Friday, June 4, 2021

types of Teaching strategy

 Teaching Strategy

cognitive strategy is used to achieved instructional objectives or educational objectives. when some strategies are used the teacher is active and student are an inactive . the competencies of  student are not developed due to this. this means there is a need of changing the teacher- centred strategy. the teacher should give maximum learning experience to the students, so that the instruction will be the expected change in behavior of the student.

Effective teachers are always on the prowl for new and exciting teaching strategies that will keep their students motivated and engaged. Whether you’re a new or experienced teacher, you may feel inundated by all of the new educational buzzwords, theories, and new strategies that are out there.


With all of the information available, it’s hard to decide which teaching strategies are right for your classroom. Sometimes, the old tried-and-true ones that you have been using in your classroom just happen to work the best, and that’s okay. Teaching strategies that are considered “new” may just not fit into your teaching style.


Here are a few teaching strategies that are a staple in most classrooms. Depending upon your style, preference, and your students, choose the ones that suit your needs.


1. Differentiated Instruction: Learning Stations

Differentiated instruction strategies allow teachers to engage each student by accommodating to their specific learning style. According to Howard Gardner’s Multiple Intelligences Theory, every person has a different mind, and therefore each person learns and understands information differently. Differentiating instruction offers a way to meet all students’ needs. One helpful strategy to differentiate instruction is learning stations. Learning stations can easily be designed to enable students with diverse learning needs to learn at their pace and readiness level. Teachers can set up each station where students will be able to complete the same task, but at the level and style that is specifically designed for them.


2. Cooperative Learning: The Jigsaw Method

Cooperative learning gives students the opportunity to work with others and see different points of view. Students learn more effectively when working together rather than apart, and it is also known to improve self-confidence in students. The jigsaw method is especially effective because each student is responsible for one another’s learning, and students find out quickly that each group member has something equally important to contribute to the group in order to make the task a successful one. Students are exposed to and use many skills throughout this strategy: communication, problem-solving skills, cognition, and critical thinking — all of which are essential for a successful academic career.


3. Utilizing Technology in the Classroom

Integrating technology into the classroom is a great way to empower students to stay connected in this technological era. Technology-rich lessons have been found to keep students motivated and engaged longer. Some examples of utilizing technology in the classroom are to create web-based lessons or multimedia presentations such as a video, animation, or some type of graphic, utilizing a tablet or an iPad, taking your class on a virtual field trip, participating in an online research project, or even creating a class website. Any of these technology integration strategies will have a positive impact on student learning.


4. Inquiry-Based Instruction

Inquiry-based learning implies involving students in the learning process so they will have a deeper understanding of what they are learning. We are born with the instinct to inquire — as babies we use our senses to make connections to our surroundings. Inquiry-based learning strategies are used to engage students to learn by asking questions, investigating, exploring, and reporting what they see. This process leads students to a deeper understanding of the content that they are learning, which helps them be able to apply these concepts in new situations. In order for our students to be able to be successful in the 21st century, they need to be able to answer complex questions and develop solutions for these problems. The inquiry-based learning strategy is a great tool to do just that.


5. Graphic Organizers

Graphic organizers are a simple and effective tool to help students brainstorm and organize their thoughts and ideas in a visual presentation. Simply put, they help students organize information so it is easier for them to comprehend. Graphic organizers can be used for any lesson, to structure writing, brainstorming, planning, problem solving, or decision making. The most popular organizers are the Venn diagram, concept map, KWL chart, and T Chart.


An experienced teacher knows that not every teaching strategy that you use will be an effective one. There will be some hits and misses, and depending upon your teaching style and the ways your students learn, you will figure out which strategies work and which do not. It may take some trial and error, but it doesn’t hurt to try them all.

Saturday, April 24, 2021

Generalisation - Introduction and examples

 

Generalisation 

Introduction 

A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims.[1][2] Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation.


Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them.


However, the parts cannot be generalized into a whole—until a common relation is established among all parts. This does not mean that the parts are unrelated, only that no common relation has been established yet for the generalization.


The concept of generalization has broad application in many connected disciplines, and might sometimes have a more specific meaning in a specialized context (e.g. generalization in psychology, generalization in learning).[2]


In general, given two related concepts A and B, A is a "generalization" of B (equiv., B is a special case of A) if and only if both of the following hold:


Every instance of concept B is also an instance of concept A.

