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“transmission, discovery and connectionist.” Transmission style involves a traditional teacher explanation followed by practice examples. In
discovery style students are tasked to discover mathematical ideas. Connectionist style emphasizes connections between pre-learned and
new mathematical ideas, and develops understanding through discussions. According to French, evidence suggests that teachers displaying
dominance in the connectionist style were found to be most effective. A reflective notebook or log with subject based ideas and reflections is
proposed as one effective means of improving instructional effectiveness. French suggests that the ability of teachers to connect between strands
of a mathematics curriculum, based upon a depth of mathematical understanding, is a more important indicator of effective teaching than a
formal mathematical qualification.
Turner (2010) describes, based on recent research, domains of mathematical literacy that teachers should be aware of, and considers challenges
that citizens may encounter in their everyday lives when faced with the need to apply mathematical knowledge.
•• Communication - Students need to understand and clarify mathematical problems and formulate a mental model of the situation
described. Supporting skills include the the abilities to read carefully and then interpret statements, questions, tasks, and diagrams.
Once solved, the student needs to present a solution and explanation.
•• Mathematising - Students need to bridge the mathematical literacy gap between real world problems and their mathematical
translations. Tasks could involve the selection of variables, taking measurements, and drawing illustrative diagrams.
•• Representation - Representations, such as “equations, formulas, graphs, tables, diagrams, pictures, textual descriptions and concrete
materials” are used to capture situations, tackle problems, and present solutions. Simple examples of this mathematical ability include
tasks such as translating text into numbers and reading values from graphs or tables of data
•• Reasoning and argument - Students use logic to piece together elements of a problem in order to infer and argue a possible solution.
Reasoning can be as simple as following instructions. Higher level reasoning can involve analysis of a problem and creation of a multi-
step argument.
•• Devising strategies - Students use strategies to “recognize, formulate and solve problems.” The strategy may be as simple as taking a
direct action based upon the task description. More complex examples may require interpretation of information or modification of
information in order to draw conclusions.
•• Using symbolic, formal, and technical language and operations - Students need to use mathematical representative symbols and the
rules and conventions needed for arithmetic expressions.
Science. Science instructional outcomes vary widely from the acquisition of core knowledge and understanding to the ability to plan
investigations, analyze data, and create hypotheses. Students can be bombarded with scientific misconceptions, such as evolutionary mismatches
in Flintstones television programs and misrepresentations of force and acceleration in Star Trek movies. Teachers require a depth of scientific
knowledge in order to avoid the reinforcement of misconceptions and to anticipate learning situations in which students may fail to assimilate
counterintuitive explanations.
Allen (2006) argues that elementary science teachers with a “solid grounding in the topic” are able to help students further their understanding
of science by building upon prior knowledge. Science teaches students to make observations and apply reasoning in order to make sense of the
world. Elementary and middle school students are commonly faced with instructional outcomes that require them to understand and make
predictions based on knowledge of natural systems of the Earth’s seasons, day, month, and year, phases of the Moon, and weather. The area of
curriculum is relevant, as an appropriate example, to one of the research findings cited by Allen. A study by the Harvard-Smithsonian Center for
Astrophysics showed that “a majority of randomly chosen Harvard University graduates, faculty, and alumni could not give correct explanations
for either the change in seasons or the phases of the moon.” (For the record, the correct explanation concerns the tilt of the earth as the major
factor.) The common misconception is that the earth follows an exaggerated elliptical orbit that places the planet much further away from the
sun in the winter.
The Association for Science Education (n.d.) explains how a deep knowledge of science subject knowledge enables teachers to anticipate common
misconceptions and challenges that science students at various ages encounter. For example, middle and high school students engaged with
chemistry content are challenged by “having to switch attention rapidly between the macro-scale (what is seen and experienced directly), the
sub-micro-scale (atoms, molecules, ions, electrons, etc. that are essentially invisible) and the representations (symbols, formulae and diagrams)
chemists use in their writing.” Student explanation of natural phenomena may be influenced by fiction and confuse facts. For example, students
frequently associate dinosaurs with cavemen, “who are given as the cause of the dinosaurs’ extinction.” Students, both young and old, will say that
“light travels further at night (or in the dark) than it does in the day.” Many more misconceptions and challenges, based upon counterintuitive
experiences or abstract ideas, are recognized in the teaching of science. The effective teacher is aware of the challenges and develops appropriate
strategies to correct misconceptions and support student learning.
English. English teaching ranges from the skills of basic communication such as persuasive language to the study and analysis of complex
works of literature. According to the National Council of Teachers of English (NCTE) “literacy education and literacy practices are in the midst
of a profound change” (Gillespie and Graham, 2012). An NCTE survey of teachers highlighted the need for students to be able to “communicate
clearly and understand complex messages” both in classroom situations and out of school. Three specifically identified academic English
language arts abilities identified were to: “seek information and make critical judgments about the veracity of sources, read and interpret many
different kinds of texts, both in print and online, and innovate and apply knowledge creatively.”
The Common Core State Standards Initiative (2012) describes English language arts expected outcomes for students in the K-12 age range.
