Two Common Approaches Used By Students To Choose Procedures When Solving Problems, And Two Things We Can Do About It.
Richard Andrew
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When students encounter an unfamiliar mathematics problem, they often find themselves at a crossroads when choosing the right formula or procedure 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’
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 prepared to undertake the work I set them.
My two main concerns are:
Students using the ‘Any Formula Will Do’ approach poorly understand the concepts underpinning any given procedure. I expand on this idea in the article?Why Maths Students Need To Understand The Concept Before We Teach Them The Procedure.
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’. Instead, 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.
What can we do about these approaches?
The following three points become apparent when encountering students who use Any Formula Will Do or Sheer Panic, Pick and Hope.?
Point 1: They have a poor understanding of the problem.
Point 2: They cannot mathematically explain WHY they chose a specific formula.
Point 3: They understand poorly the mathematical concepts underpinning the problem.
Addressing Point?#1
Ask probing questions. Commonly, students choose a formula impatiently - they see something in the question that 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:
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Note: These questions are loosely based on?Newman’s Error Analysis.
Addressing Points?#2&3 (Striving for mathematical understanding)
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).
As we have already intimated, a major reason students struggle to justify their choice of a specific formula is a lack of mathematical understanding. And Point 3 highlights a lack of understanding of related concepts. Clearly, we need to raise the level of student understanding of the concepts that underpin procedures.
Improving mathematical understanding in students?
Several articles of this Empowering Education Leaders Newsletter point to wards initiatives we can take to improve mathematical understanding. Three are:
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In summary
Two approaches that can be used to address the issue of students choosing inappropriate formulas when solving problems are:
Call to Action?
Do you relate to the above??
Are you a maths teacher and have students use the Any Formula Will Do approach? Have you observed Sheer Panic, Pick and Hope in action??
Any other takeaways? Fire away ...?
Teacher of Mathematics/Department Chair at Millburn High School - Retired
1 年Richard Andrew, C. Harun B?ke It's amazing how little effort is needed on the part of students to "select" the correct procedure if they understand what they are doing. Conceptual understanding gives clarity to the mathematics so that it isn't picking the "right" procedure as much as it is picking the "only" procedure that makes sense in the context of the problem. But, of course, I am preaching to the choir!
maths teacher | flipped learning expert | AI experimenter | content creator | teacher trainer
1 年Great read, Richard, thank you. This, again, reminds me of this IGCSE Examiner who, during training, mentioned about 'familiarisation'. If students are not familiar with a question / problem, they freeze like a rabbit in front of torch light at night!. One way of fighting this 'rabbit' problem is teaching to the test, like many teachers do. The other way is to build conceptual understanding, together with approaches to problem solving. I was explaining the Completing The Square Method today. After I solved a few examples, a student asked whether there was a formula for it, to which I answered "yes, but I am not going to show it to you. if you want to have a look, and even use it, it's in the textbook". Only a few students reached their books, others were totally confident of their understanding. And sometimes it is not so complicated; I give the example of this song that Elsa sings in Frozen 2; Do the next right thing. Especially if you have no idea how to start / what to do. You have a quadratic in an algebraic fraction expression? Well, factorise it!. Afterwards, you may notice something, like same factors that can be simplified, etc. If you have an understanding of the concept, it will probably come to you.