MATH
DAN KESTERSON
C HAPTER 1
Misconceptions Some Misconceptions About Learning Memorizing math vocabulary lists is an efficient method of learning math language. Hearing new math language results in learning new math language. Focusing on reading math is of secondary importance. Most of our college learners know how to reflect; it does not have to be learned. The math learner must learn how to do math procedures before they learn math language. There is not enough time to focus on math language and procedures. Students should have learned how to reflect and use math language in high school.
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Co-Requisite Reading/Math
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C HAPTER 2
Working Memory
Language exponentially increases the ability to categorize information and information that is not categorized challenges the limitations of working memory. If we don’t attend and reflect, we don’t learn anything new in ways that have future application. Learning math language is key. However learning math language alone, just as learning math procedures alone is not sufficient.
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Working Memory
Working Memory, Organizing Information, and Language
1. Working memory -- temporary retention of information, sensory or other, for the performance of a prospective act to solve a problem or to attain a goal. (Fuster; Prefrontal Cortex 138) It is in working memory that new information being learned is temporarily stored and manipulated. Because of the very limited nature of working memory (4 items for 10-15 seconds), the learner must learn how to mentally expand the amount of information and time they have access to in working memory at any given moment or the information is lost. This requires learning how to reflect in the learning process. 2. Chunking (categorizing information ) — information that can be coherently organized can be treated as a single element (one of the four items) in working memory. (Baars; Essential Sources, 322) In other words, by coherently organizing new information being learned (interconnected with related prior knowledge), the learner has access to that coherently organized information in working memory, thus overriding the item and time limitations of working memory. This vastly increases the amount of information the learner has access to in working memory when mentally manipulating information. 7
3. One reason human cognition is so powerful is because we have language in our brains, which exponentially increases the ability to categorize information. (LeDoux; Synaptic Self, 177) Since coherently organizing new information greatly increases the amount of information the learner has access to to manipulate at any given moment and language in our brains exponentially increases the ability to categorize information, learning math language while learning math procedures is essential to deep learning of math concepts. Working Memory, Reflection, and the Prefrontal Cortex 1. Higher cognitive functions for which the prefrontal cortex is essential -- language, intelligence, and reasoning do not reach full maturity until the third decade of life. (Fuster; Prefrontal Cortex) Math learners from ages 18 to mid thirties are still developing their prefrontal cortex, which is essential for higher functioning such as planning, prioritizing, decision making, problem solving, etc.
3. Procedural memory is stored in other structures, not in prefrontal cortex. (Fuster; Prefrontal Cortex 344) 4. Organism can produce no sustained goal-directed action, except the most automatic and routine, without the temporal integration functions of the prefrontal cortex. (Fuster; Prefrontal Cortex 335) Learning procedures alone does not route that being learned through the prefrontal cortex nor when being retrieved does it pass through the prefrontal cortex and its temporal integration functions for planning, prioritizing, decision making, problem solving, etc. (executive functions). It is reflection, an internal conversation with oneself or others, that activates and routes new learning through the prefrontal cortex. Without learning math language, working memory is limited and the executive functions of the prefrontal cortex do not come into play. (Never underestimate the power or writing to clarify and organize information as a reflection process.)
2. Planning and decision making are two of the major executive functions of the prefrontal cortex.(Fuster; Prefrontal Cortex 4)
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C HAPTER 3
First and Second Core Mental Processes
It is important that the information being learned is routed through the hippocampus and on to the prefrontal cortex if the learner wants to be able to use the information for critical thinking and problem solving. This requires that the learner stop, step back mentally and reflect. This seemingly simple mental task is the most challenging of the processes for learning in ways that transfer.
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First and Second Core Mental Processes
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Conceptual Understanding Developing Competence in Math and Transfer Learning KEY: Math rules and procedures, facts and ideas must be understood in the context of a conceptual framework if they are to become useful - applicable in new situations and make related learning easier. It is with math language that the overarching conceptual frameworks of math topics must be constructed because math language exponentially increases the ability to categorize information. Math rules and procedures which are not categorized by the brain limits the amount of information working memory can temporarily store and manipulate.