There are instances of concept A which are not instances of concept B.

For example, the concept animal is a generalization of the concept bird, since every bird is an animal, but not all animals are birds (dogs, for instance). For more, see Specialisation (biology).

Hypernym and hyponym

See also: Semantic change


The connection of generalization to specialization (or particularization) is reflected in the contrasting words hypernymand hyponym. A hypernym as a genericstands for a class or group of equally ranked items, such as the term tree which stands for equally ranked items such as peach and oak, and the term ship which stands for equally ranked items such as cruiser and steamer. In contrast, a hyponym is one of the items included in the generic, such as peach and oak which are included in tree, and cruiser and steamer which are included in ship. A hypernym is superordinate to a hyponym, and a hyponym is subordinate to a hypernym.[3]


Examples


Biological generalization




When the mind makes a generalization, it extracts the essence of a concept based on its analysis of similarities from many discrete objects. The resulting simplification enables higher-level thinking.


An animal is a generalization of a mammal, a bird, a fish, an amphibian and a reptile.


Cartographic generalization of geo-spatial data


Main article: Cartographic generalization


Generalization has a long history in cartography as an art of creating maps for different scale and purpose. Cartographic generalization is the process of selecting and representing information of a map in a way that adapts to the scale of the display medium of the map. In this way, every map has, to some extent, been generalized to match the criteria of display. This includes small cartographic scale maps, which cannot convey every detail of the real world. As a result, cartographers must decide and then adjust the content within their maps, to create a suitable and useful map that conveys the geospatial information within their representation of the world.[4]


Generalization is meant to be context-specific. That is to say, correctly generalized maps are those that emphasize the most important map elements, while still representing the world in the most faithful and recognizable way. The level of detail and importance in what is remaining on the map must outweigh the insignificance of items that were generalized—so as to preserve the distinguishing characteristics of what makes the map useful and important.


Mathematical generalizations


A polygon is a generalization of a 3-sided triangle, a 4-sided quadrilateral, and so on to n sides.


A hypercube is a generalization of a 2-dimensional square, a 3-dimensional cube, and so on to n dimensions.


A quadric, such as a hypersphere, ellipsoid, paraboloid, or hyperboloid, is a generalization of a conic section to higher dimensions.


A Taylor series is a generalization of a MacLaurin series.[1]


The binomial formula is a generalization of the formula for {\displaystyle (1+x)^{n}}.[1]



Tuesday, April 20, 2021

Teaching concepts : cognitive strategy

 





 



The Nature of Learning Tactics and Strategies


A learning strategy is a general plan that a learner formulates for achieving a somewhat distant academic goal (like getting an A on your next exam). Like all strategies, it specifies what will be done to achieve the goal, where it will be done, and when it will be done.


A learning tactic is a specific technique (like a memory aid or a form of notetaking) that a learner uses to accomplish an immediate objective (such as to understand the concepts in a textbook chapter and how they relate to one another).


As you can see, tactics have an integral connection to strategies. They are the learning tools that move you closer to your goal. Thus, they have to be chosen so as to be consistent with the goals of a strategy.


If you had to recall verbatim the preamble to the U.S. Constitution, for example, would you use a learning tactic that would help you understand the gist of each stanza or one that would allow for accurate and complete recall? It is surprising how often students fail to consider this point.


Because understanding the different types and roles of tactics will help you better understand the process of strategy formulation, we will discuss tactics first.



Top





Types of Tactics


Most learning tactics can be placed in one of two categories based on each tactic's intended primary purpose.


One category, called memory-directed tactics, contains techniques that help produce accurate storage and retrieval of information.


The second category, called comprehension-directed tactics, contains techniques that aid in understanding the meaning of ideas and their interrelationships (Levin, 1982).


Within each category there are specific tactics from which one can choose. Because of space limitations, we cannot discuss them all. Instead, we have chosen to briefly discuss a few that are either very popular with students or have been shown to be reasonably effective.


The first two, rehearsal and mnemonic devices, are memory-directed tactics. Both can take several forms and are used by students of almost every age.


The last two, notetaking and self-questioning, are comprehension-directed tactics and are used frequently by students from the upper elementary grades through college.



Top




Rehearsal


The simplest form of rehearsal, rote rehearsal, is one of the earliest tactics to appear during childhood and is used by most everyone on occasion. It is not a particularly effective tactic for long-term storage and recall because it does not produce distinct encoding or good retrieval cues (although, as discussed earlier, it is a useful tactic for purposes of short-term memory).


