Monday, 8 September 2014

How do adult learners approach numeracy problems? Updated...

How do adult learners approach numeracy problems?  This is a pivotal question relating to exactly how much we expect adult learners to learn in their respective courses.  Why?

Well, in a nutshell numeracy or mathematical problems are only useful because they help the learner LEARN something.  They have no other purpose.  If a learner can solve the problem using an existing method like an algorithm then you and the learner may be wasting your time - if your objective is to develop numeracy skills.

Do adult learners ‘use’ numeracy problems to learn, or do they treat them as micro-tests? 

The research is reasonably clear that good problem solving behaviours should include the following phases: 
  1. Discuss the problem and generate several strategies for solving it.  (What exactly is the problem? What are the variables?  How might we best solve it?) 
  2. Select the best approach based on its merits.  (Which of our ideas is the best and why?)
  3. Enact the strategy (Let’s get busy)
  4. Monitor the strategy (Is this working?)
  5. Evaluate the answer and review.  (Has the strategy worked?  How do we know?  How can we prove it?)

The above is more in line with how mathematicians approach problems.  The trick is to separate ‘calculation’ from other aspects of mathematical problem solving.  For example, have a look at the following example:

In this example if you begin calculating before examining the problem you will miss the point of the problem.  Clearly the answer is 720 because there are three of them.  We know that the answer to 720 + 720 + 720 is meaningless in this scenario.  But to 'see' this requires taking the time to work through the problem.  Engaging in ‘calculation’ is the worst thing you could do to solve this problem. The answer is evident if you take the time to look at the whole. However, imagine this problem is written as a word problem.  Suddenly it becomes harder to see the big picture.

The question then is this:  Do adult learners actually engage in the first two parts of problem solving?  Do they think about the problem situation?  Or do they simply look for the important numbers, adopt a calculation they think fits, and begin the calculations?  

The findings: 

Well, my research has found that adult learners are primarily oriented toward solving numeracy problems by applying pre-learned procedures to the quantities provided.  They skip the first two steps, attempt to identify numbers to preform calculations on and then immediately begin calculating.  Finally, they engage in very little evaluation or review.  Rather they move quickly to repeat this process with the next problem.  Below is a quick review of the behaviours used during problem solving sessions.

Lack of discussion about the problem situation or potential strategies
When a group of learners begin to analyse the numeracy problem they do not discuss the problem situation.  Rather they identify the key numbers and begin to engage in calculations immediately.  

Adoption of the first strategy proposed
Generally, the first calculation strategy proposed by a group member is adopted.  The acceptance of the strategy appears to be on the basis of ‘who’ proposed it. It may be, that being the first to generate the strategy indicates speed of comprehension, knowledge of the procedure and therefore high proficiency.   In other words, the first learner to speak must know what they are doing.  Let's go with their idea!

Focus on step-by-step procedures
Learners relied on procedural step-by-step (algorithmic) approaches to solve problems rather than engage in reasoning.  That is, answers were assumed to ‘appear’ once the correct formula had been applied.  Moreover, learners often acted as though problems could not be solved without a procedure. If a learner asks "how do you do that again?" it means they do not understand how it works.  This small sentence is an indication to you that the learner has yet to develop understanding (even if they can repeat the procedure). 

Disconnect between problems and answers
In many cases there was a disconnect in the relationship between the solution strategy used and the answers.  This was evident in several ways.  Firstly, the way answers were described as a product of a procedure and answers lacked any contextual aspect.  For example answers were described as ‘popping out’ after performing the correct procedure on the calculator and answers were often spoken as digits rather than meaningful quantities. 
 “The answer is four, two, three, eight”.
The answer was “Four-thousand, two-hundred and thirty-eight dollars” but the learner had stripped it of all meaning. 
This is a serious finding as it undermines some of the key assumptions made when teaching numeracy skills. The first being that numeracy is the bridge between mathematics and the real world.  The second that placing numeracy in a context meaningful to the learner improves outcomes.  Almost all of the learners in this study simply ignored context and engaged in the calculation of numbers.  Just like in maths class. 
Goal orientation
The primary goal of the learners was to identify the correct answer and finish the numeracy problems quickly. While this sounds reasonable to most people, it conflicts with the goal to learn from numeracy problems.
For example:  Groups would often unevenly distribute responsibility for task completion.  That means that if two or three learners were asked to solve a series of problems, they would allocate the task of calculating to the most proficient learner, the second would take responsibility for putting numbers into the calculator, and the third would write down the answers. 
Yes, they would achieve their desired goal – to finish on time and have all answers complete.  But no one necessarily learned anything new.  The learner who did the calculation was already able to do the task, and the other two learners never actually engaged in the thinking necessary to develop new understandings.
The learners in this study have developed beliefs about what mathematics is, how it is learned, and what goals and behaviours are appropriate in a maths class.  These beliefs clearly transfer to the numeracy classroom.
This is a problem.  As tutors we need to change these beliefs and the associated goals that these learners hold.  Otherwise we risk diluting the educational experience the learners are engaged in.  Do they really need another classroom experience in which they make very little progress? 

The next question is:  How do we as tutors begin to change these beliefs so that learners adopt behaviours better suited to learning?

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