This is the Hierarchy of Skills for teaching fractions given
by Mr. David Ali. The questions I pose to everyone are:
1. 1. Were you as a student taught following this
format when you were in school?
2. 2. Do you believe that teachers are aware of this
system when it pertains to teaching fractions in their classroom?
3. 3. Do you think children will understand fractions
better if they are taught following this hierarchy of skill?
Our group concluded that we were only sometimes taught using
these levels, however due to the time constraints of the lessons teachers were
often in a rush to “complete the syllabus” instead of allowing students the
opportunity to grasp understanding. We also believe that teachers are aware of
the hierarchy of skills but due to the large content and level scope they often
bypass it to accomplish the task of completing a packed syllabus in such a
short time. Honestly we also believe that teaching following accomplishment of
these different levels will ensure that students learn and UNDERSTAND the
content of fractions better.
Hierarchy of Skill for fractions.
Develop an
understanding of fractions using area models.
·
Identify wholes and parts of wholes.
·
Differentiate between equal and unequal parts of
the whole.
Become aware of the names associated with fractions to
tenths using area models.
·
Explore the relationship among concrete (area
model), pictorial and symbolic representations of fractions up to tenths.
Demonstrate an understanding of equivalent fractions.
·
Compare and order fractions by direct comparison.
·
Explore equivalent forms of fractions with
denominators up to ten.
·
Compare and order fractions using the concept of
equivalence
Extend the concept of fractions to include multiple
representations, equivalence, ordering and simple computation.
·
Explore fractions using area, linear and set
models.
·
Recognize and generate equivalent fractions
using a variety of models.
·
Use the algorithm for finding equivalent
fractions.
·
Compare and order proper fractions with unlike
denominators using equivalent forms.
·
Distinguish between proper, improper and mixed
number and convert from one form to another.
·
Add and subtract proper fractions with same
denominators.
Demonstrate an understanding of solving problems involving
fractions and the four operations.
·
Add a fraction to a whole number.
·
Subtract a fraction from a whole number.
·
Add and subtract fractions involving same
denominator and one denominator a multiple of the other.
·
Multiply fractions by whole numbers.
·
Calculate the whole given a part as a unit
fraction.
·
Divide whole numbers by fractions.
·
Solve real-life problems involving fractions and
using the algorithms developed.
Demonstrate an understanding of adding and subtracting
fractions and mixed numbers, concretely, pictorially and symbolically.
·
Develop and apply algorithms to add and subtract
fractions and mixed numbers.
·
Solve problems involving addition and
subtraction of fractions including mixed numbers.
Demonstrate an understanding of multiplying a fraction by a
whole number, multiplying fractions and mixed numbers concretely, pictorially
and symbolically.
·
Develop and apply algorithms to multiply:
ü
a fraction by a whole number
ü
fraction by fraction
ü
mixed numbers
Demonstrate an understanding of dividing whole numbers by
fractions, fractions by whole numbers and fractions concretely, pictorially and
symbolically.
·
Solve problems involving the multiplication of:
ü
a fraction by a whole number
ü
fraction by fraction
ü
mixed numbers
·
Develop and apply algorithms to divide:
ü
a whole number by a fraction
ü
a fraction by a whole number
ü
A fraction by fraction.
·
Solve problems involving the division of:
ü
a whole number by a fraction
ü
a fraction by a whole number
ü
A fraction by a fraction.
The members of my group do not recall whether or not we were taught fractions in the order above. However, we concur that most teachers are aware of it since it has been outlined (for the most part) in the former curriculum and in many text books.
ReplyDeleteThe hierarchy is very useful as it is premised on sound mathematical pedagogy. Bruner, for example, suggested the use of different modes of representation-enactive, iconic and symbolic. The hierarchy also points to the use of 'multiple representations'. Additionally, by solving real-life problems, as outlined in the hierarchy, students can discover the relevence of the use of the concept and processes. Consequently, fractions will be more meaningful to students and they will be more motivated to learn more about them. Finally, Mr. Ali's outline shows a progression that fosters what Skemp describes as rational understanding- students gradually build on previously learned concepts.
