Learning+Strategies

Some ideas and strategies that will help you learn mathematics (or any subject!):

Embrace your errors and misconceptions.

 * **Don't run away or hide from mistakes -** as painful as it may be sometimes, seeing what you are doing wrong is the best way to push the boundaries of your learning.
 * "Errors" are really just an indication of where your learning is at today - they act as valuable signposts and guides where to go to next.
 * **Keep a record of your errors and misconceptions** - and keep pushing your understanding and skills practice until you master the content.
 * **"Errors" are so valuable we should welcome them.** If your classmates are generous enough to share an error with you, thank them - don't laugh at them! They just gave the class a special gift.
 * **Being scared of errors** means you are less likely to take risks while learning. It also means you are less likely to ask the questions you need to ask to better understand the material.
 * **If you don't understand something the first time, chances are many other students in the class don't either!** In most cases, students who do understand something the first time have seen the material or something close to it before. Or maybe they aren't asking themselves deep enough questions about the new content. Do everyone a favour and ask the question!

Build your own Theory Book during the year

 * Keep a separate exercise book to maintain your own version of summaries, examples and study notes.
 * Share the contents of your theory book with your friends - and look at their theory books.
 * Make sure you include your favourite errors and misconceptions in your book.

Keep track of your learning

 * Can you organise the content of the course into your own mindmap?
 * Do you know which specific parts of the course you have and have not mastered? Check off your learning against a syllabus list, a chapter/section list from your text book, a topic standards list, or any other list you may find.
 * Are you keeping track of the your progress through the course?
 * Are you monitoring the time you spend studying? Do you need to make more or less time?

Try to get lots of sleep

 * Young adults and teenagers needs lots of sleep. Are you getting at least 8 hours per day? (Seriously!)
 * Try not to stay up late at night.
 * Ensure you have a quality sleep environment.
 * An afternoon nap of 30 minutes will help you retain your learning during long study periods. Maybe even take a kip during your lunch break?

Don't confuse your grades with your ability!

 * In most cases, your grades are a consequence of your **Effort**, **Attitude** and **Strategy** - not your ability to do mathematics.
 * **You can significantly improve your results** with careful application of Effort, Attitude and Strategy. Note the use of the word 'careful' - sometimes just applying more effort is not the answer - you may benefit from better or different learning strategies, which might actually take //less// effort.
 * **Your mathematics knowledge and skills will grow** as you apply Effort, Attitude and Strategy - your mathematics potential is not fixed. Major studies in the last few decades show that a Growth Mindset view of your ability will help your flourish.
 * **If you believe your mathematical ability is fixed, you will limit your growth.** This self limiting belief can particularly affect girls studying maths and science. See for example the research study "Is Maths a gift? Beliefs that put females at risk" from Carol Dweck. Some earlier research papers on this subject are Why do beliefs about intelligence influence learning success? A social cognitive neuroscience model. and Implicit Theories of Intelligence Predict Achievement Across an Adolescent Transition: A Longitudinal Study and an Intervention.
 * Academic studies suggest one reason students from Confucian Heritage Cultures have such a noticeable success at mathematics is that they don't believe they 'can't do maths' - students are encouraged to continuously apply effort regardless of perceived ability. How much effort should be applied is of course something we might debate - this is sometimes carried to extremes that may harm the well being of the student. See Revisiting the Chinese Learner for a summary of recent studies.

