using counter-examples in teaching & learning of … teaching-research journal online ... using...

MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE VOL 5, N 4 December 2012

1

Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other

uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York. www.hostos.cuny.edu/departments/math/mtrj

Using Counter-Examples in Teaching & Learning

of Calculus: Students’ Attitudes and Performance

Sergiy Klymchuk

Auckland University of Technology, New Zealand


2



Abstract

The paper deals with the usage of counter-examples in teaching and learning of

calculus as a pedagogical strategy. It reports on two studies. The first study

investigates students’ attitudes towards this pedagogical strategy. It is an international

study of more than 600 students from 10 universities in different countries. The

second study deals with students’ performance. It is a case study of 25 students from

a New Zealand university. An example from teaching practice is presented.

Key words: calculus, counter-examples, teaching, learning


3



Introduction and Frameworks Deciding on an assertion’s validity is important in the information age. A counter-

example can quickly and easily show that a given statement is false. One counter-

example is enough to disprove a statement. Counter-examples thus offer powerful and

effective tools for mathematicians, scientists and researchers. They can indicate that a

hypothesis is wrong. Before attempting to find proof for an assertion looking for counter-

examples may save an investigator lots of time and effort.

The search for counter-examples has been important in the history of mathematics.

Below are three famous instances. For a long time mathematicians tried to find a formula

which generated only prime numbers. Numbers of the form 122 n

, where n is a natural

number were once believed to all be prime, until Euler found a counter-example. He

showed that for n = 5 that number is composite: 67004176411252 .

Another conjecture about prime numbers is still awaiting proof or disproof. The

Goldbach-Euler conjecture, posed by Goldbach in a 1742 letter to Euler, looks


4



deceptively simple: every even integer greater than 2 is the sum of two prime numbers.

For example, 12 = 5 + 7, 20 = 3 + 17, and so on. Powerful computers were used to search

for possible counter-examples and they found none among 4, 6,…, 14104 (the latest

published result - Richstein, 2000). In 2000 the book publishing company Faber & Faber

offered a US$1 million prize to anyone who could prove this conjecture or disprove it by

a counter-example within two years. The prize went unclaimed.

In 1861 the great German mathematician Weierstrass constructed his famous counter-

example – the first known fractal – to the statement If a function is continuous on (a,b),

then it is differentiable at some points on (a,b). Many mathematicians at that time thought

that such ‘monster-functions’ that were continuous but not differentiable at any point

were absolutely useless for practical applications. About a hundred years later Norbert

Wiener, the founder of cybernetics pointed out in (1956) that such curves exist in nature –

for example, they are trajectories of particles in the Brownian motion. In recent decades

such curves were investigated in the theory of fractals –

a fast growing area with many applications.

Creating examples and counter-examples is neither algorithmic nor procedural and

requires advanced mathematical thinking which is not often taught at school (Selden &

Selden, 1998; Tall, 1991; 2001). Selden and Selden (1998) argued that “coming up with

examples requires different cognitive skills from carrying out algorithms – one needs to

look at mathematical objects in terms of their properties. To be asked for an example can

be disconcerting. Students have no prelearned algorithms to show the correct way”.


5



Many students nowadays are used to concentrate on techniques, manipulations, familiar

procedures and don’t put much attention to the concepts, conditions of the theorems,

properties of the functions, and to reasoning and justifications. Students often rely on

technology and sometimes logical thinking and conceptual understanding go

undeveloped in traditional teaching. Sometimes mathematics courses, especially at

school level, are taught in such a way that special cases are avoided and students are

exposed only to ‘nice’ functions and ‘good’ examples. This approach can create many

misconceptions that can be explained by the Tall’s generic extension principle: “If an

individual works in a restricted context in which all the examples considered have a

certain property, then, in the absence of counter-examples, the mind assumes the known

properties to be implicit in other contexts.” (Tall, 1991).

Many researchers and practitioners promote practice in constructing examples and

counter-examples as one of the best ways to learn mathematics. “Since success in

mathematics, especially at the advanced undergraduate and graduate levels appears to be

associated with the ability to generate examples and counter-examples, what is the best

way to develop this ability? One suggestion . . . is to ask students at all levels to “give

me an example of . . .”. Granted the inherent epistemological difficulties of finding

examples for oneself, are we, in a well-intentioned attempt to help students understand

newly defined concepts, ultimately hobbling them, by providing them with predigested

examples of our own? Are we inadvertently denying students the opportunity to learn to

generate examples for themselves?” (Selden & Selden, 1998).


6



“Deliberate searching for counter-examples seems an obvious way to understand and

appreciate conjectures and properties more deeply. Such a search could be within the

current example space or could promote extension beyond...In our view, learners will

inevitably encounter nonexamples of concepts and counterexamples to conjectures if they

are actively exploring and constructing their own spaces.” (Watson & Mason, 2005, p.

67). “Extreme examples…confound our expectations, encourage us to question beyond

our present experience, and prepare us for new conceptual understandings.” (Watson &

Mason, 2005, p. 7). Lakatos (1976) argued that the more students practice in creating

their own counter-examples the more likely they see them as infinite classes of examples

rather than isolated and irrelevant pathological cases. Peled and Zazlavsky (1997)

classified counter-examples into specific, semi-general and general depending on the

extent to which they provide an insight how to construct similar counter-examples or

generate an entire counter-example space. It seems obvious that the bigger collection of

examples for illustration the easier to use them as or adjust them to counter-examples for

disproving incorrect statements. Zazlavsky and Ron (1998) indicated that “students’

understanding of the role of counter-examples is influenced by their overall experiences

with examples.” Zazkis and Chernoff (2008) also suggested that “the convincing power

of counterexamples depends on the extent to which they are in accord with individuals’

example spaces.”


7



In the first study, the theoretical framework was based on Piaget’s notion of cognitive

conflict (1985) and the notion of ‘pivotal-bridging example’ introduced by Zazkis and

Chernoff (2008). The goals of the study were to investigate students’ attitudes towards

the usage of counter-examples in teaching and learning of calculus and analyse the effect

of cognitive conflict and how students resolve it from the point of view of the notion of

the pivotal-bridging example. A cognitive conflict arises when a student encounters

information containing some sort of contradiction or inconsistency with their own ideas

or experience. However, as Zazkis and Chernoff pointed out “inconsistency of ideas

presents a potential conflict, it becomes a cognitive conflict only when explicitly invoked,

usually in an instructional situation” (2008, p.196). Many studies found conflict to be

more effective than direct instruction, in particular when dealing with students’

misconceptions (Bell, 1993; Ernest, 1996; Irwin, 1997; Swedosh & Clark, 1997;

Klymchuk, 2001; Watson, 2002). Swedosh and Clark (1997) successfully used conflict in

their intervention method to help undergraduate students eliminate their misconceptions:

“the method essentially involved showing examples for which the misconception could

be seen to lead to a ridiculous conclusion, and, having established a conflict in the minds

of the students, the correct concept was taught” (p.493). Another study by Horiguchi and

Hirashima (2001) used a similar approach in creating discovery learning environment in

their mechanics classes. They showed counter-examples to their students and considered

them as a chance to learn from mistakes. They claim that for counter-examples to be

effective they “must be recognized to be meaningful and acceptable and must be

suggestive, to lead a learner to correct understanding” (Horiguchi & Hirashima, 2001).


8



Mason and Watson (2001) used a method of so-called boundary examples to help

students create examples to correct statements, theorems, techniques, and questions that

satisfied given conditions. They noticed that “when students come to apply a theorem or

technique, they often fail to check that the conditions for applying it are satisfied. We

conjecture that this is usually because they simply do not think of it, and this is because

they are not fluent in using appropriate terms, notations, properties, or do not recognise

the role of such conditions” (Mason & Watson, 2001). In this study, the students were

asked to create counter-examples to incorrect statements, i.e. the students themselves

established a conflict in their mind. However, as Zazkis and Chernoff (2008) pointed out

“researchers in mathematics education are well aware that learners may possess

contradicting ideas without facing or acknowledging a conflict. As such, a

counterexample, when presented to a learner, may not create a cognitive conflict; it may

be simply dismissed or treated as exception.” (p.197). The notion of ‘pivotal-bridging

example’ (Zazkis & Chernoff, 2008) was employed to analyze how the students resolved

their conflict (exposed by a lecturer or created by themselves). “An example is pivotal for

a learner if it creates a turning point in the learner’s cognitive perception or in his or her

problem solving approaches; such examples may introduce a conflict or may resolve it.

… When a pivotal example assists in conflict resolution we refer to it as a pivotal-

bridging example, or simply bridging example, that is, an example that serves as a bridge

from learner’s initial (naïve, incorrect or incomplete conceptions) towards appropriate

mathematical conceptions.” (p.197).


9



In the second study, practice was selected as the basis for the research framework and it

was decided “to follow conventional wisdom as understood by the people who are

stakeholders in the practice” (Zevenbergen & Begg, 1999). It was a case study on

performance of two groups of students – one group was exposed to the usage of counter-

examples on a regular basis over 8 weeks and the other group did not have that kind of

activity. Results of the two groups in a mid-semester test were compared. The goal of the

study was to check how usage of counter-examples affected students’ performance on a

test question that required conceptual understanding.

The First Study

In this study, 10 lecturers from 10 universities in different countries used counter-

examples in teaching calculus for the first 9 weeks of the first semester. The students

were first-year undergraduate students majoring in science or engineering. The lecturers

were selected using a combination of convenience and judgement sampling methods.

The countries and number of participated universities were (in an alphabetical order):

Germany (1), New Zealand (1), Poland (1), Russia (1), Spain (1), The Netherlands (1),

Ukraine (2), USA (2). An accross-countries approach was chosen to reduce the effect of

differences in education systems, curricula and cultures. After 9 weeks of using counter-

examples in teaching and learning of calculus a questionnaire was distributed among

students. The total number of students exposed to the usage of counter-examples over

the 9 week period in all 10 univesities was approximately 860. Responses to the


10



questionnaire from 612 students were received so the response rate was 71%. The

participation in the study was voluntary - it was a self-selected sample. Data was

collected in each university separately and sent to the author with translation where

needed.

Dealing with counter-examples for the first time was challenging for some students.

When they heard they could disprove a wrong statement by providing one counter-

example they thought they could “prove” a correct statement by showing an example.

Even if they knew they could not prove a theorem by providing only examples, it was

hard for them to accept the fact that a single counter-example disproves a statement.

Some students believed that a particular counter-example was just an exception to the

rule and that no other ‘pathological’ cases existed. Selden and Selden have articulated

these ideas in (1998): “Students quite often fail to see a single counter-example as

disproving a conjecture. This can happen when a counter-example is perceived as ‘the

only one that exists’, rather than being seen as generic.” A similar observarion was

reported by Zaslavsky and Ron in (1998): “Students often feel that a counter-example is

an exception that does not really refute the statement in question”. (p.4-231).

With experience though students understood the role of counter-examples and became

interested in creating them. In developing counter-examples students were forced to pay

attention to every detail in a statement – the word order, the symbols used, the shape of


11



brackets defining intervals, whether the statement applies to a point or to an interval, and

so on. Let us consider the following theorem from first-year calculus as an example that

was discussed with students:

“If a function f(x) is differentiable on (a,b) and its derivative is positive for all x in (a,b),

then the function is increasing on (a,b)”.

The students were asked to disprove the following two statements that look quite similar

to the above theorem but both are incorrect:

If a function f(x) is differentiable on (a,b) and its derivative is positive at a point

x = c in (a,b), then there is a neighborhood of the point x = c where the function is

increasing.

If a function f(x) is differentiable on its domain and its derivative is positive for all x

from its domain, then the function is increasing everywhere on its domain.

All incorrect statements given to the students were within their knowledge and often

related to their common misconceptions. Below are some more examples of such

incorrect statements that were discussed with the students:

With a continuous function, i.e. a function which has values of y which smoothly and

continuously change for all values of x, we have derivatives for all values of x.


12



If the first derivative of a function is zero then the function is neither increasing nor

decreasing.

At a maximum the second derivative of a function is negative and at a minimum

positive.

The tangent to a curve at a point is the line which touches the curve at that point but

does not cross it there.

Interestingly enough the above four statements are quotations from some textbooks on

calculus at upper secondary and university level. In this paper the issue of mistakes in

textbooks and their effect on students learning mathematics will not be discussed. There

are two reasons why this fact is mentioned here. The first reason is the importance of

enhancing students’ ability to critically analyse any information - not only printed in

newspapers but also in mathematics textbooks. The second reason is the usage of such

textbooks as a good educational resource – give students a task to find incorrect

statements on given pages and create counter-examples to them. Teaching experience

shows that such an activity boosts their confidence a lot.

