Blog Feed: keeping mathematics simple and challenging

Math knowledge for teaching fractions and decimalsNo one can teach mathematics without knowing mathematics but not everyone who knows mathematics can teach it well. Below are two tasks about teaching fractions and decimals that would give us a sense of the kind of mathematical knowledge we teachers need apart from knowledge of the content of mathematics. As teachers it is expected [...]Mathematics for Teaching - This site is NOT about making mathematics easy because it isn't. It is about ma [..] > read more

Math knowledge for teaching ā€“ counting cubes

Published on 2012-05-12 04:20:58

Math knowledge for teaching – counting cubesThis post is the second in the series of post about the Mathematical Knowledge for Teaching (MKT) where I present task/lesson that readers of this blog can discuss and reflect upon. The first is aboutĀ Tangents to Curves, aĀ Year 12 lesson. This second post is for young learners. The task How many small cubes make up [...]Mathematics for Teaching - This site is NOT about making mathematics easy because it isn't. It is about making it make se [..] > read more

The nature of math vs the nature of school math

Published on 2012-04-22 06:34:25

The nature of math vs the nature of school mathWhat is the nature of mathematics? Mathematics is human. It is part of and fits into human culture. It is NOT an abstract, timeless, tensely, objective reality… Mathematical knowledge is fallible. As in science, mathematics can advance by making mistakes and then correcting them… There are different versions of proof or rigor. Standards of rigor [...]Mathematics for Teaching - This site is NOT about making mathematics easy because it isn [..] > read more

Dancing graphs

Published on 2012-03-12 03:52:24

Mathematics for Teaching - This site is NOT about making mathematics easy because it isn't. It is about making it make sense because it does.This series of dance steps in graphs of function is not about how to do a latin dance because the hips and feet movement are not emphasized. It’s probably showing us how a mathematician or a desperate algebra 2 teacher dances:-) Do you know that you can also draw this in GeoGebra? Click here to [...]Mathematics for Teaching - This site is NOT abo [..] > read more

How to grow algebra eyes and ears

Published on 2012-03-08 05:05:11

Mathematics for Teaching - This site is NOT about making mathematics easy because it isn't. It is about making it make sense because it does.Math teachers should growĀ algebra eyes and ears.Ā Ā To have algebra eyes and ears means to be always on the lookout for opportunities for students to engage in Ā algebraic thinkingĀ which involvesĀ thinking in terms of generality and to reason in terms of relationships and structure, etc. In the postĀ Teaching algebraic thinking without the x’sĀ [..] > read more

Teaching triangle congruence through problem solving

Published on 2011-07-05 00:15:45

In this post I propose a way of teaching the concept of triangle congruence. Like most of the lessons I shared in this blog, the teaching strategy for this lesson is Ā Teaching through Problem Solving.Ā In a TtPS lesson, the lesson starts with a situ > read more

Announcement: Seminar-Workshop on GeoGebra

Published on 2011-06-27 00:24:37

The University of the Philippines National Institute for Science and Mathematics Education Development (UPNISMED) will conduct a three-SaturdayĀ seminar-workshop on using GeoGebra in the teaching and learning of high school mathematics on August 6, 1 > read more

What is a square? ā€“ 12 definitions

Published on 2011-06-23 03:58:51

Definitions of concepts in mathematics are different from definitions of concepts in otherĀ disciplineĀ or subject area. Definitions in mathematics give a list of properties of the concept. A mathematics object is an example of that concept if it fit > read more

Teaching the meaning of equal sign

Published on 2011-06-21 02:46:30

This is one of my earlier posts in this blog. I amĀ re-postingĀ it here because there are over a hundred spammers chasing this post everyday. I couldn’t think of any other way to rid the post of spammers except to delete the original post. Here > read more

What is i?

Published on 2011-06-19 06:34:33

Am i or am i not ? Khan Academy say I am, www.jamestanton.com says I’m not. It is really very tempting to deduce that because . Many have fallen victim to this. This post is part of my series of posts about common misconceptions in mathematics. > read more

Love and Tensor Algebra

Published on 2011-06-16 20:15:12

Come, let us hasten to a higher plane Where dyads tread the fairy fields of Venn, Their indices bedecked from one to n Commingled in an endless Markov chain! Come, every frustrum longs to be a cone And every vector dreams of matrices. Hark to the gen > read more

Investigating the graphs of the sum and difference of linear equations

Published on 2011-06-14 23:25:22

This short investigation about the graphs of the sum and difference of two or more linear equation may be used as an introductory activity to the lesson on solving systems of linear equation by elimination. It will provide a visual proof why the meth > read more

Math and Multimedia Blog Carnival #12

Published on 2011-06-13 02:00:26

Welcome to the 12th edition of Math and Multimedia blog carnival. Yes, you get a dozen posts this time. Ā Before we do that let’s look at some trivia about the number 12. The number 12 is strongly associated with the heavens – 12 months, > read more

How do you teach algebra in early grades?

