Mathematical Advancement in a Group of 2

by Lucy Rüttgers

 

How the choice of topic came about

Already in the letter project, Edith (now 5;4) mentioned that she really wanted to learn arithmetic next and had already started something with it at home with her mother. I promised her that we could do an arithmetic project after our letter project. She was thrilled.

In the meantime, I had also noticed Edith’s interest in music. So I talked to her again alone to clarify whether a music project would be more appropriate for her or whether we should do the arithmetic project first and then a music project? She was very sure that she definitely wanted to learn arithmetic first. She then wanted to do the music project afterwards.

The emergence of the group of two

Already in the letter project it was important for me to bring Edith and Lotte (5;10) closer together. On the one hand, they are two very different children, with different interests and caregivers, on the other hand, both are very high achievers and both lack a confidant at this level, a friend with whom they can exchange ideas. In addition, it has turned out that both of them will attend school and our after-school care centre together next summer. So it would probably do them both good to have an „ally“.

In the letter project, there were situations in which Lotte and Edith continued to work without the other children. I also noticed their similar reading levels. The cooperation in the letter project was not yet enough to build up a friendship. Before the our new project began, I said to Edith, to whom I had promised this project, „I would like Lotte to join in too. Do you agree? If so, we could ask Lotte if she wants to take part“. Edith agreed. The joint discussion with Lotte then revealed that she wanted to take part in the numeracy project.

Project description:

The mathematical area is very large and offers many possibilities. However, I only have 12 days until the report deadline for my IHVO Course due to the current staff holiday / illness situation in our kindergarten. Therefore, I can only cover a small area for now. However, I hope to be able to continue working on the project with the children beyond the deadline as long as they are interested. For the time being, I have to find out how far Edith and Lotte have come and what they are particularly interested in. I hope that this will result in good support for them.

As with the letter project, I have designed the so-called red thread. Here, too, the children themselves should design / decide the concrete course of the project – according to their interests and wishes. I can imagine this being very productive with these two children.

The contents of the project chosen by me in advance are:

Numbers:
Counting, naming / recognising numbers (reading), whereby the number range is determined by the children,
Writing numbers. Possibly learning to read and write word numbers through the children’s strength of being able to read. Assign numbers to corresponding quantities.

Types of arithmetic:
Plus, minus, possibly division and multiplication tasks.

Measurements:
Measuring objects, lengths, widths;
Units of measurement: Millimetres, centimetres and metres.
Measuring instruments: ruler, folding rule, tape measure.

Weights:
Scales,
Weight units: Gram and kilogram

Arithmetic devices:
Children’s calculator (with balls in 10s mode / abacus), arithmetic chain (self-created in 10s mode), calculator, cash register calculator that prints receipts.

Place value of numbers:
Classify units, tens and possibly hundreds.

The two project children themselves should decide on the exact course of the project, the contents and the duration of the project.

First offer

Objective: Make the selection for the first topic content.

Materials available: scales, metre measure, a game with numbers / quantities,
an apple divided into four parts, an apple divided into two parts, a knife.

Procedure:
We had a conversation about counting and numbers. Then we looked at the numbers game together. It consists of wooden cuboids in different sizes, each size corresponds to a certain amount. On each cuboid is the corresponding number in the form of the English word, for example „seven“ for the number 7. In addition, the number is represented on the cuboids by a corresponding number of lines.
The cuboids fit together in a wooden box, in a row always resulting in the number 10. Different combinations are possible.

The children worked out this system of the game and the statement of the individual cuboids.
Then we talked about the scales and the metre-measure, about their meaning and their possible uses, and the children tried them out.

Using the apples, we worked on the arithmetic operations: plus, minus and divide.

Finally, the children had to decide what they wanted to start the project with. The result was that they both agreed to measure with a metre-measure at the next meeting.

Duration: about 30 minutes. The kindergarten did not give me more time.

Observations during the offer:
Edith was disappointed about the early end, although I had announced it in advance.

Both children complemented each other in their knowledge. Edith immediately combined different plus and minus tasks with the apples. Lotte understood the division tasks faster than Edith.

Both children treated each other considerately and respectfully.

They both found the metre measure and its technique of unfolding and folding very exciting. They appeared very motivated for the next meeting.

Second offer:

Objective setting:
To get to know and use different measuring instruments, to get to know and use units of measurement.

