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Erasmus+ Project

“Effective Teaching of Mathematics”

MATHEMATICS GUIDELINES

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INDICE

Preface….………………………………………………………………………….. pag. 3

The mathematical intelligence……………………………………………pag 5

European indications….…………………………….………………………. pag. 9

Paths……………………………………………………………….………………….pag. 10

Aritmethics………………………………………………………………………….pag. 11

Logic as basis of mathematical thinking ……………………………..pag. 14

Geometry from a teaching point of view ……………………………pag. 17

Coding in primary school…………………………………………………. .pag. 19

Teach computational thinking …………………….… pag. 19

Use visual programming: “Scratc”………………… pag. 19

Let pupils work together……………………………… ..pag. 20

Use computer games and “game factories”….pag. 20

Game design applications ………………………………pag. 21

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PREFACE

The study PISA, the Program for International Students’ Assessment, in 2012, made

on all 34 member countries of OECD (Organization for Economical Cooperation and

Development) and on the 31 partner countries, showed that European students find

greater difficulties in studying maths than in studying letters and sciences.

A study of 2013 found that the amount of graduates in maths and related subjects is

decreasing and that there are poor skills in areas that require high levels of maths

knowledge; moreover, the same study found that this distrust of maths and of

learning processes related to logical thinking already start from the first cycle of

education and continue in junior and high education.

Since 2007, the great majority of European countries‘ Governments has revised

maths curriculum taking an approach based on results and on the attention to the

development of student’s skills rather than to theoretical contents; there is a

general agreement on the fact that the effective teaching of maths requires a series

of didactic methods and on the fact that some methods (like learning through

problem solving, the survey, the applications of maths onto the real world) are

particularly effective to improve results and students’ attitude towards the subject.

So the fundamental point is the lab, as a physical place as well as a moment in which

the pupils formulate their own hypothesis and control its consequences, plan and

experiment, discuss and argue choices, learn to collect data, negotiate and build

meanings, lead to temporary conclusions and to new openings through the building

of personal and collective knowledges.

Another important point is game, that plays a crucial role in communication,

education in respect of the rules, processing strategies suitable for different

contexts. Students participating in learning process by having fun have much more

chances to experience success in the subject; their involvement will grow in

proportion to their knowledges and prejudices will disappear.

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These Guidelines are the result of the whole path of EteMat Project and they

contain reflections, tips, methodological indications, for an innovative approach to

maths teaching-learning.

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The mathematical intelligence

Piaget is reputed to have formulated the first fundamental cognitive theory

concerning the formulation of number (Piaget and Szeminska, 1941).

Language is developped in the left hemisphere of the brain: the words production

involves the Broca area on the frontal lobe; listening to words depends on the

Wernicke area of the temporal lobe; looking at words involves the occipital lobe;

reading words depends on the frontal and parietal lobes.

Maths calculations happen in the left hemisphere too. In a Butterworth research,

you can note that the left hemisphere starts to work when there are some detailed

or approximate calculations. Approximations involve the parietal areas of the brain

(which control finger movements of children when making the first calculations or

during the utilization of the abacus); exact calculations involve the frontal areas of

the left hemisphere, that concern language too. Therefore, the mathematical

intelligence is based on two different “modules”, one concrete, the other abstract,

as it is for the musical intelligence.

Fabrizio Doricchi, a neuropsychologist, says that: “As soon as the human being can

see well, he develops the ability of ‘subitizing’ that lets him perceive at a glance and

in such a fast time quantities from 1 to 4. From 5 on, in general he can’t value the

exact number any more because he must count and sum up objects, what a little

child can’t do obviously.

So there must be a pre-speaking mathematical skill, innate and separate from the

linguistic-symbolic manipulation. Learning to count represents the first connection

between nature and culture.

