Tag: teaching math

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 Geometry

Teaching triangle congruence

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In this post I propose a way of teaching the concept of triangle congruence. Like most of the lessons I share in this blog, the teaching strategy for this lesson is  Teaching through Problem Solving. In a TtPS lesson, the lesson starts with a situation that students will problematize. The problems either have many correct answers or have multiple solutions and can always be solved by previously learned concepts and skills. Problems like these help students to make connections among the concepts they already know and the new concept that they will be learning in the present lesson. The ensuing discourse among students and between teacher and students during the discussions of the different solutions and answers trains students to reason and communicate mathematically and thereby help them to appreciate the power of mathematics as a language and a way of thinking. In mathematics, language is precise and concise.

Here’s the sequence of my proposed lesson:

1. Setting the Problem:

Myra draw a triangle in a 1-cm grid paper. Without showing the triangle, she challenged her friends to draw exactly the same triangle with these properties:  QR is 4 cm long. The perpendicular line from P to QR is 3 cm. 

Pose this question: Can you draw Myra’s triangle?

Give students enough time to think. When each of them already have at least one triangle, encourage the class to discuss their solutions with their seat mates. Challenge the class to draw as many triangles satisfying the properties Myra gave.

2. Processing of solutions: Ask volunteers to show their solutions on the board. Questions for discussion: (1) Which of these satisfy the information that Myra gave? (2) What is the same among all the correct answers? [They all have the same area]. Possible solutions are shown below.

triangle congruence


3. Introducing the idea of congruence: Question:  If we are going to cut-out all the triangles, which of them can be made to coincide or would fit exactly? [When done, introduce the word congruence then give the definition.]

Tell the class that Myra only drew one triangle. Show the class Myra’s drawing. Question: In order to draw a triangle congruent to Myra’s triangle, what conditions or properties of the triangle Myra should have told us?

Myra’s triangle

Possible answers:

  1. QR is 4 cm long. The perpendicular line PQ  is 3 cm.
  2. QR is 4 cm long. PQ is 3 cm and forms a right angle with QR.
  3. PQR is a right triangle with right angle at Q. QR is 4 cm and PQ is 3 cm.

4. Extending the problem solving activity: Which of the following sets of conditions will always give triangles congruent to each other?

  1. In triangle ABC, AB and BC are each 5 cm long.
  2. ABC is a right triangle. Two of its shorter sides have lengths of 4 cm and 5 cm.

I would appreciate feedback so I can improve the lesson. You feedback will inform the sequel to this lesson.. Thank you.

Posted in What is mathematics

Math is not easy to learn – that is a fact.

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I think it’s a waste of time trying to make math easy and fun to learn if your idea of fun does not involve challenge.

Mathematics is not an easy subject and it is not easy to learn it. That is a fact. The sooner the teacher accepts this, the better for her students. The challenge to us teachers is not in how we can make math easy to learn but in how we can make it makes sense and how we can make our students love the challenge that mathematics presents. Can math be challenging if students feel that what they are expected to do in the class is to follow the teacher’s method, the teacher’s way of thinking, and the teacher’s way of doing things? Where is the fun in that?

Mathematics is not fun to learn if the idea of fun is like playing bingo! However, if ‘fun’ is a function of the challenge a sport or a game presents, then indeed learning mathematics is fun. We love a sport because of the challenge it presents, the opportunities it gives us to make prediction, analyze, strategize, make our stand and defend it, etc and not because it is easy to play!

Everything in mathematics makes sense. Everything in mathematics is connected to everything else. I think this is where we teachers should be devoting our time to. And this is what this blog is about!

Posted in Math investigations

Exercises, Problems, and Math Investigations

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The quality of mathematics students learn depends on the mathematical tasks or activities we let our students engage in.

Mathematical activities/tasks can be categorized into three types: exercises, problem solving, and math investigations.

Standard exercises

These are activities with clearly defined procedure/strategy and goal. Standard exercises are used for mastery of a newly learned skill – computational, use of an instrument, and even new terms or vocabulary. These are important learning activities but must be used in moderation. If our teaching is dominated by these activities, students will begin to think mathematics is about learning facts and procedures only. This is very dangerous.

Problem solving activity

These are activities involving clearly defined goals but the solutions or strategies are not readily apparent. The student makes decision on the latter. If the students already know how to solve the problem then it is no longer a problem. It is an exercise. Click here for features of good problem solving tasks. It is said that problem solving is at the heart of mathematics. Can you imagine mathematics without problem solving?

Math investigations

These are activities that involve exploration of open-ended mathematical situation. The student is free to choose what aspects of the situation he or she would like to do and how to do it. The students pose their own problem to solve and extend it to a directions they want to pursue. In this activity, students experience how mathematicians work and how to conduct a mathematical research. I know there are some parents and teachers who don’t like math investigation. Here are some few reason why we need to let our students to go through it.

  1. Students develop questions, approaches, and results, that are, at least for them, original products
  2. Students use the same general methods used by research mathematicians. They work through cycles of data-gathering, visualization, abstraction, conjecturing and proof.
  3. It gives students the opportunity communicate mathematically: describing their thinking, writing definitions and conjectures, using symbols, justifying their conclusions, and writing and reading mathematics.
  4. When the research involves a class or group, it becomes a ‘community of mathematicians’ sharing and building on each other’s questions, conjectures and theorems.

Students need to be exposed to all these type of mathematical activities. It is unfortunate that  textbooks and  many mathematics classes are dominated by exercises rather than problem solving and investigations tasks, creating the misconception that mathematics is about mastering skills and following procedures and not a way of thinking and communicating.

Samples of these tasks are shown in the picture below:

Click here and here for a sample teaching using math investigation.