Formula

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 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.

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 ${{\sqrt {a+b}}-{{2c} \over {d+e}}}$. The second student is only allowed to pass the expression on in typed form using standard mathematical input syntax, for example: (a+b)^(1/2) - (2*c)/(d+e)

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.

Available resources

Interview Video

See also

Understanding Mathematics by Visualizing Structures, by Sabine Stank

People interested in this Bottleneck

Inna Mikhailova