Decoding the Disciplines - User contributions [en]

Quick guide

2025-06-17T10:18:20Z

Inna mik: remove "Decoding work" in title

==Adding content==
To add content, click on "New page" (which uses + as an icon) and provide the intended title of the content page. The filename of the created content will be identical to the title (with blanks replaced by underscores).

To add content you need to be a registered user.

==Editing content==
To edit content press the edit bottom (depicted by a pencil) on the page you want to edit. Keep in mind that this is a wiki. Hence, it is absolutely o.k. if you edit pages created by other people. Just stick to the Golden Rule: Edit pages created by other people as you want them to edit pages which you have created.

To edit content you need to be a registered user.

==Templates==
In order to create content you might wish to use one of the following templates:

*[[Template:Bottleneck]] for adding a bottleneck description. See [[Slip|Mechanical slip]] for an example. When contributing a bottleneck please avoid [[vague]] descriptions and follow the [[recommendations for bottleneck descriptions]].
*[[Template:Decoding work]] for adding information on Decoding work which extends the mere description of a bottleneck. See [[Group activities in interactive teaching]] for an example.
*[[Template:PublishedWork]] for adding bibliographical data about published work on Decoding. See [[PublishedWork:Decoding group activities in interactive teaching|Decoding group activities in interactive teaching]] for an example.
*[[Template:People]] for adding information about a person engaged in Decoding work (usually yourself). See [[David Pace]] for an example.
*[[Template:Resource-File]] for sharing research data related to Decoding work. See [[:File:Transcript-groupactivities.pdf]] for an example.

You don't have to adhere to the templates. Feel free to add or remove sections as needed. However, it is helpful if you take care in providing classifying information on Categories.

==Adding categories==
By adding categories you add structural information which feeds into pages like

*[[:Category:Decoding work]] which sorts Decoding work by discipline,
*[[:Category:PublishedWork]] which sorts literature by authors, discipline and type of publication,
*[[:Category:Person]] which sorts users by discipline.

Say, you have created a page describing your Decoding work which relates to the academic discipline of ''Blablaology'' and more specifically to the subfield ''Traxoline''. By adding your page to the categories "Blablaology" and "Traxoline" the page will be listed under the respective categories in [[:Category:Decoding work]].

To add your page to a category, click on the tag symbol on the top of your page when you are not in editing mode. As you type the name of the category which you want to add you will see whether this category already exists. In this case simply choose the existing category. Otherwise type return after you have fully entered the name of the category. This will create a new category.

Alternatively, editing the source code you can type <code><nowiki>[[Category:NAME]]</nowiki></code> at any place (preferably at the bottom). In the case of the above example you would type <code><nowiki>[[Category:Blablaology]]</nowiki></code> and <code><nowiki>[[Category:Traxoline]]</nowiki></code>.

Let's assume that the category ''Traxoline'' did not exist before and you have just created it. In order to tell the system that ''Traxoline'' is a subcategory of ''Blablaology'' you edit the Category-page of ''Traxoline'' (named <code>Category:Traxoline</code>) and add "Blablaology" as a category. Likewise you add "Decoding work" as a category on the page <code>Category:Blablaology</code> if you have created ''Blablaoloy'' as a new category. To edit a Category-page first click in the Category-link on the top of your content page (next to the tag icon) which will bring you to the Category-page. Then press Edit.

Technically, you always have to add the parent categories. More specifically, you add all the ancestor categories on which category pages you want your page to be listed. For instance, [[Textual descriptions in mathematics]] is categorized into [[:Category:Set theory]] (technically its parent) and [[:Category:Mathematics]] (technically its grandparent, i.e. parent of [[:Category:Set theory]]).

The current category trees of the above mentioned categories "Decoding work", "PublishedWork", and "Person" are:

<categorytree mode="categories" style="float:none; clear:right; margin-left:1ex; border:1px solid gray; padding:0.7ex; background-color:white;" depth="2">Decoding work</categorytree>
<categorytree mode="categories" hideroot="off" style="float:none; clear:none; margin-left:1ex; border:1px solid gray; padding:0.7ex; background-color:white;">PublishedWork</categorytree>
<categorytree mode="categories" hideroot="off" style="float:none; clear:right; margin-left:1ex; border:1px solid gray; padding:0.7ex; background-color:white;">Person</categorytree>Click on the arrow heads to further descend down the tree. Please try to keep the depth of the category trees small. For instance, for the case of [[:Category:Mathematics|Mathematics]] it is considered to be sufficient not to have further content related subcategories to [[:Category:Calculus|Calculus]] such as Differential Calculus or Integral Calculus.

Also note that in the categorization scheme used here a category can have more than one parent. For instance,[[:Category:History|Category:History]] is a child of (among others) [[:Category:Decoding work]] and [[:Category:Person]]. [[:Category:Group theory|Group theory]] might be a child of both [[:Category:Mathematics|Mathematics]] and [[:Category:Physics|Physics]], [[:Category:Thermodynamics|Thermodynamics]] might be a child of both [[:Category:Mechanical Engineering|Mechanical Engineering]] and [[:Category:Physics|Physics]].

DecodingWork: Load Calculation in Aviation

2025-06-17T09:06:47Z

Inna mik: (username removed) (log details removed)

#REDIRECT [[Load Calculation in Aviation]]

Load Calculation in Aviation

2025-06-17T09:06:47Z

Inna mik: (username removed) (log details removed)

==Description of bottleneck==
Students find it difficult to identify critical components, to identify and distinguish load cases and to document them in a comprehensive and systematic manner.

==Related scholarly work on this bottleneck==
The part of the Interview is analyzed here: https://youtu.be/Mp5pFmSHEfE?si=5WTbSSVTuEACJV4E
==People interested in this bottleneck==
Jutta Abulawi, Peter Riegler

==References==

[[Category:Bottleneck]] [[Category:Engineering]]

DecodingWork: Documentation of Load Calculation

2025-06-17T09:03:50Z

Inna mik: (username removed) (log details removed)

#REDIRECT [[DecodingWork: Load Calculation in Aviation]]

Load Calculation in Aviation

2025-06-17T09:03:50Z

Inna mik: (username removed) (log details removed)

Load Calculation in Aviation

2025-06-17T09:03:31Z

Inna mik: category engineering

Load Calculation in Aviation

2025-06-17T08:56:17Z

Inna mik: create a new page for load

{{Hint|text= '''How to use this template'''
# Start by describing the bottleneck in the Section '''Description of Bottleneck'''. If you like you can orient yourself on the bottleneck description [[From Derivative to Proportionality]]. If you feel able to turn the bottleneck into a (positive) learning outcome, please do also describe the intended learning outcome.
# This wiki also serves to connect people interested in certain bottlenecks. If you wish to make yourself known as interested, fill in your name under '''People interested in this Bottleneck'''.
# When done editing, save page.
# When in reading mode, add suitable tags/categories by pressing the tag symbol on the top of the page. For instance, if the bottleneck is in connection to biology, add "biology" as a tag.
# Edit once again and delete the whole Section '''How to use this Template''' and save your changes.
}}

==Description of bottleneck==
Students find it difficult to identify critical components, to identify and distinguish load cases and to document them in a comprehensive and systematic manner.