According to research reviewed by Kail (1990), most five- and six-year-olds do not rehearse at all. Seven-year-olds sometimes use the simplest form of rehearsal. By eight years of age, instead of rehearsing single pieces of information one at a time, youngsters start to rehearse several items together as a set.


A slightly more advanced version, called cumulative rehearsal, involves rehearsing a small set of items for several repetitions, dropping the item at the top of the list and adding a new one, giving the set several repetitions, dropping the item at the head of the set and adding a new one, rehearsing the set, and so on.


By early adolescence rehearsal reflects the learner's growing awareness of the organizational properties of information. When given a list of randomly arranged words from familiar categories, thirteen-year-olds will group items by category to form rehearsal sets.


 


Mnemonic Devices


A mnemonic device is a memory-directed tactic that helps a learner transform or organize information to enhance its retrievability.


Such devices can be used to learn and remember individual items of information (a name, a fact, a date), sets of information (a list of names, a list of vocabulary definitions, a sequence of events), and ideas expressed in text.


These devices range from simple, easy-to-learn techniques to somewhat complex systems that require a fair amount of practice. Since they incorporate visual and verbal forms of elaborative encoding, their effectiveness is due to the same factors that make imagery and category clustering successful--organization and meaningfulness.



Top




Self-Questioning


Since students are expected to demonstrate much of what they know by answering written test questions, self-questioning can be a valuable learning tactic.


The key to using questions profitably is to recognize that different types of questions make different cognitive demands. Some questions require little more than verbatim recall or recognition of simple facts and details.


If an exam is to stress factual recall, then it may be helpful for a student to generate such questions while studying. Other questions, however, assess comprehension, application, or synthesis of main ideas or other high&endash;level information.


Since many teachers favor higher-level test questions, we will focus on self&endash;questioning as an aid to comprehension.


Much of the research on self-questioning addresses two basic questions:


1. Can students as young as those in fourth grade be trained to write comprehension questions about the content of a reading passage?


2. And does writing such questions lead to better comprehension of the passage in comparison to students who do not write questions?


The answer to both questions is yes if certain conditions are present. Research on teaching students how to generate questions as they read (see, for example, Wong, 1985; Mevarech & Susak, 1993) suggests that the following conditions play a major role in self-questioning's effectiveness as a comprehension-directed learning tactic:


1. The amount of prior knowledge the questioner has about the topic of the passage.


2. The amount of metacognitive knowledge the questioner has compiled.


3. The clarity of instructions.


4. The instructional format.


5. The amount of practice allowed the student.


6. The length of each practice session.


 


Notetaking


As a learning tactic, notetaking comes with good news and bad.


The good news is that notetaking can benefit a student in two ways. First, the process of taking notes while listening to a lecture or reading a text leads to better retention and comprehension of the noted information than just listening or reading does.


Second, the process of reviewing notes produces additional chances to recall and comprehend the noted material. The bad news is that we know very little at the present time about the specific conditions that make notetaking an effective tactic.




Curriculum development wikipedia

Curriculum development 

 Curriculum development is a process of improving the curriculum. Various approaches have been used in developing curricula. Commonly used approaches consist of analysis (i.e. need analysis, task analysis), design (i.e. objective design), selecting (i.e. choosing appropriate learning/teaching methods and appropriate assessment methods) formation ( i.e. formation of the curriculum implementation committee / curriculum evaluation committee) and review ( i.e. curriculum review committee).


Analysis

Design

Selecting

Formation

Review

Early childhood care and education (ECCE) E

Main article: Curricula in early childhood care and education

There is no single curriculum that is 'best' for all situations. Not only does geographic location depends on the type of curriculum taught, but the demographics of the population matters as well. Some curriculums are based heavy of science and technology while another is focused mainly on the arts. However, a comparison of different curricula shows certain approaches to be generally more effective than others. Comprehensive programmes addressing health, nutrition and development have proven to be the most effective in early childhood, especially in programmes directed at very young and vulnerable children.[2][3] This requires a genuine commitment from agencies and individuals to work together, to plan projects collaboratively, and to involve parents and communities.[4]


Humanistic curriculum development

A humanistic curriculum is a curriculum based on intercultural education that allows for the plurality of society while striving to ensure a balance between pluralism and universal values. In terms of policy, this view sees curriculum frameworks as tools to bridge broad educational goals and the processes to reach them. A humanistic curriculum development perspective holds that for curriculum frameworks to be legitimate, the process of policy dialogue to define educational goals must be participatory and inclusive.[5] Central to this view is that curriculum policy and content must both be guided by the principles of social and economic justice, equality and environmental responsibility that constitute the pillars of sustainable development.

