“Do you think it would be possible to get a new ruler for the board that I can use for my lesson?”
This was a question that one of our tandem-teachers asked us just a few days ago. Even though we knew that it would take a few more weeks up to the arrival of the new Team VIII until we could make her wish come true,1 we were very pleased with her request. Seeing our tandem-teachers prepare their lessons using different material is always a little progress for all of us.
In Germany, it seems normal to have variety in lessons. We are encouraged to work with different methods and techniques, have access to abundant material as well as countless workbooks and worksheets. Here in Laos this is not the case. While the teachers follow the mandatory course book, is may also seem like there is not enough room for creative methodology or material. However, from our perspective, it is important to foster a deeper understanding of the subject by using different methods and material. They do not only make the lesson more interesting, but are desirable – or even instrumental – for facing and conquering mathematical problems. Also, it is quite possible to save time by lesson planning. Once more and more children really understand the subject matter, the less time is “wasted” in later classes, too. The time factor would level out in the long-run.
For the future it would be a major goal to introduce and start using the material from as early as Grade 1 in the primary schools so that the pupils can establish a profound mathematical understanding, which involves more than just learning formulas by heart.2 Core competences such as problem-solving and flexible thinking – which have become indispensable in the 21st century – need to be enhanced and fostered constantly.
Therefore, it is our main goal to improve the pupils’ learning environment in the local schools by encouraging the Lao teachers to foster the development of these competences by using the afore-mentioned material and techniques in their lessons. By giving them new ideas and teaching input we intend to help them in planning motivational lessons that the pupils as well as the teachers enjoy.
As we – Anja and Pauline – are working closely together with our mathematic tandem-teachers, we were eager to put all of this into one workshop, which took place on the 14th of November 2018 at Ban Phang Heng Secondary School. We were excited about welcoming four of the mathematics teachers – Ms Toukham Chanthavong, Mr Thongsing Sinnaphisith, Mr Noy Vienglakhone, and Mr Noy Sibounhueang. Apart from them, Ms Chanpen Bounyavong – former mathematics teacher but now teaching biology – participated, as well as Ms Saysamone Singhalath, who, with a constant smile on her face, translated whatever we said into Lao.
Our workshop consisted of three main parts:
- A theoretical part: Lesson planning and number line
- A practical part: Number line and games
- New material and lending system
1. Planning a lesson
The following four principles should be observed when planning a lesson.
Firstly: Activate all students!
This includes a variety of individual work, working in pairs, and group work. It is most important to get every student involved by making them calculate, think, proof, try out things, and not only copy from the board. This is particularly difficult in Lao classrooms where there are 50-60 children and it can be very challenging for the teachers. However, it is not impossible.
Secondly: Let the students explain!
Teachers usually explain subject matter to their pupils, but if the pupils try to do this it can be just as effective for their learning if not even more. If one not only listens but also verbalizes subject matter it comes to steady stabilization and a much deeper understanding.3 This way, the teacher is given the chance to check whether a pupil understood a certain point or still has difficulties working with it. While the teacher also might be afraid that a pupil might make mistakes in front of the others, mistakes actually offer great possibilities to talk about strategies, operations, algorithms, and numbers. It is not only important to make this transparent to the teacher, but also to the pupils, so that they can develop a classroom atmosphere in which every pupil feels comfortable and will not be laughed at for making mistakes.
Thirdly: Use material and visual aids!
Most subject matter is best understood when the teacher uses materials and visual aids in the lesson. It has been shown that the use of certain mathematical material like the calculating frame helps the children to construct a better understanding of the subject itself.4 Whether it is geometry, where the pupils can draw different patterns and forms using compass and triangle rulers, or arithmetics, where material such as the above-mentioned calculating frame or base ten blocks can be used to establish a number sense,5 we try to motivate all teachers to use the new material which is now stored in the Didactics Room.
