The teaching of Calculus in the secondary and postsecondary levels ...

The teaching of Calculus in the secondary and

postsecondary levels. Examples from textbook

analysis and teacher practices.

Alejandro S. González-Martín Département de Didactique

Université de Montréal

Contents – Introduction. – Some tools from ATD. – Textbook analysis: Real numbers. – Teacher practices: Numerical series. – Conclusions.

July 25 2014 CIEAEM66

Contents


– Introduction. – Some tools from ATD. – Textbook analysis: Real numbers. – Teacher practices: Numerical series. – Conclusions.

Introduction

• Mathematics and realities. – Fundamental questions about mathematics. – Their existence. – Their discovery. – Their relations with other sciences. – But also issues related to their teaching and

learning in the twenty-first century.


Introduction • “Mathematical literacy is an individual’s capacity

to formulate, employ, and interpret mathematics in a variety of contexts. It includes reasoning mathematically and using mathematical concepts, procedures, facts and tools to describe, explain and predict phenomena. It assists individuals to recognise the role that mathematics plays in the world and to make the well-founded judgments and decisions needed by constructive, engaged and reflective citizens”(PISA, 2012)


Introduction


Improper integrals Derivatives

Numerical series

Real numbers

Continuity Functions

Introduction • “Previous reports show that not only are too few students pursuing

Science, Technology, Engineering, or Mathematics (STEM) fields, but also many who originally intend to pursue these fields leave after their experiences in introductory STEM courses” (Rasmussen & Ellis, 2013)

• “In the US and elsewhere, calculus often functions as a filter, preventing large numbers of students from pursuing a career in science, technology, engineering or mathematics (STEM)” (Ellis, Kelton & Rasmussen, 2014)

• “A better understanding of the relationship between students’ instructional experiences in Calculus I and their decisions to continue taking additional calculus is needed to improve student success in calculus and their continued interest in STEM careers (Ellis, Kelton & Rasmussen, 2014)


Introduction


Research on the learning of given topics

Programs to help first-year students

Technology Engagement, motivation

Book analysis Practices

Resources Teachers’ training

Theoretical approaches

Introduction


Introduction


CERME WG14: University mathematics education

Introduction


Eds: Biza, González-Martín, Gueudet, Nardi, Winsløw

1. Anthropological Theory of the Didactic

2. Theory of Didactic Situations

3. Instrumental and Documentational Approaches

4. Communities of Practice and Enquiry

5. Commognition

Contents



Some tools from ATD

Important role of institutional choices Anthropological approach (Chevallard, 1999).


Some tools from ATD • ATD tries to develop a better understanding of

the choices made by an institution to organise the teaching of a concept, as well as the consequences of these choices for the significance given to the concepts taught and the attained learning.

• Mathematical objects are not “absolute” objects, but entities which emerge through practices in given institutions.


Some tools from ATD • These practices can be described in terms of:

– Tasks (t, being T a type of tasks). – Techniques (τ) used to complete these tasks. – Technology (θ) or discourse which justifies and explains the

techniques. – Theory (Θ) which includes the discourse.

• To understand the meaning given within an institution to “know/understand a mathematical object”, we need to identify the practices which put this object into play within the institution.


Some tools from ATD Institution (I ): a social organisation which

allows, and imposes, on its subjects (every person x who occupies any of the possible positions p offered by I ) the development of ways of doing and of thinking proper to I .

Object : any entity, material or immaterial, which exists for at least one individual; in particular, any intentional product of human activity is an object.


Some tools from ATD Every subject x has a personal relationship with any object o,

denoted as R(x, o), as a product of all the interactions that x can have with the object o. This personal relationship is created, or modified, by entering in contact with o as

it is presented in different institutions I, where x occupies a given position p. From this personal relationship, a learner will constitute what one could

designate as being ‘knowledge’, ‘know-how’, ‘conceptions’, ‘competencies’, ‘mastery’, and ‘mental images’.

The institutional relationship with o in position p, RI (p, o), is the relationship with the object o which should ideally be that of the subjects in position p within I. It is mainly forged through the exercises (or tasks), and not only through the

theoretical explanations. Also forged through the use of elements (as symbols, images…) to refer to, or to

manipulate, the mathematical notions to be constructed.