Published on 2011-06-06 09:12:36

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 Ā Teaching Algebraic Thinking Without the x’s. All the lessons in fact that I post here whether it > read more

Announcement: Japanese Grades 1- 6 Math textbooks English edition 50% off until September

Published on 2011-06-02 01:09:30

Last year I had the chance to do a few days work for aĀ JapaneseĀ Mathematics textbook series translated in English. I just received an e-mail from one of the editors that the publisher is selling the books at 50% off. It will only cost about US$75 f > read more

What is algebra? Why study it?

Published on 2011-06-01 09:56:14

I have been doing some research for my lit review and I came across this article by L.A Steen in Middle Matters. He was arguing about the Algebra for All standard in the US and part of the article includes description of what algebra is. I thought I > read more

What exactly do we do when we subtract?

Published on 2011-05-30 04:18:58

There are three ways by which subtraction can be understood: (1) Taking Away, (2) Difference, and (3) Inverse relation to addition operation. Pupilsā€™ first experience with subtraction involves taking away. The dash sign means minus and minus is tak > read more

Teaching with GeoGebra: Constructing polygons with equal area

Published on 2011-05-27 07:00:08

The power of GeoGebra lies not only for demonstrating a concept but more so on creating a situation where students are made to think, solve problems, and reason mathematically. Here is a sample lesson on how this can be done. The lesson involves > read more

Announcement: Math Carnival submission now on

Published on 2011-05-26 03:32:18

Calling all math bloggers! Mathematics for Teaching will be hosting the 12th edition of the Math and Multimedia blog carnival which will go live on 13th June 2011. Click here to send your posts/articles. > read more

What root do you want to take?

Published on 2011-05-25 02:24:51

Mathematics is already a difficult subject and most times we make it more difficult to students to make sense of with the confusing rules we tell them. For example, we insist that the square root of 36 is 6 and -6 yet we also insist thatĀ ā36 is eq > read more

Mistakes vs misconceptions in mathematics

Published on 2011-05-16 00:50:48

What is the difference between mistakes and misconceptions? While we learn from our mistakes, no one really learns a mistake or how to make one. Misconceptions on the other hand are learned. Mistakes are usually one-off, while misconceptions, the god > read more

GeoGebra applets and proofs for angles made by transversal

Published on 2011-05-11 23:18:00

Geometry is a natural area of mathematics for which students should develop reasoning and justification skills and their appreciation of the logico-deductive part of mathematics that build across the grades. Learning tasks should be so designed so th > read more

Sample problem for proving perpendicular segments

Published on 2011-05-06 11:05:12

In the figure CF = FB = FE. If C is moved along CB, describe the paths of F and E. Explain or prove that they are so. This problem can be explored using GeoGebra applet. Click this link to explore. One way to prove that FC is a straight line and perp > read more

Collapsible

Published on 2011-05-06 10:07:31

We live in a 3-dimensional world but why is it that we start teaching geometry we start from 2-dimensions to 3-dimensions? Shouldn’t it the other way around? Aren’t we supposed to start from students’Ā experiences?Ā Making connectio > read more

How is a paradigm created?

Published on 2011-05-05 15:57:53

We can change and improve our schools practices if we are aware how they were created in the first place. Paradigm Created View more presentations from Bipin Upadhyay No related posts. No related posts. > read more

Of what does mathematical knowledge consist?

Published on 2011-05-02 19:30:39

Anyone interested to understand how mathematics is learned should at first understand what mathematical knowledge consist of. The book Forms of Mathematical Knowledge: Learning and Teaching with Understanding,Ā presents and describes various types of > read more

New math curriculum

Published on 2011-05-02 03:22:01

In the video below, Tom Lehrer ‘explains’ and ‘criticizes’ the New Math curriculum in a funny way. Enjoy. What is the New Math Curriculum? New Mathematics or New Math was a brief, dramatic change in the way mathematics was tau > read more

How can we tell that mathematics education is in crisis level?