Fourthly: Practice mental maths!6
Many mathematical problems presume a good number sense and mental maths. Many pupils here tend to often count – even in higher year groups – or calculate using a written algorithm. However, in many cases it is not only faster but also makes more sense to just calculate in one’s mind.7
It is normal to use counting strategies (also using the fingers) in the first one or two grades in primary school, but they should soon be replaced by operative strategies (e.g. using analogies) and an increasing understanding and afore-mentioned number sense. There are various motivating and fun games that challenge the pupils’ mental maths skills – see a few examples below.
We chose the “number line” for our workshop as a visual aid to bring different addition and subtraction strategies to paper. As we noticed that many students in the secondary school do not really use calculating strategies, our plan was to introduce the number line to students and teachers to give them a scaffold and show easier and faster ways of adding and subtracting numbers.8
There are many different ways of solving a problem like 38 + 27 mentally, as this number line shows:
The number line offers a graphic way of making the students aware of this, showing the strategies 9 and how they relate to each other. Furthermore, letting the students find as many ways as possible for one problem offers great opportunities for talking about the strategies and the question which ones are more practical than others. They thus help the pupils practise their skills in communicating, justifying, and arguing – which we are interested in fostering also in our Lao classrooms. (In Germany, these skills are set down in Educational Curriculum for mathematics.)
During the workshop we discussed the benefits of the number line, how and when to include it in the lesson, and practised its use in several examples. To let the teacher experience the possible range of application, we included examples – both addition and subtraction – with lower (e.g. 38 + 27) as well as higher numbers (e.g. 1749 – 943). After a 10-minute period of individual work with the handout,10 the teachers presented their number lines and we talked about different problem-solving strategies.
Bingo – The game “Bingo” is an excellent game to practise mental maths and it can be adapted to any class level, degree of difficulty, and mathematical topic.11 By naming a mathematical problem instead of just giving a number, the pupils are asked to calculate. Since there is not as much time pressure as in other games, each pupil gets the chance to calculate individually. It is not always the best student who wins the game, as there is a certain amount of luck needed, too. This makes it very motivating for all students.
Trio – Another great game for practicing mental maths is the adapted classroom version of the board game “Trio”. Material needed: A big poster with the digits 0-9 (see picture below). The teacher says a certain number (e.g. 23) and the students have to form equations using three numbers which can be found horizontally, vertically or diagonally next to each other on the template (e.g. 5 x 4 + 3). Whether you succeed or not does not only depend on how fast you are at calculating mentally, but also on how fast you can find a suitable number combination. In our version, the students are divided into two or three teams and always gain points for their team, but the game can be played with all students competing individually against all others, too.
Blind passenger – This activity offers the chance to practice problem-solving and is a fantastic way of speaking about mathematics. The only material needed is either a set of flashcards (pictures) or coloured chalk. One could describe this activity of a more flexible version of “Odd One Out”. The teacher prepares several pictures/flashcards and puts/draws them onto the blackboard. The major point is that they should all have things in common but still be unique. Any of them could be the “blind passenger” – the one between the others that does not share the common features. Then the teacher tells the pupils that one picture does not fit and asks which one it could be. The pupils then have time to think about it, talk about it with their neighbours, and in the end discuss it in class. The activity can be conducted with different mathematical content, especially geometry, but also various other pictures from other subjects are suitable. It is great to play with different levels of difficulty as the teacher can decide which pictures to choose.
We played these games during our workshop so that our tandem-partners would get an idea of how to include them into their own lessons. They were really excited about the games and – this was especially nice to see – some of them started using them in their classes straightaway.
3. Didactics Room
The Didactics Room at Ban Phang Heng Secondary school was established in 2017 by Team III and is a place for studying, printing, laminating, as well as preparing and storing material. As one of the main goals of our stay was to introduce new teaching material to the teachers, we included a short visit to the Didactics Room, where we had set up a collection of maths material upon our arrival in Laos. This was inspired by the former volunteer who pioneered the work with mathematics – Fabian from Team VI. He introduced the use of triangle ruler and compass and suggested that we bring some more triangle rulers, since there were not enough for one class.12
Committed to following his suggestion and very eager to bring even more material from Germany, we started Phase one of our small project and contacted friends and family, asking whether they would be willing to donate some money for this cause.13 In the end, we arrived here with bags full of calculating frames, base ten blocks, tangrams, scissors, cubes, geoboards, small triangle rulers for the students and big triangle rulers and compasses for the teachers – material that we knew from school or that had been introduced to us during our studies at the PH Karlsruhe. Additionally, some material – measuring tapes and compasses – were bought from a local market here.
