text: User:McCormack/didactic analysis of the use of the quiz extension

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


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Quiz

See also: Test and Quiz

Welcome to the quiz learning project. To join this project, add your name to the list of confirmed participants, and then add the page to your watchlist so that you can keep up with developments. The project covers almost anything and everything connected with quizzing at Wikiversity, so if the word quiz excites you in any way at all, join.

 Quizbank under construction for OpenStax College

Aims of this learning project

 * Introduce Wikiversitarians to the quiz extension.
 * Provide for discussion and contribution to a methodology of using the quiz extension to enhance the educational value of Wikiversity pages.
 * Promote individual and collaborative development of quizzes on Wikiversity.
 * Provide feedback to the quiz extension developers.

Schools to which this project belongs

 * School of Education

Current activities

 * April 2007: most activity is over at the MediaWiki talk page for the quiz extension, where we are discussing the development of the quiz extension.
 * Late April 2007: discussion has started a lot on this Wikiversity now and is currently centered on the talk page for quiz extension documentation.
 * There is now a mobile phone program (Occleve) which can download and run quizzes offline. A forthcoming release of the software will enable new questions to be added offline on the phone, and uploaded back to Wikiversity, subject to safeguards. Please see Quiz/Mobile phone software for details.

Learning materials

 * Help:Quiz -- documentation for the quiz extension (full).
 * Help:Quiz-Simple -- a simpler and colourful documentation intended for beginners, younger users and anyone else who doesn't need the full version.

Quizzes
This section will list pages with quizzes on them. To ensure maximum possible coverage of quizzes, it might be helpful if the quiz extension automatically included a element in any pages using the quiz extension.

(Please add your page here.)
 * Wikipedia/Quizzes - please fact-check and contribute.
 * UNESCO Recommendation on Open Science This page uses multiple choice and multiple response questions.

See also: the "quizzes" category page

Articles
This section will list articles on using quizzes.

(Please add your page here.)

 * There is a table with didactic considerations for use of quizzes at Help:Quiz.

Confirmed participants
(Please add your name here.) Aashish Bhattarai
 * SimooMed
 * McCormack
 * Javier Carro 14:11, 12 April 2007 (UTC) (I'll focus on using quizzes for language learning).
 * vipulpratap 19:1107, 05 JUNE 2008 (GATECH) (I'll focus on using quizzes for language learning).
 * Cormaggio talk - Having played around just now on Help:Quiz, I think these are great, and would like to be able to learn for myself and others how they can be used within Wikiversity.
 * Rayc
 * Dionysios talk, a Participant in the Wikiversity School of Advanced General Studies, Date: 2007-07-22 (July 22, 2007) Time: 0534 UTC
 * Jtneill
 * Prakash Bebington
 * Mange01 16:34, 16 November 2008 (UTC)
 * Mbrad 00:13, 11 January 2010 (UTC)
 * rahul08 18:48, 20 August 210 (UTC) (I want to use quizzes to teach english to beginners).
 * Walkerma 16:24, 27 September 2010 (UTC)
 * User : Varnit Jain
 * Kenmayer 15:31, 4 December 2013 (UTC)
 * User:B.arjun 12:48 25 August 2016 (IST)
 * User:Knowledge sharer 18:36, 19 September 2016 (IST)
 * harjinder kaur, 13 October 2016
 * Lauren Mustard, july 27th 2018
 * ellmustard
 * User:SelfieCity
 * User:JMMaok
 * User:Anon-phys

Wikiversitarians known to be interested in quizzes

 * Kruhly
 * MichaelFrey (Switzerland)
 * Lrbabe (France; creator of the quiz extension)
 * Joe Gittings (mobile quizzes)
 * Jtneill
 * Graeme E. Smith
 * Goldenburg111 (Takes quizzes in every day life)
 * user: ukachukwu samuel (love quizzes)

Timescale
This is intended as a permanent project with no specific timeline.

Feature wishlist

 * 1) Reduce amount of blank space above the submit button.
 * 2) The ability to manually number quiz questions. // supported by GIFT format below
 * 3) Add a timer to quiz.
 * 4) Add pair-matching by means of JavaScript drag-and-drop
 * 5) Add sorting of lists by means of JavaScript drag-and-drop (partial order scoring?)
 * 6) Add "clickable maps", i.e. pointing within a certain region in an illustration for correct answer.
 * 7) Make the quizzes compatible with Moodle (an open source learning management system)  XML tags, simplifying import and export to/from Moodle. // supported by GIFT format below
 * 8) Promote score to a known Variable so that a template can manipulate it
 * 9) Can Moodle Tests be implemented as an anchor from a wiki-page? (I.E. write the test in moodle, and link it to the wikipage, then use moodle to manage it?) We could put a link to the moodle into the new Faculty portal? Students would have to register for the course, to get their marks. // supported by GIFT format
 * 10) GIFT format and extensions for assessment content in addition to Quiz table markup: import from and export to both formats (discuss)
 * 11) A simple way to hide the results for answers not selected, and only show the result for the selected answer.  This would allow the user to keep trying till they got the right answer.
 * 12) A simple way to refresh the page to clear the answers.
 * 13) The ability to create a questionnaire with totals based on response made, e.g., if there are three questions, a respondent might score 2, 4, and 1 respectively, giving a total of 7.
 * 14) Add a question type for extended responses, ie. a 3-15 line textarea that students can fill up and compare with a model answer or rubric in the feedback for the question.

title: Quiz


text:
Bloom clock project/Seen Lately (Northern Hemisphere)

The "Seen Lately" bloom clock page lists all plants that have been recorded as blooming within the last 3 weeks.

(This list is experimental)

Helleborus foetidus (Stinking Hellebore) Wikipedia | Commons | Wikibooks | Quiz page
 * --SB_Johnny | talk 14:17, 15 January 2007 (UTC)
 * --SB_Johnny | talk 18:06, 27 March 2007 (UTC)

Helleborus x hybridus (Hybrid Hellebore) Wikipedia | Commons | Wikibooks | Quiz page
 * --SB_Johnny | talk 11:40, 15 January 2007 (UTC)
 * SB Johnny-LM 11:13, 13 March 2007 (UTC)
 * --SB_Johnny | talk 18:06, 27 March 2007 (UTC)
 * SB Johnny-LM 13:22, 31 March 2007 (UTC)

Helleborus niger (Christmas Rose) Wikipedia | Commons | Wikibooks | Quiz page
 * --SB_Johnny | talk 11:40, 15 January 2007 (UTC)

Forsythia x intermedia (Garden Forsythia) Wikipedia | Commons | Wikibooks | Quiz page
 * SB Johnny-LM 13:22, 31 March 2007 (UTC)

Lamuim purpureum (Red Deadnettle) Wikipedia | Commons | Wikibooks | Quiz page
 * --SB_Johnny | talk 15:37, 24 March 2007 (UTC)
 * --SB_Johnny | talk 15:37, 24 March 2007 (UTC)

Lonicera heckrottii (Flame Honeysuckle) Wikipedia | Commons | Wikibooks | Quiz page
 * --SB_Johnny | talk 14:17, 15 January 2007 (UTC)

Magnolia x soulangeana (Saucer Magnolia) Wikipedia | Commons | Wikibooks | Quiz page
 * SB Johnny-LM 13:22, 31 March 2007 (UTC)

Magnolia stellata (Star Magnolia) Wikipedia | Commons | Wikibooks | Quiz page
 * SB Johnny-LM 13:22, 31 March 2007 (UTC)

Muscari armeniacum (Grape Hyacinth) Wikipedia | Commons | Wikibooks | Quiz page
 * --SB_Johnny | talk 22:55, 31 March 2007 (UTC)

Narcissus sp (Daffodil) Wikipedia | Commons | Wikibooks | Quiz page
 * --Rayc 31 March 2007 (UTC)

Nuytsia floribunda (Western Australian Christmas tree) Wikipedia | Commons | Wikibooks | Quiz page
 * --Gnangarra 09:42, 25 December 2006 (UTC)

Pieris japonica (Japanese Andromeda) Wikipedia | Commons | Wikibooks | Quiz page
 * SB Johnny-LM 13:22, 31 March 2007 (UTC)

Prunus 'Okame' (Okame Cherry) Wikipedia | Commons | Wikibooks | Quiz page
 * --SB_Johnny | talk 13:35, 4 April 2007 (UTC)

Ranunculus ficaria (Lesser Celandine) Wikipedia | Commons | Wikibooks | Quiz page
 * --SB_Johnny | talk 18:06, 27 March 2007 (UTC)
 * SB Johnny-LM 13:22, 31 March 2007 (UTC)

Salix discolor (Pussy Willow) Wikipedia | Commons | Wikibooks | Quiz page
 * SB Johnny-LM 13:22, 31 March 2007 (UTC)

Sanguinaria canadensis (Bloodroot) Wikipedia | Commons | Wikibooks | Quiz page
 * --SB_Johnny | talk 13:35, 4 April 2007 (UTC)

Scilla siberica (Siberian Squill) Wikipedia | Commons | Wikibooks | Quiz page
 * SB Johnny-LM 13:22, 31 March 2007 (UTC)

Senecio vulgaris (Common Groundsel) Wikipedia | Commons | Wikibooks | Quiz page
 * --SB_Johnny | talk 13:43, 16 January 2007 (UTC)
 * --SB_Johnny | talk 18:06, 27 March 2007 (UTC)

Spirea thubergii (Spirea) Wikipedia | Commons | Wikibooks | Quiz page
 * SB Johnny-LM 13:22, 31 March 2007 (UTC)

Stellaria media (Common chickweed) Wikipedia | Commons | Wikibooks | Quiz page
 * --SB_Johnny | talk 13:25, 16 January 2007 (UTC)

Symplocarpus foetidus (Eastern Skunk Cabbage) Wikipedia | Commons | Wikibooks | Quiz page
 * --SB_Johnny | talk 11:56, 15 January 2007 (UTC)
 * --SB_Johnny | talk 18:06, 27 March 2007 (UTC)

Taraxacum officinalis (Common Dandelion) Wikipedia | Commons | Wikibooks | Quiz page
 * --SB_Johnny | talk 16:04, 5 January 2007 (UTC)
 * --SB_Johnny | talk 19:35, 7 January 2007 (UTC)

Taxus bacatta (English Yew) Wikipedia | Commons | Wikibooks | Quiz page
 * SB Johnny-LM 13:22, 31 March 2007 (UTC)

Veronica persicaria (Speedwell) Wikipedia | Commons | Wikibooks | Quiz page --SB_Johnny | talk 18:06, 27 March 2007 (UTC)

Vinca minor (Periwinkle) Wikipedia | Commons | Wikibooks | Quiz page
 * --SB Johnny-LM 13:22, 31 March 2007 (UTC)
 * --Rayc 31 March 2007 (UTC)


title: Template:Bcp2


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BCP/Cornus mas

Cornus mas

title: BCP/Cornus mas


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Instructional design/Cognitive behaviors/Teaching Procedures, page 4

Instructional Design > Cognitive behaviors > Teaching Procedures > Define > Learn > Teach > Routine > Power > Steps > Try > Example

Needs Analysis
The first task of an instructional developer is to decide what to teach, if anything. This requires that a needs analysis--including a goals analysis and learner analysis--be performed (Kaufman, 1979).