==Related scholarly work on this bottleneck==

==People interested in this bottleneck==
Jutta Abulawi, Peter Riegler

==References==
The part of the Interview is analyzed here: https://youtu.be/Mp5pFmSHEfE?si=5WTbSSVTuEACJV4E
[[Category:Bottleneck]]

Reading a Formula

2025-06-17T08:22:47Z

Inna mik: (username removed) (log details removed)

#REDIRECT [[Formula]]

Formula

2025-06-17T08:22:47Z

Inna mik: (username removed) (log details removed)

Difficulties often emerge while students have to recognize a structure of a complex formula. In particular, it is difficult for students to

* use the brackets to parse the formula from handwritten form to one line form, e.g. for programming ;
* see the steps that should be done first while rearranging a formula;
* use substitutions in chain rule or while solving equations.

==Decoding work done==

===Step 1: Identification of [[bottleneck]]===
Students have difficulties while recognizing a structure of a formula, not being able to collapse and expand expressions, e.g. not seeing the difference between e^(x^2) which can be collapsed to e^t and (e^x)^2 which can be collapsed to t^2. Some students believe, that to square a+b means to square the elements a and b because they do not think that the sign + is important.

Desired outcome: Students should be able to decide on the essential structure of the formula and read it using operator names, e.g. (a+b)/(2x) is a quotient between the sum and a product. They should be able to blend out the complexity and propose a similar formula/equation with numbers, e.g. C=4 π K R_1R/( R_1−R) with respect to R is like 1=2x/(3-x) with respect to x.

===Step 2: Description of [[Mental moves|mental tasks]] needed to overcome the [[bottleneck]]===
The experts is able to see the structure by zooming in and out into the formula to blend out the details.

He encapsulates the result of one operation as an element which will be an input to the next operation. Thus he sees the operation sign as important ones e.g while reading "a+b*s" he doesnt compare the letters a,b,s but the operation signs + and * because he has to make a decision that the * goes first. This results in reading the formula as a sum of two elements.

===[[Step 3 - Modeling Mental Operations|Step 3: Modelling the tasks]]===

===[[Step 4: Give Students Practice and Feedback|Step 4: Practice and Feedback]]===

===Anticipate and lessen resistance===

===Assessment of student mastery===

===Sharing===

==Researchers involved==

==Available resources==
[https://youtu.be/p50Q3mx-VNg Interview Video]

==See also==
[https://decodingthedisciplines.org/wp-content/uploads/2020/07/OF_V_Poster-Stank_170525_Endfassung-1.pdf Understanding Mathematics by Visualizing Structures], by Sabine Stank

===People interested in this Bottleneck===
Inna Mikhailova
[[Category:Bottleneck]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Decoding work]]
[[Category:Decoding Interview]]

From Derivative to Proportionality

2025-06-17T08:21:38Z

Inna mik: (username removed) (log details removed)

#REDIRECT [[Derivative]]

Derivative

2025-06-17T08:21:38Z

Inna mik: (username removed) (log details removed)

{{DISPLAYTITLE:DecodingWork:From Derivative to Proportionality}}

===Step 1: Description of Bottleneck===
Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. x² into 2x. In case they know, that "derivative=slope", they do not know what exactly the slope means, because they do not calculate "change of function value=slope*change of argument value".

Desired learning outcome: Students should understand that the change of function value is proportional to the change of argument value, and the derivative is exactly the proportionality factor. Students should be able to construct a triangle on the graph of a function to visually represent changes in function values and the corresponding slope (rate of change).

Concrete problems may be, for example:

* A) Given that f(1)=2 and f'(1)=3 students should be able to make an estimation of f(1,01) as f(1)+f'(1)*0,01=2.003
* B) Given the entities of function and entities of argument, decide on the entities of derivative. E.g. fuel price: Argument x is in Litre, function f(x) is in Euro, follows that the derivative f'(x) is in Euro/Litre
* C) Given the text, which describes the rate of change, deduce a differential equation. E.g. the population growth: Newborns are proportional to the population number leads to the equation dN/dt=C*N

=== Step 2: Description of mental tasks needed to overcome the bottleneck ===

# While thinking about derivative, a mathematician switches his mind from "static" formula story (algebra) to "dynamic" function story (analysis). He clearly understands that the story is not about calculating one value. It is about how function value changes if argument value changes (see a related bottleneck [[Function]])
# An expert can mask from his mind the fact, that the derivative should be calculated as a limit. He works with quotient "Change of function value" over "Change of argument value" if this is useful to deduce needed information. E.g. if we are only interested in entities or if the formula f(x) is not given, but only numerical values. He deduces the needed values from the given graph of function by drawing a triangle in order to represent changes in values and the slope between two points.
# He can work with augmenting (moving to the right on the graph) and reducing (moving to the left) value of the argument, and he can split the change of the value in multiple steps if he realizes that the step is too big.
# While doing so, he is not afraid of being unprecise, because he knows, that the approximation is locally allowed and can estimate the approximation error.

=== Step 3: Modelling the tasks ===

==Available resources==
[https://youtu.be/gh-cgUDM1xI Interview Video]

===People interested in this Bottleneck===
Inna Mikhailova
[[Category:Bottleneck]]
[[Category:Mathematics]]
[[Category:Calculus]]
[[Category:Decoding work]]
[[Category:Decoding Interview]]

DecodingWork: Reading a Formula

2025-06-17T08:14:11Z

Inna mik: (username removed) (log details removed)

#REDIRECT [[Reading a Formula]]

Formula

2025-06-17T08:14:11Z

Inna mik: (username removed) (log details removed)

Formula

2025-05-29T15:42:11Z

Inna mik: mental task example

Decoding the Disciplines – A Roundtrip from Novice to Expert back to Novice

2025-05-01T16:28:46Z

Inna mik: link to DiNa

== [https://www.didaktikzentrum.de/images/cwattachments/493_46f6251eae8bd4aa1f94aeefb28ceea7.pdf Bibliographic] data ==
Riegler, P. (2020). Decoding the Disciplines – A Roundtrip from Novice to Expert back to Novice. [[Didaktiknachrichten 07-2020|Didaktik-Nachrichten (Jul. 2020)]], pp. 4-8.