Curriculum Design

What is Curriculum Design?

A crucial step in the creation of a course, CD comprises of systematically and scientifically organising the instructional blocks within the course. These blocks could include activities, readings, lessons, and assessments.


What are the various types of Curriculum Design?

There are a number of ways to categorise Curriculum Design - on the basis of learning theory, approach to design, and the curricular orientation.


How is Curriculum Design different from Instructional Design?

Curriculum Design could be explained as what a learner learns while Instructional Design determines how s/he will learn it.


What is the Curriculum Design process? Edit

Samriddhi's Curriculum Design Framework[7] includes a four step process:


Empathise

Understand the learner's context

Review Existing curriculum and past learning

Gather Insight and Inspiration

Ideate

Primary and Secondary Research

Identify New/ Relevant ID Tools

Mini-map curriculum structure

Curate and Create

Curate and Create content according to the UDL principles,[8] categorised into:


Instruction

Measurement

Engagement

Prototype

Quality Control

Seek stakeholder feedback

Refine and Improve

Monday, April 5, 2021

प्रमाण भाषा व बोली भाषा

 प्रमाण भाषा व बोली भाषा 


लहानमोठ्या कसब्यात, खेड्यात, प्रत्येक प्रदेशात विविधता आढळणारं आपलं राज्य म्हणजे महाराष्ट्र. भौगोलिक स्थितीसह इतिहास, संवादाची शैली व भाषाही बदलत असते. पण कधी विचार केला आहे का, आपण या बोलींमध्ये अधिकृत व्यवहार का करत नाही?


वरील प्रश्नाचं उत्तर म्हणजेच बोली आणि प्रमाण भाषेतला फरक… बोली भाषा ही दैनंदिन संवादाची, व्यवहाराची भाषा आहे. पण जिथे बहुभाषिक एकत्र येतात, तिथे सगळ्यांना कळावी अशी भाषा तयार करणे गरजेचे असते. त्यातूनच नियम व कायदे असणारी प्रमाण भाषा जन्माला येते. महाराष्ट्राची प्रमाण भाषा मराठी आहे. याच भाषेतून महाराष्ट्राचा व्यवहार चालतो.


परंतु इतर बोली भाषा या मराठीच्या पोटभाषा किंवा त्यातून निर्माण झाल्या आहेत, असा गैरसमज करून घेऊ नये. याउलट प्रमाण भाषेवर बोली भाषांचा प्रभाव असतो, किंबहुना तो असायलाच हवा. सध्याची आपली प्रमाण भाषा ही पुणेरी भाषेकडे जास्त झुकते.


अनेकदा आपण बोली भाषांना अशुद्ध म्हणवून हिणवतो.

आपल्याला हे लक्षात घ्यायला हवं, की बोली भाषा अगदी दैनंदिन जीवनातील, व्यवहारातील भाषा आहे. तिला नियम व बंधनं नाहीयेत. त्यामुळे ती शुद्ध किंवा अशुद्ध असण्याचा प्रश्नच येत नाही. त्यामुळे मी माझ्या बाळाला ‘बाला’, ‘बाल्या’ किंवा ‘बाळ्या’ म्हटलं तर त्यात काहीच वावगं नाही.


प्रत्येक भाषा किंवा बोली ही त्या-त्या प्रदेशाच्या संस्कृतीत रुजलेली असते. भाषा म्हणजेच संस्कृतीचा एक भाग बनलेला असतो. पण केवळ नियमांच्या चौकटीत राहणारी भाषा संस्कृतीत रुजू शकेल का?

महाराष्ट्राची संस्कृती ही कृषि संस्कृती आहे. मराठवाडा, विदर्भ, कोकणाच्या शेतकऱ्याला प्रमाण मराठी भाषा तेव्हाच जवळची वाटेल, जेव्हा त्यात त्यांच्या बोलीचा काही समावेश असेल. त्यांच्या बोली भाषेतील दैनंदिन वापरातील शब्दप्रयोग प्रमाण भाषेत समाविष्ट केल्यास ते नक्कीच स्विकारतील.