The following questionnaire was distributed among students after 9 weeks of using

counter-examples in teaching and learning of calculus as a pedagogical strategy.

Questionnaire

Question 1. Do you feel confident using counter-examples?


13



a) Yes Please give the reasons:

b) No Please give the reasons:

Question 2. Do you find this method effective?



Question 3. Would you like this kind of activity to be a part of assessment?



Students’ responses to the questionnaire are summarized in the table below:

Number

of Students

Question 1

Confident?

Yes No

Question 2

Effective?

Yes No

Question 3

Assessment?

Yes No

612

116 496 563 49 196 416

100%

19% 81% 92% 8% 32% 68%

Table 1. Summary of findings from the questionnaire.


14



The majority of the students (81%) were not familiar with the usage of counter-examples

as a method of disproof. The common comments from the students who answered ‘No’ to

question 1 on confidence were as follows:

I have never done this before; I am not familiar with this at all; I am not used to this

method; this method is unknown to me; we did not learn it at school; I heard about it but

not from my school teacher; we hardly created ourselves any examples at school.

Question 2 was the main question of the questionnaire. Obviously such notions like

‘cognitive conflict’ or ‘pivotal-bridging example’ could not be used in the questionnaire.

Instead the lecturers explained the students what was meant by “effective method”: it

makes you think about some aspects of mathematics that you never thought before, it

opens your eyes on the importance of conditions of rules/theorems and properties of

functions, it reveils your misconceptions, it forces you to pay attention to every detail, it

enhances your understanding of mathematical concepts. The vast majority of the students

(92%) reported that they found the method of using counter-examples to be effective. The

common comments from the students who answered ‘Yes’ to question 2 on effectiveness

were as follows:

Helps me to think question deeply; gives more sound knowledge of the subject; we can

understand more; it makes me think more effectively; can prevent mistakes; you gain a

better understanding; it makes the problem more clear; it boosts self-confidence; it helps


15



you retain information that you have learned; it is a good teaching tool; it teaches you to

question everything; it makes you think carefully about the concepts and how they are

applied; it makes you think critically; it supports self-control; it requires logical thinking,

not only calculations; makes problems more understandable; it is hard but it is fun; it is a

good way to select top students; I can look at maths from another angle; it is good not

only in mathematics; it really forces you to think hard; it is not a routine exercise, it is

creative.

From the subsequent interviews with selected students and discussions with several

collaborators it was revealed that showing a counter-example to an incorrect statement by

a lecturer not always created a cognitive conflict for a student. In cases when it did create

a conflict it sometimes remained unresolved. Some guidance on how to create counter-

examples was helpful for conflict resolution. Typical questions were as follows:

How can you change the statement to make it correct?

What can you change in the given function and still have it as a counter-example?

Which other wrong statements can your particular counter-example refute?

Can you use another type of function as a counter-example?

Can you construct the most general class of counter-examples?

Using the notion of pivotal-bridging examples (Zazkis & Chernoff, 2008) it can be stated

that for the majority of the students counter-examples shown by a lecturer or constructed

by themselves were pivotal examples in the sense they created “a turning point in the


16



learner’s cognitive perception or in his or her problem solving approaches”. (p.197). In

cases when counter-examples helped the students eliminate their misconceptions and

resolve their conflict they became bridging examples in a sense they served as “as a

bridge from learner’s initial (naïve, incorrect or incomplete conceptions) towards

appropriate mathematical conceptions” (p.197).

About two thirds of the students (68%) did not want that the questions on creating

counter-examples to incorrect statements would be part of assessment in contrast to the

trends pointing to the effectiveness of the method (92%). The common comments from

the students who answered ‘No’ to question 3 on assessment were as follows:

It is hard; never done this stuff before; confusing; not trained enough; complicated; not

structured; not enough time to master it; you don’t know how to start; can affect marks.

The last comment was the most common. Those students were more concerned about

their test results rather than acquiring useful skills. Apparently their attitudes towards

learning were not mature yet. The students who answered ‘Yes’ (32%) provided

comments similar to those made on the effectiveness of the method. The common

comments from the students who answered ‘Yes’ to question 3 on assessment were as

follows:


17



It provokes generalised thinking about the nature of the processes involved, as compared

to the detail of the processes; better performance test; it shows full understanding of

topic; a good way to test students’ insight; it is an extremely valuable skill; it is good to

have it in assessment otherwise we will not put much attention to it; it is challenging and

I like it; one can use this method outside university.

The Second Study

The second study was a case study. Two groups of students from the Auckland

University of Technology, New Zealand were selected for the study. The students were

majoring in science and engineering. In group A there were 14 students and in group B

(the control group) there were 11 students. All the students in both groups had similar

age and mathematics background and all were Chinese. There were 3 lectures and 1

tutorial per week in both groups. In both groups the lecturer was the same. The only

difference was - in group A one of the 3 lectures a week was given by another lecturer.

That lecturer used counter-examples in his lectures for 5-6 minutes during a 50 minute

lecture. There were 1-2 examples per lecture. There were 8 weeks before the mid-

semester test so there were 8 lectures in which counter-examples were used. The total

time for that activity over the 8 week period was about 45 minutes. After 8 weeks of

study both groups were taking the same mid-semester test that contained 11 questions:


18



the first 10 questions were on techniques and Question 11 below was on conceptual

understanding.

Question 11. Sketch a graph of a function whose graph is a continuous and smooth curve (no sharp corner) at a point but which is not differentiable at that point. What was expected from the students was a simple sketch:

Figure 1. An expected answer to Question 11. Or the students could come up with a cube root function:

420-24

1.5

1

0.5

0

-0.5

-1

-1.5

-2

420-24

1.5

1

0.5

0

-0.5

-1

-1.5

-2

420-24

1.5

1

0.5

0

-0.5

-1

-1.5

-2

420-24

1.5

1

0.5

0

-0.5

-1

-1.5

-2

Figure 2. Another possible answer to Question 11. The results of the test and Question 11 are in the table below: Passed the test Correct answer to

Question 11 Group A 13/14 = 93% 11/14 = 79%

3 xy


19



Group B 10/11 = 91% 5/11 = 45%

Table 2. Students’ results of the mid-semester test. The overall performances of the 2 groups of students on the test were very similar: 93%

of the students in group A and 91% in group B passed the test (received more that 50% of

the total marks). But there was a difference in giving the correct solution to Question 11:

79% of the students in group A versus 45% in group B. This might indicate that the

regular usage of counter-examples in group A over the 8 week period improved students’

conceptual understanding so they performed better in

Question 11.

Conclusions and Recommendations

The overwhelming statistics and numerous students’ comments from the first study

showed that the students were very positive about the usage of counter-examples in

learning and teaching of calculus. The vast majority of them (92%) reported that they

found the method of using counter-examples to be effective in the sense that it

invoked one or more of the following: it made them think about some aspects of

mathematics that they never thought before, it opened their eyes on the importance of

conditions of rules/theorems and properties of functions, it reveiled their

misconceptions, it forced them to pay attention to every detail, it enhanced their

understanding of mathematical concepts. According to (Zazkis & Chernoff, 2008) for

those students counter-examples were pivotal examples in the sense they created “a


20



turning point in the learner’s cognitive perception or in his or her problem solving

approaches” or bridging examples in the sense they served as “as a bridge from

learner’s initial (naïve, incorrect or incomplete conceptions) towards appropriate

mathematical conceptions”. (p.197). Many students made comments that the method

of using counter-examples helped them to understand concepts better, prevent

mistakes in future, develop logical thinking and question everything. A big proportion

of the students commented that creating counter-examples were closely connected

with enhancing their critical thinking skills. These skills are general ones and can be

used by the students in other areas of their life that have nothing to do with

mathematics. The ability to create counter-examples is an important instrument of

critical selection in the broader sense. Henry Perkinson, the author of the famous

book Learning from our Mistakes (1984), writes about the importance of those skills

for his theory of education (2002): “Our knowledge is imperfect, it can always get

better, improve, grow. Criticism facilitates this growth. Criticism can uncover some

of the inadequacies in our knowledge, and when we eliminate them, our knowledge

evolves and gets better…Education is a continual process of trial-and-error

elimination. Students are fallible creators who make trial conjectures and formulate

trial skills and then eliminate the errors uncovered by criticism and critical selection”.

The results of the second study indicated that the usage of counter-examples might

improve students’ conceptual understanding. Apart from that study there was another


21



indication from the same university that the usage of counter-examples was benefitial to

the students. On a routine calculus course evaluation by the students in the end of the

course the following statement was added to the form (the statement was formulated in

the same style as other statements in the evaluation form): The usage of counter-examples

in class helped me to learn. There were 54 students on that day in class who filled the

evaluation form. 52 of them (96%) ticked either ‘Agree’ or ‘Strongly Agree’ box for the

above statement.

The findings from both studies might encourage lecturers and students to use counter-

examples in teaching and learning of calculus with these goals:

to deepen conceptual understanding

to reduce or eliminate common misconceptions

to advance one’s mathematical thinking beyond the merely procedural or algorithmic

to enhance critical thinking skills – analyzing, justifying, verifying, checking, proving

– to the benefit of students in other areas of their lives

to expand a student’s ‘example space’ (Watson & Mason, 2005) of functions with

interesting properties

to make learning more active and creative.

There are many ways of using counter-examples as a pedagogical strategy:


22



giving the students a mixture of correct and incorrect statements

asking students to create their own wrong statements and counter-examples to them

making a deliberate mistake in the lecture

asking the students to spot an error on a certain page of their textbook

giving the students bonus marks towards their final grade for providing excellent

counter-examples to hard questions during the lecture

including questions that require constructing counter-examples into assignments and

tests.

A collection of incorrect statements from a first-year calculus and suggested counter-

examples to them illustrated by graphs can be found in (Klymchuk, 2010).

At more advanced level of mathematics Dahlberg and Housman (1997) suggest “it might

be beneficial to introduce students to new concepts by having them generate their own

examples or having them decide whether teacher-provided candidates are examples or

non-examples, before providing students examples and explanations”.

Appendix: An Example from Teaching Practice


23



Below are some notes from teaching experience using counter-examples with students in

both studies. The following statement is considered as an example:

Statement 1. If a function y = f(x) is defined on [a,b] and continuous on (a,b), then for

any N between f(a) and f(b) there is some point ),( bac such that f(c) = N.

The only difference between this statement and the Intermediate Value Theorem is the

shape of the brackets of the interval where the function is continuous: in Statement 1 the

function is continuous on an open interval (a,b) instead of a closed interval [a,b]. When

students are asked to disprove Statement 1 they usually come up with something like this:

Figure 3. A possible student’s counter-example to Statement 1. To create other counter-examples one can suggest the students the following exercise:

. f(b) f(a)

a b


24



Exercise 1. Give as a counter-example a graph that doesn’t have ‘white circles’. The

students may come up with something like this:

Figure 4. A possible student’s answer to Exercise 1. To generate further discussion one can suggest the students the following exercise: Exercise 2. On the graph shown on Figure 3 the statement’s conclusion is not true for any

value of N between f(a) and f(b). Modify that graph in such a way that the statement’s

conclusion is true for:

a) exactly one value of N between f(a) and f(b) ;

a b

f(a) f(a)

f(a) f(a)

f(b)


25



b) infinitely many but not all values of N between f(a) and f(b) .

One can then expect students to sketch graphs like these:

a)

Figure 5. A possible student’s answer to Exercise 2(a). b)

f(b) f(a)

a b

f(a)

f(b)


26



Figure 6. A possible student’s answer to Exercise 2(b). More practical and pedagogical insights on how to work with counter-examples in first-

year university calculus course can be found in (Mason & Klymchuk, 2009). The book by

Gelbaum and Olmsted (1964) is a classical resource for counter-examples in advanced

calculus and mathematical analysis courses.

Acknowledgements

I would like to express my sincere gratitude to Professor Rina Zazkis from Simon Fraser

University, Canada for her constructive comments and thoughtful suggestions on the

first draft of the paper. Some results of the studies were previously published in New

Zealand in a local non-refereed journal The New Zealand Mathematics Magazine.

References

Bell, A. (1993). Principles for the design of teaching. Educational Studies in

Mathematics, 24, 5–34.


27



Dahlberg, R., & Housman, D. (1997). Facilitating learning events through example

generation. Educational Studies in Mathematics, 33, 283-299.

Ernest, P. (1996). Varieties of constructivism: A framework for comparison. In L. P.

Steffe, P. Nesher, P. Cobb, G. A. Goldin & B. Greer (Eds.), Theories of mathematical

learning. Mahwah, NJ: Lawrence Erlbaum.