Goals Analysis
The goals analysis will help you identify what the learners need to be able to do after completing the training lessons.

Learner Analysis
During a learner analysis, you will assess learners’ abilities and identify any deficiencies in their skills. This will tell you how ready and able they are to reach the learning goals, and where the training lessons need to start.

Some methods for performing needs analysis include the following:

 * Survey learners
 * Interview learners or groups of learners
 * Test learners
 * Interview qualified teachers or subject matter experts who understand the needs of the learners and the procedures they need to learn

Task Analysis
After identifying the learning goals, you may need to break the goals down further into tasks, or substeps. This is known as task analysis or content analysis. Using the fractions example on the previous page, we may determine that the learner does not know what a common denominator (in step 2) is or how to find one. It would therefore be necessary to also teach the user the concepts of denominator and common denominator, as well as the task of how to find a common denominator. For more information on teaching concepts, refer to the Concept Classification lesson.

Instructional Tactics
Once we have determined what the learner needs to learn, we can focus on the instructional tactics for how to teach those tasks, concepts, or procedures to the learners. Instructional tactics are categorized as follows:

 * Routine tactics teach the basic skill or procedure
 * Power tactics enrich the learner's experience with that skill or procedure

Learning Check
{What is included in needs analysis? Select all that apply. + Determining what the learner already knows how to do. - Determining how the procedure's steps need to be arranged. + Identifying what the learner needs to be able to do. - Distinguishing between routine and power tactics.
 * type="[]"}
 * Correct! This is called "learner analysis."
 * This is not done until after the needs analysis is complete.
 * Correct! This is called "goals analysis."
 * This is not done until after the needs analysis is complete and after the procedure steps have been outlined.

{If sub-steps need to be identified, what type of analysis is used? - Goals analysis - Learner analysis - Needs analysis + Task analysis
 * type=""}
 * Goals analysis is part of needs analysis and is done before the procedure's steps and sub-steps are identified.
 * Learner analysis is part of needs analysis and is done before the procedure's steps and sub-steps are identified.
 * Needs analysis and is done before the procedure's steps and sub-steps are identified.
 * Correct! Task analysis is performed to determine the sub-steps in a procedure.

Click Next to continue.

Source and Reference

 * Procedure Using by Charles M. Reigeluth. Used by Permission.
 * Kaufman, R. (1979). Needs Assessment: Concept and Application. Englewood Cliffs, NJ: Educational Technology Publications.


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BCP/Acer negundo

Acer negundo

title: BCP/Acer negundo


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BCP/Acer platanoides

Acer platanoides

title: BCP/Acer platanoides


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Instructional design/Cognitive behaviors/Teaching Procedures, page 5

Instructional Design > Cognitive behaviors > Teaching Procedures > Define > Learn > Teach > Routine > Power > Steps > Try > Example

Routine Tactics for Teaching Procedures
Routine tactics are the commonly used instructional methods.

Consistency
The needs analysis tells us what the learner needs to be able to do as a result of the training. These are instructional goals, which are also called "post-instructional requirements." An example of a post-instructional requirement is that a learner would be able to use his or her knowledge of fractions to solve a story problem. Therefore, the training lesson would also need to include practice with fractions using story problems. This is called "consistency," "authenticity," or "performance-based learning."

Generality - Example - Practice
The typical flow of a lesson on a procedure is to:

 * G - Present the generality for the procedure
 * E - Present an example of the procedure
 * P - Provide an opportunity to practice the procedure

These steps are explained in greater detail below.

Tactics for a Generality
Since a procedure is a set of steps to achieve a goal, the generality should include three parts
 * The label (source or name) of the procedure (if any)
 * The goal
 * The steps in general terms(1)

The following is an example of generality for a procedure:

Tactics for Examples
An example of a procedure is a demonstration of the use of the steps in one particular situation. It is not enough to just show the outcome of such use; every step should actually be demonstrated. Of course, each example should be as divergent as possible from the previous examples, with one instance from each equivalence class making up each such set of divergent examples.

Consistency
The practice should be as similar as possible to the post-instructional requirements (the conditions of performance after the instruction is all over).

Dimensions of Divergence
Practice of a procedure entails "just doing it" for a case that is different from any previously encountered cases. Each such  divergent practice item should provide the goal and any necessary inputs and should require the learner to perform each step of the procedure.

Instance Pool of Exercises
Generate a number of practice items for each  dimension of divergence. This creates an "instance pool" of practice exercises that can be used for both routine and power practice.

Sequencing the Lesson
The sequence does not have to be Generality-Example-Practice (G-E-P). In fact, it is often best to give an example simultaneously with the generality. We learn many procedures through example; that is, we learn them by observing others, and gradually generalizing from a prototypical case to the full range of cases. So we might want to hold off on the generality until after many examples and practice have been done. In fact, the learner could generate her or his own generality after observing a number of cases. Furthermore, we might want to give a generality, examples, and practice for one  equivalence class, then give G, E, and P for another, and so forth until all have been mastered, rather than giving examples for all equivalence classes followed by practice for all. Alternatively, we may want to give the learner control over the sequence of the routine tactics and equivalence classes.

Click Next to continue.

Source

 * Procedure Using by Charles M. Reigeluth. Used by Permission.
 * Procedure Flash Learning Module by by Charles M. Reigeluth.


text:
Instructional design/Cognitive behaviors/Teaching Procedures, page 6

Instructional Design > Cognitive behaviors > Teaching Procedures > Define > Learn > Teach > Routine > Power > Steps > Try > Example

Power Tactics for Teaching Procedures
Power Tactics, also known as "Enrichment Tactics," are used to increase the learner's efficiency with a procedure or to encourage the user to generalize the procedure to new situations. There are power tactics for generalities, examples, practice and feedback.

Power Tactics for a Generality
Attention-focusing. For procedures the most important aspect of the generality is the steps. Therefore, we should focus attention on the actions (mental or physical) which must be performed, and the order in which they should be performed.

Alternative representation. For procedures, the most useful alternatives are usually a flowchart or a paraphrase.

Mnemonic. In some procedures, it is hard to remember the order or nature of all the steps. In such cases, a mnemonic (like SQ3R: Survey, Question, Read, Recite, Review) can be very helpful to learners.

Power Tactics for Examples
The number of examples (or demonstrations) should be increased as the difficulty of the procedure increases.

Examples should be presented in an easy-to-difficult sequence.

Attention-focusing should relate the example to the generality, either by explaining in general terms what is being done in each step of the example or by highlighting the key actions and orderings.

An alternative representation for an example will often be in the form of a flowchart.

A reminder of the mnemonic can be helpful.

Common errors are useful to warn the student about, as long as they are indeed common (otherwise you might increase the chances of the errors being made) and as long as they are explained meaningfully to the learner.

Power Tactics for Practice
The number of practice items can be increased to enrich the instruction.

An easy-to-difficult sequence should also be used, as for examples.

Prompting is often helpful on early practice items when the procedure is a difficult one. Otherwise, power should be reserved for the feedback.

Power Tactics for Feedback
Attention-focusing should be used to relate the instance to the generality by pointing out, depending on the nature of the learner's mistake, the way an action (mental or physical) should have been performed, or the order in which it should have been performed. It should be the same as attention-focusing for examples.

A variety of representations is often helpful for the correct-answer feedback when a procedure is difficult.

A reminder of the  mnemonic (if one was presented earlier) is often a very helpful aspect of feedback.

Of course, motivational enrichment can also be used: praise for correct answers and encouragement for wrong answers.

Click Next to continue.

Source
Procedure Using by Charles M. Reigeluth. Used by Permission.


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BCP/Acer rubrum

Acer rubrum

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BCP/Acer saccharum

Acer saccharum

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BCP/Cardamine hirsuta

Cardamine hirsuta

title: BCP/Cardamine hirsuta


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Instructional design/Cognitive behaviors/Teaching Procedures, page 7

Instructional Design > Cognitive behaviors > Teaching Procedures > Define > Learn > Teach > Routine > Power > Steps > Try > Example

How to Design a Procedure-Using Lesson
The following are the steps for creating a lesson that teaches a procedure.

 * 1) Identify all the steps, break them down to entry level, and write a generality that includes the label (if any), the goal, and the steps at entry level. To do this, you should observe or ask your subject matter expert (SME) about the nature of the post-instructional requirements for using the procedure and assess the entry level of the learners with regard to the steps.
 * 2) Identify and list all  dimensions of divergence and group them into equivalence classes for the procedure. Make sure the generality deals with all of them.
 * 3) Assess the difficulty level of the procedure for the target learners (on, say, a scale of 1-5).
 * 4) Create an instance pool that contains x items per dimension of divergence, where x = the difficulty level times 2.
 * 5) Create a test, using at least one item randomly selected from each dimension of divergence. If the procedure is a difficult one, increase the number of items per dimension.
 * 6) Create practice items, using at least two items randomly selected from each  dimension of divergence . If the procedure is a difficult one, increase the number of items per dimension.
 * 7) Create examples, using at least one item randomly selected from each  dimension of divergence . If the procedure is a difficult one, increase the number of items per dimension. Include appropriate power components, depending on the difficulty level of the procedure.
 * 8) Create the  feedback  for the practice items so as to be virtually identical to the enriched examples.
 * 9) Think about motivational needs and media selection, and modify the instruction as appropriate to each. (This should be done throughout the process, not just after all else is done.)

Note : In general, the number of examples and test items should be about the same, and there should be at least twice as many practice items as examples.

Click Next to continue.

Source
Procedure Using by Charles M. Reigeluth. Used by Permission.


text:
Instructional design/Cognitive behaviors/Teaching Procedures, page 8

Instructional Design > Cognitive behaviors > Teaching Procedures > Define > Learn > Teach > Routine > Power > Steps > Try > Example

Try It: Practice What You've Learned: This is your opportunity to try the skills covered in this section of the lesson. As the Wikiversity is a collaborative learning space, you will contribute your original ideas to the work of those who have come before you by enhancing and adding to the possible responses within the case scenario laid out below. Are you saying to yourself, "...but, isn't looking at the answers from a fellow learner the same as cheating?" Not in this case. In fact, contributing your original ideas while participating with fellow learners and building upon their work is the intended goal of this Skill Builder exercise. The desired outcome is to build an ever evolving working document filled with not just one possible response to each question, but a host of instructional strategies that you can use on your next instructional design project. Have fun and get creative!

