Tag: teaching mathematics

Posted in Algebra, Curriculum Reform

Teaching algebra – it pays to start early

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I believe in early algebraization. I have posted a few articles in this blog on ways it can be taught in the early grades. Check out for example  Teaching  Algebraic Thinking Without the x’s. All the lessons in fact that I post here whether it is a number or geometry or pre-algebra lesson always aim at developing students’ algebraic thinking. What do research say about early algebraization? How do can we integrate it in the grades without necessarily adding new mathematics content?

“Traditionally, most school mathematics curricula separate the study of arithmetic and algebra—arithmetic being the primary focus of elementary school mathematics and algebra the primary focus of middle and high school mathematics. There is a growing consensus, however, that this separation makes it more difficult for students to learn algebra in the later grades (Kieran 2007). Moreover, based on recent research on learning, there are many obvious and widely accepted reasons for developing algebraic ideas in the earlier grades (Cai and Knuth 2005). The field has gradually reached consensus that students can learn and should be exposed to algebraic ideas as they develop the computational proficiency emphasized in arithmetic. In addition, it is agreed that the means for developing algebraic ideas in earlier grades is not to simply push the traditional secondary school algebra curriculum down into the elementary school mathematics curriculum. Rather, developing algebraic ideas in the earlier grades requires fundamentally reforming how arithmetic should be viewed and taught as well as a better understanding of the various factors that make the transition from arithmetic to algebra difficult for students.

The transition from arithmetic to algebra is difficult for many students, even for those students who are quite proficient in arithmetic, as it often requires them to think in very different ways (Kieran 2007; Kilpatrick et al. 2001). Kieran, for example, suggested the following shifts from thinking arithmetically to thinking algebraically:

  1. A focus on relations and not merely on the calculation of a numerical answer;
  2. A focus on operations as well as their inverses, and on the related idea of doing/undoing;
  3. A focus on both representing and solving a problem rather than on merely solving it;
  4. A focus on both numbers and letters, rather than on numbers alone; and
  5. A refocusing of the meaning of the equal sign from a signifier to calculate to a symbol that denotes an equivalence relationship between quantities.
These five shifts certainly fall within the domain of arithmetic, yet, they also represent a movement toward developing ideas fundamental to the study of algebra. Thus, in this view, the boundary between arithmetic and algebra is not as distinct as often is believed to be the case.
What is algebraic thinking in earlier grades then? Algebraic thinking in earlier grades should go beyond mastery of arithmetic and computational fluency to attend to the deeper underlying structure of mathematics. The development of algebraic thinking in the earlier grades requires the development of particular ways of thinking, including analyzing relationships between quantities, noticing structure, studying change, generalizing, problem solving, modeling, justifying, proving, and predicting. That is, early algebra learning develops not only new tools to understand mathematical relationships, but also new habits of mind.”

The foregoing paragraphs were from the book Early Algebraization edited by Jinfa Cai and Eric Knuth. The book is a must read for all those doing or intending to do research about teaching algebra in the elementary grades. Educators and textbook writers should also find a wealth of ideas on how algebra can be taught and integrated in the early years. Of course it would be a great read for teachers.  The book is rather expensive but if you have the money, why not? Here are some section titles:
  • Functional thinking as a route in algebra in the elementary grades
  • Developing algebraic thinking in the early grades: Lessons from China and Singapore
  • Developing algebraic thinking in the context of arithmetic
  • Algebraic thinking with and without algebraic representation: A pathway to learning
  • Year 2 to 6 students’ ability to generalize: Models, representations, and theory for teaching and learning
  • Middle school students’ understanding of core algebraic concepts: equivalence & variable”

Check out the table of contents for more.

The following books also provide excellent materials for developing algebraic thinking.

 

 

 

 

Please share this post to those you think might find this helpful.

Posted in Mathematics education

How confident are you to teach mathematics?

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As mathematics teachers we simply cannot just stop learning and improving in our field. Reflecting on our practice is a powerful and productive way of supporting our own professional development. I found a goldmine of tools for this in the National Center for Excellence in the Teaching Mathematics (NCETM). I think this site is great for mathematics teachers who wants to keep on improving their craft. Below are some of the self-evaluation questions they have for mathematics- specific teaching strategies.

1. How confident are you that you know how and when it is appropriate to:

  • demonstrate, model and explain mathematical ideas?
  • use whole class discussion?
  • use open questions with more than one possible answer to challenge pupils and encourage them to think?
  • use higher order or more demanding questions to encourage pupils to explain, analyse and synthesise?
  • intervene in the independent work of an individual or group?
  • summarise and review the learning points in a lesson or sequence of lessons?

2. How confident are you that you can select activities for pupils that will promote your learning aims and, over time, give them opportunities to:

  • work independently as individuals or collaboratively with others?
  • engage in interesting and worthwhile mathematical activities?
  • investigate and ‘discover’ mathematics for themselves?
  • make decisions for themselves?
  • reason and develop convincing arguments?
  • visualise?
  • practise techniques and skills and remember facts in varied ways and contexts?
  • engage in peer group discussion?
  • communicate their results, methods and conclusions to different audiences?
  • appreciate the rich historical and cultural roots of mathematics?
  • understand that mathematics is used as a tool in many different contexts?