Some tools from ATD A praxeological analysis characterises the institutional

relationship with a mathematical object within a given institution. – In particular, in the analysis of tasks, we can identify the practical

block (or know-how), which is formed by the first two elements (types of tasks and techniques).

– Further a consideration of a discourse to describe, explain and justify what is done, is the focus of the knowledge block (or theoretical block) of mathematical activity (including the technology and the theory).

– Both blocks are fundamental aspects of the anthropological model of mathematical activity which can be used to describe mathematical knowledge.



Contents


Textbook analysis: Real numbers

• González-Martín, A.S., Giraldo, V. & Souto, A.M. (2013). The introduction of real numbers in secondary education: an institutional analysis of textbooks. Research in Mathematics Education, 15(3), 230-248.


Textbook analysis: Real numbers • Background

Literature has identified some difficulties in the understanding of real numbers by students (and even by teachers).

There are not many research works concerning

the introduction of this topic by the textbooks.



Even university students give incorrect definitions for irrational numbers and are unable to explain the necessity to extend the field of rationals (Soares, Ferreira & Moreira, 1999).

Some future teachers associate irrational numbers to square roots and π (Sirotic & Zazkis, 2007).

17-18 years students: real numbers as the reunion of naturals, integers, rationals and decimals, with some other numbers.



Some teachers’ conceptions about the structure of the real line and the notion of density coincide with those present in their students (Dias, 2002).

Atomistic models and almost discrete conceptions (Dias, 2002; Robinet, 1986).



Irrational numbers are seen as numbers with an infinite decimal representation (Robinet, 1986; Fischbein, Jehiam & Cohen, 1995).

Upper secondary / college students give more importance to the different writings of these numbers than to their specific properties (Robinet, 1986).


Textbook analysis: Real numbers • Background • Bronner (1997)

• Textbooks play a significant role in the construction of an institutional

relationship with mathematical topics within an institution. • Secondary school textbooks: lack of an explicit characterisation as “new

numbers” for some square roots. • The dominant approach is algebraic, dissociated from the measurement of

objects: the square root appears with a strong algebraic character, as an algebraic operator and through some aspects of algorithm for calculations (reinforced by the “√” key in calculators).

• In high school, the set of real numbers appears introduced almost as a “big bag” (p. 59), based on the initial perception of “number”, independent of their arithmetic nature; the real line is also introduced.


Textbook analysis: Real numbers • Background • Bronner (1997)

• Didactic void: lack of an explicit negotiation, both in curricula and in textbooks, for the extension from decimals or rationals to real numbers.

– to emphasise the rules for the calculation with different types of numbers (being these rules sometimes those used to define a field), but without any indication of the set of reference for these calculations, or the nature of the numbers implied:

– to present the different types of number, the different sets, and the notations for each set.

• This mathematical knowledge to be taught (Barbé, Bosch, Espinoza and Gascón 2005) plays an important role in teachers’ practices, being one of the elements which restrict the choices made by the teachers.


Textbook analysis: Real numbers • Sample

• Textbooks approved by the Ministry of Education, to be used in public schools.

• 9 titles for grades 6 to 9 (11 – 14 years)

• 5 titles for grades 10 to 12 (15 – 17 years)


Textbook analysis: Real numbers • Definitions, properties, and examples • The didactic process of the introduction of real numbers

requires a first encounter with the MO in question.

• In the case of real numbers, textbooks do not seem to be explicit enough and do not answer to questions such as: What mathematical knowledge is to be taught? Why is it important? Why is it useful?

• Instead, they seem to privilege the introduction of a set of definitions and properties.


Textbook analysis: Real numbers • Definitions, properties, and examples • Two definitions:

– DA: “Irrational number is a number which cannot be written in the form of a fraction” 5 textbooks.

– DB: “Among numbers written in decimal form, there are numbers with infinite non-periodic decimals, called irrationals” 8 textbooks.

– “Any rational or irrational number is a real number” (DC) all textbooks.


Textbook analysis: Real numbers • Definitions, properties, and examples

• 9 textbooks introduce the concept of irrational number through examples of numbers which are not rational, before the definition.