What to Teach? How to Teach It?
That friend of yours, Jennifer, from the Invariant Tasks lesson  has now been hired to tutor Sam's younger brother, Joe, in adding fractions. She remembered what wonderful advice you gave her for tutoring Sam, so she is back for more help. After you recover from the flattery, you remember that you should start with the view that the most important concerns in any instruction are "what to teach" and "how to teach it". With this in mind, what would you advise Jennifer to do first? Think about it, click "edit" for this section, and add your answer below:

 * First of all, she need to have a clear idea of the cope of the lesson. She would look at the textbook and handouts from the school teacher which indicate what to cover. Then, she would know the goal of the instruction. Next, she would text the current knowledge and skills of Joe. She can do this by getting him take a quiz on the previous contents. Finally, she can plan a lesson which fill a gap between the Joe's goal and Joe's current ability. Kei (discuss • contribs) 21:32, 20 February 2014 (UTC)

 * The mom explained what Joe needed to know regarding the fractions so we know the goal, but it would be important to conduct a learner analysis to determine what Joe's current levels are prior to instruction. She could talk to Joe or collect some of his past work to determine current levels. Together these would constitute the necessary needs analysis. Once we know what to teach, Jennifer will need to know what the steps are for calculating fractions. This may require a content analysis if she isn't familiar with the content and then specifically a task analysis to break down the steps for calculation. Sometimes job aids are used with pictures that can be taped to a notebook that include each step in a task analysis gdysard 22:06 (EST), 15 February 2015.

 * My first suggestion would be to look at Joe's homework, textbook, and class notes. This should help you to identify exactly what it is that his teacher is wanting him to learn. You then can work through the sample problems he copied in his notes and find a sequence of steps that the teacher has been using. I would then write the steps out in sequence for Joe to use so that as he is becoming accustomed to solving these problems, he can use these steps as a sort of "checklist". Providing a sample problem that shows how to utilize each step would be helpful for him to visualize the steps as well. Smccorma 03:01, 3 February 2012 (UTC)smccorma

 * If possible, I would advise Jennifer to talk to Joe's teacher and find out exactly what they're learning in class in regards to adding fractions. At the same time, I would suggest that Jennifer ask Joe's teacher what he is struggling with. This will help Jennifer figure out what she needs to focus on in their tutoring session. Hatch.nicole 02:46, 4 February 2012 (UTC)

 * We can suggest that Jennifer draw up a simple set of fraction problems to test Joe's current knowledge, employing both variable characteristics and dimensions of divergence. For example, two each of of adding, subtracting, dividing and multiplying fractions, finding common denominators and reducing fractions the lowest denominator. One Joe takes the pretest, she can interview him about what he thinks he knows and doesn't know, and what makes him uncomfortable about working with fractions. Jennifer can pair this knowledge with an interview with Joe's mother about Joe's struggles with fractions.Kevmcgra 03:49, 4 February 2012 (UTC)

 * Apparently the content that Jennifer is going to teach is adding fractions. I am hoping that in the beginning the purpose (goals) of hiring her has been made clear just like whether or not it's getting caught up with homework or preparing for standardized testing etc. If not, she should definitely ask questions in order to identify what Joe needs to be able to do after completing the tutorials and so she will be able to know at the end if it turns out successful with the tutorials. Next, Jennifer probably could speak to Joe, Joe's parents or teachers, look at Joe's existing formative and/summative evaluation items, and maybe also give some problem sets for Joe to solve in order to find out where Joe stands, what type of learner he is, what works for him, and about his abilities and deficiencies in the skills. If Jennifer's expertise is not in the specific area, she can also look for resources (i.e. from a SME) about the procedures Joe needs to learn about adding fractions. After that Jennifer can start to analyze the content/task by breaking the goals down further into substeps. By comparing all these tasks with the information from earlier steps ("learning analysis"), for example - what Joe already knows, Jennifer may determine what Joe does not know as to teach him the content, what routine tactics work for Joe in the basic skills as well as the power tactics to enhance his skills. Y.Zhang 04:43, 4 February 2012 (UTC)

 * first of first, conduct a need analysis, which includes goal analysis and learner analysis.Being aware of what Joe should be able to do about adding fractions after the tutoring and what his current knowledge or skill level is. Zhaomeng 21:18, 4 February 2012 (UTC)

 * First, Joe needs to start creating a domain map that specifies what to learn and how things are related with each other. And then, Joe needs to identify what are the goal of the task, and what needs to be learned to perform the task. And then, Joe needs to test the learners to decide what they already know and what they should learn. Dablee 21:45, 5 February 2012 (UTC)

 * If possible, I would suggest Jennifer to acquire curricular standards for Joe to learn adding the fraction. With the standards inventory that come before, at, and after the adding the fraction, she can have some diagnostic exercise with Joe to identify his current stage of knowledge and what area he needs more attention to in terms of adding the fraction instruction.Yeolhuh 22:24, 5 February 2012 (UTC)

 * I'd suggest Jennifer to check Joe's current grades and scores on the recent test that conducted in the previous semester. Jennifer can figure out what is the strength and weakness. Then, Jennifer can analyze how Joe could show a good performance on the strength part, and not a good performance on the weakness part. Songd (discuss • contribs) 14:39, 20 February 2014 (UTC)

ADD YOUR ANSWER ABOVE THIS BULLET POINT. SIGN IT WITH YOUR NAME USING THE SIGNATURE WIKICODE Phonebein 18:15, 18 December 2011 (UTC)

What are the recommended teacher actions?
Based on your advice, Jennifer has done all these analyses, so now she has truly identified "what to teach" and can proceed to think about how to teach it. You remember how useful the notion of presentation-practice-feedback was for Jennifer to teach the Presidents to Sam. Do you think that notion would help for teaching this procedural skill? Clearly, practice is important for learning a skill. We all know that "Practice makes perfect." But what should that practice be like? Joe shouldn't just do the same practice over and over again like memorization practice, should he? Think about it, click "edit" for this section, and add your answer below:

 * Presentation-practice-feedback should be good in this case. However, in order not to overwhelm Joe, the content for the presentation and practice should start easier ones then gradually become more complex. It would be good to start with adding fractions of same denominators. Jennifer can teach Joe that he has to just add numerators in cases where the denominator of the fractions are same. Then, get him to practice the calculation of adding fractions of same denominators and give him feedback. Jennifer, then, can show how to calculate adding fractions of same denominators. She should start with small numbers. Especially cases like this 2/3 + 1/6 where Joe just has to multiply the first fraction. Kei (discuss • contribs) 21:48, 20 February 2014 (UTC)

 * Joe could be making errors at one point of the task analysis. If you are using a checklist that is taped to his notebook, you can highlight this step and have him specifically practice this in different contexts. Maybe give him some authentic practical examples that he may be familiar with like cooking or measurements needed for building a dog house for his dog. gdysard 22:12, 14 February 2014

 * If Joe is struggling with this concept, I would start by using the checklist/sequence of steps and showing him how to follow it as I go back through his notes and show him how to solve each of those problems (whose final answer we already know) using the checklist/sequence of steps. In doing so, you should be able to see where he is having difficulty. After you have solved a few and Joe is starting to tell you what to do, allow him to show you how to solve a problem that you already know the correct answer to (and use that as feedback for him). Once he has correctly solved a number of problems to which he does not know the answer (this way, the tutor can provide feedback based on his steps and work), remove the checklist and see what he can recall using even more new problems that Joe does not know the answer to. Smccorma 03:05, 3 February 2012 (UTC)

 * Jennifer will want to make sure that she identifies all the dimensions of divergence for adding fractions to ensure that Joe receives enough variation in examples and practice exercises. Joe might be okay adding 1/2 + 1/2 but have difficulty solving 1/2 + 2/3. Hatch.nicole 03:18, 4 February 2012 (UTC)

 * Jennifer's analyses should have revealed how Joe will be expected to use fractions at school, and the dimensions of divergence he will be asked to demonstrate mastery of. She should match her presentation and practice to these dimensions, establishing both the generality for the procedure(s)and the equivalence classes for the dimensions. Does Joe need to know how to add simple fractions with one-number or two-number numerators and denominators, or must he also know the divergent dimensions of adding and subtracting both positive and negative fractions, or fractions with positive numerators but negative denominators? In each case, practice must encompass the dimensions in their totality, and match the complexity of the expected performance. Kevmcgra 04:06, 4 February 2012 (UTC)

I think the "presentation-practice-feedback" should work here. But if Joe has learn adding fractions in school, Jennifer may go ahead to let him practice on couple of questions first, and based how he did, present the general procedure. In this way, Joe will be more engaged and Jennifer would be able to emphasize on the steps where Joe needs more help.Zhaomeng 21:35, 4 February 2012 (UTC)
 * The practice can be very motivating when a specific context is provided. An example is as follows: Joe, imagine that several boys and girls are visiting your family. Your mother prepares a big round cake. She asks you to help her divide the cake, in order to treat those little friends. There were three kids. You planed to divide the cake equally into three pieces. Right after you finished cutting, two more boys arrived. There seems not enough cakes. So you cut each piece into two. Now there are six pieces. But we have five kids. You give the extra piece to one of them, your best friend. Finally, how much of the whole cake did your best friend eat? (The problem can be made easier or harder) Shuya Xu 07:21, 4 February 2012 (UTC)

 * Joe needs to create practices to accomodate all the divergent and variable instances and categorize them into equivalence classes. After that, the equivalence classes should be sequenced from easy to difficult. Joe needs to provide easy example followed by easy practice within the same equivalence class. If successful, Joe can proceed to more difficult equivalence class. Dablee 21:50, 5 February 2012 (UTC)

 * Jennifer should make the practice definitely align with the content, full of practice opportunities or new contexts to allow Joe to apply his skills - the right procedure dealing with fractions. The practice items can be arranged from the easiest difficulty to the hardest level.Y.Zhang 04:07, 6 February 2012 (UTC)

 * She needs to design practices with various equivalent classes examples based on divergent problems. Ranging from easy to difficult divergent problems, Joe gets to practice with various equivalent classes practice questions.Yeolhuh 15:04, 6 February 2012 (UTC) 149.159.16.5 15:03, 6 February 2012 (UTC)

 * I'd suggest Jennifer to gather instructional materials for a subject area that Joe could not get a good mark. After that, Jennifer can organize the instructional materials in order to teach Joe based on the decision on what should be the first and what should be the later. Songd (discuss • contribs) 14:42, 20 February 2014 (UTC)

