Formula - Revision history

Riegler: fixing math representation

2026-10-11T11:00:15Z

fixing math representation

Inna mik: confrontation exercise

2026-07-17T14:29:59Z

confrontation exercise

← Older revision		Revision as of 14:29, 17 July 2026
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	===[[Step 4: Give Students Practice and Feedback\|Step 4: Practice and Feedback]]===		===[[Step 4: Give Students Practice and Feedback\|Step 4: Practice and Feedback]]===
	An important outcome of decoding is the recognition that students must first master reading and writing mathematical formulas before they can successfully engage with tasks that depend on these skills, such as applying the chain rule or rearranging formulas. Consequently, explicit practice in reading and interpreting mathematical expressions is essential.		An important outcome of decoding is the recognition that students must first master reading and writing mathematical formulas before they can successfully engage with tasks that depend on these skills, such as applying the chain rule or rearranging formulas. Consequently, explicit practice in reading and interpreting mathematical expressions is essential.

			A good starting point is "confrontation" exercise: " Paul says to his teacher, "I calculated ''a'' plus ''b'' divided by two times ''c''." Write a mathematical expression that matches his statement."

			Students quickly realise that the Paul's statement is ambiguous because it does not specify how the operations are grouped. That allows to stress the role and importance of structuring and brackets.

	One possible exercise for practicing the use of brackets is a "telephone game" (known as ''Chinese whispers'' or in Germany as ''Stille Post'' ). The activity can be carried out in small groups of, for example, three students. The first student sends a neighbour a handwritten mathematical expression, such as <math>{\sqrt{a+b} - {{2c}\over{d+e}}}</math>. The second student is only allowed to pass the expression on in typed form using standard mathematical input syntax, for example: <code>(a+b)^(1/2) - (2*c)/(d+e)</code>		One possible exercise for practicing the use of brackets is a "telephone game" (known as ''Chinese whispers'' or in Germany as ''Stille Post'' ). The activity can be carried out in small groups of, for example, three students. The first student sends a neighbour a handwritten mathematical expression, such as <math>{\sqrt{a+b} - {{2c}\over{d+e}}}</math>. The second student is only allowed to pass the expression on in typed form using standard mathematical input syntax, for example: <code>(a+b)^(1/2) - (2*c)/(d+e)</code>

Inna mik: practice of "telefon game" added

2026-07-17T14:17:10Z

practice of "telefon game" added

← Older revision		Revision as of 14:17, 17 July 2026
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	===[[Step 4: Give Students Practice and Feedback\|Step 4: Practice and Feedback]]===		===[[Step 4: Give Students Practice and Feedback\|Step 4: Practice and Feedback]]===
			An important outcome of decoding is the recognition that students must first master reading and writing mathematical formulas before they can successfully engage with tasks that depend on these skills, such as applying the chain rule or rearranging formulas. Consequently, explicit practice in reading and interpreting mathematical expressions is essential.

	~~===Anticipate and lessen resistance===~~		One possible exercise for practicing the use of brackets is a "telephone game" (known as ''Chinese whispers'' or in Germany as ''Stille Post'' ). The activity can be carried out in small groups of, for example, three students. The first student sends a neighbour a handwritten mathematical expression, such as <math>{\sqrt{a+b} - {{2c}\over{d+e}}}</math>. The second student is only allowed to pass the expression on in typed form using standard mathematical input syntax, for example: <code>(a+b)^(1/2) - (2*c)/(d+e)</code>

	===Assessment of student mastery===		The third student must reconstruct the handwritten mathematical notation from the typed expression and compare it with the original formula. Differences between the original and reconstructed expressions can then be discussed to reinforce the importance of correct notation which mathes the structure of the formula.

			===Step 5: Anticipate and lessen resistance===
			Because typing mathematical formulas is an integral part of programming and prompting, the ''telephone game'' provides authentic practice and is therefore typically well received by students.