Another important aspect of teaching/learning fractions comes into play even before this hierarchy is used- it is the teaching/learning of pre-number skills. Since fractions are numbers, pre-number skills are fundamental to the understanding of fractions.
A third consideration in the teaching and learning of fractions is the strategy that is used by the teacher. Discovery learning allows students to construct their own knowledge of the topic. Similarly, by using an inductive approach, a teacher is more likely to facilitate long-term resonance of the concept. The key is to ensure that each aspect of the concept is thoroughly comprehended before moving to a more advanced level. For this reason, differentiated instructions may be necessary.
Based on the theoretical value of the hierarchy, this group suggests that after pre-number skills have been mastered, it can be taught using specific strategies to better facilitate the teaching/ learning of fractions.
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ReplyDeleteYour points are very valid, it is very sad that so many teachers in the classroom rob their students of understanding all in favor of completing a curriculum and syllabus in such short time constraints. I believe all children can learn Mathematics if they are taught following systems that cover all skills systematically.
ReplyDeleteThe Challenges of Teaching fractions in Primary School
ReplyDeleteWhy Study Mathematics?
By studying Mathematics, students develop the ability to think creatively, critically and strategically. They learn to structure and to organize, to process and communicate information and to enjoy intellectual challenge. In addition, students learn to create models and predict outcomes, to conjecture, to justify and verify, and to seek patterns and generalizations. Mathematics has a broad range of practical applications in everyday life, in other learning areas, and in the workplace.
How Is Fractions in Mathematics Structured?
"Mathematics fraction content is sequential in nature. There is a hierarchy of concepts and skills on which each major area can be built. The proper ordering of mathematical content for all learners is critical to mathematical achievement." (Mathematics Curriculum, 1999)
The guiding principles of the Mathematics curriculum content are derived from the National Council of Teachers of Mathematics standards that will allow our students to explore, discover, analyze and apply mathematics, to model and solve real world problems (NCTM.org). The NCTM standards of problem solving, reasoning, communication, representation and connections, also play an integral role in how content is delivered.
Through an integrated approach, learning fractions in the Primary Curriculum of Mathematics aims to reduce “Math anxiety” by:
1. The development of core mathematical concepts and skills by the restructuring of learning activities to enable students to see connections with their daily lives.
2. The development of appropriate dispositions that would facilitate higher order thinking skills.
3. A pedagogical approach that uses a variety of student- centred teaching techniques and strategies.
According to Adding It Up: Helping Children Learn Mathematics (2001), instructional programs must address the development of Mathematical Proficiency by focusing on the following five interwoven strands or components as prospective teachers we must engulf our student’s knowledge through these strands of:
1. Conceptual understanding: comprehension of mathematical concepts, operations and relations.
2. Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently and appropriately.
3. Strategic competence: ability to formulate, represent and solve mathematical problems.
4. Adaptive reasoning: capacity for logical thought, reflection, explanation and justification.
5. Productive disposition: habitual inclination to see Mathematics as sensible, useful and worthwhile, coupled with a belief in diligence and one’s own efficacy. It is essential that the forgoing issues are seriously considered and effectively addressed so as to create literate
My group members and I do not remember if we were taught fractions the in order that the hierarchy outlines however we do believe that most if not all teacher are aware of the hierarchy but as Cassandra pointed out many of them probably ignored it in order fulfill the demands of completing the syllabus and they short changed their students in the process. The focus of the teacher was on the content rather than the students, something they we are being taught today in our class that should not happen after all what is the point of completing the syllabus on time if what was taught to your students was not understood.