Phase two was then ready to start – the teachers would probably not use the material if they did not know that it existed in the first place or how to use it. This is why we regularly introduce different material during our tandem-lesson-preparation time and then use it during “Activity Time” so that the teachers can see how and why we use it. Especially for Mr Thongsing and Ms Chanpen, who came to our workshop but do not get tandem-time preparing lessons with us, we included the short introduction during the workshop.14
Phase three – the use of material in the maths lessons – is ongoing. Sometimes we suggest to the teachers to use certain material in their lesson, other times they come up with own ideas. Whichever way, it is always rewarding to see the lessons being enriched with new activities, new ideas, and new material.
We would like to thank the teachers for participating in our workshop and, moreover, for being so motivated and keen on applying new ideas. We are grateful that they share their teaching journey with us. Simultaneously, working with the Lao teachers is a unique learning experience for us, too, which allows us to deepen our personal, professional, and intercultural competence. Thus, we are glad that they help us to become more broad-minded and skilled, too.
Text by A. Schuler & P. Faix
Photos by M. Weis & A. Schuler
1 There are new triangle rulers in the Didactics Room.
2 Much time is spent in the maths classes for learning formulas by heart, copying exercises from the book or board, and applying the formulas with only very little transfer. It has been shown in recent LGTC exams that the graduates from our secondary school struggle applying those formulas when more transfer is required.
3 Confucius: “I hear and I forget. I see and I remember. I do and I understand.”
4 Prof. Dr. S. Wartha (PH Karlsruhe) suggests four phases for using mathematical material. At first, the pupil should calculate directly with the material while explaining what he/she is doing. Second phase: The pupil will only explain the action while the teacher conducts it. During the third phase the pupil will still explain, but no longer can see the material (the teacher can use a screen to put between the pupil and the material). The last phase is the detachement of the material – no more material is used and the pupil should now be able to use the material in his/her mind.
5 This refers to a deeper understanding of numbers and their properties as well as an understanding of how different numbers relate to each other.
6 Calculating without the help of a calculator or written algorithms, often by using mental calculating strategies that need to be acquired early on.
7 This is why we try to focus on mental maths during the Maths Club and noticed that some pupils still need a lot of practice.
8 The number line as shown in this article should be used as a documentation of those strategies only and has to be based on number understanding, which can be established and strengthened by the use of other material such as the calculating frame and the base ten blocks (cardinal number understanding) as well as the “normal” number line (ordinal number understanding).
9 The use of different colours makes it easier to keep track of the strategies. Some strategies such as for example 30+20+8+7 cannot be shown on the number line.
10 Each teacher was given a handout with all the content addressed in the workshop, including some exercises and a collection of games for the classroom).
11 Each student is given a template (3×3 or 4×4 fields), where he/she has to fill in given numbers, in whatever order he/she likes. When every student has his template filled with numbers, the teacher gives problems to solve. After calculating a problem, the student can cross out the result on his template.
12 Regular rulers can be bought here in Laos, but not the triangle rulers.
13 A big thank you to Thomas & Susanne Schuler, Almuth & Hans-Martin Schuler, Paul Schuler, Jakob Schuler, Christoph Herrmann, Angelika Weinmann, Sabine & Klaus Weis, Werner Herrmann, Alina Truöl and Simon Lang, as well as the Wissner GmbH.
14 We also set up a lending system, whereby the teachers can borrow various mathematical material for their lessons and preparation.
Padberg, F. & C. Benz (2011). Didaktik der Arithmetik. Heidelberg: Springer Spektrum.
Schulz, A. & S. Wartha (2012). Rechenproblemen vorbeugen. Berlin: Cornelsen Scriptor.