Gelbaum, B.R., & Olmsted, J.M.H. (1964). Counterexamples in Analysis. Holden-Day,

Inc., San Francisco.

Horiguchi, T., & Hirashima, T. (2001). The role of counterexamples in discovery

learning environment: Awareness of the chance for learning. Proceedings of the 1st

International Workshop on Chance Discovery (pp. 5-10). Matsue, Japan.

Irwin, K. (1997). What conflicts help students learn about decimals? In F. Biddulph & K.

Carr (Eds.), People in mathematics education - Proceedings of the 20th annual

conference of the Mathematics Education Research Group of Australasia (pp. 247-

254). Rotorua, New Zealand: MERGA.

Klymchuk, S. (2001). Counter-examples and conflicts as a remedy to eliminate

misconceptions and mistakes: A case study. In M. van den Heuvel-Panhuizen (Ed.),

Proceedings of the 25th international conference for the psychology in mathematics

education, (1, p. 326). Utrecht, The Netherlands.

Klymchuk, S. (2010). Counterexamples in calculus. Mathematical Association of

America.


28



Lakatos, I. (1976). Proofs and Refutations: The Logic of Mathematical Discovery.

Cambridge University Press: Cambridge.

Mason, J., & Klymchuk, S. (2009). Using counter-examples in calculus. UK: Imperial

College Press.

Mason, J., & Watson, A. (2001). Getting students to create boundary examples. MSOR

Connections, 1, 1, 9-11.

Peled, I., & Zaslavsky, O. (1997). Counter-examples that (only) prove and counter-

examples that (also) explain. FOCUS on Learning Problems in Mathematics, 19(3),

49–61.

Perkinson, H.J. (1984). Learning from our mistakes: A reinterpretation of twentieth-

century educational theory. Westport, CT: Greenwood Press.

Perkinson, H.J. (2002). The critical approach to social work. Journal of Social Work

Education Online 38, 3. Retrieved from:

http://www.questia.com/browselibrary/journalArticles.jsp?r=5002066025&o=2002|09|

22|038|003|Fall .

Piaget, J. (1985). The Equilibrium of Cognitive Structures. Cambridge, MA: Harvard

University Press.

Richstein, J. (2000).Verifying the Goldbach conjecture up to 4*10^14. Mathematics of

Computation. 70 (236), 1745-1749.

Selden, A., & Selden J. (1998). The role of examples in learning mathematics.

The Mathematical Association of America Online. Retrieved from:


29



http://www.maa.org/t_and_l/sampler/rs_5.html

Swedosh, P., & Clark, J. (1997). Mathematical misconceptions – can we eliminate them?

In F. Biddulph & K. Carr (Eds.), People in mathematics education - Proceedings of

the 20th annual conference of the Mathematics Education Research Group of

Australasia (pp. 492-499). Rotorua, New Zealand: MERGA.

Tall, D. (1991). The psychology of advanced mathematical thinking. In D. Tall (Ed.),

Advanced Mathematical Thinking (pp. 3-21). Kluwer: Dordrecht.

Tall, D., Gray, E., Bin Ali, M., Crowley, L., DeMarois, P., McGowen, M., … Yusof, Y.

(2001). Symbols and the bifurcation between procedural and conceptual thinking.

Canadian Journal of Science, Mathematics and Technology Education, 1, 81–104.

Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: The role of

learner-generated examples. Erlbaum: Mahwah.

Watson, J. M. (2002). Inferential reasoning and the influence of cognitive conflict.

Educational Studies in Mathematics, 51, 225–256.

Weiner, N. (1956). I am a mathematician. The M.I.T. Press.

Zaslavsky, O., & Ron, G. (1998). Students’ understanding of the role of counter-

examples. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference

of the International Group for Psychology of Mathematics Education (1, pp. 225-

232). Stellenbosch, South Africa: University of Stellenbosch.

Zazkis, R., & Chernoff, E.J. (2008). What makes a counterexample exemplary?

Educational Studies in Mathematics, 68, 195-208.


30



Zevenbergen, R., & Begg, A. (1999). Theoretical framework in educational research. In

F. Biddulph & K. Carr (Eds), SAMEpapers (pp. 170–185). New Zealand: University of

Waikato.

On Teaching Logarithms using the Socratic Pedagogy

Jenna Hirsch, Ph.D. Borough of Manhattan Community College Jessica Pfeil, Ph.D. Sacred Heart University


31



Introduction

Instructors of post-secondary remedial mathematics classes often voice similar

challenges from students: low attendance, lack of motivation, and negative attitude

toward mathematics. These challenges often result in students to low performance on

assessments. The effective mathematics teacher must first look at oneself to determine if

changes can be made to improve student motivation, attitude toward mathematics, and

class attendance. We formed a collaboration to examine our current teaching methods in

remedial classes and based on our findings, worked together to research a new pedagogy

we could adopt in our remedial classes to improve our students’ enjoyment of learning

mathematics.

Upon examination, we found we relied too greatly on teaching our remedial

students in a lecture-based style. In addition to our students being bored, we determined

that teaching mathematics primarily by lecturing may not always be effective for

remedial students. Our research began with a search for a teaching method that would


32



employ student thinking, creativity, and understanding. We found the Socratic Pedagogy

met these needs and decided to focus our joint efforts on creating lessons using this

method to teach our remedial students. We chose the topic of logarithms as we both have

found this to be a particularly challenging topic in the past. Our logarithm lesson, taught

using the Socratic Pedagogy, was designed with the objective of improving students’

overall enjoyment and classroom experience while learning the content.

The Socratic Pedagogy

The Socratic Pedagogy rests on the philosophy that knowledge and understanding

are already present within people and that this knowledge could be drawn out using

appropriate questioning, in turn advancing that very understanding. The Socratic

Pedagogy of teaching focuses on constructing questions instead of answers for the

students. This is a system of learning based on inquiry, questioning, exploration, and

discovery. This method is responsible, in part, for drawing out ideas and thoughts that

help build toward real-world construction of self-determined hypotheses.

By helping students examine their premonitions and beliefs while at the same time accepting the limitations of human thought, Socrates believed students could improve their reasoning skills and ultimately move toward more rational thinking and ideas more easily supported with logic [3, p.7]. Using the Socratic Pedagogy, teachers use questioning methods to both evaluate

and teach mathematics. Students construct both knowledge and understanding, instead of

passively receiving knowledge. The ability to understand is an emergent process that is

constantly being revised. Learners have some knowledge, and can create new knowledge

accordingly by cooperative actions, not solely based on the teachers’ effort.


33



Utilizing the Socratic method of teaching, learning is shared throughout the class;

the emphasis of the class becomes questioning rather than answering; and thinking is

valued as the essential classroom activity [1]. In lieu of overtly giving students the steps

and definitions necessary to understand systems, teachers pose questions designed to lead

students to discover how the systems work. There are a few basic tenets to the Socratic

Pedagogy. First, the objective of teaching is inquiry. Its purpose is to help modify

student argument or thought. Second, its method is dialogue, mainly between teacher and

student. The teacher’s role is to ask non-leading questions and the students’ role is to

answer questions using prior experience and/or knowledge [2].

Development of the Lesson

Our research led us to adopt the Socratic Pedagogy as an ideal teaching method to

reach our students. We chose logarithms because we both found this to be a topic our

students historically have difficulty understanding, particularly our remedial students.

Even though we were teaching at different colleges, we were both teaching similar

remedial College Algebra courses. Hence, logarithms were chosen as the vehicle to

deliver lesson plans using the Socratic Pedagogy.

We employed the Socratic Pedagogy to teach two lessons in logarithms in two

College Algebra remedial classes, with a total of 43 students. Subsequent classes were

taught using a lecture method. One was in a 2-year urban college with approximately

22,000 students in New York City while the other was in a 4-year suburban Catholic

University with approximately 6,200 students in Connecticut. In comparison, the

students at each one of these institutions were extremely heterogeneous. We maintained


34



the belief that regardless of the students’ backgrounds, their experiences in learning

logarithms using the Socratic Pedagogy would be similar.

We collaborated to develop lessons employing the Socratic Pedagogy to teach

logarithms and logarithmic properties. A deadline was set for our lesson implementation;

we committed ourselves to a regular schedule of weekly phone conversations, bi-weekly

meetings, and regular e-mail correspondence. Throughout the lesson construction phase,

the philosophy of the Socratic Pedagogy was foremost in our minds, and numerous

versions of our lesson plan were developed, revised and revisited based on our

understanding of the Socratic Pedagogy.

Three questions were investigated in response to our lesson plans:

1. Will students enjoy being taught using the Socratic Pedagogy?

2. Will students want to be taught other topics using the Socratic Pedagogy?

3. What benefits will be observed in the classroom, (either during or after), a few

lessons using the Socratic Pedagogy?

Our objectives within the context of this study included making mathematics

accessible for all learners. We took two days in our College Algebra classrooms to teach

logarithms using the Socratic Pedagogy. Lesson plans were composed to teach logarithms

as the inverse operations of exponents and to teach logarithmic properties. As a follow

up, we each conducted a comprehensive classroom discussion with the goal of anecdotal

evidence of searching for improvement in students’ overall enjoyment and classroom

experience while learning about logarithms.

Underscoring the standards of the Socratic Pedagogy, it was important that our


35



students were taught using questions rather than lecturing at them. We committed to not

“tell” the students anything, rather it was our expectation that students would be able to

discover the properties by tapping into and extending their current mathematical

knowledge, and then explain to us how each one of the properties worked. The students

weren’t told or asked to memorize anything. Instead, they were lead to discover that

“logarithms are like exponents” by using specifically designed questioning by the

teacher.

Teaching Logarithms using the Socratic Pedagogy, Two Lesson Plans:

We agreed on some basic strategies:

1. We would not explain how logarithms or their properties worked.

This meant as teachers, we were not allowed to “teach” in our standard sense. The

teacher is forced to take a back seat to our natural proclivity to tell students the

results. Instead, we had to trust our questions were good enough so that our

students could define logarithms themselves.

2. Students would be asked explain any and all of their conjectures.

Regardless of whether or not their answer was correct, it was expected that any

time a student came up with a response, we would ask them to explain it. Thinking

would be praised on any level.

3. We would give students a pattern with three examples to follow. We used three

examples because we thought that 4 examples would be too leading, and that 2

examples were not enough to establish a pattern. The pattern would then be broken

with the fourth example (a completely new problem would be posited), and we


36



would know if the majority of the class understood logarithms.

4. We would use vocabulary in context.

We would not write down definitions, or define definitions for our students.

Vocabulary would casually be mentioned in the context of our lesson (in this case,

for the different parts of logarithms), and let students pick up an understanding of

how to use the vocabulary in this manner.

5. The students would derive generalizations for logarithms.

The students would be able to tell us that “logarithms are like exponents”, or that

“adding in logarithms is the same as multiplication” (assuming the base is the

same). This is the fundamental reason we teach using the Socratic Pedagogy.

Next, we include two partial lesson plans for logarithms. The lesson plan requires

students to discover and to understand that logarithms can be expressed as exponents.

Lesson Plan #1: Logarithms are Exponent’s Inverse

Board Work Instructor Questions Possible StudentResponses

Instructor’s Response

2log 8 ? log base 2 of 8 is equal to what number? Can someone use a 2 and an 8 to find another number?

4 , 28, 16, 82

(ask the students to explain each conjecture)

Those are great answers, and all of them use the 2 and the 8, however the answer is actually 3.

2log 8 3

2log 16 ?

So, the log base 2 of 8 equals 3. I wonder how we can use all three of these numbers together? Lets’ try another, what is the log base 2 of 16?

4, 32, 14, etc. All of those answers are thoughtful as well. 4 is the answer Hmm..how can we use a 2 a 16 and a 4 to generate a mathematical answer?

2log 8 3 (at this point most students should catch on). Raise your

5, etc. So, when the base is a 2, the argument is


37



2log 16 4

2log 32 ?

hand if you think you know what the log base 2 of 32 is equal to.

a 32, the answer is a 5. Ok. Now, we are going to break the pattern. Lets try a difficult one

3log 9 ? Raise your hand if you think you know the answer to log base 3 of 9

2 Correct the answer is 2.

3log 9 2 Hmm…I wonder if someone can raise their hand and explain how logarithms work. Do they look like any other operations we have already seen? Why is log base 3 of 9 equal to 2? Does this work for the log base 2 of 8 equals three as well?

The students may say that they act like exponents. Or because 3 to the second power equals 9. And this works as well for all of the other log questions.