Organize knowledge in ways that facilitate retrieval and application. (Bransfrod, 2001) When the mental processes for deep and transfer learning that are present in this presentation, the learner increase their chances of being able to apply what they are learning in new situations, retrieve and construct new meaning easier. Higher cognitive functions for which the prefrontal cortex is essential -- language, intelligence, and reasoning do not reach full maturity until the third decade of life. (Fuster; Prefrontal Cortex)
Second Core Mental Processes
Transfer Learning Procedural memory is stored in other structures, not in prefrontal cortex. (Fuster; Prefrontal Cortex 344) If the learner wants to learn in ways that transfer to new contexts, the learner needs Develop a deep foundation of factual knowledge; in order to do so, the learner must Understand facts and ideas, rules and procedures in the context of a conceptual framework and
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Math Language and Transfer Learning Understanding the Language of Math and Reflection is but Part of Conceptual Understanding of Math Contrary to popular belief, John Bransford, learning basic facts is not a prerequisite for creative thinking and problem solving -- it's the other way around. Once you grasp the big concepts around a subject, good thinking will lead you to the important facts. (John Bransford) One can also add - learning procedures alone for working math problems are not a prerequisite for creative thinking or problem solving -- it's the other way around. Once you grasp the big concepts around a subject, good thinking will lead you to the important facts. (John Bransford). For transfer learning (apply that being learned in new contexts) the learner needs to understand language, rules, and procedures in the context of a conceptual framework. A conceptual framework is made up of systematically organized concepts.
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C HAPTER 4
Third Core Mental Process
Once the learner has reflected on what they already know and predicted what is coming next and especially figured out how new information is related to the conceptual framework, the real work of increasing the myelin around the axon to increase the transmission, processing and retrieval of that learned.
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Third Core Mental Process
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C HAPTER 5
Reading/Writing Math
Math Communication Standard “Communication includes speaking and listening, as well as reading and writing. It enables students to not only share ideas, but also to clarify their understanding. Reflection and communication are intertwined processes in mathematics learning.� (NCTM 2000, p. 61)
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Reading/Writing Math
Reflection and communication are intertwined processes in mathematics learning.” (NCTM 2000, p. 61) Mathematics programs should enable students to—
Reading/Writing Math Approach to Co-Requisite Reading/Math Courses
• organize and consolidate their mathematical thinking through communication;
Think of the learning approach of the reading course as support in the co-requisite reading/math courses as an “enhanced testing” strategy using “spaced repetition” to create the “testing effect” through increasing the interplay between reading. writing, and math through ongoing formative assessment. Through reflective processes during reading and organizing, clarifying and revising of math ideas to by using the language of mathematics to express mathematical ideas precisely employing writing, the learner’s ability to understand and communicate math concepts improves.
• communicate their mathematical thinking coherently and clearly to peers, teachers, and others; • analyze and evaluate the mathematical thinking and strategies of others; use the language of mathematics to express mathematical ideas precisely. (NCTM, 2000)
First, let’s define some of the learning standards and concepts of integrating reading/writing and math: Math Communication Standard “Communication includes speaking and listening, as well as reading and writing. It enables students to not only share ideas, but also to clarify their understanding.
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Enhanced Testing
Spaced Repetition
“Test-enhanced learning refers to the fact that taking an initial test on studied material enhances its later retention relative to simply studying the material and then taking a final test. The educational implications are that production tests (short answer or essay) and feedback soon after learning increase learning and retention. In addition, frequent testing probably has the indirect positive effects of keeping students motivated and leading them to space out periods of study.”
“Across experiments, spacing was more effective than massing for 90% of the participants, yet after the first study session, 72% of the participants believed that massing had been more effective than spacing….When they do consider spacing, they often exhibit the illusion that massed study is more effective than spaced study, even when the reverse is true” (Dunlosky & Nelson, 1994; Kornell & Bjork, 2008a; Simon & Bjork, 2001; Zechmeister & Shaughnessy, 1980). (Spaced Repetition: Literature Review)
http://www.ncbi.nlm.nih.gov/pubmed/17694901
Testing Effect “The Testing Effect is the well-established psychological effect that the mere act of testing someone’s memory will strengthen the memory, regardless of whether there is feedback. The following (Roediger et al, 2006: 181) shows the effect of a multiple choice test taken by students after reading an article on bamboo. They were tested once after a variable interval and then again the same number of days after the test. Those whose first test was after 1 day remembered more than those whose first test was 7 days or 21 days later.” (Revunote)
“Spaced repetition is a learning technique that incorporates increasing intervals of time between subsequent review of previously learned material; this exploits the psychological spacing effect. Alternative names include spaced rehearsal, expanding rehearsal, graduated intervals, repetition spacing, repetition scheduling, spaced retrieval and expanded retrieval”. (Princeton University) Formative and Summative Assessment The goal of formative assessment is to monitor student learning to provide ongoing feedback that can be used by instructors to improve their teaching and by students to improve their learning. More specifically, formative assessments: • help students identify their strengths and weaknesses and target areas that need work
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• help faculty recognize where students are struggling and address problems immediately Formative assessments are generally low stakes, which means that they have low or no point value. Examples of formative assessments include asking students to: • draw a concept map in class to represent their understanding of a topic • submit one or two sentences identifying the main point of a lecture •
turn in a research proposal for early feedback
Language, Reflection and Reading Math Language exponentially increases the ability to categorize information and information not categorized challenges the limitations of working memory. “Mathematics is the most difficult content area material to read because there are more concepts per word, per sentence, and per paragraph than any other subject: the mixture of words, numerals, letters, symbols, and graphics requires the reader to shift from one type of vocabulary to another.” (Braselton and Decker, 2008) It is only through the reflection process that language becomes useful in learning math and dealing with the difficulty of math concepts.