== External source ==
https://d-nb.info/1342433254/34

== See also ==
German translation: [https://www.didaktikzentrum.de/images/cwattachments/493_46f6251eae8bd4aa1f94aeefb28ceea7.pdf Decoding the Disciplines – vom Laien zum Experten und noch einmal zu den Anfängen zurück]
[[Category:PublishedWork]]
[[Category:Decoding]]
[[Category:Riegler, P]]
[[Category:2020]]
[[Category:Article]]

Just in Time Teaching

2025-05-01T16:14:05Z

Inna mik: Example to question about questions

The goal: ''Make student work autonomously and ask questions in order to get deeper understanding''
* How: Provide online material with test and “question about questions” ahead of the lesson ("Are there any parts of the reading that remain unclear to you? If yes, formulate a specific question on the relevant topic. If no, indicate which statement in the text you find particularly important or particularly interesting"). Before the lesson, the lecturer reads the questions and discuss them during the lesson.

* Why it works: Asking is anonymous, Student has time to formulate the question, Learning to ask questions is learning thinking!
* More Material:[https://citl.illinois.edu/citl-101/teaching-learning/resources/teaching-strategies/just-in-time-teaching Just-in-Time Teaching (JiTT)]
* Videos:
** https://youtu.be/D5ADk2gi6JM
** https://youtu.be/jzq92bHIJms (in German)
** https://youtu.be/1ImPB5ghsHw (in German)

* Reference: Novak, G. M., Patterson, E. T., Gavrin, A. D., & Christian, W. (1999). Just in time teaching. Upper Saddle River: Prentice-Hall.

Pages needed

2025-05-01T15:44:38Z

Inna mik: status bottleneck

{| class="wikitable"
|+
!Topic
!People willing to contribute
!Details
!Status
|-
|[[Limits]]
|[[User:Riegler|Peter Riegler]]
|uploading parts 4 and 5 of interview
|
|-
|<s>Language comprehension</s>
|[[User:Riegler|<s>Peter Riegler</s>]]
|<s>Summarizing the decoding work descriped in [[Lost in Language Comprehension: Decoding putatively extra-disciplinary expertise]]</s>
|done -> [[Textual descriptions in mathematics]]
|-
|[[The Decoding Clock Reading Activity|<s>The Decoding Clock Reading Activity</s>]]
|[[User:Riegler|<s>Peter Riegler</s>]]
|<s>describe mental tasks</s>
|done -> [[Reading a clock]]
|-
|Transition to College project
|David Pace
|describe the project, results etc.
|
|-
|Alternative Decoding techniques
|Joan Middendorf
|analogies, bottleneck writing tour, 3-d modeling, rubrics, game mechanics, etc
|done
|-
|[[Bottleneck]]
|
|describe flavors of bottlenecks (such as cognitive, emotional)
|Examples introduced for cognitive and emotional bottlenecks (Inna)
|-
|[[Threshold concept]]
|David Pace or Joan Middendorf
|describe relationship to bottlenecks and Decoding
|Added section comparing Decoding to TCs
|-
|Transformative Dialogues
|
|describe role of journal, link to journal, etc.
|
|-
|Graph reading
|
|Sort of a review article summarizing research work on this topic
|
|-
|Reading scholarly work
|
|Sort of a review article summarizing research work on this topic
|
|-
|[[Faculty development]]
|[[Peter Riegler]] (appreciating a helpful hand in particular when it comes to FLP etc.) -- I can help on this.
|describe how Decoding is used for faculty development; give some history (e.g. FLP); link to relevant Decoding literature
|started
|-
|[[AI Literacy as a bottle neck]]
|
|As an aspect of digital literacy and being critical users of digital tools, it is important to address AI literacy as an bottleneck for social science students
|
|-
|More development of Disrupting page
|Michelle Yeo
|More information and references could be added to the current Disrupting page
|
|-
|[[New to Decoding?]]
|David Pace
|
|
|}
See also [[Special:WantedPages]] for further pages needed.

Cognitive Bottleneck

2025-05-01T15:42:07Z

Inna mik: new line for example

A [[bottleneck]] with an emphasis on a cognitive difficulty within a lecture for the students. In these situations, students’ learning is blocked because they have failed to master particular mental operations. To help them overcome these obstacles, it is necessary to first make explicit for oneself precisely what steps are necessary to complete the work that students find so difficult.

An example of a cognitive bottleneck: [[Textual descriptions in mathematics|Textual descriptions in mathematics - Decoding the Disciplines]]

Emotional Bottleneck

2025-05-01T15:40:55Z

Inna mik: An example of an emotional bottleneck

A [[bottleneck]] with an emphasis on a negative emotional reaction to the content of the lecture or the methods used. It may be about the negative emotional reactions of students to either the processes of the course (e.g. students are upset that the work in this course does not match what they did in high school courses in the discipline) or to its subject matter (e.g. some of the findings in the discipline are at odds with things students were taught as they were growing up). 

Some bottlenecks are a combination of both emotional and [[Cognitive Bottleneck|cognitive bottlenecks]].

An example of an emotional bottleneck: [[Resistance to learning research methods|Resistance to learning research methods - Decoding the Disciplines]]

== References ==
<references />

Cognitive Bottleneck

2025-05-01T15:30:02Z

Inna mik: give an Example of cognitive Bottleneck

A [[bottleneck]] with an emphasis on a cognitive difficulty within a lecture for the students. In these situations, students’ learning is blocked because they have failed to master particular mental operations. To help them overcome these obstacles, it is necessary to first make explicit for oneself precisely what steps are necessary to complete the work that students find so difficult. E.g. [[Textual descriptions in mathematics|Textual descriptions in mathematics - Decoding the Disciplines]]

Derivative

2025-05-01T15:08:42Z

Inna mik:

Formula

2025-05-01T15:05:04Z

Inna mik:

Difficulties often emerge while students have to recognize a structure of a complex formula. In particular, it is difficult for students to

* use the brackets to parse the formula from handwritten form to one line form, e.g. for programming ;
* see the steps that should be done first while rearranging a formula;
* use substitutions in chain rule or while solving equations.

==Decoding work done==

===Step 1: Identification of [[bottleneck]]===
Students have difficulties while recognizing a structure of a formula, not being able to collapse and expand expressions, e.g. not seeing the difference between <math>e^(x^2)<math> which can be collapsed to e^t and (e^x)^2 which can be collapsed to t^2. Some students believe, that to square a+b means to square the elements a and b because they do not think that the sign + is important.

