TAL - A National Database of Questions - Classification is the Key pptx

Rather than just choose the questions on differentiation randomly we have used two additional classifica-tions to be more precise; so the first question is using differentiation to find

Introduction 2

Maths-CAA Series May 2003

TAL A National Database of Questions

-Classification is the Key

Contents

FDTL awards the idea of shared assessment resources in this form was not supported perhaps because the case was not made well enough

At present TAL is used by the University of Bristol and Liverpool John Moore’s University As part of a European Leonardo project, ATHENA, the following Universities: Università degli Studi di Perugia, CPE Lyon, University of Szeged, Universidad Politécnica de Valencia all used TAL with their students The project set a mathematics test at university entrance level and translated the test into the different languages The students then all took the same test and the results were compared Any other university interested in taking part in this project is welcome as are schools, which may find questions based on the SEFI Core Zero syllabus of interest

2 Motivation for Creating TAL

In the 1970s Bristol University staff used to mark students work every week The pressure of work eventually led to the abandonment of this practice and there was a consequent fall in the mathematical competence of students The idea of occasional progress tests was formulated to address this fall in standards but clearly if we could get a machine to mark students work every week then all would be well We conceived of a huge database of questions with students set tests that were randomly generated from the database Questions are ex-pensive to generate and need continual updating Many people however are teaching roughly the same material and so the same questions could be widely used The concept of a National Database of questions was born This concept

is much wider than mathematics and so the TAL project includes questions from many different subject areas, although the largest contributors of ques-tions are mathematics, chemistry and physiology As TAL’s home is in Engi-neering we are also generating a database of EngiEngi-neering questions stemming from a project, based in a new degree at Bristol called Engineering Design (www.fen.bris.ac.uk/engdesign) The Engineering database aims to help stu-dents understand engineering concepts; and is particularly aimed at questions that do not require detailed calculation but require a feel for the subject 2.1 Why do we use questions?

The main goal of education is that students should learn To learn mathematics they need to try their hand at problems We want students to understand the concepts of mathematics but such concepts are often quite complex When students can understand how to do a question involving a concept, the under-standing may be quite one-dimensional, i.e they can use the concept only in that sort of question In order to help students to have a richer understanding of

TAL A National Database of Questions

-Classification is the Key Jon Sims Williams and Mike Barry Dept Engineering Maths, Bristol University Email: jon.sims.williams@bristol.ac.uk, mike.barry@bris.ac.uk

Abstract: This paper discusses a CAA system called TAL TAL is unusual in that

it allows users to generate large numbers of equivalent tests from a

specifica-tion The tests are generated from a database of questions and all questions

must be classified Some of the difficulties involved in classifying questions

are discussed

1 Introduction

The Test and Learn system, TAL is a database of questions with facilities for

staff to set tests and students to take tests It was first conceived in 1994 We

built a system, which allowed students to run a programme on their PC linked

to a remote computer The system collected a test from the remote computer,

ran the test locally on the PC and then sent the results back to the host

compu-ter The system used the basic systems of computer-to-computer direct links

that were available at that time and worked very reliably A bid to JISC was

successful (Higher Education Funding Committee JISC Technology

Applica-tions Project 2/352) in 1996 With this support we built a web-based version

that was first used in 1997 The task of populating the database with a large

enough set of questions was perceived to be too onerous a task for one

univer-sity or department and so the system was built to allow several universities to

use the same set of questions in the hope that participating staff would

gradu-ally add to the database and we would all benefit This is not, of course, the

first time that the idea of sharing questions has been mooted and early in the

TAL project we visited Glyn James and Nigel Steele at Coventry University

and were given the paper results of a previous attempt to share questions

Martin Greenhow of the University of Brunel has kindly offered his set of

ques-tions, Mathletics to be included in the database The normal history of shared

sets of assessment questions is that several people are very keen and offer

questions at the outset; others are very keen to accept the sets of questions and

then the whole project dies We hope that by having the database of questions

open for all to use remotely over the Internet, this project may yet succeed;

however it really needs full support and maybe central funding by the LTSN

subject centres If the LTSN Generic Centre accepted the concept of a generic

database then that would be even better as we already support questions from

three subject areas, mathematics, chemistry and physiology In the recent

TAL - A National Database of Questions – Classification is the Key Jon Sims Williams and Mike Barry

that all the tests are similar

3 An Automated Test Generator

We first conceived of the automatic test generator just picking questions ran-domly and generating tests with about the same length and difficulty So one would choose a topic: “Differentiation” and then all the tests would be gener-ated from questions with this classification This was relatively simple to do but pedagogically it did not work – teachers need to craft their tests so that the questions are more precisely chosen The problem of random choice showed

up when some tests had two or more questions that were similar, while others had quite different questions