कोकणातून आलेल्या या मराठी माणसाने मुंबई घडवली

महाराष्ट्राच्या कानाकोपऱ्यातील शेतकऱ्यांना शिकवणी म्हणून आपल्याकडे ‘आमची माती, आमची माणसं’ हा कार्यक्रम लोकप्रिय आहे. प्रमाण मराठीत शेतीप्रक्रिया सांगितल्या जातात. पण तुम्हाला माहीत आहे का, शेतीच्या कित्येक साधनं, प्रक्रिया, अगदी पिकांसाठीही बोली भाषांमध्ये विविध शब्दप्रयोग आढळतात.


उदाहरणार्थ, ‘कुळवणी’ ही शेतीप्रक्रिया आहे. डवऱ्याने जमीन साफ केली जाते. या प्रक्रियेसाठी विदर्भात ‘डवरनी’, खानदेशात ‘कोयपनं’ आणि मराठवाड्यात ‘डूब्बनं’ म्हणतात.

मराठवाडी, अहिराणी आणि वैदर्भी बोली बोलणाऱ्या काही शेतकऱ्यांना ‘कुळवणी’ समजणं तसं गोंधळाचंच. पण हेच पर्यायी शब्द म्हणून मराठी भाषेत वापरल्यास मराठी आणखी समृद्ध नाही का होणार? याउलट आपण भाषा शुद्धीकरणाकडे जास्त लक्ष घालतो, आणि बोली भाषांकडे मात्र दुर्लक्ष करतो.


आपल्याच बोलींमधून शब्द घेतले तर मराठी काही अशुद्ध होणार नाहीये. उलट ती संपूर्ण महाराष्ट्रभर वेगाने पसरेल, आणि राज्यात व्यवहार करणं अधिक सोपं होईल. त्यामुळे मराठी वाचवायची असेल तर बोली भाषा टिकवावी लागेलच…


बोली भाषा समजून घेण्यासाठी ‘ग्रेट मराठी’ नवा उपक्रम सादर करत आहे. महाराष्ट्रातील अनेक बोली भाषांची माहिती आम्ही लेखांद्वारे आपल्यासमोर आणणार आहोत. आपल्या महाराष्ट्राच्या भाषासंस्कृतीसाठी हे एक लहानसं पाऊल.

Sunday, April 4, 2021

Summation of mathematics

 

Summation 

Introduction 

In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.


Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article.


The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need of parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0.


Very often, the elements of a sequence are defined, through regular pattern, as a function of their place in the sequence. For simple patterns, summation of long sequences may be represented with most summands replaced by ellipses. For example, summation of the first 100 natural numbers may be written as 1 + 2 + 3 + 4 + ⋅⋅⋅ + 99 + 100. Otherwise, summation is denoted by using Σ notation, where {\displaystyle \textstyle \sum } \textstyle\sum is an enlarged capital Greek letter sigma. For example, the sum of the first n natural integers can be denoted as {\displaystyle \textstyle \sum _{i=1}^{n}i.} {\displaystyle \textstyle \sum _{i=1}^{n}i.}


For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find closed-form expressions for the result. For example,[a]


{\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}}.} {\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}}.}

Although such formulas do not always exist, many summation formulas have been discovered—with some of the most common and elementary ones being listed in the remainder of this article.

Notation

Capital-sigma notation


The summation symbol

Mathematical notation uses a symbol that compactly represents summation of many similar terms: the summation symbol, {\displaystyle \textstyle \sum } \textstyle\sum, an enlarged form of the upright capital Greek letter sigma. This is defined as


{\displaystyle \sum _{i\mathop {=} m}^{n}a_{i}=a_{m}+a_{m+1}+a_{m+2}+\cdots +a_{n-1}+a_{n}} {\displaystyle \sum _{i\mathop {=} m}^{n}a_{i}=a_{m}+a_{m+1}+a_{m+2}+\cdots +a_{n-1}+a_{n}}

where i is the index of summation; ai is an indexed variable representing each term of the sum; m is the lower bound of summation, and n is the upper bound of summation. The "i = m" under the summation symbol means that the index i starts out equal to m. The index, i, is incremented by one for each successive term, stopping when i = n.[b]


This is read as "sum of ai, from i = m to n".