Desired outcome: Students should be able to decide on the essential structure of the formula and read it using operator names, e.g. (a+b)/(2x) is a quotient between the sum and a product. They should be able to blend out the complexity and propose a similar formula/equation with numbers, e.g. C=4 π K R_1R/( R_1−R) with respect to R is like 1=2x/(3-x) with respect to x.

===Step 2: Description of [[Mental moves|mental tasks]] needed to overcome the [[bottleneck]]===
The experts is able to see the structure by zooming in and out to blend out the details.

He knows the order of operation and thinks of results as

===[[Step 3 - Modeling Mental Operations|Step 3: Modelling the tasks]]===

===[[Step 4: Give Students Practice and Feedback|Step 4: Practice and Feedback]]===

===Anticipate and lessen resistance===

===Assessment of student mastery===

===Sharing===

==Researchers involved==

==Available resources==
[https://youtu.be/p50Q3mx-VNg Interview Video]

==See also==
[https://decodingthedisciplines.org/wp-content/uploads/2020/07/OF_V_Poster-Stank_170525_Endfassung-1.pdf Understanding Mathematics by Visualizing Structures], by Sabine Stank

===People interested in this Bottleneck===
Inna Mikhailova
[[Category:Bottleneck]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Decoding work]]
[[Category:Decoding Interview]]

Formula

2025-05-01T15:04:42Z

Inna mik:

Difficulties often emerge while students have to recognize a structure of a complex formula. In particular, it is difficult for students to

* use the brackets to parse the formula from handwritten form to one line form, e.g. for programming ;
* see the steps that should be done first while rearranging a formula;
* use substitutions in chain rule or while solving equations.

==Decoding work done==

===Step 1: Identification of [[bottleneck]]===
Students have difficulties while recognizing a structure of a formula, not being able to collapse and expand expressions, e.g. not seeing the difference between </math>e^(x^2)</math> which can be collapsed to e^t and (e^x)^2 which can be collapsed to t^2. Some students believe, that to square a+b means to square the elements a and b because they do not think that the sign + is important.

Desired outcome: Students should be able to decide on the essential structure of the formula and read it using operator names, e.g. (a+b)/(2x) is a quotient between the sum and a product. They should be able to blend out the complexity and propose a similar formula/equation with numbers, e.g. C=4 π K R_1R/( R_1−R) with respect to R is like 1=2x/(3-x) with respect to x.

===Step 2: Description of [[Mental moves|mental tasks]] needed to overcome the [[bottleneck]]===
The experts is able to see the structure by zooming in and out to blend out the details.

He knows the order of operation and thinks of results as

===[[Step 3 - Modeling Mental Operations|Step 3: Modelling the tasks]]===

===[[Step 4: Give Students Practice and Feedback|Step 4: Practice and Feedback]]===

===Anticipate and lessen resistance===

===Assessment of student mastery===

===Sharing===

==Researchers involved==

==Available resources==
[https://youtu.be/p50Q3mx-VNg Interview Video]

==See also==
[https://decodingthedisciplines.org/wp-content/uploads/2020/07/OF_V_Poster-Stank_170525_Endfassung-1.pdf Understanding Mathematics by Visualizing Structures], by Sabine Stank

===People interested in this Bottleneck===
Inna Mikhailova
[[Category:Bottleneck]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Decoding work]]
[[Category:Decoding Interview]]

Formula

2025-05-01T15:04:19Z

Inna mik:

Difficulties often emerge while students have to recognize a structure of a complex formula. In particular, it is difficult for students to

* use the brackets to parse the formula from handwritten form to one line form, e.g. for programming ;
* see the steps that should be done first while rearranging a formula;
* use substitutions in chain rule or while solving equations.

==Decoding work done==

===Step 1: Identification of [[bottleneck]]===
Students have difficulties while recognizing a structure of a formula, not being able to collapse and expand expressions, e.g. not seeing the difference between </math> e^(x^2) </math> which can be collapsed to e^t and (e^x)^2 which can be collapsed to t^2. Some students believe, that to square a+b means to square the elements a and b because they do not think that the sign + is important.

Desired outcome: Students should be able to decide on the essential structure of the formula and read it using operator names, e.g. (a+b)/(2x) is a quotient between the sum and a product. They should be able to blend out the complexity and propose a similar formula/equation with numbers, e.g. C=4 π K R_1R/( R_1−R) with respect to R is like 1=2x/(3-x) with respect to x.

===Step 2: Description of [[Mental moves|mental tasks]] needed to overcome the [[bottleneck]]===
The experts is able to see the structure by zooming in and out to blend out the details.

He knows the order of operation and thinks of results as

===[[Step 3 - Modeling Mental Operations|Step 3: Modelling the tasks]]===

===[[Step 4: Give Students Practice and Feedback|Step 4: Practice and Feedback]]===

===Anticipate and lessen resistance===

===Assessment of student mastery===

===Sharing===

==Researchers involved==

==Available resources==
[https://youtu.be/p50Q3mx-VNg Interview Video]

==See also==
[https://decodingthedisciplines.org/wp-content/uploads/2020/07/OF_V_Poster-Stank_170525_Endfassung-1.pdf Understanding Mathematics by Visualizing Structures], by Sabine Stank

===People interested in this Bottleneck===
Inna Mikhailova
[[Category:Bottleneck]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Decoding work]]
[[Category:Decoding Interview]]

Formula

2025-05-01T15:03:54Z

Inna mik: try math again

Difficulties often emerge while students have to recognize a structure of a complex formula. In particular, it is difficult for students to

* use the brackets to parse the formula from handwritten form to one line form, e.g. for programming ;
* see the steps that should be done first while rearranging a formula;
* use substitutions in chain rule or while solving equations.

==Decoding work done==

===Step 1: Identification of [[bottleneck]]===
Students have difficulties while recognizing a structure of a formula, not being able to collapse and expand expressions, e.g. not seeing the difference between <math> e^(x^2) </math> x^2 <math> which can be collapsed to e^t and (e^x)^2 which can be collapsed to t^2. Some students believe, that to square a+b means to square the elements a and b because they do not think that the sign + is important.

Desired outcome: Students should be able to decide on the essential structure of the formula and read it using operator names, e.g. (a+b)/(2x) is a quotient between the sum and a product. They should be able to blend out the complexity and propose a similar formula/equation with numbers, e.g. C=4 π K R_1R/( R_1−R) with respect to R is like 1=2x/(3-x) with respect to x.

===Step 2: Description of [[Mental moves|mental tasks]] needed to overcome the [[bottleneck]]===
The experts is able to see the structure by zooming in and out to blend out the details.

