Posted in Math research

Math Education Studies

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Math-ed studies is my  new blog about research in mathematics teaching and learning and teacher learning. It contains studies, books, links, etc about mathematics education. It is actually my nth attempt to organize myself. Evernote is not enough to organize me. With this blog I hope it will be easier for me to trace where I read this or that idea. I hope this will also be useful to you especially in doing a literature review for your research. For obvious reason I cannot provide access to the full paper or books just the abstract and some important ideas from them. Usually these ideas has to do with what I’m currently writing. I have included the title of the journal and publisher of the paper in each post if you want to read the full paper.

To give you an idea what the blog contains, here’s the most recent posts:

Recent Posts

If you have a paper about mathematics teaching and learning and teacher learning published online with public access, just e-mail the link to me. Of course I reserve the right to link it or not in math-ed studies blog.

Posted in Math Lessons

Math Lessons in Mathematics for Teaching

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This is a collection of math lessons posted in this blog.  Most if not all of the lessons use the strategy teaching through problem solving or through mathematical investigation. I believe that school mathematics is about teaching students how to think mathematically first and learning the mathematics second so  math lessons should be designed so that students are engaged in thinking mathematically. This is something that should not be left to chance.

  1. How to grow algebra eyes and ears
  2. How to teach the inverse function
  3. How to teach the derivative function without really trying
  4. How to scaffold problem solving in geometry
  5. What is a coordinate system?
  6. How to teach triangle congruence through problem solving
  7. Teaching the meaning of equal sign
  8. Geometry lesson: Collapsible chair model
  9. Teaching negative numbers via the numberline with a twist
  10. Introducing negative numbers
  11. Teaching with GeoGebra – Investigating coordinates of points
  12. Teaching simplifying and adding radicals
  13. Teaching with GeoGebra: Squares and Square Roots
  14. Teaching trigonometry via problem solving
  15. Introducing positive and negative numbers
  16. Teaching subtraction of integers
  17. Algebraic thinking and subtracting integers – Part 2
  18. Subtracting integers using tables- Part 1
  19. Teaching the absolute value of an integer
  20. Teaching with GeoGebra: Constructing polygons with equal area
Posted in Algebra

How to grow algebra eyes and ears

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Math teachers should grow algebra eyes and ears.  To have algebra eyes and ears means to be always on the lookout for opportunities for students to engage in  algebraic thinking which involves thinking in terms of generality and to reason in terms of relationships and structure, etc. In the post Teaching algebraic thinking without the x’s I described some tips on how to engage pupils in algebraic thinking as they learn about numbers. Likewise in Algebraic thinking and subtracting integers and Properties of Equality – do you need them to solve equation?

Here is another example. How will you use this number patterns in your algebra 1 class so students will also grow algebra eyes and ears?

Let me share how I teach this. I like to simply post this kind of patterns on the blackboard without any instruction. For a few seconds students would normally not do anything and wait for instruction but getting none would start scribbling on their notebooks. When asked what they’re doing they would tell me they are generating other examples to check if the the pattern they see works (yes, detecting patterns is a natural tendency of the mind). When I asked what’s the  pattern and how they are generating the examples I sometimes get this reasoning:  the first and the second columns increase by 1 so the next must be 5 and 6 respectively, the third and fourth columns increase first by 6, then by 8 so the next one must increase by 10 so the next numbers must be 30 and 31 respectively. That is, 5^2 + 6^2 +30^2 = 31^2. Of course this is not what I want so I would ask them if there are other ways of generating examples that does not depend on any of the previous cases.

In generating examples, students usually start with the leftmost number. I would challenge them to start from any terms in the equation. After this, if no one thought of proving that the pattern will work for all cases, then I’ll ask them to prove it. It would be easier for me and for them if I will already write the following equation at the bottom of the pattern for students to fill up and prove but this method is for the lazy and lousy teacher. A good algebra teacher never gives in to this temptation of doing the thinking of representing an unknown by a letter symbol for their students.

In proving the identity, I have observed that students will automatically simplify everything so they end up with fourth degree expressions. This is another opportunity to challenge the students: show that the left hand side and right hand side simplifies to identical second degree expressions with only their knowledge of square of the sum (a+b)^2 = a^2+2ab+b^2.

The teaching sequence I just described is consistent with the levels of understanding of equation I described in Assessing understanding of function in equation form.

Posted in Algebra

How to teach the inverse function

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In What  is an inverse function? I proposed a way of teaching this concept starting with its graphical representations using GeoGebra applets. Al-Zboun Lilliana in our Linkedin group shares her idea for introducing the inverse function. She says that the most difficult part in teaching this concept is to make it make sense to students and not so much in making the students understand its definition or teaching them the process of finding the inverse function of a given function(by a graph or by a formula)  or to “verify algebraically” that the functions are inverses.

Here’s her proposed teaching sequence starting with examples that students can relate to in Levels 1 and 2. I would suggest inserting the activities I described in What is an inverse function? before Level 3 which introduces the algebraic solution.

Examples SET(1)-Level 1:
1. If we need to call someone we are asking for her/his name on the list of our phone contacts …
2. If someone of our contacts is calling us “our phone shows who is calling” This is the job of an inverse function: “finding the name corresponding to the number”

Examples SET(2)- Level 2:

1. If George makes $100/day. We know how to answer questions such as “After 7 days, how much money has he made?” We use the function W(t)=100t
But suppose I want to ask the reverse question:
2. “If George has made $700, how many hours has he worked?” The students know the answer : Time : t(W)=W/100. Given any amount of money, divide it by 100 to find how many days he has worked.
This is the job of an inverse function. It gives the same relationship, but reverses the dependent and independent variables.

Examples SET(3)- Level 3:

In this set the teacher includes examples to show simplifying solutions of mathematical questions

Example (1): Solve log (3 x – 2) = 3
• Since logarithmic and exponential functions are inverses of each other, we can write the following.

a = log (b) if and only b = 10^a
• Use the above property of logarithmic and exponential functions to rewrite the given equation as follows.
3x – 2 = 10^3
• Solve for x to obtain.

3x = 1002
x= 1002÷3=334

Example (2): Find the Range of the function ( or any RATIONAL function) :
F(x)= (3x+1)/(3 -x) or [y=(3x+1)/(3 -x)]
• Since the RANGE of a one to one function is the DOMAIN of its inverse. Let us first show that function f given above is a one to one function.
• Hence the given function is a one to one. let us find its inverse.

• Interchange x and y and solve for y.
x =(3y+1)/(3 -y)
And find y = (3x-1)/(3+x)
The inverse g(x) of function f(x) is given by.

g(x) = (3x-1)/(3+x)
• The domain of g(x) is R except x = -3. Hence THE RANGE of f(x) is R/{-3}.