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What if the learner does it wrong?
Now, you know well that if Joe is doing a lot of practice and getting them all wrong, it could actually make things worse. His error would become ingrained to the point where it would be much more difficult for Jennifer to correct it. So feedback is clearly also important for skill learning. But what should the feedback be like? Think about it, click "edit" for this section, and add your answer below:

 * Jennifer should get Joe to show all steps of the procedure he pursued, and tell him in which step in the procedure he did wrong. Kei (discuss • contribs) 21:50, 20 February 2014 (UTC)

 * I would find out where in the task analysis of steps the error is occurring to provide him feedback with the specific step in the process. That way feedback can be provided immediately during that particular step. So, he isn't completing fraction problem after problem. The issue can be isolated and corrected gdysard 22:16, 14 February 2014

 * Initially, if you are solving problems he has already seen in his notes, the feedback is immediate as he always knows the correct answers. When he is first starting to solve them on his own (without the final answers being known), I would provide feedback after every step. Once he can make it through a few problems without needing any more feedback after every step, then allow him to solve an entire problem and provide feedback only when he has reached the final solution. Smccorma 03:07, 3 February 2012 (UTC)

 * I would make sure that Joe writes out each step he takes to solve the problem, so that it's easy to tell when a mistake was made and provide corrective feedback at that exact step. Hatch.nicole 02:56, 4 February 2012 (UTC)

 * Jennifer can stop Joe when she sees him make an error and demonstrate the correct procedure, explaining each step to him as she makes it. As she does so, she can equip him with the generality for the specific problem. For example: "When you add two fractions, always remember to look first at whether the denominators are the same. If so, you can add the numerators straight across and keep the denominator. Then you can check whether you can reduce it to a smaller fraction."Kevmcgra 04:15, 4 February 2012 (UTC)

 * Jennifer can prompt Joe by reminding him of the principles of adding fractions; or by making the fractions concrete problems, so that the task is imaginable (e.g. asking him to think about the total pieces of cake, and the number of pieces his friend ate). Shuya Xu 07:27, 4 February 2012 (UTC)

 * Once an error appears, Jennifer should correct instantaneously or provide hints to help Joe find out the way to correctly go through the whole procedure. Jennifer may also repeat the generality of the procedure, and encourage Joe "you are getting it. Follow the steps, you must be able to make it!" Zhaomeng 21:39, 4 February 2012 (UTC)

 * Jennifer needs to stop Joe and examine his misunderstanding. After she identifies the misunderstanding, she can explain the procedure and highlight what has been done wrong. Depending on the misunderstanding, she can also provide some power tactics such as giving a mnemonic device, prompts, etc. Dablee 21:53, 5 February 2012 (UTC)
 * Jennifer can point out what went wrong (by circling the mistake in another color, for example) as to have Joe's attention. In the mean time, the correct step(s) should be demonstrated again if giving prompts for correction did not work for Joe. Informative and motivational feedback should be used like it's in other scenarios (learner response => wrong/encouragement, correct/praise).Y.Zhang 03:55, 6 February 2012 (UTC)
 * To identify where Joe is having a hard time exactly or where Joe has misunderstanding, Jennifer would encourage him to think aloud when he solves a practice problem. Once she identifies the area that Joe needs feedback on, she can ask him pause the activity and revisit the area that Joe needs clarification or correction.Yeolhuh 15:07, 6 February 2012 (UTC)
 * First of all, finding specific reasons why Joe did it wrong. It is not a good idea to provide feedback that includes correct answers or right procedures. Jennifer can analyze why Joe did wrong, and then modify Joe's thought pattern based on the analysis. Songd (discuss • contribs) 14:44, 20 February 2014 (UTC)

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What else should the teacher do?
Imagine Jennifer asking Joe to add a couple of fractions. If he couldn't do it in school, it's not likely that he can do it now for Jennifer! So what other guideline should you give Jennifer? Think about it, click "edit" for this section, and add your answer below:

 * If Joe becomes overwhelmed if he tries to add three fractions, then Jennifer can get Joe concentrate on the two of the three fractions and get him add the two. Then, add the answer and the third fraction. Kei (discuss • contribs) 21:53, 20 February 2014 (UTC)

 * This example is asking Joe to do something that he couldn't do originally. Jennifer should provide a clear example by showing him how to do it by demonstrating completion of a problem successfully. She could complete and entire problem and then break it down step-by-step and have Joe repeat each step after Jennifer repeats each one. gdysard 22:15, 15 February 2014
 * Start by asking Joe to read the steps to you as you roleplay being the student and show him how you would solve a problem. Since he is reading the instructions, it is as if he is the teacher. Ask him guiding questions with everything you do so that he is engaged in "helping" you (although you are solving the problem-this allows you to quickly assess what he knows and what he doesn't know). Being that he is now the teacher and not the student, this may take some pressure off of him so that he doesn't feel that he needs to do it all perfectly. Smccorma 03:09, 3 February 2012 (UTC)smccorma
 * I would write out the steps and suggest that we complete a problem together by following the steps. Then I would complete the problem with input from Joe. I would prompt Joe for direction whenever moving onto a new step to draw his attention to the procedure. Hatch.nicole 03:23, 4 February 2012 (UTC)
 * On thing Jennifer can do is relate fractions to Joe's prior knowledge. She can tell him that if he can add, subtract, multiply and divide ordinary numbers, he can work with fractions because the skills are exactly the same. The only thing that changes is the procedure: what you do first or second, or which number you multiply or divide another by. Then she can demonstrate one or two simple examples to reinforce the generalities, and ask him to tackle similar examples. Kevmcgra 04:28, 4 February 2012 (UTC)
 * The difference between learning in school and learning from a tutor is the latter is a one-on-one type of learning, therefore, Jennifer could based on her analysis of Joe's current skill of adding fractions find out the specific obstacles which Joe is facing. For example, Joe can do pretty well when the two denominators are same and one denominator could be a multiple of the other, but he stumbles when two denominators have a common factor. If it is like this, Jennifer can tailor the presentation of the generality and practice by focusing on this type of fraction adding. Zhaomeng 22:04, 4 February 2012 (UTC)
 * She can ask Joe to explain the procedure and the principles behind them while performing the task. Dablee 21:55, 5 February 2012 (UTC)
 * Jennifer can explain the steps again and write all the steps on a piece of paper with the label of the procedure (if any) as well as the goal, and then demonstrate an example following the steps. After that Joe can try to practice again. Y.Zhang 03:46, 6 February 2012 (UTC)
 * Jennifer would create a cheat sheet type of material with all the necessary steps to follow for solving the problem. With the material, she can go over the sample problem with Joe together and let him refer to the material with steps for supplementary material.Yeolhuh 15:10, 6 February 2012 (UTC)
 * I'd suggest Jennifer to demonstrate a couple of example questions. Showing how to do it first, and then letting Joe to do that by himself. And then, Jennifer can give him feedback. Songd (discuss • contribs) 14:47, 20 February 2014 (UTC)

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When is the best time?
When is the best time to present the generality? Think about it, click "edit" for this section, and add your answer below:

 * After showing four or five examples. Compared to adults, kids tend to get the sense of generality after seeing multiple examples. They do not often get the sense of generality by themselves as kids in this age are still in the process of developing attract thinking. So, after showing several examples, Jennifer should explain about the generalities of the examples. Kei (discuss • contribs) 21:57, 20 February 2014 (UTC)

 * It may depend on Joe's prior knowledge. If he has limited knowledge, then the generality could be presented before she demonstrates the process. However, if he has some confidence and experience she could provide demonstrations and then present them, so my answer is it depends...gysard 22:21, 15 February 2015
 * If Joe is completely lost, then I would present the generality right away. If he is struggling but not entirely lost, then I would first want to see what he knows, then I would present the generality to help him keep the steps in order and provide feedback about where he is having difficulty. Smccorma 03:10, 3 February 2012 (UTC)smccorma
 * In this case, I would probably present the generality and an example at the same time. I would use the generality to work through the example. Hatch.nicole 03:03, 4 February 2012 (UTC)
 * The best time is anytime it helps the learner(s) encode the procedure. It could be paired at the front of the lesson with a prototype example, or with a demonstration, to equip a learner facing a difficult (for him or her) or complex procedure. It could be presented when the learner encounters difficulty in practice. It can be presented after practice, but before assessment.Kevmcgra 04:45, 4 February 2012 (UTC)
 * One of the best time to present generality I think is at the very beginning of teaching the procedure. The generality should be presented with variant demonstration of examples. Another time to present generality is before practice. This is like giving a kind of review. During practice, the learner may make mistake or get stuck at certain steps. At this time, the presentation of the generality should be repeated, but the focus would be on the certain challenging steps, instead of the whole process. Zhaomeng 00:26, 5 February 2012 (UTC)
 * The generality should be presented when they learn something and apply it to different contexts for the first time and when they get things wrong to correct their understanding. Dablee 22:00, 5 February 2012 (UTC)
 * One of the solutions can be to begin with presentation of specific examples and ends with the formation of generality. (showing examples again -> compare/contrast -> generality) Y.Zhang 03:36, 6 February 2012 (UTC)
 * The best time to present the generality is when they are first exposed to it, which is the beginning of the instruction. Also it can be better understood by being presented with examples at the same time.Yeolhuh 15:12, 6 February 2012 (UTC)
 * The best time to present the generality might be when Joe answers correctly a couple of times in a row. That is the point that Joe starts organizing a schema to store that knowledge. Songd (discuss • contribs) 14:49, 20 February 2014 (UTC)

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What about motivation?
How often have you heard, "I hate fractions!"? Joe is probably no exception. What can Jennifer do to keep him from getting discouraged—to keep his attitude positive and his concentration high? High motivation translates into high effort, and that means quicker and better learning. What would you recommend to Jennifer? Think about it, click "edit" for this section, and add your answer below:

 * Confidence seems to be the biggest variable decides if kids are motivated to learn fractions. Fractions look new and strange and get kids to lose their motivations. In order help Joe to gain confidence, starting with fractions with same denominators should be helpful. Jennifer should emphasize that Joe just has to add the numerators and that he would pursue the exact same procedure which he follows when he is adding integers.