Procedure:
Both were given a workbook in A4 format. They proudly labelled it with their name.

Part 1
Looking at the measuring instruments together and trying them out briefly.
They named the rows of numbers on the ruler. I know that both children can count very far – with a few gaps. Now I want to know if they can also read numbers in the 10 range in order to be able to use a measuring instrument at all. It turns out that they already can.
To deepen and extend their skills, they write the number sequences from 1-10 and from 10-20 in their notebooks, using a ruler that reaches 30 cm.

As asked to do so, they write 30, 40 and 50 under the 20. In this way, they can complete the number sequences outside of what we offer, and they have both done so.

Part 2
Using the ruler, they learn about the units of measurement millimetre and centimetre – as well as their spelling abbreviations MM and CM. I chose capital letters because they are more familiar to them.
In their notebooks they recorded the new information: 10 MM = 1 CM.

Part 3
Each of them measured small objects found in the room with the ruler.

Whoever wanted to, should write down the measurement results in the notebook. Edith did it.

Then the two of them measured large objects and distances together with the metre-measure or the tape measure, depending on their choice.

Duration up to this point: 1 hour and 15 minutes.

When Edith signs up for more activities, Lotte withdraws from the offering saying, „I can’t do it anymore“.

Observations during the offer:
Lotte tired of the theoretical part very quickly. But when it came to the practical part, she was very fit and motivated again. Edith was full of power all the time.

The teamwork of the two was very nice. Large distances that they could not measure alone, they measured together in agreement and with good arrangements. Alternately, one recorded the beginning of the measurement and the other read off the result. The wishes of the others were taken into account.

Continuation of the offer with Edith:
She wants to work with the children’s calculating machine.

Edith pushes the arithmetic balls together (in the 10s range) to form her own plus tasks, names them aloud and calculates them.

She does the same with minus tasks.

Duration of the follow-up offer: 20 minutes, total duration for Edith: 1 hour 35 minutes.

Observations during the supplementary offer:

She was still very concentrated and highly motivated and showed no signs of tiredness. At the beginning of the arithmetic exercises, she still counted each ball with her finger. After my suggestion to try it without counting, she first obviously counted with her eyes without using her fingers. Later, she also named smaller quantities without counting. My tip to use the colour gradation of the balls in steps of 5 and to continue counting only after 5 was partially implemented by her.

While she was working, she discovered the game „Rummikub“ (a number game) on the games shelf (recommended for ages 8 and up) and would have liked to have it explained to her right away. Unfortunately, this was no longer possible for organisational reasons. I promised her it for the next day.

Third offer:

Objective:
Deepening of measuring and / or introduction of the game of Rummikub.  Extension by recording what has been measured.
Material per child: 1 ruler, pencil, scissors, glue, graph paper.

When I asked the children if they would like to measure objects first or if they would rather play the Rummikub game, they unanimously answered: „Measure!

Part 1
Review of the previous day. We look back at what we have written so far in the booklet. We look at the graph paper and trace the boxes using the ruler.

Part 2
Task: Measure smaller objects of one’s own choice found in the room with the ruler. Draw the object on graph paper using the measurement results. Write the length and width of the object on the corresponding sides.

They drew two objects like this with my help. Since the task seemed a bit too difficult, the next task was:

Part 3
Draw lines and write their length next to them. I made some initial length guidelines:
10 cm, 5 cm, 8 cm, 15 cm, then their own choice. They both approached the task with great motivation.

Part 4
Cut out the drawings and stick them in the notebook. They have to think about the best way to do it. Both want to cut out each drawing individually.

Duration: 1 hour with me present, and 30 minutes without me for kindergarten  organisational reasons.

Observations during the offer:

Both girls were highly motivated and full of joy. Lotte worked very quickly and independently. Edith was slower and more insecure. Her ruler slipped more often and she had difficulties putting the 0 at the beginning of the line correctly. However, she did not seem to be frustrated by this. She confidently decided to cut out her drawings one by one, using the wave and pinking shears, where it is more difficult not to cut something off by mistake.

Lotte asked Edith if it worked well with the wave scissors, but then decided to use the normal scissors.

One difficulty arose: Because the children chose the objects to be measured themselves, the measurements resulted in decimal numbers. I explained to them that counting the remaining small millimetre lines would result in a number after a decimal point. But then they needed help again and again.

Fourth offer:

Goal setting:
The previous day we had set for this day: the introduction of the Rummikub game.
(By the way, the game is very easy to make out of cardboard).