Automatic test setting is wonderful for the teacher in that several tests are set very quickly, but more control is needed in the formation of tests The current system allows the user to set a test by specifying the topic or topics to be used for each question in the test The concept is illustrated by the short test below:

Example test Design

Setter: Jon Sims Williams Dates available: 3/5/02-15/5/02

Length of test: 11:34 +/- 2 minutes Facility of test: 60 % +/- 5%

Figure 1: A simple specification of a mathematics test.

Question No Main Topic SubTopic1 SubTopic2

1 functions understanding form of

graph

2 functions domain and/or range simple algebraic

functions

3 differentiation practical application speed

4 differentiation function of function

rule trigonometric functions

concepts, we expose them to a wide variety of questions

The goal is clear We know that students find it helpful if they can measure

themselves up against a standard to see if they are doing well, and pedagogically

we want to expose students to a rich variety of problems to reinforce their

understanding To reach this goal we need to motivate the students to do the

questions and we need to provide the questions

There are two ways of providing questions:

1 Generate questions parametrically where the questions all have the same

structure but have different numbers

2 Provide a large number of different sorts of questions on the same topic

Method 1 has two advantages:

• Students gain confidence because they can practice until they succeed on

a familiar type of question

• It is much cheaper to generate questions parametrically

Method 2 gives students lots of practice but as each question is different they

do not learn how to answer questions by drill or formula but rather gain a more

multi-dimensional understanding of the topic The disadvantages of this

ap-proach are the converse of using method 1 Most importantly, students may get

discouraged if the questions are all so different that they cannot build on the

understanding they get from successfully completing one question in the next

question they attempt Ideally the questions lead on one from another

increas-ing the scope of understandincreas-ing without beincreas-ing too challengincreas-ing A good teacher

can do this by hand selecting questions from the database Automatic support

is more difficult; it could be done but at present we only collect and use data

on the relative difficulty of questions

We use questions for motivation However just providing questions is not enough

Many students seem to need the motivation of getting some marks before they

will put the effort into trying questions As soon as you give marks, students

complain if they think that other students are in some way cheating – yet to be

most effective in supporting learning we want to offer access to the tests from

students’ homes and halls of residence Probably the most effective resolution

of this contradiction is to set a suite of tests all of which are on the same set of

topics and then tell students that one of these tests will be used for a supervised

test and the other are available for them to use for practice One of TAL’s

special strengths is that it allows you to set tests according to a specification so

4 An Approach to Classification

The design of a classification scheme is primarily influenced by the needs of users There are various publicly available classifications of mathematics [2 -6] but none of them has as its prime aim the need to classify questions for teachers to use

The users are setting a test from a large database of questions, so they must be able to find questions that suit their needs Teachers need to be able to find questions on the subject that they are teaching and at the same time they need

to know if the questions will require any other expertise than just the specified subject So if we take some examples:

1 The basic skill of differentiating a polynomial

2 The use of differentiation of a polynomial to find its maximum

3 Given a simple model of the position of a ball expressed as a polynomial, find out where the ball is travelling fastest

4 Invent a simple model for a sledge sliding down a hill with a linear resistance to motion Find the point where the sledge travels fastest Now all these problems involve differentiation of a polynomial, so they could all be classified under the same subject; however the questions require increas-ing levels of modellincreas-ing skills In the classification scheme of the previous section, these questions were classified as follows:

1 Mathematics/Calculus & Analysis/calculus/differentiation/polynomials

2 Mathematics/Calculus & Analysis/calculus/differentiation/max;min; stationary points/polynomials

3 Mathematics/Calculus & Analysis/calculus/differentiation/max;min; stationary points/Model Application/polynomials

In terms like “max;min; stationary points”, the semi-colon is used to mean

“and/or”

If the user specified that he wanted to choose questions on …/differentiation then all these questions would have been selected The user could however have excluded questions requiring the higher skills by excluding, for example, questions classified as:

Mathematics/Calculus & Analysis/calculus/differentiation/max;min; sta-tionary points