Students’ progress with advancing levels of reading comprehension as they move through grades. Knowledge and insight are gained from
reading a broad range of literature and informational texts. Students in early grades are able to write opinions and develop arguments based
upon “sound reasoning, and relevant evidence.” Research abilities to analyze and present research findings are emphasised in the writing
discovery style students are tasked to discover mathematical ideas. Connectionist style emphasizes connections between pre-learned and
new mathematical ideas, and develops understanding through discussions. According to French, evidence suggests that teachers displaying
dominance in the connectionist style were found to be most effective. A reflective notebook or log with subject based ideas and reflections is
proposed as one effective means of improving instructional effectiveness. French suggests that the ability of teachers to connect between strands
of a mathematics curriculum, based upon a depth of mathematical understanding, is a more important indicator of effective teaching than a
formal mathematical qualification.
Turner (2010) describes, based on recent research, domains of mathematical literacy that teachers should be aware of, and considers challenges
that citizens may encounter in their everyday lives when faced with the need to apply mathematical knowledge.
•• Communication - Students need to understand and clarify mathematical problems and formulate a mental model of the situation
described. Supporting skills include the the abilities to read carefully and then interpret statements, questions, tasks, and diagrams.
Once solved, the student needs to present a solution and explanation.
•• Mathematising - Students need to bridge the mathematical literacy gap between real world problems and their mathematical
translations. Tasks could involve the selection of variables, taking measurements, and drawing illustrative diagrams.
•• Representation - Representations, such as “equations, formulas, graphs, tables, diagrams, pictures, textual descriptions and concrete
materials” are used to capture situations, tackle problems, and present solutions. Simple examples of this mathematical ability include
tasks such as translating text into numbers and reading values from graphs or tables of data
•• Reasoning and argument - Students use logic to piece together elements of a problem in order to infer and argue a possible solution.
Reasoning can be as simple as following instructions. Higher level reasoning can involve analysis of a problem and creation of a multi-
step argument.
•• Devising strategies - Students use strategies to “recognize, formulate and solve problems.” The strategy may be as simple as taking a
direct action based upon the task description. More complex examples may require interpretation of information or modification of
information in order to draw conclusions.
•• Using symbolic, formal, and technical language and operations - Students need to use mathematical representative symbols and the
rules and conventions needed for arithmetic expressions.
Science. Science instructional outcomes vary widely from the acquisition of core knowledge and understanding to the ability to plan
investigations, analyze data, and create hypotheses. Students can be bombarded with scientific misconceptions, such as evolutionary mismatches
in Flintstones television programs and misrepresentations of force and acceleration in Star Trek movies. Teachers require a depth of scientific
knowledge in order to avoid the reinforcement of misconceptions and to anticipate learning situations in which students may fail to assimilate
counterintuitive explanations.
Allen (2006) argues that elementary science teachers with a “solid grounding in the topic” are able to help students further their understanding
of science by building upon prior knowledge. Science teaches students to make observations and apply reasoning in order to make sense of the
world. Elementary and middle school students are commonly faced with instructional outcomes that require them to understand and make
predictions based on knowledge of natural systems of the Earth’s seasons, day, month, and year, phases of the Moon, and weather. The area of
curriculum is relevant, as an appropriate example, to one of the research findings cited by Allen. A study by the Harvard-Smithsonian Center for
Astrophysics showed that “a majority of randomly chosen Harvard University graduates, faculty, and alumni could not give correct explanations
for either the change in seasons or the phases of the moon.” (For the record, the correct explanation concerns the tilt of the earth as the major
factor.) The common misconception is that the earth follows an exaggerated elliptical orbit that places the planet much further away from the
sun in the winter.
The Association for Science Education (n.d.) explains how a deep knowledge of science subject knowledge enables teachers to anticipate common
misconceptions and challenges that science students at various ages encounter. For example, middle and high school students engaged with
chemistry content are challenged by “having to switch attention rapidly between the macro-scale (what is seen and experienced directly), the
sub-micro-scale (atoms, molecules, ions, electrons, etc. that are essentially invisible) and the representations (symbols, formulae and diagrams)
chemists use in their writing.” Student explanation of natural phenomena may be influenced by fiction and confuse facts. For example, students
frequently associate dinosaurs with cavemen, “who are given as the cause of the dinosaurs’ extinction.” Students, both young and old, will say that
“light travels further at night (or in the dark) than it does in the day.” Many more misconceptions and challenges, based upon counterintuitive
experiences or abstract ideas, are recognized in the teaching of science. The effective teacher is aware of the challenges and develops appropriate
strategies to correct misconceptions and support student learning.
English. English teaching ranges from the skills of basic communication such as persuasive language to the study and analysis of complex
works of literature. According to the National Council of Teachers of English (NCTE) “literacy education and literacy practices are in the midst
of a profound change” (Gillespie and Graham, 2012). An NCTE survey of teachers highlighted the need for students to be able to “communicate
clearly and understand complex messages” both in classroom situations and out of school. Three specifically identified academic English
language arts abilities identified were to: “seek information and make critical judgments about the veracity of sources, read and interpret many
different kinds of texts, both in print and online, and innovate and apply knowledge creatively.”
The Common Core State Standards Initiative (2012) describes English language arts expected outcomes for students in the K-12 age range.
Students’ progress with advancing levels of reading comprehension as they move through grades. Knowledge and insight are gained from
reading a broad range of literature and informational texts. Students in early grades are able to write opinions and develop arguments based
upon “sound reasoning, and relevant evidence.” Research abilities to analyze and present research findings are emphasised in the writing