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The psychological research explains the mathematical knowledge as the set of skills

that let a child understand quantities and their transformations (Lucangeli, 1999). It

was proved that, when we are born, we are already prepared both to numeric

intelligence and to verbal one. As a matter of fact recent neurological studies have

shown that the human brain is genetically prepared to calculation and it is possible

to develop and strenghten our maths skills since the first years of our lives.

A group of researchers from Stanford University, in the United States, has found for

the first time the brain area which is involved in number identification: it is made

up of about 1-2 millions nerve cells, it is about half a centimeter large. It is in the

lower temporal circle, the superficial area of the brain external cortex, of which we

already know it is involved in visual information, and it works as soon as it sees some

numbers. It was demonstrated that, due to this particuar area of the brain, some

children get maths skills faster than others, given the same kind of teaching. The

discover, published on the Journal of Neuroscience, could help to better understand

dyslexia towards numbers and the inability to deal with numerical information

(called discalculia).

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Another research of Stanford University School of Medicine tried to explain,

through cerebral pictures, how the children brain can arrange information when

acquiring new maths knowledges and why some of them learn to recover

information from memory best than others. Indeed unlike adults, it seems that

children use the hippocampus and the prefrontal cortex, that are involved in

memory processes, during the problem solving. In a second survey, children, once

become more accurate and faster in maths problem solving, seemed to rely more

on memory than on calculation. These changes of strategy found a physiological

reply too. Moreover researchers saw that the stronger are connections between

hippocampus and the prefrontal, front temporal and parietal cortex, the bigger are

child maths skills.

Since enfant school, children should be taken to discover numbers in a funny way,

so as to develop a mathematical mind, but above all to let them “become fond” of a

logical-mathematical thinking. In particular, activities should meet interests and

curiosity children have got towards numeric symbols they see every day. Introducing

children to the maths world means, first of all, thinking about theories that are the

basis of logical and maths operations.

To know how to deal with maths structures means that a child can read, interpret,

hypothesize reality, really work on it. As Cazzago writes: “in order to achieve logical-

maths structures, the child normally goes through three steps:

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1. HE MAKES

2. HE WATCHES

3. HE SYMBOLIZES Many studies have identified five different abilities matching the main cognitive elements involved in solving problems: Understanding: it allows to find and put together the verbal and maths information to understand the text of the problem and the underlying scheme. Acquiring this skill ensures moreover to detect the meanings concerning question (syntax of the problem); Representation: it is a crucial ability to choose the correct solution, because it changes the understanding to a maths analogic system that can be expressed by figural representations or more advanced simbolic systems; Categorization: it is the ability to recognize, by means of similarities and differences, problems that can be solved in the same way, so they belong to the same category; Planning: it is the ability to choose the order of operations matching the maths order; Monitoring and Self-Assessment: it is the control on the process; it examines both the result of what produced and the perception of someone’s competence.

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EUROPEAN INDICATIONS

"Competence in mathematics has been identified at EU level as one of the key competences for personal fulfilment, active citizenship, social inclusion and employability in the knowledge society of the 21st century. Concerns about low student performance, as revealed by international surveys, led to the adoption in 2009 of an EU-wide benchmark in basic skills which states that 'by 2020 the share of 15-year-olds with insufficient abilities in reading, mathematics and science should be less than 15 %' ( 1 ). In order to achieve the target by 2020, we must identify obstacles and problem areas on the one hand and effective approaches on the other. This report, which is a comparative analysis of approaches to mathematics teaching in Europe, aims to contribute to a better understanding of these factors. The report reviews national policies for reforming mathematics curricula, promoting innovative teaching methods and assessment, and improving teacher education and training. It calls for overarching policies for mathematics education that are based on continuous monitoring, research evidence. It also argues for comprehensive support policies for teachers, a renewed focus on the various applications of mathematical knowledge and problem-solving skills, and for the implementation of a range of strategies to significantly reduce low achievement. The report also delivers recommendations on how to increase motivation to learn mathematics and encourage the take-up of mathematics-related careers. Many European countries are confronted with declining numbers of students of mathematics, science and technology, and face a poor gender balance in these disciplines. We need to urgently address this issue as shortages of specialists in mathematics and related fields can affect the competitiveness of our economies and our efforts to overcome the financial and economic crisis. I am confident that this report, which is based on the latest research and extensive country evidence, will make a timely contribution to the debate on effective mathematics education. It will be of great help to all those concerned with raising the level of mathematical competence of young people in Europe." Androulla Vassiliou Commissioner responsible for education, culture, multilinguism and youth

