Developing Patient Problem Solvers

Developing Patient Problem Solvers

Differentiation Card Game

Differentiation Card Game

In secondary school calculus, one must be proficient in differentiating algebraic expressions. One way for students to practise the power rule and get them to learn from one another is to use a card game. I have tried this last year with my sec 4 students.

Basically, the differentiation card game consists of a stack of cards and a group of players. I recommend about 50 cards and about 4 players so that each player can have 10 or more cards. It would also be good to form the group of mixed ability so that each group has at least someone who has understood the concept of power rule.

Each card consists of an algebraic expression. For instance, I have (painstakingly) printed cards with polynomial expressions such as 1, x, x^2 and so on. besides these, we can have more complicated cards such as x^2-x+1, 1/x, sqrt(x) and so on. The objective of the game is for players to take turns to throw out one card at a time, in the order of derivative. For example, Suppose player A throws x^2-x. The next player B, will have to throw 2x-1. If player B does not have this card, he/she misses his/her turn and this will continue until someone has this card. The deck can be designed such that each set of derivatives is unique except possibly having more than one "0". The player who disposes all of his/her cards correctly wins. Once "0" is reached or once no one has a card to throw, the last player who threw his/her card will have control over the game. He/she can throw what ever card he/she prefers. Hence there is obvious strategy in the game.

At the end of the game, it would be good for the teacher to do some consolidation. What do you notice about the derivative as we progress? Students will notice that polynomial expressions will eventually have derivative 0, while rational functions such as 1/x will continue indefinitely. Such thinking and reflection will help students understand power rule better. Without a firm understanding of this basic rule, students would have problems with the other rules, such as product rule and quotient rule.

Moreover, one can extend this game by including exponential and logarithmic expressions.

Of course, we will need a few e^x in the deck!

Generalising Patterns using Multiple Routes

A core focus of secondary mathematics is for students to develop inductive and deductive reasoning. In particular, this is emphasised in lower secondary mathematics where number patterns are taught. In this chapter, students learn to depict patterns using algebraic expressions and attempt to generalise. This is one route to develop inductive reasoning.

One of the central themes of the book “Developing Thinking in Algebra” (Mason, J, Graham,A, Johnston-Wilder, S, 2005) is the idea of expressing generality. Generality can be expressed in multiple ways which can help students develop algebraic thinking by approaching problems in various ways. Students often only face number pattern problems through a single approach. Here are some examples of problems that can be approached in a variety of ways:

Generalising picture sequences in the reverse manner.

A common number pattern task is to have a series of pictures and then formulate an expression for the nth picture. For instance, take a picture sequence made of sticks, perhaps in the following:

I found this example meaningful as there are many ways of finding number patterns. For instance, what is the number of sticks required to build the roof in the nth house? What is the number of sticks required for perimeter of the nth house? These questions should not be given to the students. Instead, students should come up with their own questions and try to determine the number patterns on their own. This will enable them to develop metacognition through “pattern sense-making”.

Indeed, the reverse process is even more interesting and challenging. For instance, suppose we have the sequence 2+(n-1)+(1+2n+1) and 3(n+1) for the nth house. Find ways to build structures of houses using these expressions. As mentioned by the author, this process helps to support flexibility in algebraic thinking as the student will need to interpret the algebraic expression before visualising the structure.

Cube Painting

A cube is painted on all faces. It is then cut up into 27 equal cublets by two plan cuts parallel to each face, (imagine a 1 color Rubik Cube). How many cublets are painted on how many faces? Generalise the problem.

Cube painting is but one way to expose students to inductive reasoning which is not so direct. Students may need to utilise their problem solving skills such as making a list (Creating a table for a two by two cube first, then by a three by three cube) and working backwards in order to attain the answer. Moreover, there are no clear steps to generalise the problem. Students must first think of how to extend this problem. For example, besides cutting the cube into 27 cublets, what is the next smallest number of cublets can be cut using the same operation? What if more than one color is used? If 6 different colors were to be used, like in a Rubik Cube, how many faces would have 3 colors? How many would have two? There are many questions that can be asked to extend this problem.

In summary, number pattern activities listed above require students to apply problem solving skills, metacognition, inductive and deductive reasoning. This will support students’ thinking in algebra.

Algebra and moderation

Here is a simple way to assess students' understanding of algebraic functions. Suppose you gave a test and you are about to return them the paper. Ask them this:

Let's say your marks for this paper is x, and you are giving a choice to change your makes by using a certain function. Which function will you choose? You are only given one choice!

1. x+5
2. max(60, x+10)
3. 100-x
4. 10sqrt(x)
5. 20ln(x)

Which one would you choose and in what situation would you choose each of them?

