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==Decoding work done== | ==Decoding work done== | ||
===Step 1: Identification of bottleneck=== | ===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. | 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. | 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 tasks needed to overcome the bottleneck=== | ===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. | |||
===Practice and Feedback=== | ===[[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=== | ===Anticipate and lessen resistance=== | ||
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==Available resources== | ==Available resources== | ||
[https://youtu.be/p50Q3mx-VNg Interview Video] | |||
==See also== | ==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 work]] | ||
[[Category:Decoding Interview]] | |||
Latest revision as of 10:22, 17 June 2025
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 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: Modelling the tasks
Step 4: Practice and Feedback
Anticipate and lessen resistance
Assessment of student mastery
Sharing
Researchers involved
Available resources
See also
Understanding Mathematics by Visualizing Structures, by Sabine Stank
People interested in this Bottleneck
Inna Mikhailova