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PATHS

Typical maths processes:

1. Know and work with specific contents of maths (maths objects, properties,

structures ...)

2. Know and use algorithm and processes (in arithmetics, geometry, algebra,

statistics and probability)

3. Know different kinds of representations and go from one to another (verbal,

numerical, symbolic, graphic, ...)

4. Solve problems by using tools and strategies in different fields: numerical,

geometrical - algebraic (finding and matching useful information, finding and using

solving procedures, comparing solving strategies, describing and representing the

solving procedure, ...)

5. Recognize in different fields the countable features of objects and phenomena,

using tools of measurement, measuring sizes, evaluating measures (finding the

righter unit or measuring tool, …)

6. Progressive acquire and use typical forms of the mathematical thinking

(hypothezing, arguing, checking, justifying, defining, generalizing, demonstrating…)

7. Use tools, models and representations in the information quantity settlement in

a scientific, technological, economic and social fields (describing a phenomena in

quantity terms, using maths models to describe and interpret situations and

phenomena, interpreting the description of a phenomena in quantity terms with

statistics or functions tools, ...)

8. Recognize the shapes in the space and use them to solve geometrical problems or

modeling (recognizing shapes in different representations, finding relations among

shapes, images or visual representations, visualizing tridimensional objects since a

bidimensional representation and, vice versa, representing a solid figure, being able

to understand objects properties and their positions, …)

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ARITHMETICS

Teaching maths can give vitality and mental quickness, that is one of our time

rational needs.

This requires a change in the direction of didactic methods.

In arithmetics the process acquisition and the understanding of concepts always go

together. Here it is the connection between arithmetics and other subjects, above in

the scientific-technologic area. There must be added the development of a linguistic

competence careful to correctness, to clearness and uniqueness of language, to the

use of different comunicative ways, apart from the setting of a positive attitude and

the availability to get involved also in new situations.

External assessment, national and international, on learning levels of students in

maths is moving its attention from the contents to the processes involved in

learning.

Substantially they are mechanisms used to deal with learning situations, that can

change knowledge and abilities in competences. Each of them is essential and its

lackness alters the general balance of the subject possession.

In this perspective the role the teacher assumes each time is really significant for

leading his activity; such role can increase the value of students’ knowledges, or lead

towards a more “scientific” way of thinking, or collect students’ output into steady

maths structures. It is a proper way not to propose/impose models and processes;

even maths rules must become mature and be conquered. On the opposite, for

most students the risk is the passive acceptance or the school failure.

A “frontal didactic” is not enough anymore.

So Competence is a dynamic and complex issue, it is a plot of cognitive factors

(knowledges and organization of related concepts), applied factors (actions set up

by using knowledges), affective factors (motivation and availability to get oneself

involved, giving a sense to your own knowledges and abilities).

Competence can be substantially defined as the ability to activate and combine your

own knowledges and skills, like social and/or methodological skills, your motivation

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and will, to perform a task in working or studying situations and in professional

and/or personal development.

So Arithmetics competences have got proper aspects of the subject ( topic mastery,

tools possession, procedures,…) as well as others not proper for the subject, which

can imply the cognitive and working behaviour of the person (attitudes, languages,

reasoning, argumentation, reading and interpreting of contexts and phenomena,

solution of problem situations,...)

Words used for maths competences focus on the active way of “doing maths”.

