Where Has Problem Solving Gone?

 From Wilkie Way Newsletter published with permission

 

Where has problem solving gone?

Our current curriculum seems to have pushed problem solving on to a back bench in order to prioritise the acquisition of knowledge and procedures.

Successful problem solvers require deep mathematical knowledge. Students need to be taught
procedures, for example written algorithms for the four operations. When they need to do a calculation
within a problem, they need to be able to relieve the thinking load on the brain to allow for reasoning and critical thinking.

However, knowledge and procedures alone will not create the problem solvers that the world requires to function in an ever increasingly complex world.
Problem solving is a messy business, with the solver moving forwards and backwards, reasoning, making connections and working their way towards asolution. Procedures allow students to only solve familiar problems and most problems encountered in the real world are unfamiliar problems. Technology (including AI) can be used to carry out most mathematical procedures.
However the result provided by technology must be checked to see if it is reasonable. To check if calculations are reasonable students need to be able to make estimates. A reasoned estimate requires, at the most basic level a conceptual understanding of the number system and recall of addition, subtraction, multiplication and division facts.

A major part of problem solving is in the reading and comprehension of the problem. 

Reading comprehension continues to be the biggest barrier for mathematical success. The ability to cope with a wide range of text requires more than the ability to read the words. It requires a full
understanding of the key concepts underpinning ideas and the ability to interpret a variety of symbolic, spatial and quantitative texts.
Mathematical texts require a very different approach than narrative reading. They must read closely, with attention to every word, because even the smallest words which are often insignificant in
narrative text may be important in determining the solution to the
problem.

When approaching a word problem students use two specific pathways to make sense of the text:
1. A direct translation approach
2. A problem model approach

In the direct translation approach the solution is based on key words. When the text is processed, students focus only on the numbers and the keywords ignoring most other information. 

This approach makes minimal demands on working memory and is erroneously favoured by teachers,
especially when reading comprehension is low. A keyword approach to problem solving prepares
students for solving only a small set of clear predictable problems. Unfortunately the majority of problems encountered in the real world do not come neatly presented with key words telling you how to solve them.
A problem model approach requires the student to construct a mental model of the situation focusing on
how the parts of the problem are related. It is difficult if not impossible to create a model of a problem if the context is too far outside the experience of the solver. Drawing a picture, acting it out are ways of creating a model of the situation.

In addition to reading and comprehending words, students need to know how to interpret the unique
symbols that make up the shorthand of mathematical statements, such as numerals, operation signs and
variables that stand in for numbers, and equality and inequality symbols representing the relationships
between quantities.
Using symbols with understanding is a complex process. When encountering mathematical symbols
students face a multi level decoding process: Firstly they need to recognise and separate out the
confusing mathematical symbols (+, x, - , ÷. =, ˂, ˃, ≠. ≤, ≥, ∞ ) without any phonic clues. Next they need to translate each symbol into english (or te reo), then they need to connect the symbol to the concept it represents. Finally they need to carry out any operation indicated and use any relationship indicated in the equation.
Developing the mathematical vocabulary essential for reading mathematical texts (a useful addition in our new curriculum) requires students to be involved in mathematical discussions.

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