 * It is important to give him positive reinforcement to add to his own belief that he can do it. Give him stickers next to each part of the task analysis that he gets right. This will reinforce what he does correctly and he can aim at getting his entire sheet filled out. This may give him a goal to work towards and to focus his energy on. gdysard 22:24, 15 February 2015
 * Make it a game and give him something to work toward. Each problem can be worth 3 points that he will solve on his own. For every problem that he solves correctly, he earns one point. For every problem where you didn't have to provide any feedback to him until the conclusion of the problem, he earns one more point. For every problem that he does willingly and without complaining, he earns one more point. When he reaches 30 points (10 problems solved correctly, without scaffolding feedback, and willingly...or some combination of more problems) then he earns some prize that he would enjoy: an ice cream cone, $5.00, you will order him a pizza of his choice, etc. (In my experience, this works remarkably well for high school students.) Smccorma 03:14, 3 February 2012 (UTC)
 * I think I would just try to stay positive and encouraging by telling him that most people hate fractions initially but that it will get easier. That doesn't seem as motivating as Stacey's idea... Hatch.nicole 03:25, 4 February 2012 (UTC)
 * At the moment Joe shows frustration, Jennifer can remind him of her previous appeal to prior knowledge (or present it for the first time if she has not done so), and offer encouragement: "Remember when we talked about how working with fractions is just using addition, subtraction, multiplication and division? That's all we're doing. We just need to figure out what step to take first, second or third. It's easy; you can do this!" Alternatively, she can present Joe with a procedural mnemonic, such as such as the "bow tie" method for subtracting fractions, that will give him confidence that he can complete the subtraction procedure.Kevmcgra 04:54, 4 February 2012 (UTC)
 * In order to increase motivation, it's a good way to make Joe feel that adding fraction has much to do with his experience, and that can be interesting. The cake dividing problem may work for the above purpose. It is important to prepare presentation and practices at an appropriate level, so that Joe won't feel too frustrated because of the difficulty, or feel boring with the easyness. Jennifer's praise and encouragement also values in the retention of motivation. Shuya Xu 07:38, 4 February 2012 (UTC)
 * There are several ways to motivate Joe: put the description of the procedure of adding fractions into his favorite lyrics, and sing the song with him. Always encourage him when he feels not less confident and when he wants to give up--"I believe you will make it!" Praise him when progress is made either in oral way or giving candies, ice-cream or more entertainment time. Sometimes setting challenging goals and taking away what Joe likes if he cannot achieve the goal is also a way to motivate him, but this should be applied when Jennifer is sure that he has had the capability. Switching teacher and student role, and having Joe describe the procedure he is learning is also a motivation strategy. Zhaomeng 01:02, 5 February 2012 (UTC)
 * Making him do something easier will gives him a sense of accomplishment. Also, letting him work with a partner who is eager to learn will be a good way to motivate him. A game in which learners get a positive reinforcement will be another way to motivate him. Dablee 22:02, 5 February 2012 (UTC)
 * Giving praise as positive reinforcement immediately: "Thanks for doing the practice questions! You did a great job at answering all the them correctly." Encourage Joe when he's trying to get there but struggled: "Almost!" "Joe can do it!"Y.Zhang 03:28, 6 February 2012 (UTC)
 * Using examples around his life for practice can be motivating to him. For example, Jennifer can use candies or pies for practice questions. Also sometimes using examples around him help him reconstruct his schema. For example, Joe may be familiar with batting average of major league baseball players or shooting average of NBA players. Jennifer can use that figure to explain how faction is used to get that figure. In addition, using each to difficult practice approach can help him develop high level of self-efficacy that can boost his motivation as well. Yeolhuh 15:43, 6 February 2012 (UTC)
 * It is usually good to provide real-life examples and context-related questions and situations. I would suggest Jennifer to provide Joe with some interesting real-life examples that include a concept of fraction. Songd (discuss • contribs) 14:51, 20 February 2014 (UTC)

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Exercise Synthesis
To see possible responses to this exercise (prepared by the original authors of this lesson), click here: Example. Note that these responses are purposely placed on a separate screen to provide general guidance should you become stuck, but NOT to imply a single correct response to the case scenario.

Click Next to continue.

Source
Procedure Using by Charles M. Reigeluth. Used by Permission.


text:
Instructional design/Cognitive behaviors/Teaching Procedures, page 9

Instructional Design > Cognitive behaviors > Teaching Procedures > Define > Learn > Teach > Routine > Power > Steps > Try > Example

Exercise Synthesis
The following are possible responses (prepared by the original authors of this lesson) to the exercise on the Try page. Note that these responses are purposely placed on a separate screen from the exercise to provide general guidance should you become stuck, but NOT to imply a single correct response to the case scenario.

What to teach ? How to teach it ?
Question: That friend of yours, Jennifer, from the  Invariant Tasks lesson has now been hired to tutor Sam's younger brother, Joe, in adding fractions. She remembered what wonderful advice you gave her for tutoring Sam, so she is back for more help. After you recover from the flattery, you remember that you should start with the view that the most important concerns in any instruction are "what to teach" and "how to teach it".

Possible Answer: She should do a needs analysis to make sure that Joe really needs to know how to add fractions. Rodger Kaufman (1982) has described this as identifying the gap in knowledge between what should be and what is. This requires both a  goals analysis and a  learner analysis. Joe's mother has already told Jennifer that he should know how to add fractions. Jennifer could try to confirm this by talking to Joe's math teacher, but it is probably not necessary in this situation. Jennifer could then find out what he already knows with a few questions at their first session together.

But what if she had a group of students and needed to prepare some instructional materials ahead of time ? It would be a lot harder to figure out what they already know. It might be necessary to sample the learners and conduct several interviews to identify the lowest level of entering knowledge. But then you would want to have some mechanism (like a pretest) to avoid teaching the more knowledgeable learners anything they already know. That would just waste their time and demotivate them. However, in many instructional development (ID) situations, you may not have enough time or money to sample the “target” learners, but you may be able to talk to someone who has taught the same content to the same kinds of learners. This is a more cost-effective but less reliable way to conduct a learner analysis.

Let's assume that Jennifer is confident that Joe needs to learn how to add fractions. Can we now turn Jennifer's attention to "how to teach it" ? Have we really spelled out what to teach ? Not exactly. Jennifer will need to be precise (refresh her memory) as to what the procedure is for adding fractions. This is called a task analysis (or content analysis), and entails identifying all the required steps. On your advice, after digging around in the depths of her gray matter, Jennifer comes up with a flowchart like the one on page 3. So now she is all finished with "what to teach," right ?

Not exactly. She really ought to break down some of those steps into substeps (such as steps 2, 3, and 6) until all steps are at a level that Joe can understand. Part of this analysis process is to identify any concepts that Joe may not be familiar with, such as denominator, common denominator, and numerator. Jennifer doesn't have to use those terms (unless the post-instructional situation will require their use), but some term will be needed (e.g., "top number" and "same bottom number") to communicate efficiently. These analysis activities are also called  task (or content) analysis, and the term "prerequisites analysis" is often used for identifying such substeps and concepts.

What are the recommended teacher actions ?
Question: Based on your advice, Jennifer has done all these analyses, so now she has truly identified "what to teach" and can proceed to think about how to teach it. You remember how useful the notion of presentation-practice-feedback was for Jennifer to teach the Presidents to Sam. Do you think that notion would help for teaching this procedural skill ? Clearly, practice is important for learning a skill. We all know that "Practice makes perfect." But what should that practice be like ? Joe shouldn't just do the same practice over and over again like memorization practice, should he ?

Possible Answer: The practice should be as similar as possible to the post-instructional requirements (the conditions of performance after the instruction is all over). Joe will probably be given two or more fractions and be asked to add them, in which case that's what Jennifer's practice should require him to do. But the post-instructional requirements may entail reading a story problem and figuring out what mathematical operation is needed, and then using the right procedure, in which case that's what the practice should do. This is called consistency (or authenticity or performance-based learning). Similarly, the post-instructional requirements may call for Joe to be able to handle all kinds of fractions, in which case Jennifer should give him practice adding all kinds of fractions. This is called divergence.

Furthermore, we know that some cases of adding fractions are a lot more difficult than others. To be consistent, Jennifer should include the full range of difficulty required after the instruction is over. But should she start out with difficult practice ? Clearly, it will be easier for Joe to learn from if it is an easy case, unless he already knows how to handle easy ones (in which event he wouldn't learn anything from it!), although it would help you to diagnose his entering skill. If even the easiest authentic case was quite difficult for Joe, it would probably be helpful to compromise on the consistency rule in some ways. This is a form of prompting—what behaviorists call shaping and cognitivists call scaffolding.

What if the learner does it wrong ?
Question: Now, you know well that if Joe is doing a lot of practice and getting them all wrong, it could actually make things worse. His error would become ingrained to the point where it would be much more difficult for Jennifer to correct it. So feedback is clearly also important for skill learning. But what should the feedback be like ?

Possible Answer: Errors most often occur in a single step of a procedure. So Jennifer's feedback should point out the specific step that Joe did wrong and should help him to do it right. The feedback could demonstrate the correct performance of the step, or it could give a hint that helps Joe to figure out what he was doing wrong. When Joe gets the practice right, on the other hand, Jennifer's feedback should probably just confirm that it was right. You should also remind Jennifer that motivational feedback (encouragement when wrong and praise when right) can also be useful.

What else should the teacher do ?
Question: Imagine Jennifer asking Joe to add a couple of fractions. If he couldn't do it in school, it's not likely that he can do it now for Jennifer! So what other guideline should you give Jennifer ?

Possible Answer: Jennifer should probably show Joe how to add two fractions. This is often called a demonstration or example of the skill. But some cases (demonstrations) are a lot more difficult than others. So again, it will be easier for Joe to learn if it is an easy one. Now, if Jennifer just demonstrates the procedure without saying anything, would that be good instruction ? Not exactly. So what should she say ? Think about it before you read on!

She should explain what she is doing. This explanation could be in a form that generalizes to other cases within a given equivalence class, in which event it is called a generality. Or it could be in a form that relates the generality to this particular case, which is called attention focusing.

When is the best time ?
Question: When is the best time to present the generality ?

Possible Answer: There are several options as to when Jennifer could give the generality: before the demonstrations (which is a deductive approach), after several demonstrations (which is an inductive approach), or during a demonstration. No one approach is always best.

What about motivation ?
Question: How often have you heard, "I hate fractions!" ? Joe is probably no exception. What can Jennifer do to keep him from getting discouraged—to keep his attitude positive and his concentration high ? High motivation translates into high effort, and that means quicker and better learning. What would you recommend to Jennifer ?

Possible Answer: Encourage him when he falters: "That's closer!" Give praise when he is correct: "All right! You got it this time! You're smart!"

Source
Procedure Using by Charles M. Reigeluth. Used by Permission.


text:
User:Ran Feldesh

Hi all,

I would like to check the option of uploading wikiversity lecture notes of a physics course given at Ben Gurion University, Israel, by my professor. I created the below link at my UserAccount page as a preliminary pilot. I'll link it to with its appropriate place at the phyiscs portal in case that the pilot would come out good. '''Therefore please do not alter this for the coming week. For any suggestions please write me on my talk page.'''