This test has only four questions and realistically one should never set such

short tests as their reliability is low; however here we have chosen to set two

questions on functions and two on differentiation Rather than just choose the

questions on differentiation randomly we have used two additional

classifica-tions to be more precise; so the first question is using differentiation to find a

speed in a practical application The second question specifies that the

differ-entiation requires the understanding of how to differentiate functions of

func-tions and the funcfunc-tions concerned are trigonometric funcfunc-tions So the setter of

the test has a clear idea about the sort of questions s/he is setting Lastly two

additional requirements are that the test should take 11.34 ± 2minutes and the

facility of the test should be 60% ± 5% The facility is the percentage of

students who normally can get a particular question right This is averaged

over all the questions, and each lecturer can have a different set of facilities

and time to do for each class taught The test generator searches the database

for suitable questions and generates as many tests as it can to satisfy the

speci-fication

We like to set about 20 different tests satisfying the specification and in order

to do this it may be necessary to relax the specification a little and allow a

wider range of questions to be used for each question slot The specification

can be relaxed by saying that the type of function used in the function of

function differentiation does not matter – just don’t specify it Similarly we

could allow any sort of application of differentiation not just questions based

on speed Other relaxations could be to specify another sort of question that

could be used for each question slot e.g questions on differentiation using the

chain rule could be used as well as questions on the function of a function rule

Figure 1 is illustrative of the idea of how tests are specified but MainTopic,

SubTopic1 and SubTopic2 have no particular meaning as will be shown in later

examples The whole basis of this test generator is that all the questions are

classified and the classification of questions uses a tree structure The

ques-tions slots in the example above as specified as:

1 Mathematics/Calculus & Analysis/functions/Understanding form of graph

2 Mathematics/Calculus & Analysis/functions/domain;range/simple

algebraic functions

3 Mathematics/Calculus & Analysis/calculus/differentiation/practical

application/ speed;acceleration

4 Mathematics/Calculus & Analysis/calculus/differentiation/ function of

function rule/ trigonometric functions

The ‘Mathematics’ is not redundant as the database contains questions on other

subjects A semicolon is interpreted as ‘and/or’

TAL - A National Database of Questions – Classification is the Key Jon Sims Williams and Mike Barry

tions If a teacher had taught the idea of ‘ill-conditioned’ as part of understand-ing about solvunderstand-ing equations and the errors that can occur he would want to find the question classified under terms such as ‘solution of equations’ Now the SEFI Core Zero syllabus, [2], operates at the school-university interface level and has syllabus items that can be described by the classifications:

Algebra/Linear Laws/understand the terms ‘gradient’ and ‘intercept’ with reference to straight lines

Algebra/Linear Laws/recognise when two lines are parallel Algebra/linear Laws/obtain the solution of two simultaneous equations

in two unknowns using graphical and algebraic methods These are all very much in the right area but incomplete as there is no refer-ence in [2] to ill conditioning If we introduced a classification under SEFI Core Zero syllabus about ill-conditioning then one could no longer set tests from it knowing that all questions were in the syllabus The right response is probably to say that one must know about ill-conditioned matrices to answer this question and classify the question as:

Maths/Algebra/ Linear Algebra/Matrices/Matrix

It is possible that people would want to teach about ill conditioned equations

in a way that does not involve looking at matrices at all, so a classification: Maths/Algebra/ Linear Algebra/linear equations/

may also be viable

Because of the need to be comprehensible to the target audience: teachers and university lecturers, we have adopted a twin approach to classification of math-ematics questions Firstly, we have used the SEFI Core Zero curriculum for engineering mathematics [2] edited by Leslie Mustoe and Duncan Lawson As part of a Leonardo de Vinci project called Athena 2000/1, a set of questions based on the classification in [2] was generated and a TAL test on these ques-tions was taken by students in different European countries This syllabus is useful for both school sixth forms and some first year university work Sec-ondly, we have adapted the Eric Weisstein’s World of Mathematics classifica-tion [3] This classificaclassifica-tion is aimed at helping users to find the meaning of

