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Critical Thinking in High School Mathematics
An introduction
The Role of the Tutor.
The fulltime Mathematics teachers have as many a four or fivetimes the contact hours per student available to tutors (depending upon the number of tutortimeslots assigned to the student). The tutor, however, has the advantage of a oneonone relationship with the student and the possibility of achieving significant remedial intervention. Tutors are possibly most effective when they analyze and diagnose students and then act to remedy individual problems. Such remedial intervention often significantly improves a student's performance: On occasion the effect is dramatic. This article, after examining student responses to the recent exams, discusses Critical Thinking as a possibly useful intervention which is increasingly effective with harder problems.
Students Under Stress.
A week into the new (third) term I worked with a number of students, looking at their examination question papers and their answer scripts (in some cases). There was general agreement that the examinations were hard. I suspected this was intended, and conversations with the maths staff confirmed that they were meant to be hard. They are more than just “practice exams”: They push the students to their limits, and find out where those limits are. Not surprisingly they create substantial stress for all but the most talented students. The intention was that by the time students reach the year end exams the exam papers will hold few terrors and surprises. It is so important to learn the lessons revealed by the recent exams.
I have seen the exam papers and although they were difficult in parts there is NO REASON why all of my students could not have attained between 50% and 60% or more if they accessed and applied the knowledge I know they have. They had demonstrated their knowledge and ability to me under tutorial conditions on numerous occasions. However, in the examinations that ability was frequently not exhibited. Why?
The most likely cause is the highstress environment of the examination hall ("exam stress"). In tutorials the students oftentimes failed to understand problems and started to solve the them with insufficient or faulty information and methods. Once I told them that they were on the wrong track they restarted the solution and frequently carried it through to a correct solution. This bodes ill for an examination. Several colleagues have suggested that such failure to recall material previously taught was due to failure to practice similar problems sufficiently: Lack of work or laziness was suggested as the cause. While I cannot gainsay this, it still remains true that stress can impede memory access. It is well known that under severe stress, we act as our prehistoric ancestors did when confronted by danger as epitomised by the 'Sabretoothed Cat' Smilodon. This kitten Smilodon Populator at its maximum mass of 360 kg, stood 1.4 m at the shoulder and reached a length of about 2.6metres. The human reaction was reduced to "fight or flight?". Further thought was largely precluded. A very lifelike animation of Smilodon appeared in the film "10 000 BC". Similarly the reaction of some students to problems of significant difficulty is one of panic, and normal cognition ceases.
I recognized this problem during the second term and encouraged my students to pause before starting to solve a problem and ask themselves “What have we here?” I called this “thinking analytically”. Taking time to analyse a problem certainly allows the brain time to access knowledge and skills previously learned. With several students this approach was successful but it seems that the majority, once in the exam hall, forgot to pause and think and plunged into problems with insufficient forethought. I believe there is a tendency for students to believe that when they are asked a question they must respond immediately. Even one or two seconds of thought often produce a better response. Is it considered impolite to hesitate? I ask students for Pythagoras' Theorem and some immediately reply that "the length of the hypotenuse is equal to the sum of the lengths of the shorter sides". A further few seconds' thought frequently produces the correct answer.
In the context of academic assessment Critical Thinking is an attempt to add structure to this "thinking period".
Developing Critical Thinking for Science Students
Some years with iTutor Grahamstown tutoring high school maths and teaching what I was taught, namely that one must think analytically before attempting to solve a problem left me very dissatisfied.
I have asked myself why my “analytical thought” method frequently failed students under stress. I concluded, after some research, that it lacked formalism. Telling students during my extra maths sessions that they must think (analytically) before attempting a problem provoked the response "think what!". People prefer to have an idea expressed as a formal method rather than being left to implement the idea themselves. I found that Analytical Thinking works better when it follows Socrates' cognitive method now called Critical Thinking. Socrates and other philosophers in Antiquity formalized this approach in what has become known as Socratic Questioning. Socrates and his contemporaries used the technique to develop better reasoned arguments in the the art of rhetoric. In High school maths assessment the "dialog" is essentially between the student and the examiner.
Critical Thinking.
Much of the information in this section comes from
http://en.wikipedia.org/wiki/Critical_thinking
and
http://en.wikipedia.org/wiki/Socratic_method
and
from links from these pages.
Further information comes from documentation on the “Thinking Skills” subject offered by the Cambridge University Local Examination Syndicate.
Origins, Present and Future of Critical Thinking
Critical Thinking was developed in the 'West' by the Greek philosopher/teacher Socrates and his school. In Greek (and later in classical Roman) society rhetoric (the art of discourse or formal argument) was a highly prized skill taught in many schools: Critical Thinking aimed to enhance the quality of argument by formal questioning. In the 'East' similar development occurred in the Buddhist Kalama Sutta and the Abhidharma teachings: Here Critical Thinking was used to improve the quality of religious inquiry.
Critical Thinking and the Assessment Situation
The Socratic Method is most relevant to our present issue of addressing scientific problems posed in education. Here the discourse may be considered to be between the person who set the problem or question and the student who attempts to solve the problem or answer the question. However, since the examiner is rarely available to answer questions, it falls to the questioning student to supply the answers also! Ultimately the aim is to obtain the set of stimuli can which trigger the associative memory recall needed to access the previously learned knowledge and skills: Such recall will often enable the student to solve the problem.
At this point is well to note that the student can only usefully remember what s/he has carefully learned and assiduously practised. Critical Thinking is no substitute for hard work on the part of the student. Critical Thinking strives to prevent such prodigious effort coming to nought in the examination hall.
Socratic Questioning
Our questioning may be based on the following:

What to believe?

What to do?
Given that there is a problem at the stage where the student meets the question, how should s/he proceed to analyze the question? The Greek philosopher and teacher Socrates believed that systematic questioning was the key to developing a chain of reasoning. Can we develop a battery of questions to apply to a newly stated problem? Questions should reveal the fundamental nature of the problem (what to believe) which is frequently obscured in “word problems”: In the “pen and pencil” example, narrative obscured the algebraic essence of the problem which led to the solution. The answers to the formal questions posed should stimulate the recall from memory of possible methods for solving the problem (what to do)
In Socrates' method of questioning (Socratic Questioning) the questions are, Systematic, Disciplined and Deep.
Systematic: Our questions build on each other: Sometimes questions generate questions. Is the number Prime? If not what are the factors? If the factors are not obvious, is the number a product of primes?
Disciplined: We try to avoid distractions: We stay focussed and do not give up easily. Often when helping a student with a problem I see a way to use the problem in future teaching: My job is to help the student NOW: My creativity must stay on the sidelines until the immediate job is done.
Deep: We try to reach certainty about what the examiner requires and provides. Often a trigonometric equation is to be solved on a certain interval ( find { x: 0 < x < 180) }. After wrestling with the equation we may reach the General Solution and forget to apply the 'Initial Conditions'.
In the Cambridge International subject “Thinking Skills”, Critical Thinking is again proposed as an formal and efficient means of arguing productively. Amongst other skills they suggest

Recognize a reasoned argument: What is the examiner asking? Often a multipart question forms a reasoned and structured approach to solving a problem, culminating in a solution in the final part: The "Lune of Hippocrates" problem can be structured in this way.

Identify conclusions: What kind of result is the examiner looking for? Too many marks are lost by "Answering the Wrong Question".

Recognize implicit assumptions: The instruction "Answer correct to n decimal places" implies the use if a calculator.

Recognizing the logical function of key elements in the statement: First Derivative provides "the gradient at any point" on the graph of the primitive function

Understanding terminology used: 4x2 and (4x)2 are not equivalent!