--Ran Feldesh 19:40, 8 April 2007 (UTC)

Thermodynamics Lecture Notes, BGU

title: User:Ran Feldesh


text:
User talk:Ran Feldesh

Welcome to Wikiverstiy
Hi Ran, and welcome to Wikiversity. Are the course notes you want to upload yours, or are they written by the prof? The author is the one we'd prefer to upload content, as that way they are the ones that agree to the terms of the GFDL. Thanks for stopping by, and I hope you have a great time at Wikiversity. Historybuff 19:46, 8 April 2007 (UTC)

Thank you for your interest (and for being the first one to stop by the new page). The lecture notes are written by the students of the course, in latex, I asked my professor to see if doing it through a wiki could be easier and more effective. In the pilot we'll perform for the coming few days I plan to translate the latex into a wiki-code, and upload it here. Then my professor could see it and decide what he wish to do.

Again, thanks for your interest, and I'll be very happy to any suggestions! :)

--Ran Feldesh 19:54, 8 April 2007 (UTC)

title: User talk:Ran Feldesh


text: File:Advorg teachproc.gif

title: File:Advorg teachproc.gif


text:
Statistical mechanics and thermodynamics

(The original course was based on a course summarized by Lior Yosub and Liana Diesendruck. A rewrite is in progress that brings in more detail)

Introduction
The properties of a macroscopic physical system ultimately derives from the properties of its fundamental constituents. Since an exact solution for many particle systems with non-trivial interactions is out of the question, one is forced to resort to statistical methods to make progress. Statistical methods can, of course, only be used when we know the probabilities for finding the system in a particular state. The fundamental postulate of statistical physics is that all accessible states of an isolated system are a priori equally likely. Here "state" means the exact quantum state of the system which thus provides for a full microscopic description of the system. This postulate can be motivated by Liouville's theorem and Boltzmann's H-theorem which we'll discuss later in this course.

A macroscopic property of a system is, in general, obtained from the exact state after a coarse graining. Properties such as the internal energy, temperature, pressure etc. are examples of such macroscopic properties and they all have to be understood as (derived from) coarse grained variables. In principle, however, one can make any arbitrary choice for the macroscopic properties. For any choice, one defines the so-called "macrostate" of the system to be the set of these variables. Thermodynamics is the study of the relations between the macroscopic properties of the system. In thermodynamics, the word "state" is often used for "macrostate". To avoid confusion, it is customary to call the exact physical state of the system the "microstate".

A system is said to be in thermal equilibrium if its macrostate is time independent. A major part of statistical mechanics is the study of systems in or very close to thermal equilibrium. It follows from the fundamental postulate that lacking any information about a system, the most likely macrostate is that state for which the number of microstates that corresponds to it, is maximal. Any initial macrostate should thus evolve to such a macrostate, which is thus the thermal equilibrium state. This conclusion is known as the Second Law of Thermodynamics. In the next section we'll apply the methods of statistical mechanics to isolated systems characterised by their total energy.

The Micro-Canonical Ensemble
Since the total energy of an isolated system is conserved, all accessible microstates have the same energy. The macroscopic variable corresponding to the total energy, $$E,$$ has to be understood as a coarse grained variable corresponding to the total energy of the system. We define this as follows: When $$E$$ is specified, the total energy of the system can be in the range between $$E$$ and $$E + \delta E$$, where $$\delta E$$ is assumed to be small on a macroscopic scale. We'll see later that the value of macroscopic quantities per unit mass or unit volume in the limit of infinite system size do not depend on the choice of $$\delta E$$.

We define the function $$\Omega\left(E\right)$$ to be the number of microstates with energies with an energy in the range between $$E$$ and $$E + \delta E$$. All these states are equally likely in thermal equilibrium. One can then compute macroscopic properties of systems by averaging over all the $$\Omega\left(E\right)$$ states. Put differently, one can imagine the so-called "micro-canonical ensemble " ensemble of of systems with microstates in each of the $$\Omega\left(E\right)$$ states. Averaging over this ensemble yields the macroscopic thermal variables.

Definition of temperature
Now consider two such isolated systems in internal equilibrium, but isolated from each other. They have internal energies of $$E_{1}$$ and $$E_{2}$$. The number of microstates the systems can be in are then $$\Omega_1\left(E_{1}\right)$$ and $$\Omega_{2}\left(E_{2}\right)$$, respectively. Suppose we bring the two systems into contact, such that energy can flow between the two systems. The total internal energy $$E = E_{1} + E_{2}$$ of the combined system will then be conserved. The total number of accessible states the combined system can be in, is clearly given by:

 * $$\Omega\left(E\right)= \sum_{E_{1}}\Omega_{1}\left(E_{1}\right)\Omega_{2}\left(E-E_{1}\right)$$

Here the summation variable $$E_{1}$$ increases by a steps of $$\delta E$$. This then means that the coarse grained internal energy variable $$E$$ for the combined system determines its exact system's energy to be in the range between $$E$$ and $$E + 2\delta E$$. Obviously, the most likely energy distribution over the two systems is such that the product $$\Omega_{1}\left(E_{1}\right)\Omega_{2}\left(E-E_{1}\right)$$ is maximal. When we start with any other energy distribution and bring the two systems into thermal contact, energy will flow until that particular equilibrium state is reached. While there is then still a small probability for the energy distribution to deviate from this equilibrium state, it turns out that this is exceedingly unlikely, because the product of the two omega functions has a very sharp maximum.

We can formally compute the condition for thermal equilibrium between the two subsystems by maximizing the function $$\Omega_{1}\left(E_{1}\right)\Omega_{2}\left(E-E_{1}\right)$$. It is convenient to take the logarithm of this function and equate the derivative of that w.r.t. $$E_{1}$$ to zero. This gives:

 * $$\frac{\mathrm d\log\left[\Omega_{1}\left(E_{1}\right)\right]}{\mathrm dE_{1}} +

\frac{\mathrm d\log\left[\Omega_{2}\left(E-E_{1}\right)\right]}{\mathrm dE_{1}} = 0$$

In the derivative of $$\log\left[\Omega_{2}\left(E-E_{1}\right)\right]$$ we can change variables by putting $$E_{2}=E-E_{1}$$. By the chain rule, we then have:

 * $$\frac{\mathrm d\log\left[\Omega_{1}\left(E_{1}\right)\right]}{\mathrm dE_{1}} = \frac{\mathrm d\log\left[\Omega_{2}\left(E_{2}\right)\right]}{\mathrm dE_{2}}$$

Note that the l.h.s. of the equation only refers to the property of system 1 and the r.h.s only to the properties of system 2. For any arbitrary isolated system with internal energy $$E,$$ we define the so-called temperature parameter $$\beta$$ as:

$$\beta\equiv \frac{\mathrm d\log\left[\Omega\left(E\right)\right]}{\mathrm dE}$$

The condition that two isolated systems in internal thermal equilibrium are also in thermal equilibrium with each other (i.e. that when they are brought into thermal equilibrium no energy will flow between them), is thus that the two temperature parameters of both systems are equal. The thermodynamic temperature $$T$$ is defined by:

 * $$\frac{1}{k T}\equiv\beta$$

where $$k$$ is a arbitrary constant that fixes the temperature scale. If we choose to measure the thermodynamic temperature in Kelvin, then this constant is fixed to be Boltzmann's constant. We'll see later that this definition of thermodynamic temperature implies that energy flow between two systems as a result of thermal contact is from the high temperature system to the low temperature system.

While the Omega function is not accessible at the macroscopic level, the entropy, defined as:

 * $$S = k \log\left[\Omega\left(E\right)\right]\,$$

is a thermodynamic variable that can (indirectly) be determined from the macroscopic thermal properties of the system. From the above definitions, it follows that the temperature $$T$$ can be expressed as:

 * $$\frac{1}{T} = \frac{\mathrm d S}{\mathrm d E}$$

Now, this equation is only valid for isolated systems that don't perform any work. I.e. when the energy is added it goes into the internal thermal energy of the system. In general, we deal with systems that can perform work. In general this can be described by so-called external parameters, such as the volume of the system. The energy change due to the change in the external parameters is called work. Thermal energy that flows into a system that is not due to work is called heat.

The fundamental thermodynamic relation
We will now show that in general, when heat is added to a system so slowly that it remains in internal thermal equilibrium (such slow processes are called quasistatic processes), we have:

 * $$\mathrm dS =\frac{\delta Q}{T}$$

i.e. the change in entropy depends only on the added heat to the system and not on changes in internal energy due to work. Suppose that the system has some external parameter, $$x,$$ that can be changed. In general, the energy eigenstates of the system will depend on $$x$$. According to the adiabatic theorem of quantum mechanics, in the limit of an infinitely slow change of the system's Hamiltonian, the system will stay in the same energy eigenstate and thus change its energy according to the change in energy of the energy eigenstate it is in.

The generalized force, $$X,$$ corresponding to the external variable $$x,$$ is defined such that $$X \,\mathrm dx$$ is the work performed by the system if $$x$$ is increased by an amount $$\mathrm dx\,;$$ E.g., if $$x$$ is the volume, then $$X$$ is the pressure. The generalized force for a system known to be in energy eigenstate $$E_{r}$$ is given by:

 * $$X = -\frac{\mathrm dE_{r}}{\mathrm dx}$$

Since the system can be in any energy eigenstate within an interval of $$\delta E$$, we define the generalized force for the system as the expectation value of the above expression:

 * $$X = -\left\langle\frac{\mathrm dE_{r}}{\mathrm dx}\right\rangle\,$$

To evaluate the average, we partition the $$\Omega\left(E\right)$$ energy eigenstates by counting how many of them have a value for $$\frac{\mathrm dE_{r}}{\mathrm dx}$$ within a range between $$Y$$ and $$Y + \delta Y$$. Calling this number $$\Omega_{Y}\left(E\right)$$, we have:

 * $$\Omega\left(E\right)=\sum_{Y}\Omega_{Y}\left(E\right)\,$$

The average defining the generalized force can now be written:

 * $$X = -\frac{1}{\Omega\left(E\right)}\sum_{Y} Y\Omega_{Y}\left(E\right)\,$$

We can relate this to the derivative of the entropy w.r.t. $$x$$ at constant energy $$E$$ as follows. Suppose we change $$x$$ to $$x + \mathrm dx$$. Then $$\Omega\left(E\right)$$ will change because the energy eigenstates depend on $$x\,,$$ causing energy eigenstates to move into or out of the range between $$E$$ and $$E+\delta E$$. Let's focus again on the energy eigenstates for which $$\frac{\mathrm dE_{r}}{\mathrm dx}$$ lies within the range between $$Y$$ and $$Y + \delta Y$$. Since these energy eigenstates increase in energy by $$Y\; \mathrm dx,$$ all such energy eigenstates that are in the interval ranging from $$E - Y\; \mathrm dx$$ to $$E$$ move from below $$E$$ to above $$E.$$ There are

 * $$N_{Y}\left(E\right)=\frac{\Omega_{Y}\left(E\right)}{\delta E} Y \mathrm dx\,$$

such energy eigenstates. If $$Y dx\leq\delta E$$, all these energy eigenstates will move into the range between $$E$$ and $$E+\delta E$$ and contribute to an increase in $$\Omega$$. The number of energy eigenstates that move from below $$E+\delta E$$ to above $$E+\delta E$$ is, of course, given by $$N_{Y}\left(E+\delta E\right)$$. The difference

 * $$N_{Y}\left(E\right) - N_{Y}\left(E+\delta E\right)\,$$

is thus the net contribution to the increase in $$\Omega$$. Note that if $$Y \;\mathrm dx$$ is larger than $$\delta E$$ there will be the energy eigenstates that move from below $$E$$ to above $$E+\delta E$$. They are counted in both $$N_{Y}\left(E\right)$$ and $$N_{Y}\left(E+\delta E\right)$$, therefore the above expression is also valid in that case.