The game consists of number tiles from 1 to 13, each number row is available in 4 colours and twice. There are also 2 jokers. It is played similarly to the Rommee card game.

Part 1
Each person chooses a colour and lays out the number line from 1 to 13. To help them, they were both given a ruler on which they could trace the number line and also check that it was correct.

They then did the same again with a second colour to become more confident.

Part 2
They put the same numbers in different colours together.

Part 3
I explain the use of jokers and the rule: There must always be at least 3 tiles next to each other.

Part 4
Start of the game: Each player takes 14 tiles, sorts them by colour and puts them in a number sequence. The winner is the first to get rid of all the tiles.
The player whose turn it is can lay out 3 matching tiles, i.e. either three consecutive numbers in the same colour (for example: 7, 8, 9) or one and the same number in three colours (for example 5 in blue, red and yellow).
In addition, whoever’s turn it is can put on all the tiles that match the rows of numbers that have already been laid out.
During the game, I could see the number tiles of both children and give them impulses accordingly.

I was then able to add a few levels of difficulty:
1. you can steal a number from a group (for your own use),
2. you are allowed to move rows of numbers apart to create your own, already existing numbers,
3. you can use a joker.

Duration: approx. 1 hour

Observations during the offer:

Both were very concentrated. Lotte was very quick and confident with the numbers up to 13. She did not make any mistakes when putting the numbers together. Edith was confident up to 10, above which she had difficulties. Looking at the numbers on the ruler helped her to correct it. So did repeated counting. Her difficulty was in recognising the numbers above 10. She worked more slowly and very deliberately and concentrated.

My question whether I could still explain something difficult to them (see the 3 extensions listed above) made them both excited and proud and they listened very motivated. The difficulties that were still built in were better understood and implemented by Lotte than by Edith. Edith understood the connections but needed a little more time than Lotte.

Fifth offer:

Objective setting:
Deepening the game Rummikub to give the children the opportunity to play it independently later on. Deepening the handling of ruler and metre in order to gain more confidence with it.

There was a weekend between the last offer and this one.
Both children had decided on the content of the fourth offer. However, I had the feeling that Edith was not quite as keen to repeat the Rummikub game.

Now it looked like this: Edith wanted to measure objects, Lotte wanted to play Rummikub. Her mother also plays the game and at the weekend Lotte could join in, she said proudly.
Edith and Lotte agreed to do some measurements first and then play Rummikub.

Part 1
At first, both of them were very motivated to try to measure huge distances, but then they noticed their technical and physical limits.

As their independent action was important to me and we already had time limits again, they accepted my tip to opt for smaller objects. They then wanted to draw these on graph paper again.

The decimal points again posed a problem. In addition, both of them lacked a certain spatial imagination, as they had in the first drawings. They had difficulty drawing the rectangle that lay before them. The longitudinal line was the first step and not a problem. But transferring the width in the right place overwhelmed them. They both then switched to using the rectangle as a template and simply drawing around it.
So they found a solution to their problem themselves.

Part 2
Rummikub. They both wanted me to play along.

Lotte made sure that Edith did not see her number tiles. We played the game like last time. I gave both girls impulses and suggestions when they couldn’t come up with solutions on their own. Lotte won and then supported Edith in the game against me.

Duration: 1 hour

Observations during the offer:

Lotte had more „insight“, also because she had already played the game at home. She  saw more possibilities of placing tiles than Edith and she recognized them faster. Edith had already noticed her own difficulties during the first game and was therefore not so enthusiastic about playing it again. She has a high demand on herself and doesn’t seem to like it when others can do something better than her.

In the course of the game, however, she gained confidence. With the help of my suggestions, she made appropriate moves – and she put it away very well that Lotte won, with the prospect of still being able to beat me. We then had to end the game before the second winner was determined. The game demanded a lot of concentration from both of them.

Sixth offer:

At the end of the last meeting, they both expressed the wish: to write arithmetic problems in the notebook.

Objective:
To write and use the arithmetic signs + (plus) and = (equal), and possibly – (minus) correctly.
Solve small arithmetic problems.

Part 1
To get into the mood: Both count in turn. They both manage to count up to 39 without any mistakes. After that, little help is enough to make them count even more.

