As a pre-professional student in Math Methods I have gleaned insightful knowledge and wisdom on the strategies and theories that build the foundation of Elementary Mathematics. Through various class articles, research projects, peer evaluations and discussions I have successfully compiled facts, case studies and tested theories into my own mathematical philosophy.  I consider this included information valuable due to its versatility and flexibility in various grade levels, its ability to reach unique learners and its positive impact on all classroom environments. Thus, I believe the dominant themes of early number sense, mathematical proficiency and the use of differentiated instruction are essential in fostering math confidence in all students.

According to the National Math Panel Report, students in the United States are behind in mathematical proficiency compared to other international students (pg. 18). Students are failing as they reach higher grade levels and there are major achievement gaps upon entry into college. With this information in mind, it is essential that future teachers work with children who are at risk or children with low levels of parental education so that these students have hope for their future. Not only should we begin teaching these mathematical/algebraic principles early, but it is essential that all students are taught basic skills that aide in understanding and developing early number sense.

In Bahr and Garcia’s (2010) Elementary Mathematics is Anything but Elementary, Hilde Howden, defines number sense as “good intuition about numbers and their relationships” (p.36). It is the basis in which students learn mental calculations and flexibility and fluidity of numbers. This requires early activities that help students draw relationships between numbers in different contexts and may be acquired through naturalistic opportunities, informal mathematics, or formal mathematics. Students need to be introduced to number sense through naturalistic opportunities such as completing everyday chores, practicing pretend play or simply running errands with their parents. As teachers however, we must be aware that many students do not have the opportunities at home to build number sense and thus this concept must be developed at school. Students then need to be exposed to informal mathematics which provides play materials to enhance mathematical learning.  Clement and Sarama’s (2010) article entitled “Math Play” shares exemplary ideas on how to incorporate informal math play into the classroom.  Students acquire the understanding of patterns through making necklaces, geometric and spatial reasoning through arranging dollhouses, dynamics through building blocks and strategical reasoning playing games such as Memory or Go Fish (p. 42-44). According to Tina Bruce, a Professor of London Metropolitan University, “through research on the brain, as well as in other areas of study, it is becoming continually clear that childhood needs play. Play acts as a forward feed mechanism into courageous, creative, rigorous thinking in adulthood.” All students need opportunities to experience math in their everyday life and it is the teacher’s professional responsibility to provide this ‘play’ for the students.

As students grow older, however, this number sense develops into a more formal and structured understanding of number relationships. The student then begins to develop one of the most important elements of mathematical literacy which is mathematical proficiency. The primary error among math teachers today is the inability to teach mathematical proficiency correctly. Within the definition of this word there are two specific terms: conceptual understanding and procedural fluency.  Bahr and Garcia’s also state that conceptual knowledge is described as mathematical idea that focuses on why or how something works. Procedural knowledge however, is focused on the rules and principles of carrying out a mathematical procedure (pg. 126). Unfortunately, many students are not taught the conceptual ideas of mathematics and thus they are left with information that is purely procedural. Students may understand how to carry out a specific procedure of a problem. However, without conceptual understanding, the student is unable to take this problem and solve it in different contexts or understand ‘why’ they are doing what they are doing. Without understanding, the student will become lost and confused and will be unable to solve challenging problems in the future. In Bahr and Garcia’s Elementary Mathematics is Anything but Elementary,” Piaget emphasizes equilibration or importance of the use of conceptual learning which encourages the storage of information in the student’s long term memory while fitting new information into ones schema; having a concept map of various skills makes it easier for the student to retain information of the past, present and future. (p.103). However, although conceptual understanding is important in the development of a young math student, it cannot be taught without knowledge of how to complete the problem using procedurally fluency (Bahr & Garcia, 2010, p.104). Teachers must learn to balance both conceptual and procedural knowledge, and this can be completed through the use of the Concrete, Representational and Abstract models when introducing a new lesson.

The goal of the Concrete, Representational and Abstract model is for the teacher to gracefully combine the concrete elements with the abstract elements while teaching both conceptual and procedural fluency. In Flores article entitled Teaching Subtraction With Regrouping to Students Experiencing Difficulty in Mathematics students of culturally diverse backgrounds were taught how to subtract using this method (p. 148).  After intervention using the CRA model, the students showed increased accuracy while completing their problems. For example, a student who originally had seven correct digits increased his performance to sixteen correct digits (Flores, 2009, pg. 150). In younger grades especially, it is necessary that the students begin with a very basic, visual model using base ten blocks or any other concrete object to represent the problem (Bahr & Garcia, 2010, pg. 27). Teachers must then scaffold students to begin to draw the problem by themselves and then slowly the students move into more abstract mental-math strategies. If a teacher simply moves into the abstract method of solving a problem, the students will most likely memorize the procedure and forget about understanding the problem.

Lastly, the CRA method of teaching reminds teachers of the importance of combining all mathematical content standards as well as aligned processes to create a comprehensive mathematical curriculum with differentiated instruction. Bahr and Garcia state that in schools today, inclusion is becoming more prominent in every classroom (pg. 375). “Teachers must know how to determine the individual child’s strengths and weaknesses and then maximize the strengths and minimize the weaknesses” (Karp & Howell 2004, pg. 376). Teachers are now responsible for teaching students with exceptionalities who are placed in the general education classroom and who are in need of strategies, modifications and accommodations. Thus, it is essential that teachers begin to build their toolkits early while connecting math to real-life situations. Through the use of base ten blocks, 100 charts and unifix cubes, students gain strategic competence in being able to represent and solve basic addition and subtraction problems. Also, using assessments such as journals and peer group activities as well as problem solving strategies such as the Braid Model, students gain adaptive reasoning and the ability to self-reflect on their own learning and the learning environment in the classroom. In addition, good teaching, and differentiated activities allows the student to gain a productive disposition on mathematics and focus more on their effort rather than their ability.

The world of mathematical tools and strategies is infinite. All teachers need to focus on teaching with best practice and using curriculum based assessments as early as preschool and kindergarten; the early students start working on these mathematical competencies, the better prepared they will be in later grades. It is not enough to teach one method using one tool and no modifications. Society is diverse, different homes are diverse, schools are diverse and the students are diverse in the ways in which they learn. Teachers need to provide their students with the tools necessary to have fun while developing mathematical numeracy and mathematical proficiency. The student must learn how to take ownership of these tools and use them in a way that is beneficial to their learning; as eloquently stated by VandeWalle, “mathematics must begin where the students are.”