Expressing the above expression as a derivative w.r.t. $$E$$ and summing over $$Y$$ yields the expression:

 * $$\left(\frac{\partial\Omega}{\partial x}\right)_{E} = -\sum_{Y}Y\left(\frac{\partial\Omega_{Y}}{\partial E}\right)_{x}= \left(\frac{\partial\left(\Omega X\right)}{\partial E}\right)_{x}\,$$

The logarithmic derivative of $$\Omega$$ w.r.t. $$x$$ is thus given by:

 * $$\left(\frac{\partial\log\left(\Omega\right)}{\partial x}\right)_{E} = \beta X +\left(\frac{\partial X}{\partial E}\right)_{x}\,$$

The first term is intensive, i.e. it does not scale with system size. In contrast, the last term scales as the inverse system size and will thus vanishes in the thermodynamic limit. We have thus found that:

 * $$\left(\frac{\partial S}{\partial x}\right)_{E} = \frac{X}{T}\,$$

Combining this with

 * $$\left(\frac{\partial S}{\partial E}\right)_{x} = \frac{1}{T}\,$$

Gives:

 * $$\mathrm dS = \left(\frac{\partial S}{\partial E}\right)_{x}\mathrm dE+\left(\frac{\partial S}{\partial x}\right)_{E}\mathrm dx = \frac{\mathrm dE}{T} + \frac{X}{T} \mathrm dx=\frac{\delta Q}{T}\,$$

The first part of this equation, which can be written as:

 * $$\mathrm dE = T \mathrm dS - X \mathrm dx\,$$ (1)

is called the fundamental thermodynamic relation. It is more general than the equation $$\mathrm dS = \delta Q/T$$, as the former is also valid when the changes in the system are not quasistatic. This is because the internal energy in thermal equilibrium will be completely determined by when $$S$$ and $$X$$ are specified. So, we can consider $$E$$ to be a function of $$S$$ and $$X$$ (we say that $$E$$ is a thermodynamic state function) and we always have:

 * $$\mathrm dE = \left(\frac{\partial E}{\partial S}\right)_{x}\mathrm dS + \left(\frac{\partial E}{\partial x}\right)_{S}\mathrm dx $$

The fact that (1) is valid for quasistatic changes then allows us to conclude that the partial derivatives in the above equation are given by:

 * $$\left(\frac{\partial E}{\partial S}\right)_{x} = T$$

 * $$\left(\frac{\partial E}{\partial x}\right)_{S} = -X$$

The equation (1) is thus always valid, albeit it that for non-quasistatic changes we cannot identify the two terms as heat added and work done by the system any more. Also we don't always have to interpret the infinitesimal changes as a change that actually happens in a system. E.g. one can consider two copies of the same system but with slightly different thermodynamic variables. Then $$\mathrm dE,\, \mathrm dS\, \textrm{and}\, \mathrm dx$$ can refer to the difference in internal energy, entropy, and external parameter between the two systems. These changes will satisfy (1).

Canonical Ensemble
Often, the system we want to study is not isolated, but instead is in thermal contact with an environment at some fixed temperature. Now, for a large system a system kept at some temperature will fix its internal energy with negligible energy fluctuations relative to the total internal energy. So, we could just as well treat such systems using the micro-canonical ensemble. However, it turns out that doing computations in the micro-canonical ensemble is rather inconvenient.

Since a system kept at constant temperature is in thermal contact with an environment, usually called a heat bath, it is no longer true that all microstates of the system are equally likely. We can find the probability to find the system in some microstate as follows. We imagine that the system is kept at constant temperature by a heat bath such that the system plus heat bath is an isolated system. The accessible microstates for this combined system are then all equally likely. Suppose that the combined system has a total energy of E. If the system is in some microstate with energy $$E_{r}$$, then we can express the total number of states the combined system can be in, in terms of the omega function of the heat bath as $$\Omega\left(E-E_{r}\right)$$. Since all states are equally likely, this is proportional to the probability $$P_{r}$$ of finding the system in the microstate with an energy $$E_{r}$$:

 * $$P_{r}= C \Omega\left(E-E_{r}\right)\,$$

for some constant C. Here one has to define the omega function by choosing the energy resolution $$\delta E$$ smaller than the spacing between the energy levels of the system. For the Omega function to be well defined we have to choose $$\delta E$$ to be much larger than the spacing between the energy levels in the heat bath. These conditions can be satisfied because the heat bath will have to consist of far more degrees of freedoms than the system in order to be able to be able to absorb heat from the system and yet stay at (almost) constant temperature. The more degrees of freedom a system has, the smaller the spacing between the energy levels will be.

The above expression for $$P_{r}$$ can be simplified as follows. We can write:

 * $$\log\left(P_{r}\right)=\log(C) + \log\left[\Omega\left(E-E_{r}\right)\right]= K - E_{r}\frac{\mathrm d\log\left[\Omega\left(E\right)\right]}{\mathrm dE} = K -\beta E_{r}\,$$

for some constant $$K$$. Here we have used the definition of the temperature parameter given in the previous section and that in the limit of an ideal heat bath, the higher order terms in the Taylor expansion tend to zero as they involve the change in temperature due to change in the system's energy. It thus follows that:

 * $$P_{r}= \frac{\exp\left(-\beta E_{r}\right)}{Z}$$

where the normalization constant $$Z$$ is the so-called partition function which is given by:

 * $$Z = \sum_{r}\exp\left(-\beta E_{r}\right)$$

The canonical ensemble is a hypothetical ensemble of copies of a system whose microstates are distributed according to $$P_{r}$$.

Computation of thermodynamic variables
The system's thermodynamic variables can be computed by averaging over the probability distribution $$P_{r}$$. It turns out that a knowledge of the partition function suffices to extract all of the system's thermodynamic variables. In this section, we'll derive the equations for the internal energy, generalized forces, and the entropy in terms of $$Z$$.

The internal energy is expectation value of the system's energy. This can be expressed in terms of the partition function in the following way:

 * $$E = \sum_{r}P_{r}E_{r} = \sum_{r}E_{r}\frac{\exp\left(-\beta E_{r}\right)}{Z}= -\frac{1}{Z}\frac{\partial}{\partial\beta}\sum_{r}\exp\left(-\beta E_{r}\right)=-\frac{1}{Z}\frac{\partial Z}{\partial\beta} =-\frac{\partial\log\left(Z\right)}{\partial\beta}\,$$

If the system is in the state r, then the generalized force corresponding to an external variable x is given by

 * $$X_{r} = -\frac{\partial E_{r}}{\partial x}\,$$

The expectation value of this is the thermodynamic generalized force of the system and this can be written as:

 * $$X = \sum_{r}P_{r}X_{r}=\frac{1}{\beta}\frac{\partial \log Z}{\partial x}\,$$

Suppose the system has one external variable x. Then changing the system's temperature parameter by $$d\beta$$ and the external variable by dx will lead to a change in $$\log Z$$:

 * $$\mathrm d\left(\log Z\right)= \frac{\partial\log Z}{\partial\beta}\mathrm d\beta + \frac{\partial\log Z}{\partial x}\mathrm dx = -E\,\mathrm d\beta + \beta X\,\mathrm dx\, $$

If we write $$E\,\mathrm d\beta$$ as:

 * $$E\,\mathrm d\beta = \mathrm d\left(\beta E\right) - \beta\, \mathrm dE\,$$

we get:

 * $$\mathrm d\left(\log Z\right)=-\mathrm d\left(\beta E\right) + \beta\, \mathrm dE+ \beta X \,\mathrm dx\, $$

This means that the change in the internal energy is given by:

 * $$\mathrm dE =\frac{1}{\beta}\mathrm d\left(\log Z+\beta E\right) - X\,\mathrm dx \,$$

In the thermodynamic limit, the fundamental thermodynamic relation should hold:

 * $$\mathrm dE = T\, \mathrm dS - X\, \mathrm dx\,$$

This then implies that the entropy of the system is given by:

 * $$S = k\log Z + \frac{E}{T} + c\,$$

where c is some constant. The value of c can be determined by considering the limit T → 0. In this limit the entropy becomes $$S = k \log \Omega_{0}$$ where $$\Omega_{0}$$ is the ground state degeneracy. The partition function in this limit is $$\Omega_{0}e^{-\beta E_{0}}$$ where $$E_{0}$$ is the ground state energy. We thus see that $$c = 0$$, therefore:

 * $$S = k\log Z + \frac{E}{T} \,$$

The Helmholtz free energy $$F$$ can thus be expressed as:

 * $$F = E - T S = -kT\log\left(Z\right)\,$$

Grand Canonical Ensemble
In this ensemble we attach our system to a heat-particles reservoir in order to fix the temperature and the chemical potential of our system. The chemical potential is defined as $$\mu=\left(\frac{\mathrm dF}{\mathrm dN}\right)_{V,\tau}$$. In previous ensembles we fixed number of particles $$N$$ in the system. Now we let it change, so now we should rectify the expression of the energy and all its derivatives. $$\begin{matrix} &\mathrm dU=\tau \mathrm d\sigma-p\mathrm dV+\mu \mathrm dN \\ &\mathrm dF=-\sigma \mathrm d\tau-p\mathrm dV+\mu \mathrm dN \\ &\mathrm d\sigma=\frac{\mathrm dU}{\tau}+\frac{p}{\tau}\mathrm dV -\frac{\mu}{\tau}\mathrm dN	\end{matrix}$$ We start by defining the probability of a certain state as: $$		P(N,\left\{s_{i}\right\})=\frac{1}{\zeta}e^{\frac{\mu N-\epsilon_{s}}{\tau}} $$

where $$		\zeta=\sum_{N}\sum_{\left\{s\right\}}e^{\frac{\mu N-\epsilon_{s}}{\tau}} $$

Directly from $$\zeta$$, the Grand Partition Function, we shall derive the definition of the Grand Thermodynamical Potential, $$\Omega$$. $$		\Omega=-\tau ln\zeta $$

Another form of $$\Omega$$ is: $$\Omega=F-\mu N$$. The differential form of $$\Omega$$ is: $$               \Omega=-\sigma \mathrm d\tau-p\mathrm dV- N\mathrm d\mu $$

There are other possible ensembles, some of them we shall see in this course.