Part 2
With the help of the wooden cuboids, the children do the arithmetic problems, calculate or count the solutions and write the problems with the solutions in their notebooks:
1+1=2   2+2=4   3+3=6   4+4=8.

In addition to the wooden cuboid game, both children use their fingers to count. Lotte holds out her fingers, Edith counts.

This is where an important phone call comes in for me. They continue to work independently, laying and counting.
Lotte: 1 +2=3, Edith: 2+5=6, then improves it herself.

Part 3
Edith brought a book with number pictures to the kindergarten. (With a pencil, you have to connect the numbers from 1 up to the largest number und then you can see the whole picture.) They both want to do this. Each of them choose a picture, they both decide then on the same picture and I make them a slightly enlarged copy.

The series of numbers goes from 1 to 54 and results in a vampire picture. Lotte gets to 18 on her own and then asks for my help. Edith makes it to 28 on her own.
She complains of a sore throat and her nose is running. Nevertheless, they both want to play the Pharaoh game they discovered on the shelf. (The game is called „Der zerstreute Pharao“ 〈“The Absentminded Pharaoh“〉, it is recommended for the kids from 7 to 16 years.)

Part 4
The game goes like this: Small pyramids cover up motif cards and cards without motifs. Cards that you have to draw tell you the motif you have to find under the pyramids. By moving the pyramids, you have to find the motif you are looking for without uncovering other motifs. So you have to remember both the places where the motifs are and the places where there are no motifs. There are also variations, for example, you can turn the game around 180 degrees. Lotte and Edith already know the game.

They need help where the increases in difficulty begin, as these are difficult to read from the cards.

At the end of the game, Lotte has 14 cards and 22 points, Edith has 10 cards and 18 points.

At the end, I wanted to calculate the score together with them using the calculator (abacus) and suggested that they write this calculation in their notebooks. Lotte did this. Edith didn’t want to. It must have been the flu.

Duration: 2 hours

Observations during the offer:

It was noticeable that Lotte held out for so long. Both were very nice to each other again. Lotte was allowed to draw in Edith’s notebook. Both were fully concentrated and motivated. Writing down the arithmetic problems in the notebook was a lot of work for them. Some of the numbers were written reversed. They also had to learn how to place the arithmetic signs, which was more „new territory“ for Edith than for Lotte, who has a sister in the third year of school.
The number picture was a good relaxation afterwards. I could tell that Edith was coming down with a cold. She was not quite as concentrated as usual and made untypical careless mistakes. Nevertheless, she also wanted to play the Pharaoh game until the end. The two of them spent the rest of the morning alone in the popular gym room, almost forgetting to eat. Edith then spent the afternoon asleep in the kindergarten.

Seventh offer:

Goal setting:
Short repetition of plus tasks (deepening), introduction of minus tasks (whether with or without writing them down depends on the children’s wishes).

Introduction of easy division tasks. Independent creation of a number picture.

Part 1
Short review of the last joint activity, which took place 6 days ago (weekend and staff shortage again). With the help of the calculator / calculating board, both children had to place and name a minus task with the beads one after the other. Help was needed to make this possible.

Both children were familiar with plus tasks, but they did not know much about minus tasks at first, although I had already done this with Edith during the second offer.

Edith then placed 3 beads, took 1 away, and 2 remained. The arithmetic task was to be named: 3 -1=2.
With the help of the cuboid numbers game, this could also be done. However, neither of them wanted to write it down in their notebooks.
To make it easier for them, they had to calculate the next tasks with the help of felt pens.
Lotte calculated: 2-1=1.
Edith: 5-3=2, Lotte 4-1=3, Edith again 8-4=4, Lotte didn’t want any more. So we continued with:

Part 2
Trying to make the children understand division tasks. First I tried it with the help of the felt pens.

I have eight pens and one girl. How many pens can I give her? The answer was immediately clear to Lotte, Edith still hesitated.

I have eight pens and would like to distribute them fairly to two children. How many does each child get?
I have eight pencils and would like to give four children the same number of pencils. How many does each child get?

With Lotte it somehow clicked. For Edith, I made it obvious again: we assigned places to the four children and distributed the pencils to each place in turn. That way it became clear.

In order not to let it get boring, now there must be a cake. Each of them had to paint a big round cake.

„Now, if two guests come to visit, how can you distribute the cake so that everyone gets the same amount?“ Easy for both girls! The cake was „cut“ in the middle with a line.

