Numeracy - Embedded numerical computation in the computing curriculum

Learners come to the course, with an understanding of Decimal number systems, base 10, and a variety of skills for addition. subtraction, multiplication and division. Some other complex algebraic computation is also an advantage. 

Using these basic fundamentals of numerical computation, we expand using those fundamentals to show how computers and IT systems manipulate and use other base numbering systems and techniques to explain real world tangible phenomena. E.g,

Global Connections

• Use of mathematics to model situations or events in the world;
• Explanations of how the complexity and inter relatedness of situations or events in the world are reflected in a model;
• Data generated by a model to make and defend a decision; and
• A decision or conclusion supported by the mathematics within the context of a global community.

Problem Solving

• The application of appropriate strategies to solve problems;
• The use of appropriate mathematical tools, procedures, and representations to solve the problem;
• The review and proof of a correct and reasonable mathematical solution given the context.

Communication

• The development, explanation, and justification of mathematical arguments, including concepts and procedures used;
• Coherently and clear communication using correct mathematical language and visual representations;
• The expression of mathematical ideas using the symbols and conventions of mathematics.

When teaching Spreadsheet Modelling  we can model the real world and produce what if scenarios providing a means to solve problems that have yet to happen. We can automate and make decision based on how we have coded the sheet to work. This improves the understanding of the model we are using to represent the real world, for example a business system, and allow learners to investigate the use of mathematics and numeracy and enhance those skills and knowledge application in a different context to normal numerical teaching.


E.g, Spreadsheet Formula, Functions, Arguments, Boolean logic can all be transferred and cross skilled with other disciplines being studied. Object orientated programming use the above 4 concepts and using them slightly differently to achieve a coded price of software to do that one thing. The underlying principles are the same but the use and application is different. We need to understand the use of numeracy and begin to develop the use of and application to a variety of real world phenomena. Using maths to describe the curve of a moving object on screen in a video game or piece of text, requires an understanding of arcs.

Here we see that an arc is made up of coordinate numerical data, radius, angle and directional data. This can be modeled in a spreadsheet or a C# programming example. Learners need to understand the terms, usage and application, in order to understand the predefined formula, functions and procedures in programs such as Microsoft Excel and Visual Studio.

Learners & mathematical (Numerical) anxiety :

This anxiety has been described as feelings of tension and anxiety that interfere with the

manipulation of numbers and the solving of mathematical problem. This has been known to be attributable to:

• Teacher personality and their style of teaching (existing schooling)
• Public examinations and their effect (Judgement of performance)
• Affective domain- the self-factor such as personality, perception.
• Feelings, worries, difficulties (memory, innate disability)
• Parental expectations- their aspirations and standards
• Peer group influences
• Relevance -the usage of mathematics in everyday life 

I try to show my learners that maths and the use of numeracy can aid and actually create some very rewarding experiences. In control systems I embed the idea of using robots and sensing real world (physical computing) to visually represent the numerical data being manipulated.

When sensing the real world phenomena; light, heat, pressure, moisture and humidity for instance; each electrical sensor used to measure the phenomena has a range of values it provides. this could be in the range of 1- 344, 0 through 1023 etc...

We can use a command in C/C++ on the arduino platform to change a range of values from one to another.

Example

/* Map an analog value to 8 bits (0 to 255) */
void setup() {}

void loop()
{
  int val = analogRead(0);
  val = map(val, 0, 1023, 0, 255);
  analogWrite(9, val);
}

The map function - works as below

long map(long x, long in_min, long in_max, long out_min, long out_max)
{
  return (x - in_min) * (out_max - out_min) / (in_max - in_min) + out_min;
}

Map returns a value which is of type long : 

• Long variables are extended size variables for number storage, and store 32 bits (4 bytes), from -2,147,483,648 to 2,147,483,647.
• a normal integer is 16-bit (2-byte) value. This yields a range of -32,768 to 32,767 (minimum value of -2^15 and a maximum value of (2^15) - 1). 

What we are doing in the example above is:

• Reading a value in called val.
• Taking the val and remapping it. ie 0 to 1023 and mapping it to a new range of 0 to 255.
• Val then becomes the new value based on the mapped returned result.

With regards the minimum core:

With this example I can also show ratios, the use of Integers, long and other numerical data-types. Because this is also attached to a real world instance we can then use this to manipulate other objects be it , representations on screen or using physical robots and sensing a collision in the real world and avoiding an object. 

Numeracy can be a difficult concept to comprehend and use, and many learners do not realize they are doing the computational side of things when they see a control systems in action. Playing a racing game and coding the game uses mathematical/numerical knowledge to judge distance, move  a particular amount based on a joy-pad movement.

Processing and analyzing and interpreting data is easily shown in the above 3 examples. Stealth learning again is used to mask the actual complexity being dealt with. Being interactive with numerical data and concepts allows me to make my learners feel at ease with the content and enjoy what they do.

As part of peer learning and reflection and assessment for the projects work for Object orientated programming and control systems, minimum core is satisfied for:

Personal, social and cultural factors

• Potential barriers that hinder development of numeracy skills
• The main learning difficulties and disabilities relating to numeracy skills learning and development
• The common misconceptions and confusions related to number-associated difficulties.

Communication

• Making and using decisions about understanding
• Communicating processes, and understandings.

Processes

• A knowledge of the capacity of numeracy skills to support problem solving
• Making sense of situations and representing them
• Processing and analysis
• Using numeracy skills and content knowledge
• Interpreting and evaluating results
• Communicating and reflecting on findings. 
• Use strategies to make sense of a situation requiring the application of numeracy
• Process and analyse data
• Use generic content knowledge and skills
• Make decisions concerning content knowledge and skills
• Understand the validity of different methods
• Consider accuracy, efficiency and effectiveness when solving problems and reflect on what has been learnt
• Make sense of data

Theory and my perspective:

One of the most important findings of mathematics education research carried out in 
Britain over the last twenty years has been that all pupils constantly ‘invent’ rules to 
explain the patterns they see around them. (Askew and Wiliam 1995) 

According to Malcolm Swan; Frequently, a ‘misconception’ is not wrong thinking but is a concept in embryo or a local generalisation that the pupil has made. It may in fact be a natural stage of development. (2001: 154) 

Jean Piaget and Lev Vygotsky fit well with this belief along with personal interactions and socialcultural theory.  According to Piaget, all cognitive change can be classified as one of two types: adaptation and organisation. Organisation is a largely internal process involving rearranging and linking up items of previous learning to form a “strongly 

interconnected cognitive system” (Berk 1997: 213).

A learner might superficially accept a teachers perspective without critically examining 

it, out of an unquestioning belief in the teachers authority. Piaget also asserted that 

clashing viewpoints – arguments jarring the learner into noticing a peer’s point of view 

– were necessary for peer interaction to stimulate movement towards logical thought. 

(Berk 1997: 253) 

I would say that a discovery and none acceptance of the norms and the realization from critical analysis brings a greater understanding from the learner and application of that knowledge. By making sense of the findings and making actual sense of how the maths is being applied rewards a greater understanding and a more natural motivational progression. Debate and communicating this in a peer group allows further resolution to be found.

Referances :
http://asiasociety.org/education/resources-schools/professional-learning/understanding-world-through-math

http://myweb.tiscali.co.uk/annotations/content/maths_misconceptions.pdf