You will notice that there is a citation order used here where “max;min;

sta-tionary points” comes before “Model Application” or

“SimpleModelConstruction” The normal rule in classification schemes is to

place “general” classifications before “special” ones However

“SimpleModelConstruction” is a more general classification than “max;min;

stationary points” as it could be applied to all sorts of problems not just

differ-entiation problems, but we need the citation order to relate to the way in which

the teacher will think Normally a teacher will choose to teach differentiation

and then when the subject is well established will start to ask students to use

the process of differentiation on “real” problems involving some sort of model

Rather arbitrary decisions have been made, as a teacher might want to teach

modelling and then choose what sort of mathematical techniques could

rea-sonably be used by the students At present we allow two classifications for

each question so it would be possible to implement both approaches

The classification, as you can see, is mixing together a subject classification,

“Mathematics/Calculus & Analysis/calculus/differentiation/”, with a problem

type description such as “practical application” and the sort of application,

“speed;acceleration” and then sometimes there is information about the sort of

functions that are used in the question, e.g trigonometric functions The test

setter needs to be able to specify questions using all these sorts of data, but a

single tree structure containing all this is not ideal A more general approach

to classifying questions is described in section 5

4.1 Practical classification problems

In this section example questions are displayed and then an approach to

as-signing a classification is discussed

Example 1

Two linear equations are said to be ‘ill-conditioned’ when:

Answer 1: They cannot be solved

Answer 2: They represent two parallel straight lines

Answer 3: They represent two nearly parallel straight lines

Answer 4: They represent lines of which one is parallel to either the x

or y axis

This question is posed in terms of equations and the answers are all in terms of

the geometric understanding of the equations In the mathematical literature

however, the term ‘ill-conditioned’ applies to matrices rather than sets of

equa-Answer 2: The adjoint and the minor of the element aij are identical

when i + j is odd Answer 3: The cofactor and the minor of the element aij are identical

when i + j is odd Answer 4: The cofactor and the minor of the element aij are identical

when i + j is even This question needs to be classified under both minors and adjoints, but in the classification hierarchy we have:

Algebra/Linear Algebra/Determinants/minor or cofactor and

Algebra/Linear Algebra/Matrices/Matrix Operations/Adjoint because the adjoint is the transpose of the matrix obtained by replacing each element aij by its cofactor Aij so it is a matrix operation

So once again we need to decide on a major classification and then have sub-classifications to show what additional knowledge is required This question is easy to sort out as one cannot fully understand it without understanding Adjoints and one cannot understand Adjoints without understanding cofactors, so classi-fying the question under Adjoints only is satisfactory

This example points out a problem that questions often cross many boundaries

in a simple subject classification We need to find a classification scheme that allows a teacher to pick out questions that are on the topic being taught, but then finds it easy to remove questions that require additional knowledge that has not yet been taught

It should be clear from these examples that the classification of questions is a complex process When it has been well done however, the setting of tests is easy Because the process is so complex we need a team working together rather than rely on one or two people

The classification, as you can see, is mixing together a subject classification,

“Mathematics/Calculus & Analysis/calculus/differentiation/”, with a problem type description such as “practical application” and sort of application,

“speed;acceleration” and then sometimes there is information about the sort of functions that are used in the question, e.g trigonometric functions The test setter needs to be able to specify questions using all these sorts of data, but a

terms So when we look up Linear Equation we find a definition in terms of y

= ax +b Now, a part of what a teacher wants in terms of questions about

Linear Equations will be the definition; but s/he will also want to know how to

solve them and what goes wrong when you solve them etc So we have built

upon the Weisstein’s classification in accordance with our experience of teachers

needs Part of both the SEFI and adapted Weisstein’s syllabus are shown in the

Appendices

The system allows multiple classifications so both of the interpretations (4.1.1)

and (4.1.2) are implemented If you look at the Weisstein’s classification on

the Web you will find that (4.1.2) does not exactly fit

Example 2

The equation of the straight line that passes through the point (1,2,3) and

is parallel to the line joining the points (2,3,1) and (5,4,2) is?

This question requires two ideas: the student needs to be able to form

Displace-ment Vectors, so that s/he knows the direction of the line and needs to be able

to write down the equation of a straight line using its direction and a point

through which is passes So there are two possible classifications:

Displace-ment Vectors and Equations of lines This question is really about equations of

lines and just incidentally requires one to know how to form displacement

vectors, so we would classify the question as:

Algebra/vector algebra/equations in vector form/equations of lines

But students also need to know about the displacement vectors so we put this

underneath equations of lines as other ‘required knowledge’

Algebra/vector algebra/equations in vector form/equations of

lines/dis-placement vectors

A new scheme has been invented to avoid treating ‘required knowledge’ in this

way but it is not yet fully implemented

Example 3

For an n x n matrix A, which of the following statements is true?