So can we rewrite logarithms as if they were exponents?

loga x b If I wanted to rewrite this general form of logarithms, raise your hand if you can tell me how I could rewrite log base a of x equals b using an exponential equation.

ba x

We continued with our lesson, by giving students numerous examples and practice

to sharpen their understanding of logarithms. We would place the variable in different

places, and have the students practice solving questions about logarithms, similar to ones

found in their textbook or homework. It is important to note that many students had prior

exposure to logarithms. Students have been taught many, if not all of the topics covered

in these courses while in their high school mathematics courses. Prior exposure is not an

indicator of their knowledge in these topics, but to cover the possible case that some

students did have a solid foundation in logarithms, we asked students who were familiar

with logarithms and/or their properties to ‘keep their knowledge a secret’ for a short


38



period until after the property was discovered by more of their classmates.

If we were teaching logarithms using our traditional lecture format, we would have

told the students initially that logarithms and exponents are inverse operations, and that

you can generate

loga x b by using the exponential form of ba x . Instead, we constructed questions

we know will help our students discover this fact on their own. We used our knowledge

of the mathematics to help pique the student interest in what it is we were teaching. By

using the Socratic means of teaching, we create an environment to help our students to

look for patterns, establish connections, and understand logarithms themselves. Rather

than being passive receivers of our knowledge, our students create ownership to the

lesson. They discovered the property, they found out how it worked, they looked to make

the connections themselves.

Lesson Plan #2: Addition of Logarithms (when the base is the same) acts like Addition Board Work Instructor Question Possible Student

Response Instructor Response

2 2log 2 log 4 Let’s discuss some properties of logarithms. Raise your hand if you can tell me what you believe log base 2 of 4 plus log base 2 of 8 is equal to.

3, 2log 6 , 4log 6

ask the students to explain each one of these responses

These are all interesting conjectures, lets see if we can prove what this is equal to;

2 2 2log 2 log 4 log 8

1 + 2 = 3

(the students can help with the proof, they know that log base 2 of 2 is 1, and that log base 2 of 4

Log base 2 of 8 is equal to 3.

Ok, so log base 2 of 2 plus log base 2 of 4 is equal to log base 2 of 8. I wonder if there is a


39



is 2.) I wonder how we can rewrite the number 3 as a log base 2?

short cut here?

2 2log 2 log 8 Raise your hand if you can explain what you think the answer to log base 2 of 2 plus log base 2 of 8 is equal to…

2log 16 Can you prove that as well? (have student prove and explain…

2 2 2log 2 log 8 log 16

1 + 3 = 4

Ok. So, I wonder if we really have to prove these? I wonder if there is any other way to find log base 2 of 2 plus log base 2 of 8 without proving it?

By multiplying the 2 and the 8?

Hmm…when we multiplied the arguments in our first example, did that work? Does that work for this example? I wonder if it will work for our next example?

2 2log 2 log 16 So, raise your hand if you can tell me what the log base 2 of 2 plus the log base 2 of 16 is equal to without proving it.

2log 32 Ok, so if we were to prove it, would it work?

2 2 2log 2 log 16 log 32

1 + 4 = 5

Is it true that log base 2 of 32 is equal to 5?

Yes So, maybe we have found a shortcut? Let’s try one more example, we are going to break the pattern here…

3 3log 3 log 9 Raise your hand if you can tell me what log base 3 of 3 plus log base 3 of 9 is equal to…

3log 27 Can we prove this one as well?

3 3 3log 3 log 9 log 27

1 + 2 = 3

Ok, so raise your hand if you can explain how addition works in logarithms,

You keep the base the same and multiply arguments

So, lets sum up this rule using general algebraic terms;


40



assuming the base is the same

log loga ax y If we wanted to condense log base a of x plus log base a of y to a single logarithmic expression, raise your hand if you can explain what this would equal.

log )(a xy

It is not a difficult task for a teacher to tell students, log lo (g log )a a ax y xy .

Then, the teacher engineers numerous examples for the students to work out to make sure

they understand the property. Instead, we reverse the order when using the Socratic

Pedagogy. We ask the students to justify, (and make sense of their justification) that

addition of logarithms acts like multiplication. Once again, our students are active

participants in their learning. Learning is shared and modified, and thinking and

understanding are valued.

Results

As a follow up, we each conducted a comprehensive classroom discussion

looking for anecdotal evidence of improvement in students’ overall enjoyment and

classroom experience while learning about logarithms. In response to our first question,

“Will students enjoy being taught using the Socratic pedagogy?” the majority of the

responses were positive. Many students stated during the classroom discussion that they

really enjoyed thinking about logarithms using the Socratic Pedagogy. Through informal

dialogue with our students, we received the following direct quotes:


41



“I would like to try out more ways to learn with the Socratic method.”

“Thought a little easier than some other sections.”

“I found the Socratic method helpful because instead of giving answers and rules,

we were forced to find the answers for ourselves.”

“In high school I never understood how to work with logarithms and now I

understand how to.”

“I think it was a good lesson.”

“I like it, it really made me think.”

“I really got it, for the first time, I really enjoyed learning.”

There was some negative feedback as well:

“The Socratic method just felt like being teased with information.”

“The Socratic method wasn’t really a good teaching method for me because I like

when things are explained first step by step.”

“I personally didn’t like the lesson, but I am willing to try and see how it would

work using something other than logs.”

“I’d rather you tell us the info first.”

Negative feedback may derive from lack of comfort with a new type of teaching

experience. Students may be resistant to change, relying instead on the old way of being

taught. They were not quite convinced the new way of being taught would resonate with

them. We do assert, however, if given more exposure to this method of teaching, more

students may find the Socratic method of learning to be an enjoyable experience.

In response to our second research question, “Will students want to be taught other


42



topics using the Socratic Pedagogy?” we were both informed from a majority of our

students, that they would enjoy learning other topics using this method. One of us was

sought out after class from a student who typically tends to be quiet and non-

participatory, and was asked if we could teach more lessons using this method.

An unintentional outcome of using the Socratic Pedagogy was that our classes’

participation and enthusiasm for learning mathematics grew, in part explaining our results

to our third question, “What benefits will be observed in the classroom, (either during or

after), a few lessons using the Socratic Pedagogy”. Many students who don’t normally

participate during our regular lectures did during these two classes. We speculate this

was possibly due to our students receiving positive feedback for their thinking regardless

of the precision of their answer. Therefore, students who normally don’t participate

because they are scared of giving an incorrect answer, felt safe to make mistakes during

these lessons.

After these lessons were taught using the Socratic Pedagogy, numerous benefits

were observed. One example was that students would try to convince themselves

(through proof or through numerical substitution) that some property worked. As an

example, during one of our classes, we asked students, “What is another way we can

express 3log x ?” One student stated immediately “3log x ”. When the student was

asked to justify his answer, he stated because “ 3log log log log 3logx x x x x ”.

This student discovered the property that exponentiation of an argument in logarithms is

the same as multiplying the logarithmic expression by the exponent. This property was


43



not included in our initial lesson plans to teach using the Socratic Pedagogy, however a

modified version of it will be included the next time we teach using this method.

Future Research and Implications

In extending this research, the primary objective will be to determine if teaching

using the Socratic Pedagogy will have any type of effect on how students either

understand or retain their understanding of logarithms and their properties. Future

research could assist in answering the question: Will teaching using the Socratic

Pedagogy impact student understanding of logarithms? It could also help answer the

question: Will students who have learned via the Socratic method perform better (either

on quizzes or exams) than students who were not taught using this method? In

addressing our questions about teaching using the Socratic Pedagogy, it is important to

note that we relied on anecdotal evidence from discussions with our students. There is no

quantitative data to accompany our research. We look forward to addressing the above

questions in future studies using quantitative data and statistical analysis.

Using the Socratic Pedagogy to teach logarithms resulted in numerous benefits for

researchers as well. We taught ourselves a new way to understand and learn

mathematics. We pushed ourselves out of our comfort zone and challenged ourselves to

become better teachers. We felt responsible for our student lack of enthusiasm in our

remedial classes, and we changed this by researching new methods. We also gained a

renewed enjoyment for lesson planning, choosing to create lesson plans that held onto the

ideals of the Socratic Pedagogy but also not losing the rich mathematics at the same time.

Teaching using the Socratic Pedagogy had numerous benefits both for our students


44



and ourselves. There are many topics in remedial college mathematics courses that can

be taught using the Socratic Pedagogy. It is our hope that other mathematics educators of

remedial students will gain from our study, either by using the above lesson plans or

following our process and designing their own plans using the Socratic Pedagogy and

implementing those plans in their remedial mathematics classrooms.

References:

Brogan, Bernard, and Walter Brogan (1995). The Socratic Questioner: Teaching and Learning in the Dialogical Classroom. Educational Forum 59 (3), 288-96.

Chang, Kuo-En, Mei-Ling Lin, and Sei-Wang Chen (1998). Application of the Socratic Dialogue on Corrective Learning of Subtraction. Computers & Education 31 (1), 55-68.

Copeland, M (2005). Socratic Circles: Fostering Critical and Creative Thinking in Middle and High School. Portland, MN: Stenhouse Publishers, p.7.

Seven Mysteries for your Algebra Class


45



Terence Brenner Hostos Community College, CUNY

Abstract In this paper we present seven mysteries of the week to make an algebra class more


46



interesting. The first is for an introduction to linear equations; the second is for more

complicated word problems that include consecutive numbers; the third is for a mixture

and money problems, the fourth is for equations with fractions, the fifth is for

multiplying, dividing and factoring, the sixth is for solving simultaneous linear equations

using the addition and substitution methods and the seventh is for solving more

complicated simultaneous linear equations.

Comments from my students.

“I would like to see more mysteries because it forces me to think and be able to learn the

formulas in word problems. Its useful for everyday math experiences”

“What I really like the mysteries is the fact that it makes me think more than just a

regular problem written in letters”

“I would like to see practice mysteries assignment, quiz etc not just mysteries They are

good but sometimes are hard to resolve or to see an equation”

“I like that the mysteries are challenging it put my mind to think, I dislike some are to

long and confusing because it has to much wording. I would like to see mysteries that

aren’t so hard to do when you are home”

“I like them because the help with word problems and how to sort it. Only reason I

dislike it is because it won’t show up on the test also there hard”

“What I like about the mysteries is that it makes me think a lot. I would recommend

these mysteries to other students because it really push you to study hard”


47



“Personally I like the mysteries because it makes it more interesting to learn from and it

just not numbers and numbers. Even though the problem given are not easy at all and

confusing, the point of it is to read the mysteries carefully and follow the clues given. I

would like to see more mysteries in the future, at least one every week and do it in a

group work and after discuss it in class”

“What I like about the mysteries are it helps you think about the problem and it

challenges you to go extra step to figure it out. I would recommend these mysteries to

other students so that if they are trying to become better at math they can use the

mysteries as a challenge”

“Its good because it helps you use the math skill so they stay fresh in your brain”

We present the mysteries below exactly as they are given to the students in class. Can you solve them?

THE FIRST MYSTERY OF THE WEEK

Lieutenant Columbo has just discovered that if he multiplies the number of corpses by 5

he would have 15 corpses. He also stumbled onto the fact that if he divides the number of

clues by 4 he would have 5. After a couple of days he learns that 3 more than number of

motives is 6 and 8 less than the number of suspects is 4. Finally, 3 more than twice the

number of weapons is 7. He must finish his investigation today. Please help him by

setting up the equations, solving them and showing all the work. Lieutenant Columbo

now knows that the murderer’s number is the sum of the number of corpses, clues,

motives, suspects and weapons. Who did it?


48



CORPSES CLUES MOTIVES SUSPECTS WEAPONS

number 1-15 number 16-30 number 31-45Rivas, Nancy Guillermo Quirci Ramos JudithSanchez, Amelia Hernandez Rosa Rivera Pedro JSantiago, Dolores Howard Emery L. Rodriguez Maria DAlicea,Lilian a Liranzo Ana Rosa Montano BetzaidaBurgos Jorge Mack Sandra Vassquez LuzCabrera Luis Maldonado Anibal Wilson George ACamacho , Anderson Mena Christian A Woddard DemondCedeno, Angela Molina Ana C Yeta, DanjuraCepeda, Judy Moreno Magdalena M Adeoba, AdetunjiContin Josefina Nicholas Rubia Alonzo, Orlando ECruz Rocio Nieves David Bellinger Keenan LDe La Cruz Nelly Nieves Deliris Blunt ToniDundas Natasha N Obasuyi Michael O Canario, MaritzaGomez Felina a Peralta Ramona Castellanos, MichelleGuadarrama Virginia Polanco Elaine F Cruz, Carlos M


49



THE SECOND MYSTERY OF THE WEEK

Lieutenant Hernandez , taking over the cases for Lieutenant Colombo, has just been

handed case number 457. After looking at the file he discovers that three times the

number of victims increased by seven is equal to nineteen. He also stumbled onto the fact

that twice the difference of the number of witnesses and six is 10. After a few days he

learns that the number of suspects is half the number of clues and they total 84. The

Lieutenant remembers that he had a lot of cases and recalls that case number ’s that had

the same number of corpses where three consecutive case numbers whose sum was 57.

