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Posted in Algebra

Strengths and limitations of each representation of function

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Function is defined in many textbooks as a correspondence between two sets x  and y such that for every x there corresponds a unique y. Of course there are other definition. You can check my post on the evolution of the definition of function. Knowing the definition of a concept however does not guarantee understanding the concept. As Kaput argued, “There are no absolute meanings for the mathematical word function, but rather a whole web of meanings woven out of the many physical and mental representations of functions and correspondences among representations” (Kaput 1989, p. 168). Understanding of function therefore may be done in terms of understanding of its representations. Of course it doesn’t follow that facility with the representation implies an understanding of the concept it represents. There are at least three representational systems used to study function in secondary schools. Kaput described the strengths and limitations of each of these representational systems. This is summarised below:

Tables: displays discrete, finite samples; displays information in more specific quantitative terms; changes in the values of variables are relatively explicitly available by reading horizontally or vertically when terms are arranged in order (this is not easily inferred from graph and formula).

Graphs: can display both discrete, finite samples as well as continuous infinite samples; quantities involved are automatically ordered compared to tables; condenses pairs of numbers into single points; consolidates a functional relationship into a single visual entity (while the formula also expresses the relationship into a single set of symbols, individual pair of values are not easily available for considerations unlike in the graph).

Formulas/ Equations: a shorthand rule, which can generate pairs of values (this is not easily inferred from tables and graphs); has a feature (the coefficient of x) that conveys conceptual knowledge about the constancy of the relationship across allowable values of x and y — a constancy inferable from table only if the terms are ordered and includes a full interval of integers in the x column; parameters in equation aid the modelling process since it provides explicit conceptual entities to reason with (e.g. in y = mx, m represents rate).

It is obvious that the strength of one representation is the limitation of another. A sound understanding of function therefore should include the ability to work with the different representations confidently. Furthermore, because these representations can signify the same concept, understanding of function requires being able to see the connections between the different representations since “the cognitive linking of representations creates a whole that is more than the sum of its parts” (Kaput, 1989, p. 179). Below is a sample task for assessing understanding of the link between graphs and tables. Click solutions to view sample students responses.

tables and graphs

How do you teach function? Which representation do you present first and why?

Reference

Ronda, E. (2005). A Framework of Growth Points in Students Developing Understanding of Function. Unpublished doctoral dissertation. Australian Catholic University, Melbourne, Australia.

Posted in Mathematics education

NCTM Process Standards vs CCSS Mathematical Practices

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The NCTM process standards, Adding it Up mathematical proficiency strands, and Common Core State Standards for mathematical practices are all saying the same thing but why do I get the feeling that the Mathematical Practices Standards is out to get the math teachers.

The NCTM’s process standards of problem solving, reasoning and proof, communication, representation, and connections describe for me the nature of mathematics. They are not easy to understand especially when you think that school mathematics is about stuffing students with knowledge of content of mathematics. But, over time you find yourselves slowly shifting towards structuring your teaching in a way that students will understand and appreciate the nature of mathematics.

The five strands of proficiency were also a great help to me as a teacher/ teacher-trainer because it gave me the vocabulary to describe what is important to focus on in teaching mathematics.

With the Mathematical Practices Standards I had this picture of myself in the classroom with a checklist of the standards in one hand and a lens on the other looking for evidence of proficiency. The NCTM and Adding it Up standards actually said more about math. The ones in Common Core are saying more about what students should attain. I wonder which will encourage ‘teaching to the test’. The day teachers start to ‘teach to the test’ is the beginning of the end of any education reform.

NCTM Process Standards

Five Strands of Mathematical Proficiency

CCSS Mathematical Practices

Problem Solving

  1. Build new mathematical knowledge through open-ended questions and more-extended exploration;
  2. Allow students to recognize and choose a variety of appropriate strategies to solve problems;
  3. Allow students to reflect on their own and other strategies for solving problems.

Reasoning and Proof

  1. Recognize and create conjectures based on patterns they observe;
  2. Investigate math conjectures and prove that in all cases they are true or that one counterexample shows that it is not true;
  3. Explain and justify their solutions.

Communication:

  1. Organize and consolidate their mathematical thinking in written and verbal communication;
  2. Communicate their mathematical thinking clearly to peers, teachers, and others;
  3. Use mathematical vocabulary to express mathematical ideas precisely.

Connections

  1. Understand that mathematical ideas are interconnected and that they build and support each other;
  2. Recognize and apply connections to other contents;
  3. Solve real world problems with mathematical connections.

Representation

  1. Emphasize a variety of mathematical representations including written descriptions, diagrams, equations, graphs, pictures, and tables;
  2. Select, apply, and translate among mathematical representations to solve problems;
  3. Use mathematics to model real-life problem situations.

Conceptual Understanding refers to the “integrated and functional grasp of mathematical ideas”, which “enables them [students]
to learn new ideas by connecting those ideas to what they already know.”

Procedural fluency is defined as the skill in carrying out procedures flexibly, accurately, efficiently, and
appropriately.

Strategic competence is the ability to formulate, represent, and solve mathematical problems.

Adaptive reasoning is the capacity for logical thought, reflection, explanation, and justification.

Productive disposition is the
inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

Mathematically proficient students …

  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
  • Look for and express regularity in repeated reasoning.

Image from 123RF

Posted in Algebra, Math videos

Teaching Math with Mr Khan’s Videos – Variation

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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
  1. Show the video. It’s a short one so it will be over before your class will realise it’s math.
  2. 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.
  3. 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.
  4. 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. 
  5. 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. 
  6. 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?”
  7. 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?”
    • 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!

George Polya on thinking

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.

Posted in Teaching mathematics

George Polya’s Ten Commandments for Teachers

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1. Be interested in your subject.

2. Know your subject.

3. Know about the ways of learning: The best way to learn anything is to discover it by yourself.

4. Try to read the faces of your students, try to see their expectations and difficulties, put yourself in their place.

5. Give them not only information, but “know-how”, attitudes of mind, the habit of methodical work.

teaching math

Mathematical Discovery on Understanding, Learning, and Teaching Problem Solving

6. Let them learn guessing.

7. Let them learn proving.

8. Look out for such features of the problem at hand as may be useful in solving the problems to come – try to disclose the general pattern that lies behind the present concrete situation.

9. Do not give away your whole secret at once—let the students guess before you tell it—let them find out by themselves as much as feasible.

10. Suggest it, do not force it down your throats.

I got this from the plenary talk of Bernard Hodgson titled Whither the mathematics/didactics interconnection? at ICME 12, Korea, where he highlighted the important contribution of George Polya in making stronger the interconnection between mathematics and didactics and between mathematicians and mathematics educators.

If it’s too hard to commit the 10 commandments to memory then just remember the two rules below which is also from Polya. Combine it with Four Freedoms in the Classroom and you are all set.

George Polya on teaching math

How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library)