He knows the order of operation and thinks of results as

===[[Step 3 - Modeling Mental Operations|Step 3: Modelling the tasks]]===

===[[Step 4: Give Students Practice and Feedback|Step 4: Practice and Feedback]]===

===Anticipate and lessen resistance===

===Assessment of student mastery===

===Sharing===

==Researchers involved==

==Available resources==
[https://youtu.be/p50Q3mx-VNg Interview Video]

==See also==
[https://decodingthedisciplines.org/wp-content/uploads/2020/07/OF_V_Poster-Stank_170525_Endfassung-1.pdf Understanding Mathematics by Visualizing Structures], by Sabine Stank

===People interested in this Bottleneck===
Inna Mikhailova
[[Category:Bottleneck]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Decoding work]]
[[Category:Decoding Interview]]

Formula

2025-05-01T15:03:07Z

Inna mik: try math mode

Difficulties often emerge while students have to recognize a structure of a complex formula. In particular, it is difficult for students to

* use the brackets to parse the formula from handwritten form to one line form, e.g. for programming ;
* see the steps that should be done first while rearranging a formula;
* use substitutions in chain rule or while solving equations.

==Decoding work done==

===Step 1: Identification of [[bottleneck]]===
Students have difficulties while recognizing a structure of a formula, not being able to collapse and expand expressions, e.g. not seeing the difference between <math> e^(x^2) </math> which can be collapsed to e^t and (e^x)^2 which can be collapsed to t^2. Some students believe, that to square a+b means to square the elements a and b because they do not think that the sign + is important.

Desired outcome: Students should be able to decide on the essential structure of the formula and read it using operator names, e.g. (a+b)/(2x) is a quotient between the sum and a product. They should be able to blend out the complexity and propose a similar formula/equation with numbers, e.g. C=4 π K R_1R/( R_1−R) with respect to R is like 1=2x/(3-x) with respect to x.

===Step 2: Description of [[Mental moves|mental tasks]] needed to overcome the [[bottleneck]]===
The experts is able to see the structure by zooming in and out to blend out the details.

He knows the order of operation and thinks of results as

===[[Step 3 - Modeling Mental Operations|Step 3: Modelling the tasks]]===

===[[Step 4: Give Students Practice and Feedback|Step 4: Practice and Feedback]]===

===Anticipate and lessen resistance===

===Assessment of student mastery===

===Sharing===

==Researchers involved==

==Available resources==
[https://youtu.be/p50Q3mx-VNg Interview Video]

==See also==
[https://decodingthedisciplines.org/wp-content/uploads/2020/07/OF_V_Poster-Stank_170525_Endfassung-1.pdf Understanding Mathematics by Visualizing Structures], by Sabine Stank

===People interested in this Bottleneck===
Inna Mikhailova
[[Category:Bottleneck]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Decoding work]]
[[Category:Decoding Interview]]

Formula

2025-05-01T14:56:35Z

Inna mik: test Latex

Difficulties often emerge while students have to recognize a structure of a complex formula. In particular, it is difficult for students to

* use the brackets to parse the formula from handwritten form to one line form, e.g. for programming ;
* see the steps that should be done first while rearranging a formula;
* use substitutions in chain rule or while solving equations.

==Decoding work done==

===Step 1: Identification of [[bottleneck]]===
Students have difficulties while recognizing a structure of a formula, not being able to collapse and expand expressions, e.g. not seeing the difference between \(e^(x^2)\) which can be collapsed to e^t and (e^x)^2 which can be collapsed to t^2. Some students believe, that to square a+b means to square the elements a and b because they do not think that the sign + is important.

Desired outcome: Students should be able to decide on the essential structure of the formula and read it using operator names, e.g. (a+b)/(2x) is a quotient between the sum and a product. They should be able to blend out the complexity and propose a similar formula/equation with numbers, e.g. C=4 π K R_1R/( R_1−R) with respect to R is like 1=2x/(3-x) with respect to x.

===Step 2: Description of [[Mental moves|mental tasks]] needed to overcome the [[bottleneck]]===

===Step 3: Modelling the tasks===

===[[Step 4: Give Students Practice and Feedback|Step 4: Practice and Feedback]]===

===Anticipate and lessen resistance===

===Assessment of student mastery===

===Sharing===

==Researchers involved==

==Available resources==
[https://youtu.be/p50Q3mx-VNg Interview Video]

==See also==
[https://decodingthedisciplines.org/wp-content/uploads/2020/07/OF_V_Poster-Stank_170525_Endfassung-1.pdf Understanding Mathematics by Visualizing Structures], by Sabine Stank

===People interested in this Bottleneck===
Inna Mikhailova
[[Category:Bottleneck]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Decoding work]]
[[Category:Decoding Interview]]

Just in Time Teaching

2025-05-01T14:44:06Z

Inna mik: change link

The goal: ''Make student work autonomously and ask questions in order to get deeper understanding''
* How: Provide online material with test and “question about questions” ahead of the lesson, followed by discussion during the lesson
* Why it works: Asking is anonymous, Student has time to formulate the question, Learning to ask questions is learning thinking!
* More Material:[https://citl.illinois.edu/citl-101/teaching-learning/resources/teaching-strategies/just-in-time-teaching Just-in-Time Teaching (JiTT)]
* Video: https://youtu.be/D5ADk2gi6JM

Just in Time Teaching

2025-05-01T14:37:15Z

Inna mik: First Draft JITT

The goal: ''Make student work autonomously and ask questions in order to get deeper understanding''
* How: Provide online material with test and “question about questions” ahead of the lesson, followed by discussion during the lesson
* Why it works: Asking is anonymous, Student has time to formulate the question, Learning to ask questions is learning thinking!
* More Material:[https://cft.vanderbilt.edu/guides-sub-pages/just-in-time-teaching-jitt/ Just-in-Time Teaching (JiTT) | Center for Teaching | Vanderbilt University]
* Video: https://youtu.be/D5ADk2gi6JM

Derivative

2025-05-01T07:11:12Z

Inna mik: Video Link added

{{DISPLAYTITLE:DecodingWork:From Derivative to Proportionality}}

===Step 1: Description of Bottleneck===
Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. x² into 2x. In case they know, that "derivative=slope", they do not know what exactly the slope means, because they do not calculate "change of function value=slope*change of argument value".

Desired learning outcome: Students should understand that the change of function value is proportional to the change of argument value, and the derivative is exactly the proportionality factor. Students should be able to construct a triangle on the graph of a function to visually represent changes in function values and the corresponding slope (rate of change).