Published on 2011-04-28 20:48:08

We do not need a nationalĀ examination to know the state of ourĀ mathematics education. We only need to go to the classroom. By the following signs you can tell if we are already in the crisis level. Click link below Ten signs that mathematics educat > read more

How confident are you to teach mathematics?

Published on 2011-04-26 00:40:40

As mathematics teachers we simply cannot just stop learning and improving in our field. One of the best practice for continuing professional development is to engage in self-evaluation. It is a powerful and productive way of supporting our own profes > read more

The Pi Song

Published on 2011-04-24 23:54:31

Memorizing without understanding is a no-no in this blog. But just for fun you can try this one on pi. For my thoughts regarding teaching irrational numbers you may want to read Teaching irrational numbers: break it to me gently. > read more

What is mathematical modeling?

Published on 2011-04-16 03:51:32

While there is no consensus yet as to a precise definition of this term, mathematical modeling is generally understood as the process of applying mathematics to a real world problem with a view of understanding the latter. One can argue that mathemat > read more

Tips and sample lessons for scaffolding understanding of algebra

Published on 2011-04-14 09:27:25

One way to make algebra make sense to students is to show where those mathematics Ā objects (e.g. algebraic expressions/equations/formula) come from. It will even makes sense more to students if they themselves can generate those objects. As John Mas > read more

Why it is bad habit to introduce a concept by defining it

Published on 2011-04-06 20:09:09

This post is related to my earlier post on the meaning of understanding. Here I present why it is bad habit for K-12 mathematics teachers to introduce a concept through its definition. The excerpt is from my dissertation which you can download here. > read more

Introducing the negative number via the numberline

Published on 2011-04-05 03:47:19

One popular way of introducing the negative numbers is through the number line. Most textbooks Ā 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 int > read more

Common errors in algebra

Published on 2011-04-03 20:13:37

Mathematics is indeed a universal language. Even errors are universal. Here is a funny video of common algebra mistakes. It would be fun for students to identify, explain, and correct the errors. If they have their math portfolio they can even make t > read more

Stories about mathematics

Published on 2011-04-02 10:01:10

Like all other forms of languages, mathematics developed Ā because of the necessity to communicate ideas. In the case of numbers, it was to express and record ‘countables’. Below is a little video that can explain this to pupils in the pr > read more

Stories about mathematics

Published on 2011-04-01 10:01:10

Announcement: GeoGebra seminar-workshop at UP NISMED

Published on 2011-03-23 23:21:41

UP NISMED will run a two-day seminar-workshop in the use of GeoGebra in the teaching and learning of mathematics. Click image for details. Filed under: geogebra Tagged: GeoGebra, technology and teaching > read more

Doing problem solving

Published on 2011-03-20 08:20:18

I love these graphs by Jason Dryer which show how a mathematician and a typical student solve a problem. The graphs were from the postĀ “Some research discoveries”. The last one is mine, on teachers time-line graph in doing problem solvin > read more

What does it mean to understand mathematics?

Published on 2011-03-16 23:38:22

I’m sharing in this post some of the “theories” underpinning the lessons, learning tasks, and math teaching ideas that I blog here. I thought, rather than talking about them in each post I make I will just refer the reader to this a > read more

Sorting number expressions

Published on 2011-03-15 03:07:55

To understand is to make connection. The stronger the connection the most powerful the understanding. One of the ways to help students to make connections among concepts is to give them problem solving tasks that have many correct solutions or answer > read more

Making sense of negative numbers

Published on 2011-03-15 03:07:55

To understand is to make connection. The stronger the connection the most powerful the understanding. One of the ways to help students to make connections among concepts it to provide them with problem solving tasks that have many correct solutions o > read more

From PCK to TPCK

Published on 2011-03-13 05:18:03

In the 80′s, Lee Shulman introduced the concept of pedagogical content knowledge (PCK) to differentiate it from content knowledge (CK). PCK is teachersā€™ knowledge of how a particular subject matter is best taught and learned. Since Shul > read more

New name, new domain

Published on 2011-03-12 07:55:46

I thought that because this blog is dedicated to discussing mathematical ideas in the context of teaching them to k-12 audience I should rename it to “Mathematics for Teaching”. It’s old title is Keeping Mathematics Simple. Note tha > read more