– A problematisation for the introduction or the existence of these “new” numbers is absent.

– 6 use √2 as the introductory example.

– 3 use examples of numbers with non-periodic decimal expansions.



• In all the textbooks: examples concerning irrational numbers are mainly used to illustrate definitions and properties.

– The decimal expansions of √2, √3, and π are assumed to be known.

– In general, there is no justification to show that the decimal writings of these numbers do not have a period.



• In order to define irrational numbers, both DA and DB suppose the existence of another type of number which is not rational.

• DC comes after the definition of irrational and also depends on it.

• DB is based on the assumption that every real number admits a decimal representation.



• Examples are mainly used to illustrate properties.

• None of the properties is proved or discussed.

• Idea that a property can be proved through the verification of some examples?

• 7 textbooks state the density property of rational numbers, 3 textbooks the density of real numbers, 0 textbook the density of irrational numbers.



• Transparent rules.

• The raison d’être of this set of properties is absent, and serves only to solve some tasks which are given to practise them; the choice is to institutionalise them as transparent rules.

• Also, elements of what Bronner (1997) identified as the second strategy to reduce the institutional didactic void: to emphasise the rules for the calculation with these numbers.


Textbook analysis: Real numbers • Tasks

• Classifying a given number as rational or irrational (TCR) • Determining a fraction equal to a rational number given in decimal

writing (TFR) • Finding a rational or irrational number between two given numbers

(TBET) • Finding a finite decimal approximation for a given irrational number

(TAPP) • Ordering real numbers (TORD) • Representing numbers in the real line (TRL)



• Absence of questions trying to promote a conceptualisation of real and irrational numbers.

• Real numbers appear to be reduced to a list of properties and to a definition which does not question their existence.



• The majority of tasks (t) favour the reproduction of given techniques (τ, which are previously exemplified) and of algorithms to get solutions.

• The justification θ of these techniques is often out of reach for this school level.

• The list of properties and techniques which is given is not proved nor justified in any way.

• Incomplete praxeology.


Textbook analysis: Real numbers • Comments

• Definitions are formally inconsistent.

• The justification of properties is based exclusively on the use of examples.

• Mathematical argumentation is almost absent.

• The coordination of different representations is almost absent.

• Little discussion about the theoretical need for the construction of real numbers, as what type of problem rationals are unable to solve.



• We do not intend to advocate a formal approach to real numbers.

– The mere presentation of procedures and representations, without any kind of argumentation, is not likely to contribute to students’ understanding.

• Instructional approaches should adopt a more problematic approach.

– Focus more on mathematical problems that engender the concept of real numbers.

– Not just trying to formalise, to avoid, or to take things for granted.



• We think our sample of Brazilian textbooks share characteristics of secondary textbooks in other countries.

• If teachers simply follow the approaches proposed in the textbooks, then students will be unlikely to be able to build an adequate structure to deal with real numbers and their properties in undergraduate studies.



Contents


Teacher practices: Numerical series • González-Martín, A.S., Nardi, E. & Biza, I. (2011).

Conceptually-driven and visually-rich tasks in texts and teaching practice: the case of infinite series, International Journal of Mathematical Education in Science and Technology, 42 (5), pp. 565-589.

• González-Martín, A.S. (2010). The concept of series: teachers’ conceptions and teaching practices. In M.M.F. Pinto & T.F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (vol. 3, pp. 33-40). Belo Horizonte (Brazil): PME.


Teacher practices: Numerical series • Background

Few research reports focusing the notion of series. Diversity of approaches.


• Exercises used during teaching practices do not guide students to build an adequate notion for the convergence of numerical series.

• The use of visual reasoning could be advantageous for the students.

• Calculus textbooks show little or not relation between improper integrals and infinite series.

• The use of historic examples could help to improve the teaching of infinite series and help to overcome the idea that “infinite terms imply an infinite sum”.

Robert (1982)

Alcock & Simpson (2004) Bagni (2000, 2005)

Codes & Sierra (2007)

Fay & Webster (1985)

Bagni (2000, 2005)

Teacher practices: Numerical series


• Duality process – object.

• Potential infinity vs. Actual infinity.

• Concept definition vs. Concept image.

• Symbolic notation.