Answer 1: The adjoint and minus the minor of the element aij are

identi-cal when i + j is even

TAL - A National Database of Questions – Classification is the Key Jon Sims Williams and Mike Barry

Application Area - this tree allows the editor to describe the application

area e.g physics/motion/speed;velocity

In addition we use the following lists, where only one selection is made from each list:

1 Modelling level: this classification only applies to certain questions We distinguish the following modelling levels:

• Simple Model Construction - a simple model must be constructed to complete the question

• Model Application - a model is provided as part of the question and must be used

• Practical Application - this indicates that the student is expected to be able to use the particular mathematics described in the Subject tree pointer in a practical situation Implicitly this will usually mean that they have to use a standard model used in this sort of application

2 Evaluation Type: This takes one of two values: Numeric or Algebraic, and tells us if the question requires the answer in a numeric or algebraic form

3 Bloom’s Educational Objectives: http://www.mathematicsweb.org/

mathematicsweb/show/Index.htm are a classification of the type of question thus:

• Knowledge: states that the question simply checks if the student knows something factual

• Comprehension: usually refers to a question in which data is given and the question checks if the student understands what it means

• Application: The use of previously learned information in new and concrete situations to solve problems that have single or best answers

• Analysis: The breaking down of informational materials into their component parts, examining (and trying to understand the organizational structure of) such information to develop divergent conclusions by identifying motives or causes, making inferences, and/or finding evidence to support generalizations

• Synthesis: Creatively or divergently applying prior knowledge and skills

to produce a new or original whole

• Evaluation: Judging the value of material based on personal values/ opinions, resulting in an end product, with a given purpose, without real right or wrong answers

Lastly the numeric field is used to describe how theoretical the question is This is very important, as a question posed in an abstract manner on a simple

tree structure containing all this is not ideal A more general approach to

classifying questions is described in the next section

5 A General Classification Scheme

TAL is conceived as a question database for many different subjects, so we

have developed a database structure that allows each subject area to have its

own subject classification Essentially each question is classified under one or

more top level subjects e.g mathematics and medicine Then for each of

these top-level subjects the subject editor is able to specify three lists, three

trees and a numeric to describe questions The questions can have multiple

descriptors taken from these lists and trees Each list has a subject specific title

so that the users understand the interface in the language suited to the subject

area So in mathematics we are interested in the set of function types used in

a question, but in chemistry they are not Instead a list of question descriptors

has been proposed:

Question_descriptor = { Law or definition, Nomenclature, Qualitative test,

Reactions & reactivity, Structure, etc}

Rather than go into the details of the computer representation of this, the scheme

will be described from a mathematical viewpoint The scheme described

be-low has been designed and built by a student but as yet it is not implemented

in the production system

5.1 The Mathematics Subject Scheme

For mathematics we have selected three trees called:

Subject tree - for the subject tree we have adopted two schemes: one

based on Eric Weisstein’s World of Mathematics classification [3], this is

the same as has been described in most of the examples above There are

considerable benefits from having a classification that is related to a

syl-labus and so in the ATHENA project we used the SEFI sylsyl-labus, [2] If you

use the SEFI syllabus you know that the questions you find are in the

syllabus

Function Type - this allows questions to be described in terms of the types

of functions used e.g {simple algebraic functions, trigonometric

func-tions, exponential/log funcfunc-tions, inverse/hyperbolic functions} Since this

structure is a tree it allows a fine description of function types although

only types given in the list are used in classification at present

place questions that the user does not find appropriate However it would be desirable to be able to control the ‘facility’ and perhaps the ‘time_to_do’ of some of the question slots in the test; so we propose to allow the user to specify the minimum facility and maximum length of questions in a question slot Finally the test setter can select the average facility and length of the whole test and how accurately these two conditions must be satisfied So the test can

be specified as running for an hour plus or minus 5 minutes and similarly the average facility can be required to lie in the range (55,65) for example The test compiler will then find all the questions that match the specifications for each question slot and try to generate the number of tests requested

6 Conclusions

The TAL system has been running since 1997 Initially a very small number of questions were available to set mathematics tests but now we have nearly

