). The dyscalculia toolkit: Supporting learning difficulties in maths. Thousand Oaks, CA: SAGE Publications.

Bruner, J. S. (1960). The process of education. Cambridge, MA: Belknap Press.

Gujarati, J. (2013). Deepening mathematics teaching and learning through the

concrete-pictorial-abstract approach. Strategies for Successful Learning,

6(2). Retrieved from

http://www.ldworldwide.org/educators/strategies-for-successful-

learning/1096-deepening-mathematics-teaching-and-learning-

through-the-concrete-pictorial-abstract-approach

Hannell, G. (2013). Dyscalculia: Action plans for successful learning in mathematics. New York, NY: Routledge.

National Center for Learning Disabilities. (2014). What is dyscalculia? Retrieved from

http://www.ncld.org/types-learning-disabilities/dyscalculia/what-is-dyscalculia

Sousa, D. A. (2008). How the brain learns mathematics. Thousand Oaks, CA: Corwin Press.

Stowe, C.M. (2005). Understanding special education: A helpful handbook for classroom teachers. New York, NY: Scholastic Inc.

Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2013). Elementary and middle school mathematics: Teaching developmentally (8th ed.). Upper Saddle River, NJ: Pearson Education, Inc.

Joan Gujarati, Ed.D., is the Associate Dean for Accreditation and Technology in the School of Education at Manhattanville College in Purchase, New York where she taught the childhood mathematics methods courses. She is a former elementary school teacher and Math Teacher Leader. Dr. Gujarati has presented at numerous professional conferences and has published in the field of childhood mathematics education. Dr. Gujarati’s research interests include early childhood and elementary mathematics education, teacher beliefs and identity, teacher quality, effectiveness, and retention, and curriculum development. Dr. Gujarati can be reached at joan.gujarati@mville.edu