3. How confident are you that you know how and when you might provide:

  • alternative or supplementary activities for pupils who experience minor difficulties with learning?
  • mathematical activities designed to respond to pupils’ diverse learning needs, including special educational needs?
  • suitable activities for mathematically gifted pupils?
  • suitable homework?

4. How confident are you that you are familiar with a range of equipment and practical resources to support mathematics teaching and learning, such as:

  • structural apparatus and other models for teaching number?
  • measuring equipment?
  • resources to support the teaching of geometrical ideas?
  • board games and puzzles?
  • resources to support and stimulate data handling activities?
  • calculators?
  • ICT and relevant software?

Here are sample questions for self evaluation about mathematics content knowledge. Go to the NCETM.org site for other topics.

1. How confident are you that you know and can explain the properties of:

  • the sine function?
  • the cosine function?
  • the tangent function?

2. How confident are you that you can explain:

  • why sin ? / cos ? = tan ? and use this to solve simple trigonometric equations?
  • why sin² ? + cos² ? = 1 and use this to solve simple trigonometric equations?

Please share this with your co-teachers.

 

Posted in Elementary School Math, Number Sense

Teaching negative numbers via the numberline with a twist

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One popular way of introducing the negative numbers is through the number line. Most textbooks start with the whole number on the number line and then show to the students that the number is decreasing by 1. From there, the negative numbers are introduced. This seems to be something easy for students to understand but I found out that even if students already know about the existence of negative numbers having used them to represent situations like 3 degrees below zero as -3, they would not think of -1 as the next number at the left of zero when it is presented in the number line. They would suggest another negative number and some will even suggest the number 1, then 2, then so on, thinking that maybe the numbers are mirror images.

Here is an alternative activity that I found effective in introducing the number line and the existence of negative numbers.  The purpose of the activity is to introduce the number line, provide students another context where negative numbers can be produced (the first is in the activity on Sorting Situations and the second is in the task Sorting Number Expressions), and get them to reason and make connections. The task looks simple but for students who have not been taught integers or the number line the task was a problem solving activity.

Question: Arrange from lowest to highest value

When I asked the class to show their answers on the board, two arrangements were presented. Half of the class presented the first solution and the other half of the students, the second solution. Continue reading “Teaching negative numbers via the numberline with a twist” →

Posted in Algebra

sorting pi, e, and root 2

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Mathematicians, always economical,   love to categorize numbers according to their properties. This is because numbers belonging to the same category behave in the same way. You don’t have to deal with each one! That’s an economical way of preserving the energy demand of brain cells.  In the grades we give pupils tasks that involve sorting numbers. Whole numbers  can be sorted out as odd or even, prime or composite, for example. This is a very good way of giving the students a sense of how strict definitions are in mathematics and in understanding the nature of numbers. In the higher grades they meet other numbers which they can categorize as imaginary or real, transcendental or algebraic. The same mathematical thinking is used.

\pi is one of the most widely known irrational number. Ask a student or a teacher to give an example of an irrational number, the chances are they will give \pi as the first example or the second one, after square root of 2.  And of course at a distance third is the number e. Now, although they belong to the same set of numbers, the irrationals, they don’t really belong to the same category. For example, \pi and e are both irrationals but pi is transcendental and square root of 2 is algebraic.  The number e is also transcendental. Here’s a short and simple explanation.

 

 

A  transcendental number is one that cannot be expressed as a solution of ax^n+bx^(n-1)+…+cx^0=0 where all coefficients are integers and n is finite. For example, x=sqrt(2), which is irrational, can be expressed as x^2-2=0. This shows that the square root of 2 is nontranscendental, or algebraic.

It is very easy to prove that a number is not transcendental, but it is extremely difficult to prove that it is transcendental. This feat was finally accomplished for ? by Ferdinand von Lindemann in 1882. He based his proof on the works of two other mathematicians: Charles Hermite and Euler.

In 1873, Hermite proved that the constant e was transcendental. Combining this with Euler’s famous equation e^(i*?)+1=0, Lindemann proved that since e^x+1=0, x is required to be transcendental. Since it was accepted that i was algebraic, ? had to be transcendental in order to make i*? transcendental. Click here for source.

Of course understanding the proof of pi as a transcendental number is beyond the level of basic mathematics and hey, we don’t even talk about transcendental numbers before Grade 10. But students at this level can understand the expression ax^n+bx^(n-1)+…+cx^0=0 where all coefficients are integers and n is finite. With proper scaffolding or if they have been exposed to similar task of sorting numbers before  students can make sense of the logic and reasoning shown above which characterizes most of the thinking in mathematics.

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Image from http://studenthacks.org/wp-content/uploads/2007/10/pumpkin-pi.jpg