Dear teachers, you are now reading the guide we published together with a printed textbook. Nonetheless it will serve you fully as a guide for your online textbook in H-edu as well:

**Objectives of the textbook **

The objective of the textbooks this teacher’s guide is a part of is to support constructivist education. Its main features are: **maximum emphasis on the pupil’s autonomy, support and moderation of pupils’ discussions, effective work with mistakes** (we do not punish for a mistake, pupils are encouraged to look for the source of the mistake). Mathematical concepts and principles are discovered by the pupils on their own in discussions on the solution of a carefully thought-out series of graded problems.

Having made the decision to follow a constructivist way of teaching, the teacher will have to overcome deep-rooted behavioural patterns such as explaining, correcting mistakes, giving advice on how to solve things. The reward for moving away from the spotlight, side-lining their own personality and encouragement of pupils’ initiative is the increased pupils’ interest. It is essential that all pupils enjoy work they do.

That is why the textbook presents simple, more difficult and very demanding problems. **Each pupil can solve a problem adequate to their level and thus is neither bored, nor frustrated by mathematics.**

We try to build on the pupil’s life and mathematical experience as much as possible. The tool for this are various environments (Stepping, Origami, Spider webs, Cubic solids, ...) that enable us to pose problems with adjustable difficulty, touching on various areas of mathematics. The basic mathematical objects, number and shape, are grasped by pupils through their own movement and manipulative activities. **Psychology informs us that this experience leaves the deepest footprints in the pupil’s ideas.**

**Communication**

Communication among pupils is the key activity in the constructivist approach. The pupil learns to:

- formulate their own ideas and thus their mathematical thinking and language are cultivated,
- understand sometimes clumsy formulations of their classmates, which generally contributes to an increase of empathy,
- regard discussion as a means of collective search for the truth,
- evaluate classmates’ opinions critically.

Moreover, the pupil tends not to analyse knowledge they get from the teacher – the authority in depth but to memorize it. Knowledge acquired from their classmates is, in contrast, tested against their own experience.

Friendly and selfless work atmosphere in the class is prerequisite to successful communication. This atmosphere is characterized by the ability of every pupil to praise ideas of other classmates and by the shared joy from a successful solution.

**Textbook design**

The design of the textbook builds on knowledge of child developmental psychology. There are three major differences between how an adult and a child learn:

- An adult stores knowledge in a structured way. It helps them to remember. A child can store their knowledge as a set of episodic data because a child of a younger school age has the ability to absorb and connect a great variety of stimuli.
- An adult is able to work on one problem for a relatively long time. A child needs a frequent change of activity.
- An adult can put off the need to learn and discover, a child has to have this need fulfilled immediately.

Children manage their learning process effectively. They focus intensively on the stimuli that they perceive as acute. These stimuli contribute to their personal development. When the need of this activity is over, the repeated stimulus does not contribute to development so strongly. That is why children turn their attention to another stimulus whose contribution to growth is perceived as higher. The teacher or the parent sometimes interpret this volatility as shallowness, disobedience, or even defiance. In contrast, the textbook accommodates this volatility in children of a young school age and often changes the stimuli.

This also contributes to individualization of lessons because one problem has a high motivation potential for pupil A, while another problem for pupil B. And what happens is concurrence of various solving processes where the pupils’ intensive involvement is essential.

The same type of problem is used again and again. It is modified both in its difficulty and in its connection to different environments.

Thus each pupil gets deeper and deeper insight into the environment. They see more links and relations, mathematical ideas. Each pupil can get to understand something at a different time, following a different path. Our experience shows that this way of learning allows the pupils to join in even after a longer period of absence at school.

*Story: One colleague complained that using this approach she was not able to follow what was going on in the class. However, she confirmed that the pupils had created groups on their own and it they needed help they asked for it. Half a year later, the same teacher spoke really highly of it. The fact that she could follow only part of what was going on did not worry her any longer. She appreciated the pupils’ diligence and enthusiasm. *

*Think of a holiday on which you visited several castles and palaces in a very short period. Recollect how your brain was shutting down and you gradually stopped listening to the guide. Almost nothing was interesting in the third and the fourth castles. If you visit one castle at a time and another one some time later, you remember much more about both and you enjoy the visits much more. *

*Changing environments allows children with different preferences to get access to knowledge. One pupil may be happy working with colour pencils, another one prefers manipulating and building. Some need to move, others to discuss, solve problems on their own etc.*

**Structure of the textbooks**

The learning content of the 1st grade is presented in three parts of the textbook. All the parts have the same design and structure. Thus teachers and pupils should find it easy to work with the textbook. The graphic design corresponds – the problems are clearly structured and well-arranged. This should help also special needs pupils to find their way through the textbook.

