Tag: GeoGebra

Posted in Math videos, Mathematics education

Should we do away with calculation?

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We don’t need to spend much time with calculation. Technology can do that for us. We can use the extra time saved for engaging in ‘higher-level’ thinking. Mr. Conrad Wolfram in his TEd Education Talk makes a lot of sense in this video. But, I have my reservations but not because I’m for teaching a lot of calculations.

We also said the same during the era of calculator. Did it improve the math education of our youth? So, what makes us think it will happen in the age of Mathematica, GeoGebra, Sketchpad? There is something in our math classes or math education that’s stuck in the middle ages, that’s not keeping with time. Unless we find and address this, no amount of technology can help us.

Posted in Math blogs

Top 20 Math Posts and Pages in 2012

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The thinker

I blog in order to organise what I think. And I don’t think I’m succeeding judging from the range of topics that I have so far written since I started Math for Teaching blog in 2010. Here’s the twenty most popular math posts and pages in this blog for the year 2012. It’s a mix of curricular issues, lessons, and teaching tips.

  1. What is mathematical investigation? – Mathematical investigation refers to the sustained exploration of a mathematical situation. It distinguishes itself from problem solving because it is open-ended….
  2. Exercises, Problems, and Math Investigations – The quality of mathematics students learn depends on the mathematical tasks or activities we let our students engage in….
  3. What is mathematical literacy? – Mathematical literacy involves more than executing mathematical procedures and possessions of basic knowledge that would allow a citizen to get by. Mathematical literacy is mathematical knowledge, methods,…
  4. My issues with Understanding by Design (UbD) – Everybody is jumping into this new education bandwagon like it is something that is new indeed. Here are some issues I want to raise about UbD…
  5. Curriculum change and Understanding by Design, what are they solving? – Not many teachers make an issue about curriculum framework or standards in this part of the globe. The only time I remember teachers raised an issue about it was in 1989, when the mathematics curriculum moved …
  6. Math investigation lesson on polygons and algebraic expressions – Understanding is about making connection. The extent to which a concept is understood is a function of the strength of its connection with other concepts. An isolated piece of knowledge is not powerful…
  7. Mathematics is an art – Whether we are conscious of it or not, the way we teach mathematics is very much influenced by what we conceive mathematics is and what is important knowing about it…
  8. Mathematical habits of mind – Learning mathematics is not just about knowing, understanding, and applying its concepts, principles and all the associated mathematical procedures and algorithms. It’s not just even about  acquiring the capacity to solve problem,  to reason, and to communicate…
  9. Subtracting integers using numberline – why it doesn’t help the learning – I have reasons to suspect that for a good percentage of students, the end of their mathematics career begin when they are introduced to subtracting integers. Well, for some, it’s when the x‘s start dropping from the sky without warning…
  10. Teaching positive and negative numbers – Here’s my proposed activity for teaching positive and negative numbers that engages students in higher-level thinking…
  11. Trigonometry – why study triangles – What is so special about triangles? Why did mathematicians created a branch of mathematics devoted to the study of it? Why not quadrinometry? Quadrilaterals, by its variety are far more interesting….
  12. Teaching the concept of function – Mathematics is not just about the study of numbers and shapes but also about the study of patterns and relationships. Function, which can define some of these relationships, is an indispensable tool in its study…
  13. Algebraic thinking and subtracting integers – Part 2 – Algebraic thinking is about recognizing, analyzing, and developing generalizations about patterns in numbers, number operations, and relationships among quantities and their representations.  It doesn’t have to involve working with the x‘s and other stuff of algebra….
  14. Patterns in the tables of integers – Mathematics is said to be the science of patterns. Activities that involve pattern searching is a great way to engage students in mathematical thinking. Here are some of my favorites …
  15. Making generalizations in mathematics – 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 …
  16. Teaching with GeoGebra: Squares and Square Roots – This post outlines a teaching sequence for introducing the concept of square roots in a GeoGebra environment. Of course you can do the same activity using grid papers, ruler and calculator….
  17. Algebra vs Arithmetic Thinking – One of the solutions to help students understand algebra in high school is to start the study of algebra earlier hence the elementary school curriculum incorporated some content topics traditionally studied in high school. However,…
  18. Teaching with GeoGebra – Educational technology like GeoGebra can only facilitate understanding if the students themselves use it. This page contains a list of my posts …
  19. Teaching combining algebraic expressions with conceptual understanding – In this post, I will share some ideas on how the simple investigation of drawing polygons with the same area can be used as an introductory lesson to teach operations with algebraic expressions with meaning and understanding.
  20. Mistakes and Misconceptions in Mathematics – Misconceptions are very different from the mistakes students make. Mistakes are not consciously made. Misconceptions are. Mistakes are usually one-off, while misconceptions, the gods forbid, could be for keeps….
Posted in Algebra, Mathematics education