„Now, if 4 guests come to visit, how can you distribute the cake so that everyone gets the same amount?“ Lotte immediately draws the second centre line through the cake. Edith ponders longer, looking at her cake.

Then 8 guests!
Lotte draws the corresponding diagonal lines, also in the right place. The cake pieces are almost the same size this way. Edith draws them too, although she can also draw from Lotte. To illustrate the cake pieces, they should use different colours for the lines and number the eight pieces.

Lotte managed 16 pieces of cake well. With Edith, the lines went over each other, resulting in very small uneven pieces.

Writing down these division tasks, they both did not want to learn that – at least not at that time. But they announced very motivatedly that they wanted to paint the cake: with strawberries and chocolate …

Before that, we shared a square cake. As with the round cake, Edith first drew the vertical and horizontal lines. Then, to get 8 parts, the diagonals. Lotte, on the other hand, drew the diagonals first and divided them again later – but in such a way that unequal pieces were created. I think the task with the same size was not so clear to her. But you can only see the result when the lines are already on the paper, maybe she wanted to try something out.

Part 3
After the wonderful colouring of the cake, I asked them if we should stop.

I made a new worksheet for Edith. It contained – written from top to bottom – the numbers from 1 to 10 written as a number and next to it the corresponding number words.
Edith read the numbers to me first, but she „cheated“ and didn’t read the number word, but the number. I then wrote down five more number words for her – not in order and not with the number next to it. She then read them to me and enthusiastically started copying them. Very carefully and accurately!

In the meantime, Lotte very motivatedly drew a fruit skewer (this was Lotte’s own idea), which is ideal for three children to share. Then she also drew a vegetable skewer. After that, she wanted the same worksheets as Edith. While Lotte then read the numbers and copied them down, Edith then drew skewers. Lotte copied the numbers very quickly and did not pay attention to the size of the letters or to writing straight (unlike Edith).

Duration: 1 hour and 20 minutes

Observations during the offer:

The two children were not only very nice to each other, but a funny atmosphere developed in between, where both of them really „fraternised“ against me. It was also noticeable that Lotte did not find an end this time either, although she seemed quite tired at first, during the minus tasks. The fact that Lotte is usually quicker with her answers or with her work doesn’t seem to bother Edith. I think each child was able to profit from the other and learn something.

Somehow we didn’t get around to making a number picture ourselves. They both understood the task of sharing well.

Read also: Basic Ideas of Mathematics.

Final thoughts

About the project:

The content of the project was very well chosen for both children and is far from being  exhausted. Both of them were always happy when I collected them in the morning, even though they then, so that we had enough time, could not take part in the „morning circle“. Unfortunately, due to a lack of staff, I often had to stop the activities earlier than it would have been suitable for the children. This was especially true for Edith, who never really wanted to stop.

I tried to explain it to the children in a way they could understand and I think I succeeded. During the sessions, both children were very friendly and respectful with each other. They gave each other tips and help and were very patient with each other. (Lotte, for example, had to wait more often when Edith was not yet finished because of her thoroughness and accuracy).
Neither grumbled or complained about the other. They clearly enjoyed their privilege of doing these offerings together with me alone.

I hope that I will be able to continue this project with them. Apart from the great knowledge potential of the two, which needs to be fostered, and apart from Edith’s huge intrinsic motivation, this is also the best way to enable the two to become friends. Apart from the situation in the gym, where the two of them played alone for a very long time, I don’t know of any other situation where they played together intensively. They both have their own play partners. But the project is really only at the beginning, and I believe that there is still a lot that can develop.

At least they will be able to draw on their shared experiences when they both attend the same school class and experience going to school together.

To Edith (my „observation child“ in the IHVO Course):

Edith’s intrinsic motivation is also huge when it comes to arithmetic. However, she seems to be even more interested in language. It was good for me to have Lotte there for comparison. Lotte is known as an intelligent child – but will soon be 6 years old and has a sister in third grade. Edith is only 5;4 years old, she is an only child, both parents work and she is usually picked up very late from kindergarten.

When I look at all this together, I think that Edith is certainly far more gifted than average and needs appropriate support.

Postscript

Six years later, contact between Edith and Lotte still continues and is supported by the parents. Edith has since been tested, showing high intelligence in the language area.

Read more about Edith:

Project „Letters“ – Activity for Small Groups 

 

Date of publication in German: September 2012
Copyright © Lucy Rüttgers, see Imprint.