Developing Fluency Requires A Balance And Connection Between Conceptual Understanding And Computation Proficiency

In order to improve my instructional practices, I analyzed instructional data from district math diagnostic and proficiency assessments. The most recent assessment assessed student’s abilities to count, add and subtract, and their understanding of place value. My students scored below not only the other first grade students at the school, but also all first grade students in the district. 81.6% of my students could count, read, and write numbers to 120. This was an improvement from their diagnostic assessment. However, only 66.7% could relate counting to addition and subtraction, and only 45% demonstrated understanding of place value in two digit numbers.

From birth, it is important for practitioners to support the early years’ mathematical development. Children learn emergent maths which is a “term used to describe children construct mathematics from birth” (Geist, 2010). The Early Years Statuary Frameworks (EYFS) (Department of Education) states that maths is one of the specific areas.

Algebra is a critical aspect of mathematics which provides the means to calculate unknown values. According to Bednarz, Kieran and Lee (as cited in Chick & Harris, 2007), there are three basic concepts of simple algebra: the generalisation of patterns, the understanding of numerical laws and functional situations. The understanding of these concepts by children will have an enormous bearing on their future mathematical capacity. However, conveying these algebraic concepts to children can be difficult due to the abstract symbolic nature of the math that will initially be foreign to the children. Furthermore, each child’s ability to recall learned numerical laws is vital to their proficiency in problem solving and mathematical confidence. It is obvious that teaching algebra is not a simple task. Therefore, the importance of quality early exposure to fundamental algebraic concepts is of significant importance to allow all

Van de Walle, J, Karp, K. S. & Bay-Williams, J. M. (2015). Elementary and Middle School Mathematics Teaching Developmentally. (9th ed.). England: Pearson Education Limited.

The math concepts taught in this lesson are teaching the students how to use certain math formulas, and practice addition and multiplication. It is beneficial for students to know what tools to use for capturing and displaying information that is important to them (Davis, 2011). The science concepts taught in this

This program is appropriate in a diverse, 4th grade general education classroom. The modules are made up of “Topics” and “Lessons” that are aligned to Common Core State Standards (CCSS). Each module provides the foundational standards needed for the lessons (i.e. CCSS from the previous grade), as well as the focus grade level standards. The first module introduces concepts which are then spiraled within the next module’s focus. While the modules are thematic and based on each mathematics domain (base ten numbers, geometry, fractions, data, algebraic thinking), some standards are seen across topics and lessons. Each lesson has allocated time to four major components: fluency practice, concept development, application problems, and student debrief.

At what point does a student, in all intents and purposes, experience something mathematical? Does it symbolise a student that can remember a formula, write down symbols, see a pattern or solve a problem? I believe in enriching and empowering students’ mathematical experiences that fundamentally stems from a Piagetian genetic epistemological constructivist model that allow students to scaffold their learning through cognitive processes that are facilitated by teaching in a resource rich and collaborative environment (Thompson, 1994, p.69).

The National curriculum states that in Mathematics teachers should use every relevant subject to develop pupils’ mathematical fluency. Confidence in numeracy and other mathematical skills is

“Making Sense: teaching and learning mathematics with understanding” is a book that focuses on learning for understanding, rather than having students blindly memorize topics that are deemed important. If students understand and interact with what they are learning, they are more likely to enjoy it and retain the concepts for further use. “Making Sense” outlines five dimensions teachers can use to create this type of classroom. These dimensions are the nature of classroom tasks, role of the teacher, social culture of the classroom, the use of mathematical tools as learning supports, and equity and accessibility. Hiebert writes, “All five dimensions and the critical features within each are needed for the system to work” (Hiebert, 12, 1997).

Adopting the actor-oriented perspective, Ellis (2007) developed a student-centered generalization taxonomy to describe the different types of generalizations that students create when reasoning abstractly. The taxonomy distinguishes between students’ generalizing actions and the product of generalizing (i.e. reflection generalizations). Generalizing actions include relating, searching and extending. When relating, students form an association between two or more problems, situations, ideas, or mathematical objects by recalling a prior situation, inventing a new one, or focusing on similar properties or forms of mathematical objects. When searching, students engage in a repeated mathematical action, such as calculating a ratio, to find an invariance

Children at this age also have different skills related to numeracy. Fostering numeracy skills in early childhood education not only predicts for later Math achievement but also for greater literacy skills (NAEYC, 2009). Since mathematical concepts are often intertwined with so many areas such as science, literacy, dramatic play, block building, and more, math learning centers offering interdisciplinary materials and activities ensure that young children grow to understand and appreciate the real life applications of mathematical concepts (Fu, 2010).

Multiplicative thinking is imperative to a child’s understanding of important mathematical concepts and is seen as the ‘big idea’ in number that links multiple key ideas and strategies (Vergnaud, as cited in Siemon, 2011). Commonly, children have a procedural based view of multiplicative thinking which can hinder progress, as opposed to a more conceptual view which is a far better learning framework (Hurst & Hurrell, 2016). If teachers are to maximise a child’s learning, they must acknowledge this and help children maintain a conceptual understanding of multiplicative thinking and emphasise this much more so than procedural rules. Several key ideas and strategies underpin the success of multiplicative thinking and a greater conceptual understanding.

It is imperative that learners have a rich understanding of numerical concepts. Many of the later math disciplines and skills that students will encounter are going to require knowledge and application of basic numeracy skills. Merely following steps in a process (e.g. multiplying fractions) or using an algorithm without knowing what is happening to the numbers, will leave the student void of true understanding and in turn, inhibit the path of learning. For example, imagine a scenario where a student who was never able to understand primary concepts related to prime and composite numbers finds him or herself in a classroom years later that requires factoring trinomials. Teachers may never be able to make math easy for their students, but they can support students in a way that enables them to be successful by ensuring they teach it in a way that promotes genuine understanding and knowledge of numerical content.

Maths is ubiquitous in our lives, but depending on the learning received as a child it could inspire or frighten. If a child has a negative experience in mathematics, that experience has the ability to affect his/her attitude toward mathematics as an adult. Solso (2009) explains that math has the ability to confuse, frighten, and frustrate learners of all ages; Math also has the ability to inspire, encourage and achieve. Almost all daily activities include some form of mathematical procedure, whether people are aware of it or not. Possessing a solid learning foundation for math is vital to ensure a lifelong understanding of math. This essay will discuss why it is crucial to develop in children the ability to tackle problems with initiative and confidence (Anghileri, 2006, p. 2) and why mathematics has changed from careful rehearsal of standard procedures to a focus on mathematical thinking and communication to prepare them for the world of tomorrow (Anghileri).

Students continue to struggle to understand rational numbers. We need a system for identifying students’ strengths and weaknesses dealing with rational numbers in order to jump the hurdles that impede instruction. We need a model for describing learning behavior related to rational numbers – prerequisite skills and development of rational number sense – that is dynamic and allows for continuous growth and change. It would inform us of the important background knowledge that students bring with them and the prior experiences that influence their level of understanding. It would further enable us to assess students’