Inference and Deduction:
 Inference: Logical conclusion from premises (axioms) known or assumed
to be true. A famous example:
All men are mortal
Socrates is a man
Socrates is mortal

But

True premises can lead to a false conclusion

All flying feathered creatures are birds (true)

Ostriches are birds (true)

Ostriches are flying feathered creatures (untrue)
Also,
For Real Numbers a, b, and c
a=c (take as a premise)
b=c (take as a premise)
a=b (Conclusion)
With acknowledgement to Euclid.
Also Abraham Lincoln and Daniel DayLewis (in the movie "Lincoln") who quoted Euclid.
"Things that are equal to the same thing are equal to each other"
This is extended generality and so strictly this example is Deduction.
 Deduction: An inference in which the conclusion is of no greater generality than the premise (first two examples above).
 Premise: A premise is a statement that can be argued to a conclusion. (An axiom is a selfevident premise).
To these I would add, Analyse and Evaluate the information provided. In geometry / trigonometry two sides of an nonright angled triangle and the included angle suggest the use of the Cosine Rule.
2 premises plus one conclusion are considered by some to be the basic unit of argument.
The first word or words words of each of the above skills suggest the types of thought processes required. Recognise, Identify, Understand, Infer, Deduce and Analyse.
Given a mathematical problem stated by an examiner or whoever, we question her/his statements even though we have to try to discover the answers within our selves.
Formalizing Critical Thinking
for Science Examinations:
The Tool
What to Believe?
What to do?
What to Believe?
It will help to write down the essence of the question, even if this is in the briefest of shorthand form. This is particularly important for “word” questions where the 'précis' or condensing process can make the problem manageable. In the case of geometry or trigonometry problems it really helps to copy the diagram from the question paper to your answer sheet. Enter all the given information in the diagram. If a graph is provided copy that and enter all information. This is also important for some Physics and Chemistry questions.
We need to recognise (or identify) the essentials of the question. What type of problem is it? Can we identify the question as being similar to previous questions which we have encountered and solved? If there are similarities and differences, note them. What information is supplied? Are there patterns in the supplied data? In multipart questions the parts themselves may reveal a critically thoughtthrough path to the final part of the question. Such questions can teach us the types of 'Socratic' questions to ask.
When we look at the memorandum for a question after failing to answer correctly, we can learn what questions we should have asked.
These questions should supply key words for our brain to use in tracking down related information.
What to do?
Hopefully, the stimuli provided by the preceding analytical section will have produced a number of ideas for solving the problem. Some may be incomplete and others may be inappropriate. Try to see where various ideas may lead. You may see non right angled triangles and decide to try the Cosine Rule. You have an angle and 2 sides. Are they the correct two sides? Is the angle the 'included' angle.
We may take guidance from military theory. Military problems are amorphous (of uncertain form) and constantly changing, at least in detail. Great leaders tended to think in terms of a 'master plan' (strategy) which addressed the essence of the problem and then a means of implementing the strategy, known as tactics.
In WW2 allied seaborne supply lines across the North Atlantic were threatened by enemy submarines. At one stage Britain's food stocks fell to less than a week's requirements. Allied strategy involved collecting supply ships into groups called convoys where they could be protected by the limited number of antisubmarine vessels available and also patrolling the sealanes with antisubmarine aircraft to keep the enemy submarines from surfacing to charge their propulsion batteries. Part of the Allied tactics involved breaking the enemy communications codes and hence tracking the submarines, improved submarine detection using soundbased “underwater radar” or SONAR and deploying small “escort”aircraft carriers which could provide air support in midAtlantic when ships were beyond the reach of landbased aircraft.
The Tool
The is the first attempt to put the above thoughts to work on actual test and examination questions
Using the Critical Thinking Tool (CTT)
 It is not necessary to write everything down but it will help to do so while you are getting used to making an effective CTT.
 If a diagram or a graph are involved always make your own sketch and add all the given data.
 After you solve the problem, update the tool you created in step one. This way you learn how to make better tools.
It is time to put the CTT to work
The Critical Thinking Tool
Examples of Applications
to South African High School Maths.
iTutor Grahamstown developed this tool from the ideas of Socrates (as presented by Wikipedia and others on the www), the ideas and enthusiasm of Mr Andrew Maffessanti at St Andrew's College in Grahamstown and the syllabus of the “Thinking Skills” subject offered by the Cambridge University Local Examination Syndicate.
Examples
Consider: Grades 10 to 12
The figure shows f(x) =4x, g(x) = 4x, h(x)=log4(x) and y=1(for interest)
Fact 1: f(x) = ax,
Fact 2: f and g are symmetrical about the y axis,
Fact 3: f and h are symmetrical about y=x
Fact 4: f passes through (1;4) (given).