**Specific features:**

There are icons accompanying to some problems that provide additional information about the problem:

- describes a problem where manipulation will be needed for the solution,
- describes a problem where dramatization will be needed for the solution,
- describes a problem for group work,
- describes a problem which is supplemented by a card with graded problems.

### Graded cards

The teacher can use extra cards with additional problems to individualize work. These cards offer easier problems for pupils who need to work at a slower pace, problems of the same difficulty as the problems presented in the textbook for those who need more practice. There is also “food” for pupils who perform better and need extra, more demanding problems. Of course, the level of difficulty is only an estimation. These cards can be purchased at www.h-ucebnice.cz/1rocnik.

**Environments**

As stated above, pupils learn mathematics by working, solving problems, discussing their solutions in various mathematical environments. Each environment brings some deep mathematical ideas and each has its pitfalls and limits.

Most of these environments have been known for a long time. However, experience from schools showed that new environments were needed to make the scale of perspectives and approaches fuller. The learning process of a pupil has the following sequence: experience → its evidence → its organization → discovering a relation → naming the relation → recording the relation.

The new environment Trains gives pupils experience with the number as a quantity and a simple tool for argumentation – manipulation.

The new environment Children’s park develops experience in the area of work with data in the familiar context of a playground. Usually, there are playgrounds close to schools and thus work in this environment can be done with the hand-on experience in a real playground.

Pupils’ numeracy is developed in the new environment Abacus. Pupils reinforce their calculative links in an enjoyable way. Moreover, the Abacus team offers worksheets with methodological comments. These are supported by electronic applications and a number of aids. All this can be purchased at www.h-ucebnice.cz.

**The teacher’s role**

The teacher’s role is namely to:

- create favourable work atmosphere,
- set adequate, differentiated problems to pupils in a way to ensure the each pupil achieves their personal best in every area,
- moderate the class discussion.

The teacher does not explain the subject matter. The teacher talking time is limited to minimum. The teacher does not make decisions on whether the solution is correct. That is the pupils’ responsibility. The teacher organises activities, encourages pupils to propose different ideas and moderates their discussion. It may happen that the class agree on a wrong solution or an untruth. The teacher does not correct the mistake but tries to guide their pupils to the right solution with the help of a well-chosen problem. It is not necessary to react at once. It is enough to say that the class would get back to the problem later. This gives the teacher the time needed to think about the adequate problems they can bring to the following lessons. Another possibility is to inform the class at the end of the lesson that one untruth was agreed on. Very likely, there will be at least one pupil who discovers the mistake until the next lesson.

**Aids**

Information about where the aids can be purchased are available at www.h-mat.cz/pomucky.

The list of the needed aids:

- cubes – a set for the class
- wooden sticks (natural or coloured, wooden stirrers or wooden ice-cream sticks) – a set for the class
- coins (a set for pupils + large for the blackboard)
- tiles – cardboard, magnetic or cut out and laminated
- stepping belt (a huge one under the blackboard + a small one to be stuck on the pupils’ desks)
- stairs – stepping belt with numbers (a huge one under the whiteboard + a small one to be stuck on the pupils’ desks)trains – set of prisms in colour
- bus – a box as a bus and tokens as passengers, possibly labels for different stops (the pupil prepares with the pupils)
- ards with numbers and symbols +, −, <, >, = (photocopiable sheets can be used)
- dice – large or the usual size – a set for the class
- objects for manipulation (tokens, bottle caps, ...) – brought by the teacher
- square paper (at least 10×10 cm)
- dry wipe board with a grid 1 cm, 2,5 cm
- squared paper – 1 cm, 2,5 cm (photocopiable sheets can be used)
- clock (face) – a set for a class (photocopiable sheets can be used)

#### A list of recommended aids:

- poster – Introduction, Introduction – Spot the differences, Bus, Children’s park
- wooden sticks with magnets for a magnetic board
- stamps with templates – addition triangle (photocopiable sheets can be used)

#### Photocopiable templates

Photocopialbe templates can be downloaded by teachers free from www.h-ucebnice.cz/predlohy. This is the cheap variant for schools. E.g. it is possible to substitute coloured prisms (aids) by printed out and laminated coloured rectangles (photocopiable sheets). The templates will be updated.

#### A set for the first-grader

It is possible to buy a set of mathematical aids for the first grader. It includes: wooden trains for primary school in a bag, dice (red, blue, green), stickers to be put on the desk or A4 notebook covers – stepping belt and stairs, A4 notebooks with 10 mm square grid on one side and blank sheet on the other side of each page, erasable dry wipe board – grid, ergonomically shaped dry-erase marker, paper coins, cardboard tiles, clock face, A4 notebook cover. It can be purchased at www.h-ucebnice.cz/1rocnik.

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