What is odd is the case numbers that had the same number of weapons is the same for

three consecutive odd case numbers, where twice the smallest case number added to the

largest is 67. By accident, he realizes that the case numbers that had the same number of

clues is the same for three consecutive even case numbers where twice the largest added

to the smallest is 74. PLEASE HELP!!!!!!!!!!!!!!!!! He must finish his investigation

today. Set up the equations, solve them and show the work. Wait a minute, if he finds the

average of the number of victims, witnesses, clues and the smallest number of case

numbers for corpses, weapons and clues he will have the murderer’s number. Who did it?

VICTIMS WITNESSES


50



SUSPECTS AND CLUES

CASE NUMBERS FOR CORPSES CASE NUMBERS FOR WEAPONS

CASE NUMBERS FOR CLUES


51



THE THIRD MYSTERY OF THE WEEK

Lieutenant Polanco was just handed the case file from Lieutenant Hernandez. After

reading the file, Lieutenant Polanco discovers that each victim had exactly $3.35.

Investigating further, she notices that for each victim they had twice as many quarters as

dimes and two more nickels than dimes. Exactly how many nickels, dimes, and quarters

did each victim have? She needs to take fingerprints from the coins, but needs a 40%

solution of pure ink. She has 55 liters that is 60% pure ink. Her partner, Detective Del La

Cruz tells her that if she adds some 30% solution of pure ink she can make the 40%

solution of pure ink that she needs. How much of the 30% solution of pure ink does she

need? After many days of investigating, she now knows that 25% of the total: nickels,

dimes, quarters and 30% solution of pure ink is the murder’s number. Who did it?

Please help her by setting up the equations solving them and showing all the work.

NICKELS, DIMES AND QUARTERS



52



30% SOLUTION OF PURE INK



53



THE FOURTH MYSTERY OF THE WEEK

Detectives Gomez and Sanchez stumble over an interesting fact. From all their previous

cases the number of weapons was four less than the number of clues. Even stranger, one-

half the number of clues is three more than one-third the number of weapons. How many

weapons and clues were there? The two detectives were handed 258 case files.

Detective Gomez remembered that it took her 10 hours to review that many files, while

detective Sanchez remembered it took him 40 hours to review that many files. To save

time, they decide to work together. How long will it take them working together to

review all the files? Captain Jeter is on vacation, but gets a call from the detectives at

1:30pm. The captain and the detectives realize they are 160 miles apart. At 2:00pm the

captain leaves his vacation spot and the detectives also leave at the same time. They

travel towards each other and arrive at 4:00pm at the crime scene. The captain’s speed is

two-thirds of the detectives. What were the speed of the captain and the speed of the

detectives? Finally, after all their hard work they discover that the sum of: one-fourth the


54



number of hours the detectives worked together , one-half the number of clues, one-third

the number of weapons and one-sixteenth the speed of the detectives is the murderer’s

number. Who did it? The detectives do not like fractions, who does? Please help them

by setting up the equations, solving them and show all your work.

Weapons and clues working together

Their speeds

Captain Jeter

Detectives Gomez and Sanchez



55



THE FIFTH MYSTERY OF THE WEEK

Murder at Con Ed Detective Sandy Freeze found a power line wrapped around the president of Con Ed’s

neck and found the following important fact: 12 % of the number of suspects is three

less than the number of weapons. How many suspects were there? After a few days she

learns that the number of weapons is 62.5% of the number of clues. Is 62.5% a decimal

number? How many weapons are there? Wait a minute; she recalls that the ratio of

fingerprints to clues is four to five. She found nineteen good fingerprints, how many

clues are there? She remembers that she has to round off all her answers to the nearest

whole number because the reports she files do not have decimal numbers. Detective

Sandy Freeze likes doing “higher Math-Calculus”, so she wonders what percent of the

suspects is the number of weapons? What is the percentage? Finally, she discovers that

if she subtracts 33 % of the number of weapons from 25% of the sum of the number of

suspects and the number of clues, she will have the murderer’s number. Who did it?

Please help her, even though she doesn’t want to find who killed the President of Con Ed.

She still has to file the report and needs a lot of help in filling it out. Set up the equations

and show her all your work while you’re in a well-lit and warm room.

SUSPECTS WEAPONS


56



CLUES



57



SIXTH MYSTERY OF THE WEEK

Detective Rodriquez is trying to learn how Lieutenant Columbo solves his cases. After

questioning the suspects many times Detective Rodriquez discovered that if she subtracts

the number of corpses from twice the number of weapons she ends up with 4. She then

stumbled onto the fact that if she adds the number of weapons to three times the number

of corpses she now has 9.After thinking about this for a few days she learns that the

number of clues is one less than triple the motives. After further investigation she

discovers that the sum of three times the motives and clues is 11.Can all the mystery’s be

this complicated? Please help her by setting up the equations and solving them (show her

all your work).She finally realizes that five times the average of the number of weapons,

corpse, motives and clues is the murderer’s number. Who is the murderer? WEAPONS, CORPSES MOTIVES, CLUES


58





59



THE SEVENTH MYSTERY OF THE WEEK

ANOTHER MURDER AT CON-ED

Detective Sandy Freeze, the hero from the famous murder case at CON-ED, found The

Vice President of CON-ED in his nice warm lit office with internet and cable TV with a

utility pole on his head and a broken light bulb in his left hand. Could the utility pole be

the murder weapon? Or is this a strange, but satisfying coincidence? Detective Sandy

Freeze finds the following strange information: the sum of the number of clues squared

and six more than five times the number of clues divided by the difference of the number

of clues squared and nine is then divided by the sum of number of clues squared and two

more than three times the number of clues divided by the sum of the clues squared and

six more than seven times the number of clues. This is equal to four. How many clues

are there? Even stranger, thirteen times the number of suspects less than three times the


60



number of suspects squared is ten. How many suspects are there? Wait a minute, eight

times the number of motives cubed less than four times the number of motives to the

fourth is somehow thirty two times the number of motives squared. Is this any clearer

than the information you get from the CON-ED phone number? After all this hard work

Detective Freeze finally realizes that if she adds the number of clues, the number of

suspects and the number of motives she has the murderer’s number. Who did it? Please

help her by setting up the equations, solving them and show all your work.

Clues Suspects

Motives



61



Triad of Piaget and Garcia, Fairy Tales and Learning Trajectories

Bronislaw Czarnocha Hostos Community College, CUNY, NYC

Abstract


62



The presentation formulates a conjecture concerning the generality of learning

trajectories, whose framework underlines the design of the Common Core Standards in

Mathematics to be introduced into practice in the Fall 2014 in USA. The conjecture is

based on the unusual similarity between observed elementary trajectories of learning in

mathematics and learning progressions identified in a class of fairy tales. Both cases are

explicated with the help of the Triad of Piaget and Garcia, (1989), clarifying their

structural and interpretative similarity. The “learning triple” thus identified in both

learning situations is conjectured to be one of the simplest progressions of learning,

which can underline the general structure of learning trajectories in mathematics.

Key words: learning trajectories, Triad of Piaget and Garcia, learning triple.


63



Introduction.

The presentation addresses itself to some of the unresolved issues present in the highly

dynamic framework of learning trajectories (LT) and learning progressions, whose

importance significantly increased when it became the underlying principle for the

construction of new Common Core Standards in Mathematics (CCSS, 2012) in USA

about to enter into instruction in the Fall 2014.

The number of different definitions of this concept abundant in the literature suggests that

its ontological and epistemological nature is far from clear.

A Hypothetical Learning Trajectory consists of the [mathematical] goal of student

learning, the Mathematical Tasks that will be used to promote student learning, and

hypotheses about student learning (Simon,1995).

[Actual] learning trajectory of children includes a model of their initial concepts and

operations, an account of observable changes in those concepts and operations...and the

account of mathematical interactions that were involved in these changes (Steffe, L,

2004).

For Clements and Sarama (2009) learning trajectories have three parts: a goal (that is an

aspect of a mathematical domain children should learn), a developmental progression, or

a learning path through which children move through levels of thinking, and the

instruction that helps them to move along that path.(p.17) However, in the paper by the

same authors (Clements and Sarama, 2004) a more complex definition appears: “the

simultaneous consideration of mathematics goals, models of children’s thinking,


64



teachers’ and researchers’ models of children’s thinking, sequences of instructional

tasks, and the interaction of these at a detailed level of analysis of processes” (p. 87)

Confrey and colleagues (2009) defined them as “researcher conjectured,

empirically-supported description[s] of the ordered network of

experiences a student encounters through instruction … in order to move from

informal ideas … towards increasingly complex concepts over time” (p. 2).

The comparison of these definitions reveals several axes of disagreement; are LT’s

hypothetical or actual, or if both of them make sense, then what is the relationship

between them? Should they be conjectured by teachers or researchers, or both and what

is the relation between these options? An essential question underlying these differences

as well as the critique of new Standards is to what degree the generality of a particular

learning trajectory can be asserted. It translates itself into the question to what degree the

LT’s framework can justly be considered a basis for national standards of mathematics

education. Susan Emmons, (2011) points out that the concept of LT’s is not new and

asks: “Why then talk about learning trajectories now? The metaphor emphasizes the

orderly development of children’s thinking and draws our attention to learning targets

and possible milestones along the way. To what extent is this kind of assumption about

learning warranted? That is, in what sense does children’s mathematics learning follow

predictable trajectories?”


65



In response to these questions it is being conjectured (Section 1) that an elementary

trajectory of learning observed, among others, in classes of algebra, called here a

“learning triple” is a basic building block for learning trajectories addressing the

development of some elementary mathematical concepts. It is a characteristic feature of

the learning triple that its first two events are at the same level of generality while the

third one is at the higher level.

The motivation guiding the pursuit of this inquiry is the observation that such triples of

learning can be observed in two very different cultural domains, that of learning

mathematics and that of certain class of fairy tales where the heroine or a hero reaches the

task in three steps.

Our methodology is to explicate both cases with the help of the Triad of Piaget and

Garcia showing their structural and interpretative similarity (Sections 2, 3). As a result,

we obtain thus two different domains of human learning activity where similar processes

of learning are taking place. The self-consistency of the learning triple construct leads us

to conjecture that the “learning triple” found in the individual process of learning

mathematics and in pathways of fairy tales’ heroines and heroes representing the

socio/cultural wisdom of the folklore, is a manifestation of a general strategy of human

learning available to an individual, which may not depend on the actual context.

Therefore, the learning triple may become the basic, general structure of learning

trajectories once the nature of its facilitation is thoroughly understood by further research.

The dialog below from the intermediate algebra class is the composition of such three

triples of learning.


66



The learning triple of generalization.

0 The teacher asked the students during the review: “Can all real values of be used for

the domain of the function 3X ?”

1 Student: “No, negative X ’s cannot be used.” (The student is confusing here the

general rule which states that for the function X only positive-valued can be used as

the domain of definition, with the particular application of this rule to 3X .)

2 Teacher: “How about 5X ?”

3 Student: “No good.”


5 Student: “No good either.”


7 Student, after a minute of thought: “It works here.”


9 Student: “It works here too.”

A moment later

10 Student adds:” Those X ’s which are smaller than 3 can’t be used here.”


12 Student, after a minute of thought: “Smaller than 1 can’t be used.”

13 Teacher: “In that case, how about aX ?”

14 Student: “Smaller than a can’t be used.”


67



The strategy of the teacher is clear: upon detecting a standard student’s error he leads the

student toward the cognitive conflict (lines 2-7) through the careful choice of numerical

values of the variable to be checked by the student. By adding third properly chosen

numerical value of the parameter of the function, X=-2, the teacher facilitates a moment

of understanding during a short reflection, through which the student generalizes two

previous numerical cases to the general class of numbers with the property X>-3 in the

line 10. The next question, line 11, of the teacher provides sufficient information for the

student concerning the domain of the function 1X , so that at the last question, line13,

the student is able to make one more step in the process of generalization, to the general

class of numbers with the property X>a. We see here two examples of an interesting

triple, for which (*) two first events are on the same level of generality, while (**) the

third one is on the higher level of generality than the previous two. The first instance

(lines 6-10) - the student reaches first generalization; the second instance (lines 0,10-14)

– the student reaches second generalization. It is important, in further research to

investigate precisely what are the elements of learning environment which facilitate the

learning triple’s occurrence.