Concrete problems may be, for example:

* A) Given that f(1)=2 and f'(1)=3 students should be able to make an estimation of f(1,01) as f(1)+f'(1)*0,01=2.003
* B) Given the entities of function and entities of argument, decide on the entities of derivative. E.g. fuel price: Argument x is in Litre, function f(x) is in Euro, follows that the derivative f'(x) is in Euro/Litre
* C) Given the text, which describes the rate of change, deduce a differential equation. E.g. the population growth: Newborns are proportional to the population number leads to the equation dN/dt=C*N

=== Step 2: Description of mental tasks needed to overcome the bottleneck ===

# While thinking about derivative, a mathematician switches his mind from "static" formula story (algebra) to "dynamic" function story (analysis). He clearly understands that the story is not about calculating one value. It is about how function value changes if argument value changes (see a related Bottleneck [[Function]])
# An expert can mask from his mind the fact, that the derivative should be calculated as a limit. He works with quotient "Change of function value" over "Change of Argument value" if this is useful to deduce needed information. E.g. if we are only interested in entities or if the formula f(x) is not given, but only numerical values. He deduces the needed values from the given graph of function by drawing a triangle in order to represent changes in values and the slope between two points.
# He can work with augmenting (moving to the right on the graph) and reducing (moving to the left) value of argument, and he can split the change of the value in multiple steps if he realizes that the step is too big.
# While doing so, he is not afraid of being unprecise, because he knows, that the approximation is locally allowed and can estimate the approximation error.

=== Step 3: Modelling the tasks ===

==Available resources==
[https://youtu.be/gh-cgUDM1xI Interview Video]

===People interested in this Bottleneck===
Inna Mikhailova
[[Category:Bottleneck]]
[[Category:Mathematics]]
[[Category:Calculus]]
[[Category:Decoding work]]
[[Category:Decoding Interview]]

Threshold concept

2025-05-01T06:03:19Z

Inna mik: Link Step 4 repaired

[[File:Treshhold concept 1743096802788.png|thumb|300x300px|A threshold concept it akin to a portal or a liminal space, opening up a new and previously inaccessible way of thinking about something.]]
A '''threshold concept''' is a core concepts which, once understood, transforms perception of a given subject, phenomenon, or experience. Its acquisition is inherently difficult. Metaphorically it involves being stuck by repeatedly bumping into a threshold until one manages to overcome the threshold. Other metaphors used are portal and liminal space, opening up a new and previously inaccessible way of thinking about something. The term threshold concept was coined by Jan Meyer and Ray Land.<ref name=":0">Meyer J H F and Land R 2003 "Threshold Concepts and Troublesome Knowledge: Linkages to Ways of Thinking and Practising" in ''Improving Student Learning: Ten Years On''. C. Rust (Ed), OCSLD, Oxford.</ref>

Like [[Decoding the Disciplines]] and Perkin's notion of troublesome knowledge, threshold concepts are a theory of difficulty, i.e. a theory explaining aspects of learning and teaching by focussing on difficulties inherent to them.<ref>Pace, D. (2017): [[The Decoding the Disciplines Paradigm: Seven Steps to Increased Student Learning]]. Bloomington: Indiana University Press, p. 21</ref>

== Properties of threshold concepts ==
Threshold concepts typically have the following properties:<ref name=":0" />

* ''transformative'': Once a threshold concept has been understood, it can change the perception of a subject or part of it.
* ''irreversible'': The change in perspective that the acquisition of a threshold concept entails is difficult to forget or can only be unlearned again with considerable effort.
* ''integrative'': Threshold concepts reveal previously hidden connections.
* ''bounded'': Every concept has boundaries with thresholds to neighboring new conceptual areas.
* ''troublesome'': The internalisation of concepts is troublesome for a variety of reasons. Concepts can seem strange, implicit, conceptually difficult, counter-intuitive or characterized by over-complexity.

Threshold concepts are not only discussed for students who are familiarising themselves with a subject, but also for teachers and their understanding of teaching processes. According to Meyer and Land the idea of threshold concepts is a threshold concept by itself.<ref>J. H. F. Meyer, R. Land: ''Threshold concepts and troublesome knowledge (2): Epistemological considerations and a conceptual framework for teaching and learning.'' In: ''Higher Education.'' Band 49, Nr. 3, 2005, S. 373–388</ref>

Give example of threshold concept

== Relation to Decoding the Disciplines ==
Threshold Concepts and [[Step 1 - Identify a Bottleneck to Learning|Step 1 Identify bottlenecks]] of ''Decoding the Disciplines'' both use theories of difficulty (Perkins, 2007) and focus on the hurdles of content. They both foreground the aspects of learning that prove consistently troublesome to uncover the tacit or “secret knowledge” of the discipline. Theories of difficulty design the challenge, but do not solve them. The main way the two theories differ is that ''Decoding the Disciplines'' from the start had a second theoretical component. Decoding Steps [[Step 3 - Modeling Mental Operations|3]]-[[Step 4: Give Students Practice and Feedback|4]]-[[Step 5 - Motivate and lessen resistance|5]]-[[Step 6 - Assess Student Mastery|6]] navigate the problem using pedagogical theory. These Steps organize teaching and learning processes for good results, scaffolding instructional techniques to foster learning (Perkins, 2007). “The great strength of decoding is the methodology that undergirds the theory…,” (p. 41, Shopkow & Middendorf, 2019).

In categorial terms every threshold concept is a bottleneck while not every bottleneck is a threshold concept.

Shopkow <ref>Shopkow, Leah (2010). “[[What ‘Decoding the Disciplines’ has to offer ‘Threshold Concepts]],’” in Threshold Concepts and Transformational Learning, ed. Jan H. F. Meyer, Ray Land, & Catherine Baillie (Rotterdam: Sense Publications), 317-32.</ref>. Threshold concepts are the paradigm-shifting kinds of bottlenecks (Middendorf, 2025).

Shopkow, L., & Middendorf, J. (2019). CAUTION! THEORIES AT PLAY! Threshold concepts and Decoding the Disciplines. In J.A Timmermans & R. Land (Eds.) ''Threshold concepts on the edge.'' Leiden: Brill/Sense, pp. 37-50.

Middendorf, J (2025 in press). The Theory Bottleneck and Decoding the Disciplines. Die Hoschudt

Perkins, D. (2007). Theories of difficulty. In N. Entwistle, & P. Tomlinson (Eds.), ''Student learning and university teaching'' (pp. 31–48). Leicester: British Psychological Society.

