Learn Implement Share

Two Common Approaches Used By Students To Choose Formulas When Solving Problems, And Two Things
We Can Do About It.

Richard Andrew  
When students come across an unfamiliar mathematics problem, they often find themselves at a crossroad when choosing the right formula to solve it. This leads to the adoption of several problematic approaches by students when choosing formulas to solve problems. This article deals with two such approaches. ​

Approach #1: ‘Any Formula Will Do’ (an example)

My Year 9 maths students are working on their next set of questions. As per usual, the students are not particularly engaged, however, they are relatively compliant and on a good day are mostly willing to undertake the work I set them.

My two main concerns are:
  1. Many of the students resist checking their answers and when forced to, show little concern for those that are incorrect. Their first priority seems to be to finish their work. However, they view the checking of their answers as a burden. My informed guess is that they don’t want to have to make second attempts because THAT equates to more work. They’d rather not risk discovering they’ve made mistakes in the first place!  
  2. The primary reason behind their incorrect answers is an incorrect choice of formulas resulting from what we are aptly naming the ‘Any Formula Will Do’ approach. They see a new problem, (almost) randomly choose a formula - one that they have a vague recollection of using in a seemingly similar situation - and then apply it. They gain an answer, which sufficiently pleases them because they are now one question closer to completing their assigned work and, more importantly, one question closer to walking out the door.
Students using the ‘Any Formula Will Do’ approach clearly have a poor attitude towards mathematics. They may be compliant and relatively cooperative, yet they are a long, long way from operating with any genuine mathematical endeavour. Their aim is to gain answers regardless of whether they are correct or not and to stay out of trouble. They endure the 45-minute lesson before delightedly bolting out the door. 

Approach #2: ‘Sheer Panic, Pick and Hope’

In an online discussion on this topic, Patrick Sinclair, an Australian mathematics teacher, mentioned that the approach he witnesses most commonly isn’t ‘Any Formula Will Do’. Rather, the approach most regularly observed he coined ‘Sheer Panic, Pick and Hope’. On reflection, I suspect Sheer Panic, Pick and Hope is probably the more common approach of the two.

By implication, students who use Sheer Panic, Pick and Hope are more conscientious than those using Any Formula Will Do. Due to the words ‘Panic’ and ‘Hope’ we can assume the students care more about gaining a correct answer than their Any Formula Will Do counterparts.

What can we do about these approaches?

In my experience, upon encountering students who use either Any Formula Will Do or Sheer Panic, Pick and Hope, the following three points tend to become apparent. 

Point 1: They have a poor understanding of the problem.
Point 2: They cannot explain mathematically WHY they chose a specific formula.
Point 3: They have a poor understanding of the mathematical concepts underpinning the problem.

Addressing Point #1

To address their lack of understanding of the problem, a good strategy is to ask probing questions. Too often students want to jump straight to choosing a formula - they see something in the question which causes them to assume “Ahh, it must be that formula we saw the other day”, yet they do not fully understand the problem. Using questions and statements like the following will help guide the student into understanding the problem:
  • Read the problem to me.
  • Re-state the problem using your own words.
  • What do we know?
  • What do we need to find out?
  • What information do we need in order to gain the answer?
  • What should we do first?
  • Is there a formula we can use to help us?

Note: These questions are loosely based on Newman’s Error Analysis.

Addressing Points #2&3 (Striving for mathematical understanding)

We need to see that both Points 2 and 3 stem from a lack of mathematical understanding, although Point 2 (cannot justify their choice of formula) may also relate to Point 1 (not understanding the problem).

Clearly, a major reason why students struggle to justify their choice of a specific formula is a lack of mathematical understanding. And Point 3 specifically highlights a lack of understanding of related concepts. Clearly, it is therefore the students’ lack of understanding that needs to be addressed.

Three examples of addressing the ‘lack of understanding’ issue ​

In the article ‘Why We Need Students to Understand The Maths They Are Working On For The Majority Of Lesson Time’ three examples are provided that shine some light on providing students increased opportunites to understand maths for extended periods of lesson time. Rather than replicate parts of that article here, read it via this link. 

The following graphic is taken from the article referenced above.

In summary

Two approaches that can be used to address the issue of students choosing inappropriate formulas when solving problems are:
  1. The use of Newman Error Analysis-type questioning
  2. Present mathematics in a way that has students understanding the activities they are working through for the majority of lesson time.
​Disclaimer: Learn Implement Share articles do not aim to be academic. Written using ‘plain-speak’, they are mostly experience-based, aim to be classroom-useful, hopefully, provoke thought and try to shed light on pedagogical issues in fresh ways.
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Call to Action 

As a maths teacher, do you relate to the above scenario? 
Do you have students who use the Any Formula Will Do approach? 
Have you observed Sheer Panic, Pick and Hope in action? 
Do you agree in the merits of an Understanding-first approach?