Reading/Writing and Math “Deep, practical learning depends upon the reciprocal “interplay of numbers and words, especially on expressing quantitative relationships in meaningful sentences” (Steen, 2007, p.10). This connection between math and literacy becomes clearer in Steen’s simple formulation that all mathematics has two fundamental aspects: calculation and interpretation. The greater of these is interpretation. For numbers and algorithms to have meaning and worth, they must be interpreted and applied to “serve human purposes.” ( Schmoker, 2011)
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C HAPTER 6
Lessons “Largest gains ever recorded” in the history of educational research (Popham) * Clear learning objective/target (“Revise for word choice”; “Demonstrate knowledge of mean, median & mode”) * Anticipatory set/background/purpose/ preview of lesson * Teach; model/”think aloud” *Guided practice--& lots of think/pair/ share * Multiple checks for understanding (“formative assessment”) * Independent practice/assessment
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Lessons IMPACT OF FOLLOWING LESSONS “Largest gains ever recorded” in the history of educational research (Popham) * Clear learning objective/target (“Revise for word choice”; “Demonstrate knowledge of mean, median & mode”)
Clear Learning Objective/Target The single most important method for routinely sharing learning targets is using assignments that match—really match— the learning goal. It is in the assignment that the teacher translates the learning goal into action for the student. The student will strive to do the assignment, not the abstract goal. When we say an assignment or activity must "embody" the learning goal, we mean that the assignment or activity is such a close match with the goal that the student would be able to think, "If I can do [this assignment], then I can do [the learning objective]."
* Anticipatory set/background/purpose/preview of lesson * Teach; model/”think aloud” *Guided practice--& lots of think/pair/share * Multiple checks for understanding (“formative assessment”) * Independent practice/assessment Paul Black and Dylan Wiliam. On the basis of a careful appraisal of about 250 published research investigations, these two British researchers concluded, "The research reported here shows conclusively that formative assessment does improve student learning" (p. 61). They also reported that the learning gains triggered by formative assessment were "amongst the largest ever reported for educational interventions" (p. 61).
The purpose of the Anticipatory Set is to: •Provide continuity from previous lessons, if applicable •Allude to familiar concepts and vocabulary as a reminder and refresher •Tell the students briefly what the lesson will be about •Gauge the students' level of collective background knowledge of the subject to help inform your instruction
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Think Alouds
Think/Pair/Share
"Think-alouds" (Davey) help students understand the kind of thinking required by a specific task. The teacher models her thinking process by verbalizing her thoughts as she reads, processes information, or performs some learning task. Students see how the teacher attempts to construct meaning for unfamiliar vocabulary, engages in dialogue with the author, or recognizes when she isn't comprehending and selects a fix-up strategy that addresses a problem she is having. Ineffective readers especially benefit from observing what skilled readers think about while reading.
The Think-Pair-Share strategy is designed to differentiate instruction by providing students time and structure for thinking on a given topic, enabling them to formulate individual ideas and share these ideas with a peer. This learning strategy promotes classroom participation by encouraging a high degree of pupil response, rather than using a basic recitation method in which a teacher poses a question and one student offers a response. Additionally, this strategy provides an opportunity for all students to share their thinking with at least one other student which, in turn, increases their sense of involvement in classroom learning. Think-Pair-Share can also be used as in information assessment tool; as students discuss their ideas, the teacher can circulate and listen to the conversations taking place and respond accordingly.
Guided Practice Guided Practice is interactive instruction between teacher and students. After the teacher introduces new learning, he/she begins the student practice process by engaging students in a similar task to what they will complete later in the lesson independently. Students and teacher collaboratively complete the task as a model. The teacher leads the activity but solicits help from students at predetermined points along the way. Through the completion of the guided practice task, the teacher gradually releases more and more responsibility of the thinking to students, and offers less assistance from the teacher. Teachers should be using this time to recognize any need to re-teach portions from the Introduction to New Learning and to determine when/if students are ready to work independently.
Formative assessment The goal of formative assessment is to monitor student learning to provide ongoing feedback that can be used by instructors to improve their teaching and by students to improve their learning. More specifically, formative assessments: • help students identify their strengths and weaknesses and target areas that need work • help faculty recognize where students are struggling and address problems immediately 25
Note: “We are talking about a test supported process, not a test. Formative assessment is not a test. Rather, it is an ongoing process in which teachers use test-elicited evidence to adjust their instruction or students use it to adjust their learning tactics.� (Popham, 2009 Independent Practice Through Independent Practice, students have a chance to reinforce skills and synthesize their new knowledge by completing a task on their own and away from the teacher's guidance.
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