One may add that the lines 2-6 also contain a similar triple of teacher’s questions and

student’s answers leading to the cognitive conflict in lines 6,7. Student correct responses

in lines 3 and 5 provide two first events of the triple, while the line 7, where the student

solves the cognitive conflict is the third, transcending event, which at the same time is the

beginning of the next triple. This way we see here a sequence of triples joined together so


68



that every third of the previous triple is at the same time the first of the next triple. Such a

structure can also be encountered within the subclass of fairy tales under discussion.

The Triad of Piaget and Garcia.

The Triad is a mechanism of thinking leading to concept formation formulated on the

basis of the thorough comparative analysis of the development of physical and

mathematical ideas in history of science on one hand, and the psychogenetic development

of these concepts in a child, on the other (PG,1989). It is defined in the chapter on the

history of algebra development, as the passage through intra-operational, inter-

operational and trans-operational stages. “Intra-operational stages are characterized by

intra-operational relations, which manifest themselves in forms that can be isolated”.

The intra-operational stage for algebra of equations had lasted till the time of Lagrange,

who for the first time asked for a general method of solution for equations of degree 3

and degree 4. Till then, every equation was seen as separate, isolated case whose solution

required its own method related only to the properties of that particular equation. It was

with Lagrange’s inquiry (PG,p.150) into general common methods of solving equations

of 3rd and 4th degree, when the developing concept entered the inter-operational stage,

that is a stage, which is “characterized by correspondencies and transformations among

the forms that can be isolated at previous levels…”. Note that one of the fundamental

tools of inter-operational thinking is the search for similarity between two or more

instances of the forms.


69



Similarly, the number -3 of line 6, and the number -2 of line 8 by themselves are isolated

instances of the numbers in the domain of the function f(x) – thus they represent the intra-

operational reasoning. However, when in the line 10, upon visible reflection on the

previous answers, the student, possibly through noticing common qualities of the two

numbers (their negativity and good fit into the domain of the function), enters the inter-

operational stage of her thinking and reaches the trans-operational stage by declaring that

” Those X’s which are smaller than -3 can’t be used here.”

According to (PG, 1989) “The trans-operational stages are characterized by the

evolution of structures whose internal relationships correspond to inter - operational

transformations.” In history of algebra of equations, trans-operational stage was reached

with the formulation of the group concept by Galois; in child’s pre-algebraic

development it can be understanding of multiplication as the generalized addition (“I’ve

taken m times n.”(PG,p.182).

In the elementary situation when only two different individual cases (or two classes of

cases) exist, the trans-operational transformation coalesce into one with the inter-

operational observation , because in this case the defining structure of the trans-

operational transformation is at the same time the structure of the inter-operational one.

The trans-operational statement ” Those X’s which are smaller than -3 can’t be used

here” is a generalization of the inter-operational observation that “-3” and “-2”, the

elements of the domain of the given function, are larger or equal (not smaller than) to -

3”. Thus the sequence of the first triple: two cases isolated from each other, which allow

for inter-operational comparison of similarity, followed by the third event, which raises


70



understanding to the higher, more general level. It is very interesting that the first

generalization triple develops into the second one, where by single isolated cases in this

case we take (1) the lines 0,10 – the first case of 3 XXf , (2) line 11 - the

second case for f(X)= 1X and the third, (3) the generalization step for f(X) = aX

- line 13. Two individual cases of the square root function followed by its generalization

to aX . Note that whereas the historical inter-operational period in algebra lasted

around 60 years, child development inter-operational period lasts several years into

adolescence, the inter-operational period within given teaching moment can last just a

moment or two.

It is critical to understand why such a rather complicated structure as the schema obtained

by the Triad construction can have such a simple, three step sequence. The simplicity is

due to the fact that we have only two separate instances at the intra-operational level. Had

we had instead three separate instances of intra-operational level, we would have to, in

general, take into consideration at least 3 different relationships between them in the

inter-operational level, whose composition at the trans-operational level would have be

more complex than each of them separately. On the other hand, when we have exactly

two different instances of intra-operational nature, they have only one relationship (or

one class of relationships) between them discovered at the inter-operational level, which

by its simplicity and uniqueness leads directly to the trans-operational level through

generalization. It is obvious that just one case of intra-operational instance is not enough

to give rise to the inter-operational level, and consequently to the trans-operational level.


71



Thus our considerations can be seen from the point of the Triad as the response to the

question: what is the minimal number of intra-operational instances for the intra-, inter-,

trans- process to be possible? The answer is – two such instances are sufficient and

necessary.

Learning Progressions of Fairy Tales.

Many fairy tales around the world contain series of three events, the triples

(or trebling), which suggests the presence of triples of learning. Whereas the

full description and characterizations of fairy tales with trebling is contained

in the paper Mathematics of Fairy Tales to appear in the upcoming issues of

Mathematics Teaching-Research Journal (mtrj) on line at

www.hostos.cuny.edu/mtrj, here we provide several illustrations of what can

be called learning progressions of fairy tales. The structure of trebling has

been identified by (Propp, 1975) as one of the general morphological or

structural components of a fairy tale. It is a triple repetition of similar type of

event, usually with the first two attempts of the hero (or two older brothers

or sisters) being either negative or positive but insufficient for the success,

generally of the intra-operational type, whose inter-operational comparison

allows for the third event, which transcends the scope, the meaning and the

importance of the previous two.


72



As an example, we will discuss the triples of the classical Persian folktale The Padishah

and his Three Daughters (Guppy, 2009) – the first one in the collection of Magical Tales

from Classical Persia:

One day, he [the Padishah] sat on his throne, flanked on either side by his two vizirs and

surrounded by his courtiers and summoned his daughters. Presently they were ushered

in, dazzling in their fineries and jewelry, and the audience gasp at their beauty and

grace. “The moon had divided into three!”: they whispered. The nightingale would not

know which rose to choose.!” And other such complements.

The Padishah turned to his eldest daughter and said: “Tell me Shahrock, is it the lining

that protects the coat, or the coat that protects the lining?”

“The coat protects its lining, Crowned Father” – Shahrock replied.

“very well answered” – Said the Shah. “I see that your education has not been in vein.

You are thoughtful and discerning, and you deserve a good husband. I will give you to my

Right Hand Visir.”

Now the Padishah turned to his second daughter “Does the coat protect its lining or vice

versa?” The coat protects its lining, Sire, naturally.” Responded Makrokh without

hesitation.

Well done my dear. Your worthy of no less a good man than my Left Hand Visier.

Finally the king called forward Golrokh, the youngest and the loveliest of his daughters,

and put the same question to her.

“I believe its lining protects the coat” answered Golrokh.


73



“Surely you don’t mean that!” Frowned her father. “When it rains or snows, if there is a

fight or some other accident, it is the coat that takes the brunt and gets damaged, while

the lining

But Golrokh was adamant “I still think beloved Crowned Father that it is the lining that

supports and safeguards the coat not the other way around.”

The Shah said that Golrokh was not only ignorant, but stubborn and opinionated, that

her education had been wasted. “Never let me see your face again” he roared, and

banished his beloved daughter forever. He ordered his servants to search the city and

find the lowest, poorest, most unworthy man, and let him marry and take away his

youngest daughter.

We see here a triple, whose first two events differ significantly from the third one. After

her two sisters answer the essential question of the king correctly and are awarded by

marrying two viziers, Golrokh, the youngest one chooses the second alternative of the

question as her answer and is punished for that. Her punishment turned out to be the

invitation for adventure during which her lowly husband Hassan, guided by Golrokh,

matures through another triple to become the inheritor of his mentor – the merchant

Ahmed’s wealth, respect and leadership. He is invited to join the trade caravan during

which he demonstrates an excellent sense of business by buying citrus fruit despite the

opposition of his mentor, and selling it at high profit during the sea crossing. The second

event of the Hassan’s triple takes place during, or rather before the sudden storm in the

desert when Hassan decides to make an encampment in a high ground, avoiding


74



destruction that comes to other caravans later with the pouring of the rain. The third,

transcendental event takes place when his mentor merchant, Ahmed Haji, through inter-

operational comparison recognizing Hassan’s extraordinary quality of good sense in two

cases at the intra-operational level, writes his will bequeathing his business, the Grand

Merchant title and the wealth to Hassan, Golrokh’s husband.

The process is highly similar manner to the conclusion taken by the student after the first

two intra-operational events –“ all those X’s which are smaller than -3 are not good

here”. Clearly, Ahmed learned after the two initial events that Hassan is trustworthy and

will know what to do while taking care of and developing the bequeathed inheritance.

Two events and their natural inter-operational comparison were enough for Ahmed to

reach his general conclusion and write an appropriate will as the transformation of

transcendence.

The two triples of Golrokh and of her husband are constructed to keep the fairy tale

within their framework, just like two generalization triples compose into student

understanding of the domain for aX . However, whereas in the algebra class, the

first generalization triple takes place within the first event of the second generalization

triple, the Hassan triple is constructed within the last event of the Golrokh’s triple, whose

meaning is revealed at her wedding to Hassan accompanied by King’s blessing: ”Didn’t I

tell you, Beloved Father, that the lining protects the coat not the other way around.

Woman is the lining, man is coat, and it is the woman that makes the man, not other way

around ” - says Golrokh to her father. After two older sisters provided expected answers


75



to the King and were married off to visirs, she may have concluded that to avoid this fate,

she had to contradict the father.

In general, fairy tales leave it to the reader to discover meaning of the triple’s learning

progression, however, the Hungarian fairy tale Ilonka found in Crimson Fairy Tale

collection (A.Lang, 2006) gives us an explicit argument supporting the conjecture of the

fairy tale triples as learning progressions.

Ilonka, the heroine is cheated by the swineherd’s daughter from marrying the King who

is in love with her. She receives a magical spinning spindle, which can produce unlimited

amount of silk. The new Queen, the swineherd’s daughter, desires the magical spindle;

however Ilonka asks in exchange to be able to spend one night in the King’s room. The

queen agrees, however, she gives the king a sleeping draught, and Ilonka’s efforts to be

recognized by the King are in vain. This repeats on the second attempt. After those two

attempts “…some King’s servants had taken a note of the situation, and warned their

master not to eat nor drink anything that the queen offered him, as for two nights running

she had given him a sleeping draught.” The fairy tale asserts in these words, that the two

first events are enough for servants to learn from the situation and to warn their master,

what of course resolves the story into the third, transcendental event of marriage between

Ilonka and the king. The first two events in such a learning progression are enough to

make it possible by the heroine (hero) or her well wishers to make the inter-operational

comparison and to draw the lesson/conclusion for the construction of the third

transcending event.


76



Conclusions

If one accepts the general nature of the argument presented in previous sections, then we

are confronted with an interesting but not unfamiliar situation: two significantly different

domains of human experience, a mathematics classroom and a fairy tale share similar

developmental process, a learning triple. If we apply that learning triple structure to the

situation we are confronting then we have to conclude, moving to the trans-operational

stage of the argument, that the learning triple is a general process of learning, and,

possibly it can become the objective basis for learning trajectories in mathematics, among

others.

The presented argument naturally bears strong resemblance to the design of the

presentation of the Triad by Piaget and Garcia, (1989), whose significance rests on the

inter-operational comparison of exactly two different domains of human development,

history of science and child’s intellectual development. As aside argument, it’s

interesting to note that the Triad is formally introduced in the chapter on Algebra, after

two chapters on Pre-Newtonian Physics and Development of Geometry, where its

particular, intra-operational formulations are made.

The triple of learning had been successfully utilized and observed in different courses of

mathematics, among others, in calculus, while scaffolding students’ understanding of the

limit with the help of Achilles and Tortoise paradox (Prabhu, Czarnocha, 2004) through

mathematics essay writing, and in logic, while conducting classroom investigations of

students’ understanding of universal and existential quantifiers (Ye, Czarnocha, 2012).


77



The triple is also effective in the design of short teaching sequences, or short learning

trajectories (Baker et al, 2012)

REFERENCES

Baker, W., Czarnocha, B., Dias, O., Prabhu,V. (2012) Problem Solving in Remedial

Arithmetic: Jumpstart to Reform. ICME 12, PP Presentation, TSG 15 Problem

Solving, Seoul, South Korea, July 2012.

Clements, D.H. and Sarama J. (2009) Early Childhood Education Research; Learning.

(ds) Trajectories for Young Children, Rutledge, Taylor and Francis group, New

York ,London.

Clements, D.H. and Sarama,J. (2004) Learning Trajectories in Mathematics Education. In

Clements, D.H. and Sarama,J. (eds), Special Issue: Hypothetical Learning

Trajectories. Mathematical Thinking and Learning, V6,N2,2004.

Confrey, J., Maloney, A., Nguyen, K., Mojica, G., & Myers, M. (2009).