== See also ==

* [[:Category:Threshold concept|List of publications related to Decoding and Treshold Concepts]]
* [https://www.ee.ucl.ac.uk/mflanaga/thresholds.html#spectop List of Threshold Concepts]

== References ==

Template:Decoding work

2025-04-24T16:20:56Z

Inna mik: Links Step 6-7

short summary

==Decoding work done==

===Identification of [[bottleneck]]===

===Description of [[Question 2: Uncovering the tacit mental moves|mental tasks]] needed to overcome the bottleneck===

===[[Step 3 - Modeling Mental Operations|Modelling the tasks]]===

===[[Step 4: Give Students Practice and Feedback|Practice and Feedback]]===

===[[Step 5 - Motivate and lessen resistance|Anticipate and lessen resistance]]===

===[[Step 6 - Assess Student Mastery|Assessment of student mastery]]===

===[[Step 7 - Share What Has Been Learned Through the Decoding Process|Sharing]]===

==Researchers involved==

==Available resources==

==See also==

==Notes==

==References==

[[Category:Decoding work]]

Template:Decoding work

2025-04-24T16:19:16Z

Inna mik: Link Step 2

short summary

==Decoding work done==

===Identification of [[bottleneck]]===

===Description of [[Question 2: Uncovering the tacit mental moves|mental tasks]] needed to overcome the bottleneck===

===[[Step 3 - Modeling Mental Operations|Modelling the tasks]]===

===[[Step 4: Give Students Practice and Feedback|Practice and Feedback]]===

===Anticipate and lessen resistance===

===Assessment of student mastery===

===Sharing===

==Researchers involved==

==Available resources==

==See also==

==Notes==

==References==

[[Category:Decoding work]]

Template:Decoding work

2025-04-24T16:18:08Z

Inna mik: Links step 3-4

short summary

==Decoding work done==

===Identification of [[bottleneck]]===

===Description of [[mental tasks]] needed to overcome the bottleneck===

===[[Step 3 - Modeling Mental Operations|Modelling the tasks]]===

===[[Step 4: Give Students Practice and Feedback|Practice and Feedback]]===

===Anticipate and lessen resistance===

===Assessment of student mastery===

===Sharing===

==Researchers involved==

==Available resources==

==See also==

==Notes==

==References==

[[Category:Decoding work]]

Template:Decoding work

2025-04-24T16:14:51Z

Inna mik: Links Steps

short summary

==Decoding work done==

===Identification of [[bottleneck]]===

===Description of [[mental tasks]] needed to overcome the bottleneck===

===[[Modelling the tasks]]===

===[[Practice and Feedback]]===

===Anticipate and lessen resistance===

===Assessment of student mastery===

===Sharing===

==Researchers involved==

==Available resources==

==See also==

==Notes==

==References==

[[Category:Decoding work]]

Template:Decoding work

2025-04-24T15:59:47Z

Inna mik: Link to Bottleneck

short summary

==Decoding work done==

===Identification of [[bottleneck]]===

===Description of mental tasks needed to overcome the bottleneck===

===Modelling the tasks===

===Practice and Feedback===

===Anticipate and lessen resistance===

===Assessment of student mastery===

===Sharing===

==Researchers involved==

==Available resources==

==See also==

==Notes==

==References==

[[Category:Decoding work]]

Formula

2025-04-24T15:18:12Z

Inna mik: Categories

Difficulties often emerge while students have to recognize a structure of a complex formula. In particular, it is difficult for students to

* use the brackets to parse the formula from handwritten form to one line form, e.g. for programming ;
* see the steps that should be done first while rearranging a formula;
* use substitutions in chain rule or while solving equations.

==Decoding work done==

===Step 1: Identification of [[bottleneck]]===
Students have difficulties while recognizing a structure of a formula, not being able to collapse and expand expressions, e.g. not seeing the difference between e^(x^2) which can be collapsed to e^t and (e^x)^2 which can be collapsed to t^2. Some students believe, that to square a+b means to square the elements a and b because they do not think that the sign + is important.

Desired outcome: Students should be able to decide on the essential structure of the formula and read it using operator names, e.g. (a+b)/(2x) is a quotient between the sum and a product. They should be able to blend out the complexity and propose a similar formula/equation with numbers, e.g. C=4 π K R_1R/( R_1−R) with respect to R is like 1=2x/(3-x) with respect to x.

===Step 2: Description of [[Mental moves|mental tasks]] needed to overcome the [[bottleneck]]===

===Step 3: Modelling the tasks===

===Practice and Feedback===

===Anticipate and lessen resistance===

===Assessment of student mastery===

===Sharing===

==Researchers involved==

==Available resources==

==See also==
[https://decodingthedisciplines.org/wp-content/uploads/2020/07/OF_V_Poster-Stank_170525_Endfassung-1.pdf Understanding Mathematics by Visualizing Structures], by Sabine Stank

===People interested in this Bottleneck===
Inna Mikhailova
[[Category:Bottleneck]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Decoding work]]
[[Category:Decoding Interview]]

Formula

2025-04-24T15:12:10Z

Inna mik: Öinks added

Formula

2025-04-24T10:12:46Z

Inna mik: Description

Derivative

2025-04-24T08:01:40Z

Inna mik: Link to Function + less Steps

{{DISPLAYTITLE:DecodingWork:From Derivative to Proportionality}}

===Step 1: Description of Bottleneck===
Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. x² into 2x. In case they know, that "derivative=slope", they do not know what exactly the slope means, because they do not calculate "change of function value=slope*change of argument value".

Desired learning outcome: Students should understand that the change of function value is proportional to the change of argument value, and the derivative is exactly the proportionality factor. Students should be able to construct a triangle on the graph of a function to visually represent changes in function values and the corresponding slope (rate of change).

Concrete problems may be, for example:

* A) Given that f(1)=2 and f'(1)=3 students should be able to make an estimation of f(1,01) as f(1)+f'(1)*0,01=2.003
* B) Given the entities of function and entities of argument, decide on the entities of derivative. E.g. fuel price: Argument x is in Litre, function f(x) is in Euro, follows that the derivative f'(x) is in Euro/Litre
* C) Given the text, which describes the rate of change, deduce a differential equation. E.g. the population growth: Newborns are proportional to the population number leads to the equation dN/dt=C*N

=== Step 2: Description of mental tasks needed to overcome the bottleneck ===

# While thinking about derivative, a mathematician switches his mind from "static" formula story (algebra) to "dynamic" function story (analysis). He clearly understands that the story is not about calculating one value. It is about how function value changes if argument value changes (see a related Bottleneck [[Function]])
# An expert can mask from his mind the fact, that the derivative should be calculated as a limit. He works with quotient "Change of function value" over "Change of Argument value" if this is useful to deduce needed information. E.g. if we are only interested in entities or if the formula f(x) is not given, but only numerical values. He deduces the needed values from the given graph of function by drawing a triangle in order to represent changes in values and the slope between two points.
# He can work with augmenting (moving to the right on the graph) and reducing (moving to the left) value of argument, and he can split the change of the value in multiple steps if he realizes that the step is too big.
# While doing so, he is not afraid of being unprecise, because he knows, that the approximation is locally allowed and can estimate the approximation error.