The Role of Memorisation and Practice

 * **Memorisation and skills practice are an important part of building 'fluency'.** If you remember how to do a procedure and can do it without too much thought, you free up your mental processing to focus on harder aspects of the work at hand. (You are 'reducing the cognitive load' and 'freeing up working memory' to use the current terminology.) So for example, if you can't factorise a quadratic, you will be struggling to solve curve sketching problems that rely on having the factors handy.
 * **Make time to practice and revise so you develop automaticity.**
 * **Your success at memorisation will be greatly enhanced if you have deep understanding of the material.** Make sure you really do understand the ideas and processes you are memorising - otherwise you likely to forget the content as soon as you no longer need it. (Ask your parents if they can remember how to solve quadratic equations).
 * **Plan to regularly revise skills.**
 * **Some formulas are just not worth remembering as a formula** - it may be better to remember a key idea or diagram behind the formula. For example, I never remember the formula for the area of sector - I remember the area of a circle (back from Year 8) and I just know if the sector is angle theta, the area will be theta divided by 360 times the circle area. I also don't bother remembering the formula for the angles in a polygon - instead I visualise a diagram of triangles inside the polygon. When I need that formula (which is rarely), I reconstruct it. On the other hand, while I could derive the quadratic formula if I forgot it, I keep it memorised because it is so darned handy any time I work with a quadratic - especially since it has the formula for the determinant and the axis of symmetry inside it!

Handy helpers: GeoGebra and Wolfram Alpha
You have at your disposal two extremely powerful mathematics helpers:

The simplest way to use these tools is to just use them to find answers to your problems. Need to find the roots of quadratic equation? Graph it in GeoGebra and look at the x-intercepts, or ask Wolfram Alpha to "solve" the equation for you.
 * GeoGebra** - on your school laptop, and freely available at [|www.geobera.org] for download.
 * GeoGebra is the perfect tool to see the links between algebraic expressions/functions and their graphical representation. SInce everything is dynamic, you can experiment on the fly.
 * GeoGebra can solve equations graphically for you - use the Point Intersection tool to see where graphs cross the axes.
 * Wolfram Alpha** [|www.wolframapha.com] through your internet browser. Wolfram has a powerful CAS (Computer Algebra System) behind it which can do most if not all of the mathematics in your course!
 * Use the keyword "factor" to ask Wolfram Alpha to factorise complex algebraic expression
 * Use the keyword "solve" to solve equations
 * Wolfram Alpha knows how to do calculs
 * Or just type in an expression and see what happens


 * So why bother doing maths exercises by hand?** The point of doing exercises is //not// to "get the answer" - the point is to practice techniques, and depending on the complexity of the exercises, explore new ideas or mathematical properties that follow from what is being learnt or practiced. The "Development" sections in exercise sets usually offer this opportunity. So don't just dive into GeoGebra or Wolfram Alpha to get the answer - try it yourself first - and //then// see if the tools agree with your answer. If not - //why// not?

More powerful ways to support your learning by using these tools include:
 * **Think about the answer** - is it what you expected? If not - why not?
 * **Compare the graphical results to the algebraic results** - there is usually something interesting to observe, or a key idea to reinforce.
 * **Explore what happens if you change the problem slightly**. Now that you have solved one version of the problem, how does the answer change if you vary the parameters? When you have powerful computational tool under your fingers, you can ask lots of challenging questions without getting bogged down in the details.
 * **Examine all the different things Wolfram Alpha did with your algebraic expression** - there is usually at least one interesting diversion because Wolfram Alpha is throwing all its tricks at the problem, Ask yourself if your level of mathematics is at a stage you should be able to do what Wolfram Alpha did - or consider how it may have done it.

See the Wikipedia page at @http://en.wikipedia.org/wiki/Computer_algebra_system These two papers by Bernhardt Kutlzer take an interesting view on the CAS discussion: The Algebraic Calculator as a Tool for Teaching Mathematics Indispensable Manual Calculation Skills in a CAS-Environment //You might like to have a discussion with your teacher about the role of CAS in learning mathematics - it's still a hot topic in mathematics education today. What do you think?//
 * More about Computer Algebra Systems**

As a final comment, don't forget : you won't have CAS to help you during your exams! Use these tools wisely to support and extend your learning, not to do your thinking!

In 2011, Wolfram Alpha made cheap versions of its CAS software available for iPhones and iPad @http://products.wolframalpha.com/courseassistants/
 * **Getting CAS software**

Wolfram also make the powerful (and expensive) Mathematica software - one of the industry leading tools.

Many free and open source CAS packages are available on the net - see the wikipedia page []. GeoGebra version 4.2 is expected to offer fully functioning CAS window. || ||