Teaching must gradually introduce students, starting from significant situations, to

use maths language and reasoning as tools to understand reality and not only

propose an abstract load of notions. Many difficult aspects of maths teaching come

from a prevalent working on given models. On the opposite, working on models

building starting from concrete cases, without deepening analogies with similar

situations or acquiring the necessary way of working, interfere with further

developments, creating superficial and passing learning.

The student must be an active subject, who develops his own strategies. The

teacher is not a shepherd anymore, but a trainer. As each student has got his own

skills and learning rythm and everyone must reach a right competence level, a lab

methodology is necessary and appropriate, that is the activation of paths and

requests, in which everyone is followed, stimulated and assessed. It is a strategy

that involves the centrality of the learning environment, with various multiple paths

to work on choices, assess them and justify: that is the right context to show off the

experience and students’ knowledges, to encourage curiosity, exploration and

discover, to promote the awareness of their own attitudes and way of learning, in

order to “learn to learn” in an orienting function.

The class is not a homogenous group of people that go on together, but people that

move together or in small groups towards different goals, coherently aligned. So

there could be group synergies, integration and sharing of conquests and learnings.

Broadly “lab” is a kind of work that strenghten the problem making and the

planning; it involves students in thinking-realizing-assessing shared activities; it

requires a flexible use of usual school places and/or availability of fitted places. At

each school grade, lab didactics is not a simple doing, because it uses scientific ways

to get high results from the linguistic (you explain the reason of a choice) and

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conceptual point of view (thought and work to deal with problem situations). So the

subject gives meaning to competence.

In this respect technologies give the right support: from taking part to individual,

single or interactive lessons, to the critical research of answers to curiosities or

problems, to the consolidation of knowledges and working techniques in the models

developed. Students can start from a unitary stimulation to go on, alone or in

smaller/larger groups, under guidance of the teacher, and exchange ideas to

compare each other. As recent experiences with Interactive Whiteboards show, the

utilization of new technologies allows teachers to keep control of such a complex

situation and children to be helped by familiar tools.

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LOGIC AS BASIS OF MATHEMATICAL THINKING

Even maths has got high levels of formalization, that is its power entering different

subjects, it is necessary to know that these levels must be conquered gradually. So

in the teaching you must have clear aims to reach progressively, to let students have

a full subject competence.

Therefore it is better to give students the sense and meaning of operations, to

develop mental calculation, to evaluate the approximation of the result and to

determine the greatness order. This promotes the possession of tools to solve

problems, to evaluate the result got and to check its reliability.

In the Ministerial Programs of the different countries involved in ETEMAT Project

they explicitly refer to logic issues.

Acquiring logic knowledges has become fundamental, considering the role they play

both in learning all school subjects and in the daily life, above all.

The aim of Logical Education is to stimulate the cognitive development of kids, by

helping them to build reasonings, to understand, to interpret, to communicate

information, to formulate hypothesis and conjectures, to generalize, to relate, to

represent; besides it gives us a tool to promptly discover possible learning

difficulties.

The Logical Education is not to be considered a sort of didactic unit to develop

during a specific topic, but rather a continuous reflection on each aspect of the

subject.

The habit to reason by diagrams helps to give a clear and effective methodology

that will be a precious baggage for future studies.

The aim of activities proposed will be letting kids understand that Logic is placed

above all in thinking and language: indeed, often, first difficulties can arise from

linguistic fields: in following a reasoning, in decoding a text, in organizing an

explanation, in describing objects and situations, in giving definitions.

Logic can let us understand any language, study its structures (syntax) and interpret

its meanings (semantics). From now on there will be the skill to solve problem

situations, from the easiest to the most difficult ones (Problem Solving).

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General aims

· Understand, interpret, communicate information.

· Formulate and check hypothesis and conjectures.

· Generalize; relate; represent.

· Learn to discuss from peer to peer avoiding excesses and conflicts

· Recognizing own and other role

· Learn from own mistakes

Skills’ targets

· Be able to recognize situations, phenomena and processes.