Here is an example showing the summation of squares:


{\displaystyle \sum _{i=3}^{6}i^{2}=3^{2}+4^{2}+5^{2}+6^{2}=86.} {\displaystyle \sum _{i=3}^{6}i^{2}=3^{2}+4^{2}+5^{2}+6^{2}=86.}

In general, while any variable can be used as the index of summation (provided that no ambiguity is incurred), some of the most common ones include letters such as {\displaystyle i} i, {\displaystyle j} j and {\displaystyle k} k.[1]


Alternatively, index and bounds of summation are sometimes omitted from the definition of summation if the context is sufficiently clear. This applies particularly when the index runs from 1 to n.[2] For example, one might write that:


{\displaystyle \sum a_{i}^{2}=\sum _{i\mathop {=} 1}^{n}a_{i}^{2}.} \sum a_{i}^{2}=\sum _{i{\mathop {=}}1}^{n}a_{i}^{2}.

One often sees generalizations of this notation in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition. 

For example:


{\displaystyle \sum _{0\leq k<100}f(k)} \sum _{0\leq k<100}f(k)

is the sum of {\displaystyle f(k)} f(k) over all (integers) {\displaystyle k} k in the specified range,


{\displaystyle \sum _{x\mathop {\in } S}f(x)} \sum _{x{\mathop {\in }}S}f(x)

is the sum of {\displaystyle f(x)} f(x) over all elements {\displaystyle x} x in the set {\displaystyle S} S, and


{\displaystyle \sum _{d|n}\;\mu (d)} \sum _{d|n}\;\mu (d)

is the sum of {\displaystyle \mu (d)} \mu (d) over all positive integers {\displaystyle d} d dividing {\displaystyle n} n.[c]


There are also ways to generalize the use of many sigma signs.

 For example,


{\displaystyle \sum _{i,j}} {\displaystyle \sum _{i,j}}

is the same as


{\displaystyle \sum _{i}\sum _{j}.} {\displaystyle \sum _{i}\sum _{j}.}

A similar notation is applied when it comes to denoting the product of a sequence, which is similar to its summation, but which uses the multiplication operation instead of addition (and gives 1 for an empty sequence instead of 0). The same basic structure is used, with {\displaystyle \textstyle \prod } {\displaystyle \textstyle \prod }, an enlarged form of the Greek capital letter pi, replacing the {\displaystyle \textstyle \sum } \textstyle\sum.


Special cases

It is possible to sum fewer than 2 numbers:


If the summation has one summand {\displaystyle x} x, then the evaluated sum is {\displaystyle x} x.

If the summation has no summands, then the evaluated sum is zero, because zero is the identity for addition. This is known as the empty sum.

These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case. For example, if {\displaystyle n=m} n=m in the definition above, then there is only one term in the sum; if {\displaystyle n=m-1} n=m-1, then there is none.

Formal definition


Summation may be defined recursively as follows:


{\displaystyle \sum _{i=a}^{b}g(i)=0}, for b < a;{\displaystyle \sum _{i=a}^{b}g(i)=g(b)+\sum _{i=a}^{b-1}g(i)}, for b ≥ a.


Tuesday, March 30, 2021

Historical perspective of teaching

 HISTORICAL OVERVIEW

Can teaching be taught? Do individuals learn to teach or are they endowed with an innate gift for pedagogy? Are certain individuals born teachers? Do individuals learn about teaching from copying others, from listening to lectures, from reading about it? Are some ways of preparing teachers better than others? These and related questions about teaching and teacher education persist.



 

Didactic and Evocative Teaching

Joseph Axelrod describes two types of teaching as "the didactic modes, employed by teacher-craftsmen, and the evocative modes, employed by teacher-artists" (p. 5). Didactic teaching implies passing on traditional knowledge or lore, or teaching how to do something. Teachers use lecture to inculcate knowledge or demonstration to model actions, after which students demonstrate they have learned what was taught either by reciting or writing the material or by repeating the demonstration, as in a science class experiment. Much state and national testing relies on rote recall of material. In this context, learning means being able to reproduce what has been taught or demonstrated. For example, students should recall key facts of American history such as the order of the American presidents. Emphases are often on learning facts and conditions, not on understanding complexity and drawing conclusions.