2000 questions mainly suited to first year university mathematics There are another 2000 questions on other subjects Automatic test setting is wonderful

in that it saves considerable staff time and the availability of multiple tests allows students to get practice and be motivated It is however easy to be sloppy about setting tests automatically and we have proposed some addi-tional controls in section 5 to make it easier to build appropriate tests For the TAL database to be a really good resource it needs editors for every sub- area of mathematics to maintain the standards and encourage development of new questions It is doubtful if any university can really maintain such a resource and so a collaborative effort is needed with several universities collaborating

in its development

Appendix A - A part of the SEFI classification Analysis and Calculus

Rates of change and differentiation

Average & instantaneous rates of change Definition of derivative at point

Derivative as instantaneous change rate Derivative as gradient at a point Difference between derivative & derived function Use notations: dy/dx, f(x), y etc

Use table of simple derived functions Recall derived functions for simple fns Use multiple & sum differentiation rules

subject may be quite inappropriate The scheme used for this Theoretical

Nu-meric is:

0 or 1 a pure numbers question

2 the question is about basic rules or nomenclature of the subject

3 or 4 there is a simple parameter in the question

5 or 6 the question requires a simple theoretical understanding of the

subject

7,8 or 9 the question requires a proof of some theoretical property

We have had very little experience in using these although quite a number of

questions have been classified on this scale

5.2 How to specify a question slot

Section 3 describes the way tests are set with the existing system In this

section we describe the use of the new classification scheme Tests are set by

specifying a number of question slots sufficient to form a test of the required

length If the questions used have an average time_to_do of 5 minutes then 12

question slots will be needed to form a test that is an hour long Each question

slot is defined by specifying:

• The subject or a disjunction of subjects from the subject tree

• The set of permitted function types

• If the question is to be an application problem then choose the application

area

• If the problem is a modelling type of question then choose the level of

modelling

• The Bloom category for the question and the enumeration type {numerical

or algebraic} can be set

• Finally a constraint on the level of the theoretical numeric can be set

When the question slot has been specified the database is searched to find all

the questions that satisfy the specification and the additional subject

knowl-edge that is implicit in the set of questions are displayed The test setter may

then select any of these subject areas to exclude them the test

From our experience of using automatic test setting we find that just

control-ling the content of question slots is not sufficient for good test design At

present the system allows the user to look at each question in each of the tests

set and, provided there are spare questions, to swap spare questions in to

re-TAL - A National Database of Questions – Classification is the Key Jon Sims Williams and Mike Barry

Theory Total differential Vector Derivative Increasing & Decreasing Monotone Decreasing Monotone Increasing Monotonic Function Etc

Mean-Value Theorems etc

9 References

[1] TAL – the Bristol University “Test and Learn System” at www.tal.bris.ac.uk

[2] The SEFI core curriculum for engineering mathematics http://

learn.lboro.ac.uk/mwg/core.html [3] Eric Weisstein’s World of Mathematics classification http://

mathworld.wolfram.com/

[4] MathematicsWeb - http://www.mathematicsweb.org/mathematicsweb/ show/Index.htt this is basically about Journals but has a 3 level classification e.g maths/applied maths/numerical analysis

[5] Zentralblatt MATH - http://www.emis.de/ZMATH/

[6] 2000 Mathematics Subject Classification http://www.ams.org/msc/ [7] Bloom’s Educational Objectives http://www.mathematicsweb.org/

mathematicsweb/show/Index.htt

Use the product rules of differentiation

Use the quotient rules of differentiation

Chain rule for differentiation

Relation of gradient and derivative

Equation of tangent & normal to graph

Stationary points, maximum and minimum values

Find if function is.increasing using differential

Define a stationary point of a function

Distinguish turning and stationary point

Locate a turning point using derivative

Classify Turning Points by 1st derivative

Find second derivative of simple functions

Classify stationary Points by 2nd derivative

Appendix B - A part of the General Classification Scheme

Calculus and Analysis

Calculus

Differentiation

Chain;product;ratio Rules

Differentiability

Directional derivative

Equations of Lines

Function of function rule

Gradients;slopes in R2

higher derivatives

Implicit;log differentiation

Introductory

Maclaurin series

Max;min or stationary points

Global Maximum

Inflexion point

Local Minimum

Saddle point

Numerical differentiation

Parametric equations

Partial Derivative

Second derivative test

Tangents;normals

Taylors series