· Be able to collect data concerning real experiences and classify them.

· Be able to elaborate a logic reasoning.

· Be able to show accuracy of a thesis.

· Be able to elaborate a summary.

· Be able to generalize logical criteria tested.

Methodological choices

· Conversations

· Exercises at different level of difficulty

· Real experiences

· Direct observations

· Analysis of teaching materials

· Proposals of individual or group activities

· Lab activities

· Playing activities

The consequence is a responsible behaviour

· Ability to self manage study and general learning processes

· Knowledge and ability to self setting the studying process

· Be able to use failure to correct own performance

· Recognize typical reasons of mistakes

· Be able to connect what studied

· Be able to predict consequences of own actions

· Keep control on own emotions

· Trust own ability to succeed

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External valuations, national and international, on students’ learning levels in maths

are paying attention to processes involved in learning rather than to contents.

Substantially it is about mechanisms dealing with learning situations, able to change

knowledges and skills to competences. Each of them is essential and its lack alters

the general balance of the subject teaching-learning process.

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GEOMETRY FROM A TEACHING POINT OF VIEW

The world around us is full of pictures which remind us geometry elements:

architecture, decorating arts, painting, sculpture are only some fields of application

of geometry. Geometrical definitions help to represent space, to develop logical thinking, to

reason. Being able to use geometry elements can assure the connection to other

areas of maths, but also to other subjects, like IT (information technology).

Nowadays almost everyone admits that the theoretical aspect is more important

than the experimental one, but if we consider geometry from a teaching point of

view, connected to the teaching-learning process, the relation between intuitions

matched to experience and the geometry reasoning remains fundamental.

So the evolution of geometrical thinking is to be searched starting from the first

spatial experiences of children until the most modern theories.

In the first school grades this subject is applied on organizing visual, tactile, physical

students’ experience, paying attention to some space features of objects, and then

organizing rationally in a more and more autonomous way. That is, at the beginning

geometry deals with feelings, experiences and external observations of a physical

kind, then it goes on rationalizing these first observations. Throughout this evolution

the natural language has got a fundamental role, that gives advices to manage

observations and interpret the world. Children try to develop starting geometry

concepts, on one hand by organizing perceptions, on the other hand by using

language. Then they should begin to set and rationalize geometry knowledges,

which will continue in a more and more critical and deeper way and which will have

to take into account that the educational value of such subject will incorporate the

different possible approaches.

So the geometrical organization is to be built didactically in an active way by the

student, rather than given as a finished product.

Many teachers introduce this subject starting from topics like the point, the line and

the plane, important for a rational discussion, but far from the student’s experience

or from definitions that should be considered as a finishing point of a student’s

personal learning path.

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Moreover, that choice involves starting only from plane geometry, followed only

after many years by the spatial geometry; but from a didactic point of view different

experimentations emphasized that the three-dimensional geometry (3D) is a more

intuitive interpretation of reality for a child, due to being nearer to his experiences.

According to the teacher’s style and creativity, you can use your favourite

methodological structure and the interactive methods that you need to stimulate

learning and personal development even in very young children. These methods are

modern tools to make easier the children’s cooperation, the direct and active

involvement, the concept planning, experiences, knowledge. Along the different

moments of the lesson, you can propose different techniques.

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CODING IN PRIMARY SCHOOL It is predicted that over the next 10 years programming will become one of the

fastest growing occupations. It is estimated that there will be about 1.4 million new

IT specialists needed on the job market in the world, with only 400 thousand new

graduates available. Some specialists predict that in 2020 there will be 900 thousand

new jobs for programmers in European Union only!

Learning to write computer programmes not only stretches the mind, but leads to

developing useful strategies in solving different problems. It will help in learning

different subjects and let the children use computers consciously to their needs and

solving problems.

Teach computational thinking

It involves the problem analysis, logic, solving problems. Indeed programming is

solving problems by using a computer. Therefore we should teach solving problems

by using the computer, that is by using the algorithms.

