In the post Algebra vs Arithmetic, I distinguished between arithmetic and algebra by arguing that it has nothing to do with the use of letters. That algebra is about letters and arithmetic is about numbers is an oversimplified view of algebra and can create misconceptions. Here are more ways of characterizing algebra. Continue reading “When is it algebra and when is it arithmetic?”
Tag: algebraic thinking
Top 20 Math Posts and Pages in 2012
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.
- 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….
- Exercises, Problems, and Math Investigations – The quality of mathematics students learn depends on the mathematical tasks or activities we let our students engage in….
- 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,…
- 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…
- 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 …
- 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…
- 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…
- 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…
- 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…
- Teaching positive and negative numbers – Here’s my proposed activity for teaching positive and negative numbers that engages students in higher-level thinking…
- 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….
- 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…
- 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….
- 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 …
- 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 …
- 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….
- 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,…
- 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 …
- 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.
- 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….
Develop your ‘Variable Sense’
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.
Math Education Studies
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
- Historical objections against the number line
- Structure sense
- Extension of number concepts to integers
- Intuitive nonexamples: the case of triangles
- Stages in the history of algebra
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.