• Confusion between sequence and series.

Kidron (2002)

Kidron (2002) Mamona (1990)

Mamona (1990)



• Background

• How is organised the teaching of the notion of series?

• Does this teaching take into account the difficulties identified by research?



REVISION OF LITERATURE

Does the teaching of series take into account learning

difficulties?

Textbook analysis

Teaching practices



• Context

• School system in Québec: – Primary (6 – 11 years). – Secondary (12 – 16 years).

– Collégial (17 – 18 years). – University.



A B C D E F Robert (1992)

Ayres & Mendelson

(1993)

Charron & Parent (1993)

Beaudoin & Laforest

(1994)

Wild (1995)

Swokowski (1995)

G H I J K L

Anton (1996)

Massé (1997)


Ouellet (2000)

Hughes-Hallet & Gleason (2001)

Bradley et al (2002)

M N O P Q

Dominguez & Rouquès

(2002)

Thomas et al (2002)


Ross (2006)

Amyotte (2008)


• Methodology Textbook analysis

17 textbooks from 1993 to 2008 (González-Martín, Nardi & Biza, 2011):

R1: Series are usually introduced through organisations which do not lead to a questioning about their applications or their importance (raison d’être).

R2: Organisations tend to introduce series as a tool in order to later introduce functional series, but the importance of series per se is usually absent.

R3: These organisations tend to ignore some of the main difficulties in learning series identified by research.

R4: The vast majority of tasks concerning series are related to the application of convergence criteria, or to the application of algorithmic procedures.



• Methodology

• 5 teachers from different postsecondary establishments in Montreal. – T1: 5 years of experience (all the 5 with series). – T2: 20 years of experience (4 with series). – T3: 32 years of experience (30 with series). – T4: 6 years of experience (4 with series). – T5: 7 years of experience (2 with series).



• Data analysis • The five stated that the textbook they use fills their

teaching needs.

• T1, T3, and T4 agree their textbook fills students’ needs. – T2: “there are difficulties for students with series […], but

until we know what we want to do, we won’t find a textbook that is convenient”.

– T5 specified that the textbook he uses is good, but students do not have enough time to do all the exercises.



• Data analysis

• Examples of applications (especially of everyday life) offered the first day to introduce the notion:

– T1: [idea] that certain functions are not integrable […] Two windows and a ray of light is reflected, but not completely, a part passes through and the other part is reflected.

– T2: Zeno’s paradox […] Numerical series, it’s not sure that they are interesting, it’s mainly Taylor and Mclaurin developments.

– T3: Ah, from everyday life. Yes, there’s an example we give for the geometrical series. It’s the ball that bounces. And at every bounce, it reaches a fraction of the preceding height […] There are no other examples that come to my mind.

– T4: numerical calculations… how a calculator will do to calculate a square root […] At the start, I think that the idea of the cheese [we eat first a half, later a half of what remains…] is okay. …

– T5: Maybe at the end of the semester, we get to see the applications of series […]: Taylor series.



• Data analysis

• Other examples of applications during the teaching of series: – T1: maybe I give one or two from time to time, but I don’t give

many examples. – T2: finite developments… – T3: interests. So, in finances, when you place some money,

there are periodic interests. – T4: the calculation of transcendent values… – T5: No.



• Data analysis

• Do they ever ask their students to produce a visual representation for series? – T1, T2, T4, and T5: no. – T3 said that sometimes they go to the computer lab to

work with Maple (however, he did not mention any specific activity to visualise series).



• Data analysis

• Ideas their students should really develop about series:

T1 Taylor – The idea (usually well understood through integrals) that an infinite sum may have a finite result – Integrals

T2 With numerical series, that arithmetic laws do not work as with finite sums (understand the difference between convergence and divergence) – Taylor

T3 Care: infinite sums are not the same as finite sums – To verify whether the conditions for the tests are respected

T4 The sequence of partial sums – Comparison – Feel that an infinite sum might have a finite result

T5 Take some time before concluding convergence or divergence – To use the criteria – Power series



T1 Students do not understand them well, they perform routine exercises – They mix sequences, series, and partial sums