Equi-partitioning/splitting as a foundation of rational number reasoning. In

M. Tzekaki, M. Kaldrimidou & C. Sakonidis (Eds.) Proceedings of the 33rd

Conference of the International Group for the Psychology of Mathematics

Education, Vol. 1. Thessaloniki, Greece: PME.

Czarnocha, B. and Prabhu, V. (2004) Teaching-Research and Design Experiment – Two

methodologies of Integrating Research and Classroom Practice. Proceedings of

the epiSTEME -1, Goa, India.


78



Empson, S. On the idea of learning trajectories: Promises and Pitfalls. The Mathematics

Enthusiast, ISSN 1551-3440, Vol. 8, no.3, pp.571-596.

2011©Dept of Mathematical Sciences- The University of Montana Information

Age Publishing.

Grimm, William and Jacob (1992) The Complete Fairy Tales of the Brothers Grimm.

Jack Zipes (ed), Bantam Books

Guppy, S. (2009) The Secret of Laughter. Magical tales from Classical Persia.Tauris

Back Paperbacks.

Lang, A. (2006) Traditional Folk Tales and Fairy Stories From Around the World.

Published by Shoes and Ships and Sealing Wax Ltd. Omnibus edition.

Piaget, J. and Garcia, R. (1989) Psychogenesis and the History of Science. Columbia

University Press, New York.

Propp, V. (1975) Morphology of the Folktale. University of Texas Press, Austin and

London; (translation from first Russian edition of 1927).

Simon, M.A. (1995) Reconstructing mathematics pedagogy from a constructivist

perspective. Journal for Research in Mathematics Education, 26, 114-145.

Steffe, L.P. (2004) The case of commensurate fractions. In Mathematical Thinking and

Learning,V6,N2, 2004.

Ye, R. And Czarnocha, B. (2012) Universal and Existential Quantifiers Revisited in Tai

Yih-Tso (ed) Proceedings of the 36 Conference of the International Group for

Psychology of Mathematics Education, July 2012, Taipei, Taiwan


79



TEACHING ELEMENTARY and SECONDARY TEACHERS in UGANDA: MATHEMATICAL PROBLEM SOLVING, MATHEMATICS for UNDERSTANDING, and REDUCTING MATHEMATICS ANXIETY

Brian R. Evans Janet Mulvey Pace University Pace University New York, United States New York, United States


80



Abstract

This purpose of this article is to present the work two professors conducted with

elementary and secondary teachers in Uganda. Mathematical problem solving,

mathematics for understanding, and reducing mathematics anxiety techniques were

taught to elementary and secondary school teachers in the last two weeks of August 2011.

The authors found this experience to be very rewarding, and believed it was rewarding

for the teachers with whom the authors worked. It was found that while teachers had

fairly strong to very strong basic content knowledge in mathematics, many struggled with

problem solving and conceptual understanding of mathematics.


81




82



Introduction

The purpose of this article is to present the work of two university professors in a

remote village in Uganda, a country on the continent of Africa. We based our work on

current benchmarks for elementary and secondary mathematical standards, a

methodology for teaching and learning in Uganda. Researching practices implemented in

the rural and often unequipped classrooms, we presented a more authentic way to explore

the conceptual base for higher level thinking and to teach mathematical problem solving

and understanding.

Teachers from all levels within the village attended the two week hands-on

seminar and participated in practices aimed at reducing mathematics anxiety and

establishing authentic learning experiences. We outline the process and procedures used

to motivate the teacher participants, to simulate student experiences, and to alter rote

practices into more relevant problem solving methodologies.

This article follows the reluctance and gradual acceptance and excitement in

teaching and learning in mathematics for the elementary teachers. The participant

secondary teachers, who had fairly strong basic procedural content knowledge in

mathematics, began to learn and embrace problem solving and conceptual understanding

of mathematics through authentic learning experiences.

The arrangement for working with teachers in Uganda was made through a

relationship with a charity, Educate Uganda, founded in 2006. A previous visit had

familiarized the second author with the needs of the community. It was determined that

working with established teachers within the schools would have potential for the greatest


83



impact. There were three goals applied for this current visit: work with teachers to use

mathematical problem solving strategies in place of pure rote learning; reinforce and

solidify conceptual understanding of mathematics for the teachers themselves; and apply

techniques that would reduce mathematics anxiety for the students in the schools.

Background

Uganda, slightly smaller than the U.S. state of Oregon, is bordered by Kenya,

Tanzania, Rwanda, the Democratic Republic of the Congo, and South Sudan. The

country is located to the north of Lake Victoria and has been called the “Pearl of Africa”

by Winston Churchill (Children of Uganda, 2011a). The population of over 34 million

people is made up of numerous tribes who live in mostly rural areas (CIA World

Factbook, 2011). Agriculture, mineral mining, and oil production make up the natural

resources and industry for the Ugandan economy (CIA World Factbook, 2011). Most of

the rural residents farm their own small land, and raise animals for self-sustenance and

trade. Many in these areas rely on water collected from rain barrels, rivers, or village

wells.

Uganda gained its independence from the United Kingdom in 1962 (Children of

Uganda, 2011a). From 1971 to 1979, Uganda was ruled by the brutal dictator, Idi Amin,

whose rule resulted in the death of approximately 300,000 to 500,000 Ugandans

(Keatley, 2003). Amin expelled approximately 35,000 Asians from Uganda in 1972,

which led to economic hardship for the country (Keatley, 2003). Since 1987, the Lord’s

Resistance Army (LRA) has engaged in armed rebellion in northern Uganda (Children of


84



Uganda, 2011a). While life in Uganda has stabilized and the economy has begun to

improve, there is a high level of poverty, especially in rural areas.

The poverty rate has resulted in a high mortality rate among children less than

five years of age from malaria, respiratory illness, and diarrhea. Polio is also prominent,

leaving many children with physical impairments and little care. The average life

expectancy is 52 years for women and 51 years for men (Build Africa, 2011). There are

20,000 babies infected by HIV annually through mother-to-child transmission (UNICEF,

2011). Thus, the fate of children’s health for future growth is a major factor in societal

education, social stability, and development (Children of Uganda, 2011b).

Travel to Uganda

In the middle of August 2011 we left New York for the Entebbe airport on Lake

Victoria in central Uganda, which is about 20 miles west of the capital, Kampala. From

there we made our way to the destination, the town of Nkokonjeru, which has a

population somewhat under 15,000 people and is also in central Uganda near Lake

Victoria about 30 miles (50 kilometers) east of Kampala. Nkokonjeru is a beautiful town

situated in the green rolling hills of central Uganda. Like much of Uganda, Nkokonjeru is

not affluent by Western standards. Per capita GDP in Uganda is approximately $1300,

which makes it one of the poorest countries in the world (CIA World Factbook, 2011).

We were able to determine that a teacher’s yearly salary is approximately equal to the per

capita GDP in Uganda. Most people in the town do not have running water in their

houses, but rather carry water in jerry-cans from a water source or collected rainwater.

Many of the residents do not have electricity in their houses, and during the visit


85



electricity was unavailable, for those who have access to electricity, most of the time.

Several severe rain storms struck the town during our stay, which inevitably kept all

electricity off for several days at a time.

Education in Uganda

The education systems consists of primary school (P1-P8) and secondary school

(S9- S12) with the primary school completion rate of 56 percent, according to 2009

statistics, while the average amount of schooling for the population is 3.5 years (Nation

Master, 2011). Education has become a priority for the Ugandan government, according

to the Build Africa Organization (2011). The country has begun to realize the importance

of prioritizing education mandating primary education for all and funding secondary

schooling. Funding for education has become the largest budget item for the government

allowing access for many more children in all areas of the country. The amount of

funding is, however, insufficient to afford adequate resources, teachers and schools

(Build Africa, 2011).

Education in Nkokonjeru

Resources at the schools were limited as well. The schools are public under the

Ugandan Ministry of Education and Sports, but are overseen by the Catholic Church in

Nkokonjeru. In the first week we taught in a school that consisted of one large classroom.

There was no plumbing or electricity infrastructure for this building. Any available

bathrooms are outside and consist of pit toilets. In the second week of instruction, the

school building had inconsistent electricity for basic lighting, but no other infrastructure


86



in the building. The teachers informed us that their classes range between 30 and 100

students, but average about 60 per class.

Near the schools are a hospital and an orphanage both under the name, the

Providence Home. The Providence Home is administered by the Catholic Church, but has

poor conditions that would not be found in Western hospitals and orphanages. Visiting

both hospital and orphanage resulted in an understanding of the difficulties for the

children living in the town. Many of the children in the orphanage have been abandoned

by their parents due to illness or disability, both physical and psychological, greatly

affecting their ability to learn.

Attendance in schools is affected by the means to purchase mandated uniforms,

books, and material. Shortage of teachers, distance to schools in rural areas, cost of

materials, and the fact that one in five children is an orphan are reasons for poor

attendance. Health issues and hunger among school age children affect attendance and

learning. According to an interview with a primary school teacher in the village of

Nkokonjeru, some children often get their only sustenance at school, causing them to be

listless and unfocused during instruction time.

The large size of the classes impedes any real opportunity for individual attention

or differentiation of learning needs. Exploration of concepts and number sense in

mathematics is difficult to achieve in such large classes and that teachers themselves have

not been educated in concept development, which results in rote learning and

memorization. Teacher training begins immediately following secondary school

graduation in a two year post-secondary program. Foundations for learning in Uganda are


87



based on outdated systems founded in religious school experiences of structured

repetition and memory. Counting is automatic without emphasis on one-to-one

correspondence and number sense. Addition and subtraction are taught from rote and

repeatedly practiced, multiplication tables are memorized without understanding the

concepts, and formulas are utilized at the secondary levels with little or no application.

The planning for the development of conceptual learning through problem solving

was based upon the prior year’s experience in 2010. Prospective teachers in the post-

secondary local college program in Nkokonjeru attended class to prepare them for

mathematics teaching in their future classrooms. It was evident after a few days that their

ability to employ problem-solving strategies through mathematics stories was not fully

developed. Constant modeling and scaffolding brought some results by the end of the

second week and students were able to create simple mathematics word problems based

on their own understanding and application. It was, however, unclear they were ready to

employ these strategies in the classroom.

The plan for the 2011 teaching trip, the primary focus of this article, used the

knowledge from the prior year to build effective learning strategies for the teachers who

would be taught by us in 2011. Stories relating to the lives of the teachers in the town

brought meaning and understanding to the development of theory in concept

development.

Problem Solving

The primarily focus of instruction was on mathematical problem solving,

mathematics for understanding, and techniques for reducing mathematics anxiety. The


88



first week was spent with the elementary teachers and the second week with the

secondary teachers, although some attended both sessions. The attending teachers lived in

or near Nkokonjeru.

Problem solving continues to be of high importance in mathematics education

(National Council of Teachers of Mathematics [NCTM], 2000; Posamentier & Krulik,

2008; Posamentier, Smith, & Stepelman, 2008). It is one of the five NCTM process

standards (NCTM, 2000), and is important in how students best learn mathematics

(Posamentier et al., 2008). The National Council of Supervisors of Mathematics (NCSM)

has considered problem solving to be the principal reason for studying mathematics

(NCSM, 1978), and it is recommended that mathematics should be taught through a

problem solving approach (NCTM, 2000; Schoenfeld, 1985).

To understand problem solving, first the definition of a mathematical “problem”

must be understood. Charles and Lester (1982) defined a mathematical problem as task in

which (a) The person confronting it wants or needs to find a solution; (b) The person has

no readily available procedure for finding the solution; and (c) The person must make an

attempt to find a solution. According to Krulik and Rudnick (1989), problem solving is a

process in which an individual uses previously acquired knowledge, skills, and

understanding to satisfy the demands of an unfamiliar situation. Polya (1945), in his

seminal work How to Solve It, outlined a general problem solving strategy that consisted

of (a) Understanding the problem; (b) Making a plan; (c) Carrying out the plan; and (d)

Looking back. We guided teachers to develop the general problem solving method

published by Polya, in addition to the problem solving strategies such as look for a


89



pattern, work backwards, guess and check, draw a picture, orderly listing, use a model,

and solve a simpler problem.

The Uganda Ministry of Education and Sports places emphasis on problem

solving, though many teachers have learned and therefore teach through rote learning and

memorization. The Ministry of Education and Sports published the Mathematics

Teaching Syllabus (2008) for secondary school mathematics teaching and said,

Practical problem solving should be an everyday part of the mathematics

curriculum. Problems should be chosen as to link the concepts and skills acquired

in the mathematics lesson with their applications in problem situations that arise

in the environment of the students. This not only gives the learner a chance to

consolidate the new concepts and techniques that have been learned but also

allows him or her to appreciate the power of the subject as a tool to understand,

interpret and control the environment. (p. x)

We observed that the teachers had fairly strong to very strong basic procedural content

knowledge in mathematics, but struggled with problem solving and conceptual

understanding of mathematics.