=== Step 3: Modelling the tasks ===

===People interested in this Bottleneck===
Inna Mikhailova
[[Category:Bottleneck]]
[[Category:Mathematics]]
[[Category:Calculus]]
[[Category:Decoding work]]
[[Category:Decoding Interview]]

Derivative

2025-04-16T12:00:52Z

Inna mik: minor changes Mental Task

{{DISPLAYTITLE:DecodingWork:From Derivative to Proportionality}}

===Description of Bottleneck===
Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. x² into 2x. They do not have an idea of "derivative=slope" or in case they have it they do not know what exactly the slope means, because they do not calculate "change of function value=slope*change of argument value"

Desired learning outcome: Students should understand that the change of function value is proportional to the change of argument value, and the derivative is exactly the proportionality factor. They should be able to draw for a given graph of function a triangle to illustrate the changes in values and the slope

Concrete problems may be, for example:

* A) Given that f(1)=2 and f'(1)=3 students should be able to make an estimation of f(1,01) as f(1)+f'(1)*0,01=2.003
* B) Given the entities of function and entities of argument, decide on the entities of derivative. E.g. fuel price: Argument x is in Litre, function f(x) is in Euro, follows that the derivative f'(x) is in Euro/Litre
* C) Given the text, which describes the rate of change, deduce a differential equation. E.g. the population growth: Newborns are proportional to the population number leads to the equation dN/dt=C*N

=== Description of mental tasks needed to overcome the bottleneck ===

# While thinking about derivative, mathematician switches his mind from "static" formula story (algebra) to "dynamic" function story (analysis)
# He clearly understands that the story is not about calculating one value, but about the process and how function value depends on argument value
# He knows, that he can look onto small local changes if he wants to calculate the concrete change rate, and he knows where to place the changes into the graph of function.
# He can forget, that the derivative should be calculated as a limit, if the fact that derivative is a quotient df/dx is enough to deduce needed information or is the only possibility. E.g. if we are only interested in entities or if the formula f(x) is not given, but only numerical values df and dx
# He is not afraid of being unprecise, because he knows, that the approximation by tangent is allowed

=== Modelling the tasks ===

===People interested in this Bottleneck===
Inna Mikhailova
[[Category:Bottleneck]]
[[Category:Mathematics]]
[[Category:Calculus]]

Derivative

2025-04-16T11:51:51Z

Inna mik: Describe mental tasks

{{DISPLAYTITLE:DecodingWork:From Derivative to Proportionality}}

===Description of Bottleneck===
Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. x² into 2x. They do not have an idea of "derivative=slope" or in case they have it they do not know what exactly the slope means, because they do not calculate "change of function value=slope*change of argument value"

Desired learning outcome: Students should understand that the change of function value is proportional to the change of argument value, and the derivative is exactly the proportionality factor. They should be able to draw for a given graph of function a triangle to illustrate the changes in values and the slope

Concrete problems may be, for example:

* A) Given that f(1)=2 and f'(1)=3 students should be able to make an estimation of f(1,01) as f(1)+f'(1)*0,01=2.003
* B) Given the entities of function and entities of argument, decide on the entities of derivative. E.g. fuel price: Argument x is in Litre, function f(x) is in Euro, follows that the derivative f'(x) is in Euro/Litre
* C) Given the text, which describes the rate of change, deduce a differential equation. E.g. the population growth: Newborns are proportional to the population number leads to the equation dN/dt=C*N

=== Description of mental tasks needed to overcome the bottleneck ===

# While thinking about derivative, mathematician switches his mind from "static" formula story (algebra) to "dynamic" function story (analysis)
# He clearly understands that the story is not about calculating one value, but about the process and how function value depends on argument value
# He knows, that he can look onto small local changes if he wants to calculate the concrete change rate and he knows were to place the changes into the graph of function.
# He can forget, that the derivative should be calculated as a limit, if the fact that derivative is a quotient df/dx is enough to deduce needed information e.g. the entities
# He is not afraid of being unprecise, because he knows, that the approximation by tangent is allowed

=== Modelling the tasks ===

===People interested in this Bottleneck===
Inna Mikhailova
[[Category:Bottleneck]]
[[Category:Mathematics]]
[[Category:Calculus]]

Derivative

2025-04-16T10:44:03Z

Inna mik: Concretization of bottleneck

User:Inna Mikhailova

2025-03-05T17:18:16Z

Inna mik: Describe next steps

==Inna Mikhailova ==
==Field of Interest (Discipline or Area of Teaching)==
Mathematics

==Type of Approach (i.e. students as co-investigators)==
Just -in-Time-Teaching & Peer Instruction & Project-based

==Contact==
inna.mikhailova@h-da.de

== Why writing? ==
Thanks to Decoding the Disciplines, I realized that what I do as an "expert" in my subject differs considerably from what I teach in the lecture, precisely at the points that are difficult for my students. I would like to share with my colleagues by writing two different sorts of experience: specific mathematical content, and general change of attitude towards teaching produced by decoding praxis.

== Next Steps ==
Describe Bottlenecks 1) about Formulae reading (Zoom in-out)2) Derivative is not magic transformation
[[Category:People]]

User:Inna Mikhailova

2025-03-05T16:23:44Z

Inna mik: move text down

User:Inna Mikhailova

2025-03-05T16:20:22Z

Inna mik: user-page with template

==Inna Mikhailova ==

==Field of Interest (Discipline or Area of Teaching): Mathematics==

==Type of Approach (i.e. students as co-investigators): Just -in-Time-Teaching & Peer Instruction & Project-based==

==Contact: inna.mikhailova@h-da.de==
[[Category:People]]

Derivative

2024-07-19T13:03:39Z

Inna mik: Change Error in Proportionality, remove template

Derivative

2024-07-19T12:27:00Z

Inna mik: added bottleneck about derivative and proportionality

===How to use this Template===

#On the top ... add the discipline the bottleneck is belonging to
#Under the headline "Description of Bottleneck" add your description
#...
#Delete the whole Section "Hot to use this Template
#Save

===Description of Bottleneck===
Students have difficulties to use derivative to calculate the change of function value. They believe the derivative is a magic transformation of one formula into another, e.g. x^2 into 2*x. They do not have an idea of Derivative=slope and change of Value=slope*change of argument

Desired learning outcome: given that f(1)=2 and f'(1)=3 the students should be able to make an estimation of f(1,01) as f(1)+f'(1)*0,01=2.003

===People interested in this Bottleneck===
[[Category:Bottleneck]]
[[Category:Mathematics]]