Important parts of computational thinking are:

- analysis,

- decomposing problem into smaller parts,

- abstraction (removing irrelevant details) and finally:

- creating algorithm.

Use visual programming

Children will benefit from visual programming environments,e.g. Scratch. Visual

programming lets children understand the most fundamental programming

structures like loops or conditional instructions without writing a word.

Scratch

Scratch is a free app and it is translated in 40 different languages. It is constantly

being further developed with many different iterations of it. There is a huge

community behind it ( http://scratch.mit.edu.)

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It is an application useful for developing critical and creative languageand any way it

can be used in every subject. Lastly all projects can be shared by web pages, blogs,

virtual bulletin board etc.

It can be used to create presentations, games, stories, animations, cards, projects,

geometrical games.

Geometrical game:

https://scratch.mit.edu/projects/106231069/?fromexplore=true

Moreover a website has been created http://studio.code.org that offers many tasks

, which seem to be funny and interesting for children; it is arranged in a gradual way

relating to users’ age. Besides there are many more visual programming

environments – one of the most important advantages of using them is that making

syntax errors is virtually impossible. Therefore, using visual programming, children

can focus on computational thinking and solving the problems.

We often start teaching coding by asking pupils to write new programmes. However,

showing them existing programmes as examples and letting pupils analyze and

modify them is a good way to start. It does not only provide student with an easy

way to start programming (by making small changes to an existing algorithm) but

also resembles the real life practice. Also, looking for mistakes in the given

algorithms is a very instructive task.

Let pupils work together

Group working is important in today’s world, not only for future programmers, but

virtually for everyone. Making pupils work in pairs or larger groups let hem develop

communication skills and improves motivation. It also forces children to learn how

to divide problems into smaller parts and how to share their work.

Regardless of the kind of lesson (individual or group work) each

student should be able to do his/her part of the task on the individual

computer. Practising to acquire programming skills involves the “doing”, what will

become “being able to do”.

Use computer games and “game factories”

Programming computer games may provide huge motivation for pupils. They can

easily start learning how to make simple computer games in environments like

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Scratch or the so called “Game factories”. Programming simple computer games will

let children understand fundamental computing concepts and have fun at the same

time.

Game design applications

In order to choose the right tool, we should consider the aims:

• easeness of use

• age target

• level target

• programmino paradigm

• availability on various platforms and Opening Systems

• Limits on the kind of games to be created

KODU http://www.kodugamelab.com/

Kodu lets children create games on PC and Xbox by an easy visual programming

language. Kodu can be used to teach creativity, problem solving, storytelling, as well

as programming. Anyone can use Kodu to make a game, young children as well as

adults with no design or programming skills.

The important thing to know about Kodu is the visual programming language to use,

is a language specificly invented for this product and it operates at a very high level

of abstraction. Students love Kodu for the fantastic rich games that can be

produced in a short time. It is easy to create visually impressive 3D game

environments.

ALICE: http://www.alice.org/index.php?page=what_is_alice/what_is_alice

Alice is an innovative programming environment in 3D that makes easy create

animations for a story, playing, sharing a video. It is a freely available teaching tool

designed to be a student`s first exposure to object-oriented programming, that`s

why it is used for creating geometry shapes and bodies. It allows students to learn

fundamental programming concepts in the context of creating animated movies and

simple video games. In alice, 3D objects (e.g., people, animals, shapes and vehicles)

populate a virtual world and stundents create a program to animate the objects. In

Alice`s interactive interface, student drag and drop graphic tiles to create a program,

where the instructions correspond to standard statements in a production oriented

programming language, such as Java.

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Alice allows students to immediately see how their animation programs run,

enabling them to easily understand the relationship between the programming

statements and the behaviour of objects in their animation. By manipulating the

objects in their virtual world, students gain experience with all the programming

constructs typically taught in an introductory programming course.