T2 Wanting to extend finite arithmetic rules to infinite series – The level of abstraction – Even if we can determine convergence, we cannot determine the sum in many cases – It is necessary to do many exercises to understand

all the convergence tests

T3 Counterintuitive – The general term can tend to zero, but we cannot say anything about the sum

T4 Abstraction and mathematical formalism

T5 Confusion between sequence and series

• Data analysis

• Main difficulties to understand series:



T1 Students do not understand them well, they perform routine exercises – They mix sequences, series, and partial sums

• Data analysis


• Type of tasks they offer to their students:

T1 Practically all the problems [from the textbook] Mostly based on the convergence tests… understand the convergence tests to

eventually being able to speak about convergence intervals and all that.



T2 Wanting to extend finite arithmetic rules to infinite series – The level of abstraction – Even if we can determine convergence, we cannot determine the sum in many cases – It is necessary to do many exercises to understand

all the convergence tests

T2 A first element is being able to identify the general term […] many exercises about that.

Then […] the study of the convergence criteria […] They’ll see many exercises, identify whether a series is convergent, if it doesn’t converge… and through

which criterion we can get there… Many, many exercises about convergence.



• Data analysis



T3 Counterintuitive – The general term can tend to zero, but we cannot say anything about the sum

T3 Do exercises about geometric series […] They’re asked to recognise, among different series, a geometric series. Then, if they’ve identified it, they’re asked to be able to

correctly evaluate to what it converges. Then, in an application, being able of formulating correctly, mathematise correctly the model [only the bouncing ball was

mentioned]… There are examples were the convergence of series of positive terms and alternate series is

worked, they must identify which is the most suitable criterion to find if it converges or not.



• Data analysis



T4 Abstraction and mathematical formalism

T4 You must selective about the exercises that you ask them to solve in general and they’re at the end of the semester…

They like the course, you can see it, but it doesn’t appear in the exam. I try to make a realistic selection. This way, there’s a lot of algorithmic, tell if

the series converges. Well, it’s clear that at the beginning it’s more conceptual, I try to make them feel the sequence of partial sums, but without

many applications, there are not many at their reach in the textbook I use.



• Data analysis



T5 Confusion between sequence and series

T5 The writing of a series, it’s worked a bit, the explicit writing… the writing with the general term.

Also, questions about the convergence of a series. After having familiarised the students with the writing of a series, we shift to the

convergence of a series. And there, all the criteria… for the convergence. Then, there’s a great part about convergence. This way, practice is more based upon the criteria than upon the concept of series. Because the criterion helps to be more at

ease with the series. It’s more there than we make exercises about the criteria, we spend more time with it.



• Data analysis



• Comments

• The practices seem to follow what is done in the textbooks.

• The institutional relationship of the teachers seems to mirror the mathematical organisations found in the textbooks.

• Is this institutional relationship enough to take into account the students’ difficulties to learn series, or to introduce series in a more significant way? (PCK)



• Comments

• “Know”: these teachers’ knowledge about series seems to mirror the mathematical organisation found in textbooks.

• “Know-how”: Practices seem to follow what is done in the textbooks.

• “Conceptions”: Their vision of series seems to reduce them to a tool for introducing other notions, mirroring what appears in the official programs.

• Their position regarding the didactic void is similar to that found in the textbooks.

• The personal relationship of these teachers with series seems to suffer many restrictions which, surely, are linked to the position series have in MTC.




Contents


• The institution does not seem to sufficiently take into account the

difficulties to learn real numbers or series.

• The institutional relationship of the institution with real numbers and series seems to privilege incomplete praxeologies which make them lack of a raison-d’être and to appear just as a prerequisite for other notions.

• This institutional relationship of the institution seems to have a strong influence on the teachers’ personal relationship with mathematical notions.

• Which are the consequences for students’ learning? And for teachers’ training?

• Which recommendations could be given?

Conclusions


Conclusions


Research on the learning of given topics

Programs to help first-year students

Technology Engagement, motivation

Book analysis Practices

Resources Teachers’ training

Theoretical approaches

WG14: University mathematics education

GT7: Enseignement des mathématiques aux niveaux post-secondaire et supérieur

TSG16: Teaching and learning of Calculus

The teaching of Calculus in the secondary and postsecondary levels ...

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