			===Step 6: Assessment of student mastery===

	===Sharing===		===Sharing===

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

2025-06-17T08:22:47Z

(username removed) (log details removed)

← Older revision	Revision as of 08:22, 17 June 2025
(No difference)

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

2025-06-17T08:14:11Z

(username removed) (log details removed)

← Older revision	Revision as of 08:14, 17 June 2025
(No difference)

Inna mik: mental task example

2025-05-29T15:42:11Z

mental task example

← Older revision		Revision as of 15:42, 29 May 2025
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	===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 ~~<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.		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 moves\|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 to blend out the details.		The experts is able to see the structure by zooming in and out into the formula to blend out the details.

	He ~~knows~~ the ~~order~~ of operation and ~~thinks of~~ results as		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+bs" 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 3 - Modeling Mental Operations\|Step 3: Modelling the tasks]]===

Inna mik at 15:05, 1 May 2025

2025-05-01T15:05:04Z

← Older revision		Revision as of 15:05, 1 May 2025
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	===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 </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.		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.		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.

Inna mik at 15:04, 1 May 2025

2025-05-01T15:04:42Z

← Older revision		Revision as of 15:04, 1 May 2025
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	===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 </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.		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.		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.

Inna mik at 15:04, 1 May 2025

2025-05-01T15:04:19Z

← Older revision		Revision as of 15:04, 1 May 2025
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	===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 <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.		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.		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.

Inna mik: try math again

2025-05-01T15:03:54Z

try math again

← Older revision		Revision as of 15:03, 1 May 2025
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	===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 <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.		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.		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.

← Older revision		Revision as of 11:00, 11 October 2026
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	===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 <math>e^(x^2)</math> which can be collapsed to e^t and <math>(e^x)^2</math> which can be collapsed to <math>t^2</math>. 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. <math>(a+b)/(2x)</math> 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. <math>{C=4 \pi K R_1 R / (R_1 - R) }</math> with respect to <math>R</math> is like <math>1=2x/(3-x)</math> with respect to <math>x</math>.

	===Step 2: Description of [[Mental moves\|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.		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+bs" 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.		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+bs" 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 3 - Modeling Mental Operations\|Step 3: Modelling the tasks]]===
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	An important outcome of decoding is the recognition that students must first master reading and writing mathematical formulas before they can successfully engage with tasks that depend on these skills, such as applying the chain rule or rearranging formulas. Consequently, explicit practice in reading and interpreting mathematical expressions is essential.		An important outcome of decoding is the recognition that students must first master reading and writing mathematical formulas before they can successfully engage with tasks that depend on these skills, such as applying the chain rule or rearranging formulas. Consequently, explicit practice in reading and interpreting mathematical expressions is essential.

	A good starting point is "confrontation" exercise: " Paul says to his teacher, "I calculated ''a'' plus ''b'' divided by two times ''c''." Write a mathematical expression that matches his statement."		A good starting point is "confrontation" exercise: " Paul says to his teacher, "I calculated <math>a</math> plus <math>b</math> ' divided by two times <math>c</math> ." Write a mathematical expression that matches his statement."

	Students quickly realise that the Paul's statement is ambiguous because it does not specify how the operations are grouped. That allows to stress the role and importance of structuring and brackets.		Students quickly realise that the Paul's statement is ambiguous because it does not specify how the operations are grouped. That allows to stress the role and importance of structuring and brackets.
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	The third student must reconstruct the handwritten mathematical notation from the typed expression and compare it with the original formula. Differences between the original and reconstructed expressions can then be discussed to reinforce the importance of correct notation which mathes the structure of the formula.		The third student must reconstruct the handwritten mathematical notation from the typed expression and compare it with the original formula. Differences between the original and reconstructed expressions can then be discussed to reinforce the importance of correct notation which mathes the structure of the formula.

	===Step 5: Anticipate and lessen resistance===		=== Step 5: Anticipate and lessen resistance===
	Because typing mathematical formulas is an integral part of programming and prompting, the ''telephone game'' provides authentic practice and is therefore typically well received by students.		Because typing mathematical formulas is an integral part of programming and prompting, the ''telephone game'' provides authentic practice and is therefore typically well received by students.