Early in human history, most teaching was didactic. Poets recited ancient myths and stories and a few listeners learned them by rote. Individuals acquired skills by observing their elders who were fishers, artisans, lawyers, or anything else, and emulating what they saw. Seeing teaching as a process of passing on knowledge has persisted. Paul Woodring argues that "The oldest form of teacher education is the observation and emulation of a master. Plato learned to teach by sitting at the feet of Socrates. Aristotle, in turn, learned from Plato" (p. 1).


Much observation and emulation still go on. In The Teacher Educator's Handbook, Sharon Feiman-Nemser and Janine Remillard note that "Like much of our society, prospective teachers believe that teaching is a process of passing knowledge from teacher to student and that learning involves absorbing or memorizing information and practicing skills. Students wait like empty vessels to be filled and teachers do the filling" (p. 70).


Most teaching in early America was highly didactic. Teachers taught both the processes of learning to read and the morals attendant to a proper life through moralistic texts. Children learning their letters in the early nineteenth century read in the New England Primer under the letter A, "In Adam's fall, we sinned all"; under the letter F, "The idle fool Is wipt at school"; and under the letter J, "Job feels the rod Yet blesses God" (pp. 12–13). Students thus simultaneously learned their letters, religious lessons, and injunctions about behavior.


Not all teaching in the past was didactic; not all learning was rote. Socrates relied on the relationship between himself and his students to arrive at truths of human existence; he was, in Axelrod's sense, an evocative teacher. Socrates corrected occasionally and enjoined his students, but rarely taught didactically. The Socratic, or evocative, method places responsibility for knowledge growth on the students.


Teacher Education Faculty


Who teaches the teachers? Who is a teacher educator? The broadest conception of who is a teacher educator includes everyone who teaches prospective and practicing teachers, from their freshman English professors and those who teach special methods courses to those who supervise student teaching. Teacher educators may be defined specifically as "those who hold tenure-line positions in teacher preparation in higher education institutions, teach beginning and advanced students in teacher education, and conduct research or engage in scholarly studies germane to teacher education" (Ducharme, p. 6).


Research on teacher educators began in the 1980s as Heather Carter, Edward Ducharme and Russell Agne, Judith Lanier and Judith Little, and others began publishing research studies of teacher education faculty. In The Handbook of Research on Teacher Education, published in 1996, Nancy Zimpher and Julie Sherrill describe the teacher education professoriate as majority male and more than 90 percent Anglo. Summarizing several studies, they note that males dominate in the higher ranks, publish more than females, and work less in schools. Ducharme offers the observation that "there is a contradiction between a commitment to prepare a professional cadre of students, a majority of whom are female, to become powerful teachers and effective advocates for youth in which the female faculty are in roles and positions implying an inequity between the genders" (p. 120).


The ethnic makeup of the teacher education professoriate is heavily skewed toward white males. The Anglo population of the professoriate is between 91 and 93 percent. Candidates for teaching remain heavily white. With the exception of faculty in the historically black colleges, there are few black or other minority professors in teacher education. As the schools become more and more multicultural, those who teach teachers remain majority white and male; those who teach children in elementary schools remain mostly female and white; those who teach adolescents remain majority female and mostly white.


Teacher Education Themes


Many teacher education programs have defining characteristics. Programs generally lean toward one of several thematic patterns: behaviorist or competency-based, humanistic, and developmental. The 1960s and 1970s were the heydays of the competency-based teacher education (CBTE) and performance-based teacher education (PBTE) programs. In CBTE, researchers attempted to isolate what they perceived as the discrete tasks of teaching, develop protocols for training teachers to master the tasks, and produce tests to assess whether or not the teachers could perform the tasks. The CBTE movement soon degenerated into lists of hundreds of competencies as proponents attempted to outdo one another through elaborate lists. Instead of a system designed to help manage teacher education, it became an unmanageable process.


In Teacher Education (1975), N. L. Gage and Philip Winne defined PBTE as "teacher training in which the prospective or inservice teacher acquires, to a prespecified degree, performance tendencies and capabilities that promote student achievement of educational objectives" (p. 146). Both CBTE and PBTE derived from beliefs in relationships between teaching practices and student learning. Intensely behaviorist, both CBTE and PBTE grew in part from a desire for accountability in education, a concern that has persisted into the twenty-first century. Although the nomenclature of CBTE and PBTE has largely vanished from higher education teacher education syllabi, the concerns for accountability and the premises underlying the movements persist.





types of Teaching strategy

 Teaching Strategy cognitive strategy is used to achieved instructional objectives or educational objectives. when some strategies are used ...