Tag: teaching algebra

Posted in Humor

Things you learn in math education forums

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You always get good ideas from forums (or fora), whatever form they are. If you want great insights about math and science education, try attending a PhD forum or seminar. I’ve just been to one. Following are some of the things I learned from the spirited discussion during the question and answer portion from these serious educators.

  1. We complain that our learners are not doing well in their Maths especially in secondary schools. These students are now engineers, doctors, lawyers, and politicians. We trust them anyway (except the politicians).
  2. It is only in math that 1+1 = 2. In real-life, it doesn’t work that way. For example, when two churches combine, you get 3 – the new one and the two old ones. This also applies to political parties.

    number theory
    number theory
  3. On the question of the relevance of your PhD to science education. Short answer by the speaker: I am now relevant to the science education. They now have one learned participant in the science education discourse.
  4. Why do we always expect the teachers to know all their Maths? Answer: It is probably because of our experience of our teachers in first grade as all-knowing. We believe everything teacher say and it was important for us then to have believe them. I think we need to grow up.
  5. Tell me, “Do you know of a mathematician who know all their mathematics?” Why should a math teacher know all their math? This is not fair to teachers. Do you complain in the media when a doctor misdiagnose your illness?math teachers
  6. “My conclusion in my review of literature why, despite the extent of research about teaching and learning algebra we still have not solved the difficulty of learning it, is that because algebra is a moving target.”
  7. “I initially thought to explore the reasons of students absenteeism in lectures. But then I thought, why should they when they can find great lectures in the net. Now I do not know how to proceed from here. Will anybody suggest a research question that’s not in the net?”
  8. “In my interview with teachers, most of them said that they don’t really know why students are not getting the test. When they teach them, they seem to understand everything they are discussing and solving. My interview with students confirms this. The students said that they understand everything during the lectures but they couldn’t answer the same questions and problems in the test.”
Posted in Algebra

What Makes Algebra Difficult is the Equal Sign – Part 1 of x

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Algebra is one of the most researched topics in mathematics education. And most of these studies are about students understanding of algebraic concepts, particularly equations and the 24th letter of the English alphabet. With the volume of studies, one wonders why until now algebra many learners still have difficulty with the subject. I read a remark somewhere comparing the search for effective means of teaching/learning algebra similar to that of the quest for the holy grail.

I’m not about to offer in this post a way of making learning algebra easier. I have not found it myself. But let me offer an explanation why algebra is illusive to many first time learners of the subject. I adhere to the belief that once you know where the problem is, you have solved half of it. Sometimes, it could turn out of course that the solution of the other half of the problem is learning to live with it.

Consider the following familiar symbols we write in our blackboard. I will label each string of symbols, A and B.

Equivalence

What do the math symbols in A and B mean? How does A differ from B? How are they similar?

Let’s start with the ‘visual’ similarity. They both have an equal sign. They both show equality. Are they both equations? The statement 12+4x=4(3+x) is an equivalence. It means that the right hand side is a transformation of the left hand side. This transformation is called factoring, using the division operation. The transformation from right to left is called getting the product and you do this by multiplication.

Would you consider statement B an equivalence? It certainly not. You can test this in two ways. One, try to think of an transformation you can do. Two, you can test a few values of x for both sides of the equality sign to check if it will generate equal values. You will find that only x=-5.5 will yield the same result. This means that statement B is not an equivalence but a conditional equation. They are only true for certain values of x. This is what we commonly call equation.

I have shown that we have used the ‘=’ sign in two ways: to denote an equivalence and an equation. How important are the distinctions between the two? Is it so much of a big deal? Are they really that different? Let’s fast forward the lesson and say you are now dealing with function (some curriculum starts with function). Let f:x?12+4x, g:x?4(3+x), and h:x?2x+1. Their graphs are show below. Note that functions f and g coincide at all points while function h intersect them at one point only.

intersecting lines

The graphical representation clearly show how different statements A and B are and that the ‘=’ sign denotes two different things here. Now, if you notice the graphs above, the function notation also use the ‘=’ sign. Is it use the same as  in A and B? Try transforming. Try solving. It’s different isn’t it?  In function notation such as f(x) = 12 + 4x, ‘=’ is used to denote a label or name for the function that maps x to (12 + 4×0. This meaning should be very clear to students. Studies have shown that learners misinterprets f(x) as f times x and tried to solve for x in the equation.

In 13 – 5 =____, what does ‘=’ equal sign mean? Ask any primary school learner and they would tell you it means ‘take way’ or ‘do the operation’. You may be interested to read What Pupils Think About the Equal Sign and Teaching the Meaning of Equal Sign.

I have presented four meanings of ‘=’ in mathematics: equivalence, equation, to denote a name for a function, and to do the operation. My point is that one of the factors that make algebra difficult is the multiple meaning of symbols used. We also use of the word equation to everything with ‘=’. Students need to be able to discern the meaning of these in the context to which they are used if we want our learners to make sense of and do algebra.

In Part 2, I talk about the multiple meanings of the letter symbols as source of students difficulties in algebra. You may also want to read Making Sense of Equivalent Equations and Expressions and Equations, Equations, Equations. If you want some references for Algebra teaching you can try Fostering Algebraic Thinking.

Posted in Algebra

Tough Algebra Questions about Equations and Expressions

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Here are some questions your students have been wanting to ask you in your algebra class. Daniel Chazan and Michal Yerushalmy in their article On Appreciating the Cognitive Complexity of School Algebra posed these questions about equivalence of equations , solving equations, and equivalence of expressions for us teachers to ponder upon.

function_notationHow will you answer the following questions? What explanation will you give to the students?

Continue reading “Tough Algebra Questions about Equations and Expressions” →

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