Problems were selected from literature unfamiliar to the teachers representing

authentic problem solving. Problems were selected so that concepts behind them were at

an elementary level, thus teachers could give these problems to their students and with

the proper scaffolding students could solve the problems using their critical thinking

skills and prior knowledge to satisfy the demands of the unfamiliar situation. Problems


90



were adapted for the experiences of the teachers in Uganda. For example, we used the

classic “locker” problem.

A school has exactly 1000 lockers and 1000 students. The students decide that the first

student will enter the school and open all of the lockers. The second student will then

enter the school and close every locker with an even number (2, 4, 6, 8,…). The third

student will enter the school and reverse every third locker (3, 6, 9,…). In other words, if

the locker is closed the student will open it. If open, the student will close it. The fourth

student will then reverse every fourth locker, and so on until every student in turn has

entered the school and reversed the proper lockers. Which lockers will remain open?

Adapting to the physical structure of schools in Nkokonjeru, doors were

substituted for lockers. We were able to physically demonstrate the basis behind the

problem using the several doors in the classroom. It was required that teachers solve the

problems using several methods or representations. In this problem teachers solved a

simpler problem of 10 “doors.” Many used a table to record the number of doors that

remained open. Another method that was tried was to have 10 teachers stand in front of

the class to represent the “doors” in which other teachers could turn the “doors”

accordingly. A teacher facing the class was an open door and a teacher facing the board

was a closed door. Many teachers successfully found that doors 1, 4, and 9 remained

opened, which are the perfect squares and have an odd number of factors. An odd number

of factors meant that an odd number of people “reversed” the doors. Since doors started

closed, an odd number of “reversing” led to a closed door in the end. Teachers benefited


91



from having to think critically and explore multiple representations for solving this

problem.

Good problem solving practice requires collaborative thinking and sharing.

Teachers were grouped randomly to work together to solve problems. The teachers did

not have experience working in collaborative groups, and found it difficult at first.

However, as time and problems progressed they seemed to become more comfortable and

supportive within the groups. The Ministry of Education and Sports (2008) said,

Learners should be encouraged as much as possible to work together in solving

problems and conducting mathematical investigations to become familiar with the

processes in mathematics. This will help to eradicate anxiety and promote

cooperation within the class as a whole. (p. x)

Through observations we found that the teachers adapted to working together fairly

quickly. We altered the classroom from single rows to grouped desks thus making

collaborative groups more natural and collegial. Constant encouragement was given for

shared ideas and multiple methods of solving problems.

We presented several problems to the teacher groups to solve and asked each

group to share and demonstrate the methods they used. Following several opportunities

for practice, we required the teachers to develop their own problems in their groups to

present and use as lessons for the rest of the class. At first the teachers imitated the

problems we had presented. Gradually the problems became more original and

sophisticated, and the teachers began developing authentic problems related to their own

environment and experiences.


92



Working in collaborative groups, teachers who were at first reluctant and unsure

improved in their understanding of problem development and were more able to create

authentic problems for their students. At the end of the two weeks all teachers had

improved their problem solving abilities and were able to create their own authentic

problems for their students to solve.

Mathematics for Understanding

Ball, Hill, and Bass (2005) proposed conceptual Mathematical Knowledge for

Teaching (MKT). Ball et al. (2005) defined MKT as “a kind of professional knowledge

of mathematics different from that demanded by other mathematically intensive

occupations, such as engineering, physics, accounting, or carpentry” (p. 17). If teachers

possess strong conceptual MKT, they are better able to use their content mathematics

knowledge to support the conceptual learning of their students. For example, Ball et al.

(2005) said it is insufficient for an elementary school teacher to simply know how to

multiply, but rather the teacher also needs to know how to explain multiplication to

students and to recognize student mistakes.

Teachers were provided with problems that enabled them to help their students

understand the concepts behind the mathematics. For example, the class discussed the

reasoning behind the area formulas for various geometric shapes and the reasons they

hold true, including the triangle, trapezoid, and circle. The class also discussed the

reasoning behind the algorithms for addition, subtraction, and multiplication and how

they work based upon number place value.


93



Teacher preparation in towns similar to Nkokonjeru too often rely on lecture, rote

learning, and memorization for their students. Exploring the underlying concepts and

reasons that concepts work the way they do, teachers were able to gain a better

understanding of deeper mathematics and be better equipped to teach their students in

this manner. We acknowledge that a shift in thinking like this takes much time, practice,

and support. It is hoped that we were able to expose the teachers to a new way of

thinking, teaching, and learning.

Mathematics Anxiety

Mathematics anxiety is defined by Richardson and Suinn (1972) as the feeling of

“tension and anxiety that interfere[s] with the manipulation of numbers and the solving of

mathematical problems in a wide variety of life and academic settings” (Suinn &

Winston, 2003, p. 167). Rossnan (2006) defined mathematics anxiety “as feelings of

tension and anxiety that interfere with the manipulation of numbers and the solving of

mathematical problems in a wide variety of ordinary life and academic situations” (p. 1). Mathematics anxiety can cause a reduction in self-confidence (Rossan, 2006; Tobias,

1993), and students with low level knowledge of mathematics can have anxiety that can

lead to avoidance of higher level mathematics courses and hinder career opportunities

(Ma, 1999; Tobias, 1993). Alleviating mathematics anxiety in students could lead to

better economic opportunities and better mental health (Ma, 1999).

Teachers seemed unfamiliar with methods to reduce mathematics anxiety among

their students, so the topic of reducing mathematics anxiety for students was addressed

during our time with the teachers. Witnessing the teachers’ own anxieties, we helped


94



them develop methods for reducing anxiety for their students in the classroom.

Additionally, we brought researched-based strategies from the literature that addressed

mathematical understanding in contrast to memorization and rote learning. Removing the

pressure of examinations, connecting mathematics to the real world, using students’

interests in creating problems, and demonstrating enthusiasm and enjoyment toward

mathematics will improve student learning and help reduce mathematics anxiety.

Conclusion

At end of the two weeks one teacher from each collaborative group was selected

to publicly express gratitude for the work that was done with them. It was quite touching

to hear their appreciation and acknowledgement of how much they had learned during the

time spent together. We learned much from the teachers as well. We learned about the

methodology implemented in mathematics classes in Ugandan schools and the

approaches teachers take to teaching. We recognized that while classroom practice is

often based upon lecture, rote, and memorization, Ugandan standards emphasize real

world connections, problem solving, and work in collaborative groups. We also

recognized that this situation is not so different than the situation in many classrooms in

the United States. While U.S. standards also focus on real world connections, problem

solving, and collaborative group work, many U.S. teachers also rely heavily on lecture,

rote, and memorization.

Teachers often have to contend with very limited resources and low salaries.

Many teachers in Uganda earn only $3 to $4 per day. And while many U.S. teachers in

less affluent urban and rural districts have to struggle with limited resources and


95



inadequate pay, teachers in Uganda confront a system that is much more lacking in

resources than in the United States.

The gender gap in Nkokonjeru is pronounced and part of the culture. Most

females are not educated, have many children, and are primary care-takers in the home.

Females who teach do so primarily at the elementary level, while males, who are

culturally dominant, teach at the secondary level, particularly in mathematics. We

acknowledge that while less pronounced, elementary school teachers in the United States

are primarily female as well and many male teachers reside in secondary mathematics

classrooms. The first author of this article is male and the second female. The second

week was spent exclusively with male secondary mathematics teachers, and their

discomfort with a female instructor was obvious. It took several days proving

competence and knowledge for the male teachers to accept a university female instructor.

Several of the teachers told us that this was not a situation in which they were familiar.

Teachers in the program had a strong understanding in fundamental basic

procedural mathematics. The need was to integrate problem solving strategies and

mathematics for understanding. The lack of fundamental mathematics knowledge of

teachers in the United States has been widely known for some time (Ma, 2010). Perhaps

more investigation into the Ugandan teachers’ acquisition of fundamental mathematics

can help inform U.S. teacher education.

Prior to our departure, we were asked to return for follow-up and continuation.

We hope to return with university students to work both with teachers and students in the

classroom. We believe there are mutual benefits from this experience for the children and


96



teachers in Nkojkonjeru and for university students from the United States.

Understanding the process of teaching/learning from a global perspective brings critical

insight and understanding in cultural differences and the development of learning.

Integrating viewpoints, cultural influences, and methodologies open a new paradigm in

on both sides of the globe. As the world quickly moves toward increased globalization,

diverse perspectives can be adopted by teachers in both countries, benefiting both

Ugandan and U.S. school students.

The plan to return when school is in session allows access into classrooms with

the teachers to mentor and to assist them in implementing the ideas generated during our

previous work together. We would like to expand our work to include the orphans in the

Providence Home. The children at the orphanage are in need of special assistance and by

recruiting special education faculty and graduate students we hope to provide an

educational process that can be supported and continued year after year.

Working with these hardworking, dedicated, and bright teachers was a

wonderful and fulfilling experience. We feel it has been a transformative experience for

the Nkokonjeru teachers and indeed for us as well. Children in the village classroom

benefit from a more authentic and comprehensive practice in learning mathematics and

we benefit from challenging our own perceptions of education from diverse backgrounds.

Note: The first author’s background is in mathematics education at both the elementary

and secondary school levels. The second author’s background is elementary education,

special education, and educational leadership.


97



References

Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who

knows mathematics well enough to teach third grade, and how can we decide? American

Educator, 14-17, 20-22, & 43-46.

Build Africa. (2011). Uganda facts and figures. Retrieved from

www.build-africa.org/pages/stories-from-uganda.html

Charles, R. I., & Lester, F. K., Jr. (1982). Teaching problem solving: What, why, & how.

Palo Alto, CA: Seymour.

Children of Uganda. (2011a). About Uganda. Retrieved from

www.uccf.org/about/about_children_of_uganda.cfm

Children of Uganda. (2011b). Children of Uganda: Facts about the children of Uganda.

Retrieved from www.uccf.org/facts/facts_about_the_children_of_uganda.cfm

CIA World Factbook. (2011). CIA world factbook Uganda. Retrieved from

https://www.cia.gov/library/publications/the-world-factbook/geos/ug.html.

Keatley, P. (2003, August 17). Idi Amin. The Guardian. Retrieved from

http://www.guardian.co.uk/news/2003/aug/18/guardianobituaries

Krulik, S., & Rudnick, J. A. (1989). Problem solving. Boston, MA: Allyn and Bacon.

Ma, L. (2010). Knowing and teaching elementary mathematics: Teachers’ understanding

of fundamental mathematic in China and the United States. New York, NY:

Routledge.


98



Ma, X. (1999). A meta-analysis of the relationship between anxiety toward mathematics

and achievement in mathematics. Journal for Research in Mathematics

Education, 30(5), 520-540.

Ministry of Education and Sports. (2008). Mathematics teaching syllabus: Uganda

certificate of education. Kampala, Uganda: National Curriculum Development

Center.

Nation Master. (2011). Africa Uganda education. Retrieved from

http://www.nationmaster.com/country/ug-uganda/edu-education

National Council of Supervisors of Mathematics. (1978). Position paper on basic

mathematical skills. Mathematics Teacher, 71(2), 147-152.

National Council of Teachers of Mathematics. (2000). Principles and standards for

school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Richardson, F., & Suinn, R. N. (1972). The Mathematics Anxiety Rating Scale:

Psychometric data. Journal of Counseling Psychology, 19(6), 551-554.

Rossan, S. (2006). Overcoming math anxiety. Mathitudes, 1(1), 1-4.

Schoenfeld, A. (1985). Mathematical problem solving. New York, NY: Academic Press.

Suinn, R. M., & Winston, E. H. (2003). The mathematics anxiety rating scale, a brief

version: Psychometric data. Psychological Reports, 92(1), 167-173.

Polya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press.

Posamentier, A. S., & Krulik, S. (2008). Problem-solving strategies for efficient and

elegant solutions grades 6-12. Thousand Oakes, CA: Corwin Press.


99



Posamentier, A. S., Smith, B. S., & Stepelman, J. (2008) Teaching secondary

mathematics: Teaching and enrichment units (8th ed.). Boston, MA: Allyn &

Bacon.

Tobias, S. (1993). Overcoming math anxiety. New York, NY: W. W. Norton Company.

UNICEF. (2011). Information by country: Uganda. Retrieved from

www.unicef.org/infobycountry/uganda_statistics.html

using counter-examples in teaching & learning of … teaching-research journal online ... using...

Documents