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: arithmetic
What are the Three Worlds of Mathematics?
There are three worlds of mathematics according to David Tall: the world of conceptual embodiment, the world of symbolic calculation and manipulation, and the world of axiomatic formalism. This classification is based on how mathematical concepts/objects developed. It is important for us teachers to be at least aware of these three worlds. It tells us that math ideas are not formed in the same way therefore we can’t teach all math topics in the same way. The use of real-life contexts, the use of concrete materials, may afford learning of some concepts but may hinder the learning of others.
World of Conceptual Embodiment
According to Tall, the world of conceptual embodiment grows out of our perceptions of the world and consists of our thinking about things that we perceive and sense, not only in the physical world, but in our own mental world of meaning. The world includes the conceptual development of Euclidean geometry and other geometries that can be conceptually embodied such as non-Euclidean geometries and any other mathematical concept that is conceived in visuo- spatial and other sensory ways. A large part of arithmetical concepts also developed via conceptual embodiment (see Figure below).
World of Symbolic Calculation
The second world is the world of symbols that is used for calculation and manipulation in arithmetic, algebra, calculus and so on. The ‘development’ of the objects of this world begin with actions (such as pointing and counting) that are encapsulated as concepts by using symbol that allow us to switch effortlessly from processes to do mathematics to concepts to think about. But the focus on the properties of the symbols and the relationship between them moves away from the physical meaning to a symbolic activity in arithmetic. My post Levels of understanding of function in equation form describes the development of the idea of equation from action to object conception.
World of axiomatic formalism
The third world is based on properties, expressed in terms of formal definitions that are used as axioms to specify mathematical structures (such as ‘group’, ‘field’, ‘vector space’, ‘topological space’ and so on). It turns previous experiences on their heads, working not with familiar objects of experience, but with axioms that are carefully formulated to define mathematical structures in terms of specified properties. Other properties are then deduced by formal proof to build a sequence of theorems. The formal world arises from a combination of embodied conceptions and symbolic manipulation, but the reverse can, and does, happen.
Read the full paper Introducing the Three Worlds of Mathematics by David Tall.
Teaching Math with Mr Khan’s Videos – Variation
I’ve yet to read a math educator’s blog that endorses Khan Academy materials. Well, this blog does. Yes, you read it right. This blog endorses Mr. Khan’s materials for teaching mathematics. No, not by simply viewing the video but using the Mr Khan’s lecture as the object of investigation. Let’s take the video on direct variation. In the video, Mr Khan started with “varies directly” like it’s the simplest thing in the world to understand. Mr Khan then gave the sample problem and solved it as shown in the image below. Mr. Khan’s method is deductive and he uses lecture method. Click here to view the video in YouTube then read on below to see how the same video can be used to develop the concept of direct variation with conceptual understanding by linking it to students previously learned knowledge about proportion and then as context to introduce or review the concept of function.
How to use Mr Khan’s videos in teaching math
- Show the video. It’s a short one so it will be over before your class will realise it’s math.
- Ask the class if they can solve the same problem without using Mr Khan’s solution. The problem is elementary school level so students can solve it using arithmetic. Since a gallon of gas costs 2.25 so all they need to do is to find how many 2.25 in 18. They can continue to add 2.25 until they get to 18; continue taking away 2.25 from 18; or just divide 18 by 2.25.
- Ask for another solution. Didn’t they do ratio and proportion in 5th/6th grade? So, with a little scaffolding, students can set up 1:2.25 = n:18. I’m not a fan of product of the means is equal to the product of the extremes since it has nothing to do with proportional reasoning but I’ll allow it this time.
- Ask for another solution. Again with a little scaffolding questions like “If 1 gallon costs 2.25, how much would 2 gallons cost? 3 gallons? Can you organise those data in tables? It’s important that at 4 gallons you asked the students to solve the problem. There’s no need to continue all the way to 18$. Asking students to predict will make them consider the relationship between pairs of values. This is an important habit of thinking and it is crucial to appreciating and understanding algebra.
- Ask for another solution. With a little scaffolding again like “What do you notice about the values in the table? Can you imagine the arrangement of the points if you plot the values on the Cartesian plane? How will you use the graph to solve the problem?” Again there’s no need to plot the points all the way to 18. Students should think of extending the line to make the prediction.
- Now, go back to Mr Khan. “Study Mr Khan’s solution. What are those x and y that he’s talking about? What does y = kx mean in relation to your graph? Where is it in your table? Anyone can explain what Mr Khan mean by varies directly?”
- Assessment/ Assignment/ Further discussion: “The following are questions other students posted in Mr. Khan’s direct variation video in YouTube. How would you answer them?”
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- Sorry if this question seems basic, but I don’t understand how this example relates to functions…could someone please explain? Thanks!
- What is K in general?
- Why do we always have to set x?
- The practice for this video includes inverse variations, which are not yet covered. It would be great if there was practice specifically for direct variation only. Thanks!
This style of teaching is called teaching math through problem solving. If you enjoyed Teaching Math with Mr Khan, don’t forget to subscribe to this site. I will try to develop more lessons where I will be co-teaching math with Mr Khan’s videos.
Math Teachers at Play at Math Mama
Math Teacher at Play (MTAP) #51 is now live in Math Mama Writes …. Really great collection of 51 posts from teachers, lecturers, professors, bloggers, … in the following categories: Arithmetic, Patterns and Logic, Visual Math, Algebra-Geometry-Trigonometry, Puzzles and Games, Notations and Logic, and Breaking News, and may favorite, Teaching Mathematics. I even got two posts in the carnival. Thanks Sue.
The next MTAP carnival will be hosted in Let’s Play Math. Submit your post first week of July using the MTAP submission form.