Gibbs Potential
We should now elaborate a specific characteristic of physical parameters of a thermodynamic system. An extensive parameter is a parameter whose value is proportional to the size of the system (number of particles), an intensive parameter is a parameter that is independent of the system size.

Let's summarize the thermodynamic potential we have seen: We have expressed all of these potentials in a differential form, where the number of elements depends on the number of ways to change the system's energy. Each term involves one extensive parameter and one intensive parameter. The pairs we have encountered so far are $$(\tau,\sigma)$$, ($$p,V$$), ($$\mu,N$$), ($$B,M$$), where the first element of each pair is the intensive parameter and the second is the extensive one. The parameters under differential sign have a special meaning: when they are fixed, this thermo potential tends to minimum/maximum. For example:
 * $$U$$ - Energy
 * $$F$$ - Free Energy
 * $$H$$ - Enthalpy
 * $$\Omega$$ - Grand Potential

$$d\sigma=\frac{dU}{\tau}+\frac{p}{\tau}dV -\frac{\mu}{\tau}dN \Rightarrow \sigma\uparrow \,\textrm{max}$$, when $$U,V,N =\textrm{const}$$. or

$$\mathrm dF=-\sigma \mathrm d\tau-p\mathrm dV+\mu\mathrm dN \Rightarrow F\downarrow\; \textrm{min}$$ when $$\tau,V,N = \textrm{const}.$$

In order to get from one potential to another one we simply need to add the proper pair of parameters to the first potential. For example, let's construct an ensemble where we fix the temperature, the number of particles and the pressure of the system. Like in the previous ensembles we'll look for a matching potential for our new ensemble and we shall see that our matching potential tends to a minimum. $$\begin{matrix} &G(\tau,p,N)=U(V,\sigma,N)+pV-\tau\sigma=H(p,\sigma,N)-\tau\sigma=F(V,\tau,N)+pV \\ &\mathrm dG=-\sigma \mathrm d\tau-Vdp+\mu \mathrm dN \end{matrix}$$ We attach to the system a heat-particle reservoir and will show that Gibbs potential is minimum at $$\tau$$, $$p$$ and $$N$$ - constant.

The total entropy $$\sigma_{R+S}$$ increase since $$R+S$$ is a closed system.

We use Taylor expansion: $$\begin{matrix} &\sigma_{R}(U_{0}-U,V_{0}-V)+\sigma_{S}(U,V)\ \uparrow\, \textrm{max} \\ &\sigma_{R}(U_{0},V_{0})-\frac{\partial \sigma_{R}}{\partial U}|_{U_{0}}\ U-\frac{\partial \sigma_{R}}{\partial V}|_{V_{0}}\ V+\sigma_{S}\ \uparrow\, \textrm{max} \end{matrix}$$ Using Maxwell relations we get: $$\begin{matrix} &\left(\frac{\partial\sigma_{R}}{\partial U}\right)_{N,V}=\frac{1}{\tau}\\ &\left(\frac{\partial\sigma_{R}}{\partial V}\right)_{N,U}=\frac{p}{\tau} \\ &\underbrace{\sigma_{R}(U_{0},V_{0})}_\textrm{constant}-\frac{U}{\tau}-p\frac{V}{\tau}+\sigma_{S}\ \uparrow \,\textrm{max} \end{matrix}$$ Now we multiply by $$-\tau$$, so the expression should tend to minimum: $$		G=U+pV-\tau\sigma\ \downarrow \,\textrm{min} $$ We call this potential Gibbs Free Energy.

Intensive Potentials
All of our potentials so far are extensive quantities. Sometimes we want to characterize the system by an intensive quantity with a meaning of thermodynamic potential. The easiest way define it is to divide the potential by an extensive parameter. If we divide the potential by, for example, N we'll get the potential per particle, which is an intensive size. We shall see now that in the special case of the potential G we'll get: $$		\mu=\left(\frac{\partial G}{\partial N} \right)_{p,\tau}=\frac{G}{N} $$ i.e. $$\mu$$ is Gibbs free energy per particle

We start with the Energy (U): $$		U=(\sigma,V,N)=NU\left(\frac{\sigma}{N},\frac{V}{N},1\right) $$ The above equation simply states that the energy of a system is equal to the number of particles times the energy of one particle.

The same can be done with the Free Energy (F): $$		F(\tau,V,N)=NF\left(\tau,\frac{V}{N},1\right)=Nf_{1}\left(\tau,\frac{V}{N}\right) $$ (The temperature is an intensive quantity so it's the same for every particle) Now we do the same to Gibbs Free Energy: $$		G(\tau,p,N)=NG(\tau,p,1)=Ng_{1}(\tau,p) $$ From the differential form of G we calculate the Chemical Potential: $$		\mu=\left(\frac{\partial G}{\partial N}\right)_{p,\tau}=g_{1}(\tau,p)\ \Rightarrow\ \mu=\frac{G}{N} $$ From here we receive: $$\begin{matrix} &G=F+pV=U+pV-\tau\sigma=\mu N \\ &\Omega=F-\mu N=F-G=-pV \end{matrix}$$

Phase transitions:
(example of liquid-vapor transition)

Consider vessel which contain water vapor with volume $$V$$ and at low pressure so that the gas is an ideal gas. We start decrease the volume so the pressure increases: $$\left(p\approx\frac{1}{V}\right)$$.

In a certain point a small layer of water will appear and from this point, the following decreasing of volume will not change the pressure, it will be constant. So what happens in the system? If we continue to reduce the volume, the water level will rise, but the pressure will remain constant. When all the vapor turn into water, reducing the volume will increase the pressure drastically and the proportion between the pressure and volume will change. In range of constant pressure we observe coexistence of two phases: liquid and gas.

The process we described above is done at a constant temperature. If we raise the temperature, the range of phase separation will shrink. But, if we raise the temperature high enough, we will have only a point of constant pressure. This temperature is called critical temperature, and for a phase transformation process at this temperature we don't have phase separation, we have everytime coexistence of gas and liquid, this named - fluid.

Theory of phase transition
Despite the complicated mathematical description like for example the BCS theory all known phase transitions like superconductivity, superfluidity or ferromagnetism are the result of the competition between the ordering interaction and the thermal fluctuations and they are similar to frequent attempts of falling out of the ping-pong ball in the hemispherical bowl or the box excited for example by the stream of  air with the cooling fan or the vacuum cleaner. After exceeding the certain excitation intensity i.e. for the true case the critical temperature with the sufficiently strong stream of air the ball will stop randomly bounce from the walls of the bowl and it will fall out i.e. its order parameter which is its localization in or outside the bowl will undergo the phase transition. It is therefore the result of strong non-linearity of motion which shows up when the system is randomly excited. The similar mechanism is responsible for "melting" of the ferromagnet i.e. for its total demagnetization with the thermal excitation above the Curie temperature when the interaction with the thermal reservoir is causing that the spin from small oscillations around the parallel direction of other spins starts randomly oscillate averaging to zero and ignoring the other spins around or when the thermal oscillations cause the ionization of Cooper pairs in the superconductor. For the Heisenberg model for example it is the pendular cosine scalar product interaction between the spins and the separatrix between two types of pendular spin motion in the external field which is responsible for the transition between the oscillatory (ordered) phase and the demagnetized or rotational phase with the critical temperature $$k T_c \approx {\sup}_{\phi} J (\cos \phi + 1) = 2 J$$ or the energy between the pendulum up and down, close to its exact value from two dimensional Ising model given implicitly by $$2 \tanh^2(2 J/k T_c)=1$$ or $$k T_c \approx 2.269  J$$. Considering the spin as pendulum with the potential of the interaction with two other spins (with one above in the third dimension) one gets $$k T_c = 4 J$$ close to its numerical value $$k T_c=4.512 J$$ obtained from the numerical Monte Carlo calculations and generally as interacting with one more for extra dimension $$k T_c = 6 J$$ for four dimensions close to numerical $$k T_c = 6.666 J$$ and $$k T_c = 2(d-1) J$$ close to Bethe-Peierls-Weiss theory in $$d$$ dimensions where it is $$k T_c = 2 d J$$ for the very large $$d$$.


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Talk:Statistical mechanics and thermodynamics

Hi all,

I would like to check the option of uploading wikiversity lecture notes of a physics course given at Ben Gurion University, Israel, by my professor. I'll link it to with its appropriate place at the phyiscs portal in case that the pilot would come out good.

Therefore please do not alter this for the coming week. For any suggestions please write me on my talk page.

--Ran Feldesh 19:40, 8 April 2007 (UTC)

Removed speedy deletion tag
There was a speedy deletion tag on this resource, stating:
 * orphaned page, no apparent purpose, according to the resource creator's talk page, this is a composite from the students lecture notes. If so, multiple copyrights may have been violated. Creator stated professor would agree to this. No such statement available.

I have removed the tag, because there was extensive work on the page from other users. While there might have been a possible problem with the original page, copyright on that would be unclear. Given all the other work, this should be a discussed deletion if we need to delete it at all. We are not Commons. If an original has been published, we could look at rights and permissions and consider any of various possible ways of handling this. --Abd (discuss • contribs) 21:11, 27 September 2014 (UTC)


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BCP/Crocus vernus

Crocus vernus

title: BCP/Crocus vernus


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BCP/Erythronium americanum

Erythronium americanum

title: BCP/Erythronium americanum


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BCP/Alnus rugosa

Alnus rugosa

title: BCP/Alnus rugosa


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BCP/Citrus sinensis

Citrus sinensis

title: BCP/Citrus sinensis


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BCP/Corylopsis spicata

Corylopsis spicata

title: BCP/Corylopsis spicata


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User:Doce~enwikiversity

For a brief while, I've been thinking about trying to bring K-12 students and skilled educators together to create an open K-12 reference work and an open K-12 curriculum, to parallel the achievements of other Free Culture movements and institutions. However, as I thought about these projects, and sketched out the work that would be involved in realizing them, I realized that they would take a very long time. The goals of this site are slightly different from mine, but I think I will save time by learning some management lessons from the bottom up as I work within its framework to learn some "content-area" knowledge.

I currently teach just math at all levels (k-college), but will soon be teaching all subjects at just one elementary grade level. Therefore, I am especially knowledgable about mathematics and mathematics instruction, but am interested in compensating for my relative deficincies in all other subject areas.

title: User:Doce~enwikiversity