ReplyDeleteGroup members: Rishmattie Maharaj, Christine Jianath, Angalie Maraj, Maya Dass
ReplyDeleteMathematics Advisory Panel, 2008; Council of Chief State School Officers [CCSSO] and National Governors Association [NGA], 2010). Because of the importance of fraction understanding, documents such as Principles and Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 2000), Foundations for Success (National Mathematics Advisory Panel, 2008), and the Common Core State Standards for Mathematics (CCSSO and NGA, 2010) recommend intense focus on fractions from fourth through eighth grades. However, many students struggle with basic fraction and rational number concepts in these grade levels (Lamon, 2007; Wu, 2005). A lack of visualization skills offers one explanation for students’ difficulties with fractions. The visualization of mathematical concepts plays a pivotal role in how well students apply their fraction understanding to novel situations (Arcavi, 2003).
NCTM states, “The ways in which mathematical ideas are represented are fundamental to how people can understand and use those ideas” (2000, p. 67). Therefore, as learners develop clear and sophisticated visualizations of mathematical concepts, they will have a deep understanding of those concepts, and develop what Tall and Vinner (1981) refer to as a concept image. In this study, we define a visual static model, as a still picture that is either printed or drawn on a page to represent mathematical concepts. In this study, we adopt Arcavi’s (2003) definition of mathematical visualization: the ability to create, use, interpret, and reflect on images in the mind or on paper. Therefore, students use and create visual static models as they develop mathematical visualization skills. These visualizations support meaningful connections with different types of representations and abstract mathematical concepts. Lesh, Post, and Behr (1987) identify five types of mathematical representations: static pictures, manipulative models, written symbols, real-life situations, and spoken language. Understanding a mathematical concept involves a) recognizing the concept among different types of representation, b) flexibly manipulating the concept within a type of representation, and c) translating the concept from one type of representation to another. Static pictures are of particular interest to this study because static models are what students often develop when problem solving, and are often what students see on tests, worksheets, and in textbooks during typical mathematics instruction (Yeh & McTigue, 2009).
References:
Abrams, J. P. (2001). Teaching mathematical modeling and the skills of representation. In A. A. Cuoco & F. R. Curcio (Eds.), The Roles of Representation in School Mathematics, 2001 Yearbook (pp. 269-282).
Reston, VA: National Council of Teachers of Mathematics.
Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), 215–241.
Group members: Rishmattie Maharaj, Christine Jianath, Angalie Maraj, Maya Dass
ReplyDeleteVisual representations alleviate cognitive load during problem solving (Clark, Nguyen, & Sweller, 2006) and allow learners to mentally work on one part of the model without having to keep track of the entire model in their minds (Woleck, 2001). For example, many students automatically picture a square divided equally into three parts, two of which are shaded, when they hear or see the symbol, 2/3. This visual model enables learners to maintain the part-whole meaning of the fraction. Findings by van Garderen (2006) also indicate that visualization skills correlate significantly with students’ ability to understand mathematics. High-achieving students often display the highest level of spatial visualization. Likewise, low-achieving students benefit from working with given visual static models (Moyer-Packenham, Ulmer, & Anderson, 2012).
Visual models provide a scaffold for students as they develop their own visualization skills. But these models can only be useful to students when the students are able to create an accurate model themselves or interpret a given model and use the model effectively for problem solving.
When students interpret and create visual static models, they develop new knowledge that can be applied to other problem solving situations. Researchers emphasize the importance of engaging students in real-world mathematics (Baruk, 1985, Greer, 1993; Verschaffel, De Corte, & Lasure, 1994; Verschaffel, Greer, & De Corte, 2007). Model generation, selection, and interpretation become key factors in students’ success in solving mathematical problems (Martin, Svihla, & Petrick Smith, 2012; Moseley & Okamoto, 2008; Ng & Lee, 2009). In his coordination class theory, diSessa (2002) argues that a student’s interpretation of a problem situation is connected to his or her readout (i.e. consistently identifying the important information in a problem situation in order to enact a solution strategy).
References:
Abrams, J. P. (2001). Teaching mathematical modeling and the skills of representation. In A. A. Cuoco & F. R. Curcio (Eds.), The Roles of Representation in School Mathematics, 2001 Yearbook (pp. 269-282).
Reston, VA: National Council of Teachers of Mathematics.
Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), 215–241.