Write down the equation of the graph of f.

How are f and g related?

How are f and h related?

Write down the equations of the graphs of g and h
Critical Thinking:
What to Believe? (What have we here?)
Graphs of similar shapes. Related to each other?? Asymptotes. Axes intercepts,
graph intercepts. Four Facts given.
What to do?
Strategy: Identify the graphs. Exponential? Hyperbolic? Log?
Find their interrelationships
Tactics: Write down the canonical forms of the graphs. Deduce using the graph and
otherwise the parameters of the graphs. Answer the questions.
******************************************************
Consider: Grades 8 and 9
Background: The Speed of a body expresses the distance that a body would
travel in a given time period.
A car traveling at 120 kph (kilometres per hour) will cover 120 kilometres
in one hour.
At 120 kph:

How long will it take to cover 240 km?

How long will it take to travel 60 kilometres?

How far will the car travel in one minute?
Critical Thinking:
What to Believe? (What have we here?)
A definition of speed. An example. The question has to do with proportion and ratio.
What to do?
Strategy: Use the definition and the example to determine a relationship between speed, distance and time.
Tactics: Use proportion and ratio to answer questions 1 to 3.
Consider: Grades 8 – 9 and higher
A bullet travelling horizontally at 600 metres per second will travel,
ignoring the effects of airresistance, gravity and breaking the soundbarrier,
600 metres in one second.

How long will it take the bullet to reach a target 1000 metres away?

If I double the constant speed to 1200 mps how long will it take the bullet to reach the target 1000m away?

Consider question 2. above. If the bullet is fired at sealevel and sound travels at 343.2 m/s at sea level, how long after the the bullet reaches the target will the sound of the gunshot reach the target?
Critical thinking:
What to believe? (What have we here?)
A similar set of problems to questions 1 – 3 on the preceding page. The units are different. Now (question 3, directly above) we have two moving entities,
the bullet and sound energy, moving through the air.
What to do?
Strategy: (question 6). We have two items traveling the same course at different constant speeds.
Tactics: Compute the travel times for the bullet and for the sound energy.
Compute the difference.
Consider: The Lune of Hippocrates Grade 8 and higher
Hippocrates died in 370BCE, 14 years after Aristotle was born. Hippocrates pursued one of the mathematical puzzles of his age; how to square the circle. How to construct (using only an ungraduated straight edge and a compass (the drawing variety) a square of the same area as a given circle. We now know that this task is impossible since the constant ? (pi) is a transcendental1 number (See EndNote1). A transcendental number is not rational. It cannot be expressed as the as a ratio of integers. One can divide a line in a number of ratios using a compass, but to construct a line of length ?(?)r in not possible. The Lune was part of a fruitless investigation but left us a delightful problem which may look difficult, but can in fact be solved with the knowledge taught in Grade 8.
Draw a circle of radius r. Use two radii with an included angle of 90 degrees to create a quarter circle. Join the ends of the radii to form a chord and an isosceles righttriangle. By construction or otherwise bisect the chord and use the midpoint to constrict a circle having the chord as diameter. The region of this small circle lying outside the large circle has the shape of a roughly quartercycle moon and is called a “lune”. Show that the area of the lune is equal to the area of the triangle.
Critical Thinking:
What to Believe? What have we here?
Circles and a triangle. A rightangled isosceles triangle. We are asked for an area, the area of the lune. Areas of circles and triangles are important.
What to do?
Strategy: If I can find the area of the minor segment AB, I can subtract this from the area of half the small circle and find the area of the Lune.
Tactics: Find the areas of the circles and the triangle. Quarter the area of the large circle; Find the area of the triangle and the length of its hypotenuse Halve the area of the small circle.
Possible question:

Calculate the area of the large circle in terms of its radius, r. (1)

Calculate the area of the triangle AOB in terms of (i.t.o.) r. (2)

Calculate the length of the hypotenuse of triangle AOB i.t.o. r (3)

Calculate the area of the small circle i.t.o. the radius of
the large circle, r. (1) 
Calculate the area of the minor segment AB i.t.o. r. (3)

Calculate the area of the Lune i.t.o. r. (4)
(14)
Now consider: Grades 11 and 12. Of interest to Grade 10 students.
Sin (2x) = Cos (3x)
Solve for x (General Solution)
Critical Thinking:
What to Believe: (What have we here?)
Sine and Cosine, no numbers. Two multiplied variables. Sine will cycle twice and
Cosine three times on 0<=x<=360. Expect 6 solutions.
Notice Cos(3x)=Cos(3x): It might be useful.
What to do?
Strategy:
Convert Cosine to Sine using complementary angle formula and solve.
Tactics: Develop solutions for Cos (3x) and Cos(3x)
sin (2x) = cos (3x)
Sin (2x) = Sin (903x)
2x = 90 – 3x
5x = 90 = RA
5x = 90 +360k k iamo Z
x = 18 + 72k k iamo Z
x= 18, 90, 162, 234, 306
Now make the argument of Cos negative
sin (2x) = cos (3x)
Sin (2x) = Sin (90+3x)
2x = 90 + 3x
x = 90 = RA
x = 90 + 360k k iamo Z
x = 90 (270)
Six Solutions ====>
1A transcendental number is not a root of a nonconstant polynomial
equation with rational coefficients
iTutor Grahamstown developed this tool from the ideas of Socrates (as presented by Wikipedia and others on the www), the ideas and enthusiasm of Mr Andrew Maffessanti at St Andrew's College in Grahamstown and the syllabus of the “Thinking Skills” subject offered by the Cambridge University Local Examination Syndicate.
Examples
Consider: Grades 10 to 12
The figure shows f(x) =4x, g(x) = 4x, h(x)=log4(x) and y=1(for interest)
Fact 1: f(x) = ax,
Fact 2: f and g are symmetrical about the y axis,
Fact 3: f and h are symmetrical about y=x
Fact 4: f passes through (1;4) (given).

Write down the equation of the graph of f.

How are f and g related?

How are f and h related?

Write down the equations of the graphs of g and h
Critical Thinking:
What to Believe? (What have we here?)
Graphs of similar shapes. Related to each other?? Asymptotes. Axes intercepts,
graph intercepts. Four Facts given.
What to do?
Strategy: Identify the graphs. Exponential? Hyperbolic? Log?
Find their interrelationships
Tactics: Write down the canonical forms of the graphs. Deduce using the graph and
otherwise the parameters of the graphs. Answer the questions.
******************************************************
Consider: Grades 8 and 9
Background: The Speed of a body expresses the distance that a body would
travel in a given time period.
A car traveling at 120 kph (kilometres per hour) will cover 120 kilometres
in one hour.
At 120 kph:

How long will it take to cover 240 km?

How long will it take to travel 60 kilometres?

How far will the car travel in one minute?
Critical Thinking:
What to Believe? (What have we here?)
A definition of speed. An example. The question has to do with proportion and ratio.
What to do?
Strategy: Use the definition and the example to determine a relationship between speed, distance and time.
Tactics: Use proportion and ratio to answer questions 1 to 3.
Consider: Grades 8 – 9 and higher
A bullet travelling horizontally at 600 metres per second will travel,
ignoring the effects of airresistance, gravity and breaking the soundbarrier,
600 metres in one second.

How long will it take the bullet to reach a target 1000 metres away?

If I double the constant speed to 1200 mps how long will it take the bullet to reach the target 1000m away?