Announcement: GeoGebra seminar-workshop at UP NISMED

Published on 2011-02-28 22:21:41

UP NISMED will run a two-day seminar-workshop in the use of GeoGebra in the teaching and learning of mathematics. Click image for details. > read more

sorting pi, e, and root 2

Published on 2011-02-10 23:00:01

Mathematicians, always economical, Ā 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!Ā That’s an economical way > read more

the irrationals pi, e, and root 2

Published on 2011-02-10 23:00:01

Teachers, principals, DepEd, this oneā€™s for you

Published on 2011-02-06 09:31:51

I got this comment from my post Curriculum change and understanding by design: what are they solving? It’s from a student with very legitimate complain, concern, and challenge. Sometimes we are so focussed on finishing the syllabus, followingĀ > read more

Teachers, principals, DepEd, this oneā€™s for you

Published on 2011-02-06 09:31:51

Math and Multimedia Carnival #7

Published on 2011-01-31 11:53:23

Welcome to the 7th edition of Mathematics and Multimedia blog carnival. Before we begin Carnival 7, letā€™s look at some of the trivias about the number seven: TheĀ Millennium Prize Problems are seven problems inĀ mathematics that were stated by the > read more

Math and Multimedia Carnival # 7

Published on 2011-01-31 11:53:23

Teaching algebraic expressions: Counting smileys

Published on 2011-01-29 00:49:09

I love this task for introducing the concept of variable and algebraic expressions through problem solving. It combines numerical, geometric, and algebraic thinking. Ā The figure below shows the standard version of the task. Of course some easier ver > read more

Counting smileys

Published on 2011-01-29 00:49:09

Attention Math Bloggers

Published on 2011-01-17 23:50:10

This blog will be hosting the 7th edition of Math and Multimedia Blog Carnival. It will be posted on 31st January 2011. This special edition carnival will focus on teaching and learning algebra at all levels. Click the link to send your posts/ articl > read more

Attention Math Bloggers

Published on 2011-01-17 23:50:10

Art and Mathematics

Published on 2010-10-31 22:22:14

This post is actually one of the comments in my Mathematics is an Art post. I thought it would be a waste to just leave it in the comments section. It’s very informative and well written so I’m sharing it here. That way more people will h > read more

Art and Mathematics

Published on 2010-10-31 22:22:14

(Mis)Understanding by Design

Published on 2010-09-23 20:24:22

The country’s schools are now implementing ‘Understanding by Design (UbD) curriculum.’ Some private schools are implementing it at all levels while all the public schools are on its first year of implementation starting with first y > read more

(Mis)Understanding by Design

Published on 2010-09-23 20:24:22

Teaching irrational numbers ā€“ break it to me gently

Published on 2010-09-15 08:00:31

Numbers generally emerged from the practical need to express measurement. From counting numbers to whole numbers, to the set of integers, and to the rational numbers, we have always been able to use numbers to express measures. Up to the set of ratio > read more

Teaching irrational numbers ā€“ break it to me gently

Published on 2010-09-15 08:00:31

Conference on Assessing Learning

Published on 2010-09-13 00:51:15

The conference is open to high school mathematics and science teachers, department heads and coordinators, supervisors, tertiary and graduate students and lecturers, researchers, and curriculum developers in science and mathematics. Plenary Topics an > read more

Conference on Assessing Learning

Published on 2010-09-13 00:51:15

GeoGebra and Mathematics: Investigating coordinates

Published on 2010-09-08 22:32:16

The most basic mathematics students need to know to understand GeoGebra is the coordinate axes and the grid. Must you teach students how to plot points and interpret coordinates of points before they use GeoGebra or the other way around? I think, at > read more

GeoGebra and Mathematics: Investigating coordinates

Published on 2010-09-08 22:32:16

GeoGebra and Mathematics: Simplifying and Adding Radicals

Published on 2010-09-08 02:21:40

The square root of a number is usually introduced via an activity that involves getting the side of a square with the given area. For example the side of a square with area 25 sq unit is 5 unit because 5 x 5 = 25. To introduce the existence of , a sq > read more

GeoGebra and Mathematics: Squares and Square Roots

Published on 2010-09-02 22:26:41

This is the second in the series of posts about integrating the teaching of GeoGebra andĀ Mathematics in lower secondary school. The first post was about teaching the point tool and investigating coordinates of points in a Cartesian plane. This post > read more

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Blog Feed: keeping mathematics simple and challenging

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