Develop your ‘Variable Sense’

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Number sense refers to a person’s general understanding of number and operations along with the ability to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for solving complex problems (Burton, 1993; Reys, 1991). Researchers note that number sense develops gradually, and varies as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms (Howden, 1989).” – NCTM

If there’s number sense, there must also be also ‘variable sense’. Number sense is associated with arithmetic and basic to numeracy while variable sense (also called function sense) is associated with algebra. I collected the following set of tasks I believe will develop variable sense. Having a sense or feel of variables helps develop algebraic thinking and functional thinking.

Task 1

What must be true about the numbers in the blanks so that the following equation is always true?

____ + -2 = ____ + -4

Task 2

The following integers are arranged from lowest to highest:

n+1, 2n, n^2.

Do you agree? Explain why.

Task 3

What is the effect of increasing a on the value of x in each of the following equations?

1) x ? a = 0

2) ax = 1

3) ax = a

4) x/a = 1

Reason without solving.

Task 4

Drag the red point. Describe the relationship among the lengths of the line segments in each of the figure below. It would be nice if you can come up with an equation for each.

[iframe https://math4teaching.com/wp-content/uploads/2012/10/meaning_of_variable.html 600 360]

Tasks 1 and 2 are common problems. Task 3 is from a research paper I read more than five years ago. I could not anymore trace the paper and its author. Task 4 is  from Working Mathematically on Teaching Mathematics: Preparing Graduates to Teach Secondary Mathematics by Ann Watson and Liz Bills from the book Constructing Knowledge for Teaching Secondary Mathematics: Tasks to enhance prospective and practicing teacher learning (Mathematics Teacher Education). I just made it dynamic using GeoGebra.

Posted in Math blogs

Math and Multimedia Blog Carnival #17

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WELCOME  to the 17th edition of Math and Multimedia Blog Carnival!

As is the tradition, a math blog carnival should introduce the mathematical significance of n in its nth edition. I was lucky to host this 17th edition. I didn’t have to look further for the significance of the number 17. The 17 beautiful patterns of wall paper groups is enough introduction for your eyes.

A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. There are 17 possible distinct groups. – from Wikipedia.

Now, on to the 17 math posts from your favorite math blogs and bloggers.

1. Guillermo P. Bautista Jr. who started the Math and Multimedia Carnival presents The man who conned a mathematician at Mathematics and Multimedia.

2, John Cook presents Poor Mercator posted at The Endeavour, saying, “Mathematics behind the Mercator projection”.

3. Romeo Vitelli presents The Boltzmann Mystery (Part 1) posted at Providentia, saying, “A two-part post on the strange life and death of mathematician and physicist, Ludwig Boltzmann”

4. Amy Broadmoore presents 10 Children?s Books About Math posted at Delightful Children’s Books, saying, “Here are ten excellent picture books about math. These books are both entertaining and helpful for teaching kids about addition, subtraction, multiplication, division, measuring, fractions, graphing, very large numbers, and roman numerals.”

5. Colleen Young presents Top 100 Tools for Learning 2011 posted at Mathematics, Learning and Web 2.0, saying, “The tools I use from Jane Hart’s top 100” and Top 10 Mathematics Websites posted at Mathematics, Learning and Web 2.0, saying, “Another possibility – not sure which of these posts best for the carnival!”

6. IMACS  presents Alternatives to Math Competitions for the Dreamer Child and An Introduction to Modular Arithmetic posted, saying, “When choosing activities to engage a math-talented child, think about what makes that particular child thrive.”

7. Earl Samuelson of samuelson mathxp’s posterous submitted a bunch

8. Edmundo Gurza presents Writing with LaTex – Cure For Some Headaches posted at Reconstructing Climate.

9. William Dvorak presents Amortized Analysis posted at Deterministic Programming”

And here’s for Geogebra enthusiasts:

John Golden presents 2nd Fundamental Theorem of Calculus posted at Math Hombre, saying, “A GeoGebra activity to better understand functions defined as an integral, as in the 2nd fundamental theorem. Two other Calc 2 sketches at http://mathhombre.tumblr.com/post/11912609932/by-parts-dynamic-text and http://mathhombre.tumblr.com/post/11910861558/by-parts-and-product”

Guillermo Bautista presents  The Pantograph at GeoGebra Applet Central and Sanjay Gulati presents Maximum area of a rectangle inside a triangle at Mathematics Academy.

That’s all for now. See you next time. I think I host every fifth of this math blog carnival since hosting the Math and Multimedia Blog carnival #7.