Consider question 2. above. If the bullet is fired at sealevel and sound travels at 343.2 m/s at sea level, how long after the the bullet reaches the target will the sound of the gunshot reach the target?
Critical thinking:
What to believe? (What have we here?)
A similar set of problems to questions 1 – 3 on the preceding page. The units are different. Now (question 3, directly above) we have two moving entities,
the bullet and sound energy, moving through the air.
What to do?
Strategy: (question 6). We have two items traveling the same course at different constant speeds.
Tactics: Compute the travel times for the bullet and for the sound energy.
Compute the difference.
Consider: The Lune of Hippocrates Grade 8 and higher
Hippocrates died in 370BCE, 14 years after Aristotle was born. Hippocrates pursued one of the mathematical puzzles of his age; how to square the circle. How to construct (using only an ungraduated straight edge and a compass (the drawing variety) a square of the same area as a given circle. We now know that this task is impossible since the constant ? (pi) is a transcendental1 number (See EndNote1). A transcendental number is not rational. It cannot be expressed as the as a ratio of integers. One can divide a line in a number of ratios using a compass, but to construct a line of length ?(?)r in not possible. The Lune was part of a fruitless investigation but left us a delightful problem which may look difficult, but can in fact be solved with the knowledge taught in Grade 8.
Draw a circle of radius r. Use two radii with an included angle of 90 degrees to create a quarter circle. Join the ends of the radii to form a chord and an isosceles righttriangle. By construction or otherwise bisect the chord and use the midpoint to constrict a circle having the chord as diameter. The region of this small circle lying outside the large circle has the shape of a roughly quartercycle moon and is called a “lune”. Show that the area of the lune is equal to the area of the triangle.
Critical Thinking:
What to Believe? What have we here?
Circles and a triangle. A rightangled isosceles triangle. We are asked for an area, the area of the lune. Areas of circles and triangles are important.
What to do?
Strategy: If I can find the area of the minor segment AB, I can subtract this from the area of half the small circle and find the area of the Lune.
Tactics: Find the areas of the circles and the triangle. Quarter the area of the large circle; Find the area of the triangle and the length of its hypotenuse Halve the area of the small circle.
Possible questions?
1. Calculate the area of the triangle AOB in terms of (i.t.o.) r.
2. (Calculate the length of the hypotenuse of triangle AOB i.t.o. r
 Calculate the area of the small circle i.t.o. the radius of the large circle, r. 1)

Calculate the area of the minor segment AB i.t.o. r. (3)

Calculate the area of the Lune i.t.o. r. (4)
(14)
Now consider: Grades 11 and 12. Of interest to Grade 10 students.
Sin (2x) = Cos (3x)
Solve for x (General Solution)
Critical Thinking:
What to Believe: (What have we here?)
Sine and Cosine, no numbers. Two multiplied variables. Sine will cycle twice and
Cosine three times on 0<=x<=360. Expect 6 solutions.
Notice Cos(3x)=Cos(3x): It might be useful.
What to do?
Strategy:
Convert Cosine to Sine using a complementary angle formula and solve.
Tactics: Develop solutions for Cos (3x) and Cos(3x)
sin (2x) = cos (3x)
Sin (2x) = Sin (903x)
2x = 90 – 3x
5x = 90 = RA
5x = 90 +360k k iamo Z
x = 18 + 72k k iamo Z
x= 18, 90, 162, 234, 306
Now make the argument of Cos negative
sin (2x) = cos (3x)
Sin (2x) = Sin (90+3x)
2x = 90 + 3x
x = 90 = RA
x = 90 + 360k k iamo Z
x = 90 (270)
Six Solutions ====>
1A transcendental number is not a root of a nonconstant polynomial
equation with rational coefficients
Critical Thinking  Useful Texts
An Excellent History of Critical Thinking
{ Taken from the California Teacher Preparation for Instruction in Critical Thinking: Research Findings and Policy Recommendations: State of California, California Commission on Teacher Credentialing, Sacramento, CA, March 1997. Principal authors: Richard Paul, Linda Elder, and Ted Bartell }