Of course you can extend this problem into a two variable function. Suppose paper 1 is x and paper 2 is y. Which one would you choose as your marks?

1. (x+y)/2
2. sqrt(x^2+y^2)
3. sqrt(xy)
4. max(x , y)

You can come up with many other functions to assess the students' higher order thinking skills.

Activities for TI Graphing Calculators

The following website holds many mathematics activities using Texas Instruments (TI) GCs as pedagogical tools.
You can find the following topics in this website:

Algebra, Advance Algebra, Geometry, Pre-Calculus (ie Binomial Theorem, Trigonometry) and Calculus.

URL is http://www.timath.com/algebra1/archive/ti84-plus

Besides support for TI84, there seems to be more activites for TI inspire as well. TI inspire is also another great pedagogical tool for the classrooom. For more information on TI GCs, visit the following URL:

http://education.ti.com/educationportal/sites/US/productDetail/us_ti84p.html

3 Levels in Mathematics Education

In a recent mathematics conference, I had the honour of attending a talk by Japanese educators. Prof Yoshinori Shimizu shared about the different stages of mathematical development and how students can master a particular topic. In particular, we can structure learning that is interesting and insightful. The following are 3 levels where we can structure a mathematic topic:

Level 1: 守 (Shu) Follow the form.

Following the form requires the student to apply standard techniques to standard questions that are probably similar to what they have encountered.

Level 2:破 (Har) Break the form.

Breaking the form requires the student to apply standard techniques to solve problems that they have not encountered before. This requires students to apply higher order thinking skills. Moreover, they must possess creativity and problem solving skills in order to apply prior knowledge to a new situation. This will assess their metacognition and habits of mind.

Level 3: 离 (Ri) Extend away from the form.

This may require students to extend what they have learnt to deeper concepts. For instance, if they have only learnt how to solve quadratic equations, students can question how they can extend this to cubic or quartic equations? Is there a standard formula to solve quartic or quintic as well? Also, another view is how students can extend what they have learnt to other disciplines and how the concepts can be applied to authentic situations.

In many cases, teachers often complain that students are competent to solve math problems that they have done before. However, when faced with new problems, students either avoid the question or simply raise their hands in defeat. By structuring lessons in the above 3 forms, students should be able to master their content using problem solving skills.

"Set" Card Game

I came across this card game called "Set" which is used for cooperative learning in class. It is a simple card game where a group of students try to identify sets of cards from a deck with different attributes, such as shape and color. A "set" of cards is one where either each attributes is the same or different.

Try this Flash tutorial to understand the mechanics of the game.
http://www.setgame.com/set/index.html

There is alot of potential in this game for teaching and learning in the following areas:

Permutation and Combination
Eg: What is the total number of sets that can be formed from the deck?
Set Theory
Eg: If A is the set of red cards and B is the set of diamond shaped cards, what does the set A union B mean?
Probability
Eg: What is the probability that the game will end with 3 cards?
Problem Solving
Eg: What is the maximum number of cards that can be picked before a set must be found?

More information can be found in the following two URLs:
http://www.setgame.com/set/article_nctm.htm
http://www.setgame.com/set/index.html

Whatcha Know About Math?

Kinematics and Calculus Made Easy

Geometry from the Incas

This is one of the most comprehensive sets on geometry that I 've seen. It contains dozens of examples of different results in geometry from Marion Walter's Theorem to Menalus' Theorem. A wonder site for exploration in the world of Geometry.


The URL is
http://www.gogeometry.com/index.html

A paradox in logic

Mr Lee, the teacher, announces to the class that there will be a surprise test next week. It can occur on any day from Monday to Friday afternoon. Alex, the smartest student in the class, stood up and pointed out the following to Mr Lee.


"Mr Lee"

"Yes, Alex?"

"Well there can't be a spring test on any day next week."

"Why not Alex?"

Alex begins to explain his argument:

"Let's assume you are telling the truth and the test is indeed a surprise. If there is no test by Friday morning, then it is no surprise that the test will be on Friday afternoon. Hence the test cannot occur on Friday, so it must occur between Monday and Thursday. However, if the test does not occur on Friday, then we would have known by Thursday morning that the test must be on Thursday afternoon, which again is no surprise. After ruling out Thursday and Friday, we can apply a similar argument to the rest of the days. Hence there cannot be a surprise test from Monday to Friday"

"Very well Alex, we shall see how it goes next week."

In the following week, a test was given out on Wednesday when the students did not expect it. It was indeed a surprise test.

Analyse Alex's argument using Paul's Wheel of Reasoning:

1. What is the problem in the question?
2. What information is given by Mr Lee?
3. What are the assumptions made by Alex?
4. What are the implications and consequences of his assumptions?
5. Whose perspectives should be considered?
6. What are the main concepts in this paradox?