Posted in Algebra

Top Ten Reasons To Become A Teacher

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1.  Work with Children and Young Adults – Children are a joy to work with. You will laugh almost every day at silly jokes that you come up with as you teach or the funny statements they can make. Children’s energy, imaginations, and fresh insight will be certain to keep you forever young.

2. Lifelong Learning – If you love learning, teaching is a career that uses that passion. You will become an expert in your subject area(s) because there’s no better way to learn a subject than to teach it. Students will ask you interesting questions that you haven’t thought about before, challenging you to learn even more to stay ahead of them.

3. Variety – While most people have jobs that require them to do the same thing every day, teaching offers a lot of variety. You will work with a completely different group of students each year. As you gain experience, you may decide to adopt new teaching methods or to teach a new topic. Best of all you will learn new things from your students every day, not to mention a great deal about yourself.

4. Health Benefits – Many teachers get great health and pension plans. In fact, teachers’ health, vision, and dental benefits are often among the best compared to other industries.

5. Reasonable Hours – Since teachers work on school hours, it’s a great job for anyone with kids. You’ll have more time with them before and after school, which means that you don’t have to worry about childcare or missing their next soccer game.

6. Interpersonal Interaction – Forget sitting in a cubicle all day. Teachers get to interact with students, parents, and other educators on a daily basis. You’ll also have many opportunities to get involved in the community through field trips, after-school activities, and service projects.

7. Job Security – Teachers are always needed. Once you’ve established yourself, you should always be able to find a job. Downsizing and layoffs don’t happen nearly as often in schools as they do in other industries. And contrary to popular belief, teacher salaries aren’t as low as you think, and they can increase significantly over time.

8. Autonomy – Even if you have to teach a specific curriculum, you get to decide what happens in the classroom each day. Every teacher teaches differently based on his/her personality, creativity, knowledge, and beliefs.

9. Vacation Time – You’ll have not only two weeks of vacation for the holidays, but several other breaks throughout the year for public holidays, mid-term breaks, and of course, summer! Even if you teach at a year-round-school, plenty of vacation time will be scheduled throughout the year. This is a great time to travel the world, pursue a hobby, further your education, or even have a seasonal career at a summer camp or national park.

10. Make a Difference – Teachers impact students’ lives in profound ways. Just think back to some of your favorite teachers and how much they meant to you. Teachers can help students discover their passions and develop skills to reach their full potential. You will even get to see the difference you make every day in the classroom as students get excited about learning and improve throughout the year.

This article is by Andrea Erins. Andrea has been teaching  13 years and enjoys writing about a variety of topics related to education. She is the owner of the site Masters in Education.

Readers are invited to submit articles to this site. Click here for details about guest posts.

Posted in Teaching mathematics

What is the role of visualization in mathematics?

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Like abstraction and generalization which I described in my earlier posts here and here,visualization is central to the learning and understanding of mathematics. (Note that these processes are also natural human mental dispositions and so we can claim that doing mathematics is a natural human activity.)

Visualization used to be considered only for illustrating otherwise abstract ideas of mathematics but now visualization has become a key component of mathematical processes such as reasoning, problem solving, and even proving.

What is visualization?

Synthesizing the definitions of visualization offered by Zimmermann and Cunningham (1991, p. 3) and Hershkowitz (1989, Abraham Arcavi proposes the following definition:

Visualization is the ability, the process and the product of, creation, interpretation, use of and reflection upon pictures, images, diagrams, in our minds, on paper or with technological tools, with the purpose of depicting and communicating information, thinking about and developing previously unknown ideas and advancing understandings– Abraham Arcavi, ESM, 2003

What are examples of use of visualization in mathematics?
  1. For communicating information, the graph is perhaps the most recognizable of the visual representations of mathematics.
  2. For proving, visual proofs are already accepted as legitimate proofs. For example, here’s a visual proof of the Pythagorean Theorem. Click here for source of movie. [iframe 350 500]
  3. Of course, visuals can also be used to challenge students to reason and explain in words and symbols. For example teachers can show the visual in #2 then ask the students what the visual is telling them about the relationships between the areas of the three squares and about the sides of right triangles. Students should be asked to support their claim.
  4. Visualization tasks also trains students mind to ‘think outside the box’. Click here for an example of a problem solving tasks which can be solved by visualizing possible arrangements. Patterning activity like Counting Hexagons are great activities not only for generating formulas and algebraic expressions but trains the mind to look for relationships, an important component in algebraic thinking.
  5. Because what we see usually depends on what we know, visuals can also be used as context for assessing students knowledge of mathematics. Click here for an example on how to assess understanding by asking students to construct test items.
Posted in Teaching mathematics

Making generalizations in mathematics

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Making generalizations is fundamental to mathematics. Developing the skill of making generalizations and making it part of the students’ mental disposition or  habits of mind in learning and dealing with mathematics is one of the important goals of mathematics education.  Making generalizations is a skill, vital in the functioning of society. It is one of the reasons why mathematics is in the curriculum. Learning mathematics (if taught properly) is the best context for developing the skill of making generalizations.

What is generalization?

There are three meanings attached to generalization from the literature. The first is as a synonym for abstraction. That is, the process of generalization is the process of “finding and singling out [of properties] in a whole class of similar objects. In this sense it is a synonym for abstraction (click here to read my post about abstraction). The second meaning includes extension (empirical or mathematical) of existing concept  or a mathematical invention. Perhaps the most famous example of the latter is the invention of non-euclidean geometry. The third meaning defines generalization in terms of its product. If the product of abstraction is a concept, the product of generalization is a statement relating the concepts, that is, a theorem.

For further discussion on these meanings, read Michael Mitchelmore paper The role of abstraction and generalization in the development of mathematical knowledge. For discussion about the importance of generalization and some example of giving emphasis to it in teaching algebra, the book Approaches to Algebra – Perspectives for Research and Teaching is highly recommended. There is a chapter about making generalizations and with sample tasks that help promote this attitude.

Sample lessons

Mathematical investigations and open-ended problem solving tasks are ways of engaging students in making generalizations. The following posts describes lessons of this type:

  1. Sorting number expressions
  2. Lesson study: Teaching subtraction of integers
  3. Math investigation lesson: polygons and algebraic expressions
  4. Polygons and teaching operations on algebraic expressions

Of course it is not just the type of tasks or the design of the lesson but also the classroom environment that will help promote making generalization and make it part of classroom culture. Students will need a classroom environment that allows them time for exploration and reinvention. They will need an environment where a questioning attitude is promoted: “Does that always work?” ,”How do I know it works”? They will need an environment that accords respect for their ideas, simple or differing they may be.