# How would you do the math mentally

## Development of surgical presentations

### General information on the presentation of the operation

Children with difficulties in learning math usually have a one-sided understanding of operations. For them, mathematics is sometimes a "world full of mysterious digits and symbols that have to be regularly linked to one another in an even more mysterious way." times and shared.

If children can perform arithmetic procedures, but have difficulty translating a mathematical task into pictures or actions, this indicates that they have not yet understood what is actually meant by the operating symbol. They do not understand an operation as an action, but rather the sign as an algorithm, as a mere invitation to calculate. For the development of basic ideas, the "basic idea [...] is that concrete actions on suitable materials are converted into mental operations." (Wartha / Schulz 2011, p. 11)

### How do you recognize unsustainable ideas about operations?

A possible diagnosis for one-sided and unsustainably developed surgical concepts are tasks in which children should explain a bill, e.g. in the form of an "Indian story". Here a child should explain to an Indian child who does not speak or understand the German language and knows neither numbers nor arithmetic operations, e.g. what is 2 · 5. The representations that children choose say a lot about their vision of the operation. (see Schipper 2005, p. 53ff.)
An example of one-sided operation presentations is that children fall back on the pure calculation (see Fig. 1). Here a child draws both the multiplier (2) and the multiplicand (5) as a set for the representation and connects them as a painting task with a symbol. The result cannot be seen in this illustration. Fig. 1: Student representation for the task 2 · 5
(Based on: Schipper 2005, p. 55)

### Flexible and sustainable operation concepts

Solid operational concepts are important, since arithmetic is not just about practicing arithmetic rules that have not been understood. This mechanical side of arithmetic must also be learned, but the arithmetic operations must also be understood and ideas about what they mean must be developed.
Practicing and ultimately automating tasks, e.g. the multiplication table as memorized, retrievable tasks, is justified, but for a deep mathematical understanding it must be understood in advance which dynamic action or static arrangement stands behind 3 5 before the children can Name result 15 because you have automated the result. With an idea of ​​the importance of arithmetic operations and an understanding of the relationships between tasks, children can learn and remember arithmetic tasks more easily and permanently. (cf. Gaidoschik 2007, p. 69f.)

The various basic ideas about the four operations can be static (including combining, comparing) or dynamic (including adding, removing, adding).
If children are to take on the role of teacher explaining what “plus” means, much can be inferred about how the children's surgical ideas are developed. If the child shows his fingers and counts fingers accordingly, this can be referred to as a dynamic idea. If objects are placed next to each other and united, the idea is more static. On the other hand, it can also happen that children place the numbers next to each other with objects in the form of a calculation (see Fig. 1). It can be assumed that the concept of the operation is not sufficiently developed.
(see The Computing Institute for the Promotion of Mathematical Thinking, no date)

### Development of the surgical concept

The development of the basic ideas can be achieved based on actions on the material, the flexible interpretation of representations and on the basis of subject contexts. An understanding of arithmetic symbols is built up through “translations” of specific actions (in subject contexts) or using suitable visual aids and writing and vice versa. However, an act with material alone does not develop a viable idea, but only a reflection on it. In addition, the action with material must be supported by the teacher, as otherwise children may only count on the materials and not make use of the existing structures. Working with hidden material and describing it verbally - that is, using mental constructions for internalized actions - develop ideas about operations. (see Häsel-Weide / Nührenbörger 2012. In: Bartnitzky / Hecker / Lassek, p. 29)

Developing different basic ideas about an operation is also important in the classroom so that, for example, not only dynamic, but also static situations can be penetrated. If, for example, the subtraction is understood exclusively as taking away, this can become a problem if, for example, material tasks address a different basic idea. By getting to know factual tasks or arithmetic stories with diverse contexts, this can be counteracted. (see ibid., p. 29) Fig. 2: Ambiguous figure. Instead of 7-3, 3 + 4 and 7-4 can also be recognized.
(Based on: Gaidoschik 2007, p. 79)

In addition to actions, discussing, describing, and interpreting representations and actions relating to invoices promote sound operational ideas (cf. ibid., P. 30). E.g. the arithmetic problem 5 + 4 can be done step by step with the sentences “First I have 5 fingers, then I add 4 fingers. In total I then have 9th “to be linked. The connection of mathematical symbols with real situations deepens the understanding of the plot. (cf. Gaidoschik 2007, p. 74f.)

Furthermore, illustrations are suitable for the construction of surgical concepts. However, representations cannot stand on their own and are even more ambiguous (see Fig. 2). Children will find a wide variety of invoices for an illustration, all of which can be justified with a corresponding justification.

That is why it makes sense to let children interpret illustrations themselves and under no circumstances to pretend an interpretation as "the correct one". Making images of your own can also give children a deeper understanding of an operation. (cf. Gaidoschik 2007, p. 79f.)

### Equations as a reason to calculate Fig. 3: Typical faulty solution.
(Based on: Gaidoschik 2007, p. 155)

The equals sign has a special meaning in connection with surgical concepts. While the sign is often interpreted primarily as an invitation to do arithmetic, it is important for children to understand the meaning of the equals sign even in elementary school. The equivalence of both sides of the equation can be illustrated, for example, as a scale. The fact that an invoice is not called up immediately for every equals sign is necessary for a sustainable operation concept. (cf. ibid., p. 154ff.)

### Exchange with other children

For the development of ideas about all operations it is important that children with and without difficulties in learning mathematics develop ideas about the operations and the actions associated with them. You must then be shown further ideas that also exist for the operations. This can be done through an exchange with other children who have an alternative idea. In doing so, they expand their own knowledge and also verbalize their own ways of thinking. (see Häsel-Weide / Nührenbörger 2012. In: Bartnitzky / Hecker / Lassek, p. 37)
Otherwise, children may learn an operation as an algorithm without understanding the operation. The goal of the math class is the flexible representation and presentation of the operations. (see ibid., p. 29f.)

### Literature:

The computing institute for the promotion of mathematical thinking (ed.) (O.J.): Typical difficulties in the 2nd school year.
http://www.recheninstitut.at/mathematische-lernschwierbaren/merkmale/typische-schwierbaren-im-2-schuljahr/ [07.05.2017]

Gaidoschik, Michael (2007): Preventing arithmetic weaknesses - First school year: From counting to arithmetic. Vienna: ÖBV HPT.

Häsel-Weide, Uta / Nührenbörger, Marcus (2012): Support in mathematics lessons. In: Bartnitzky, Horst / Hecker, Ulrich / Lassek, Maresi (eds.): Individual support - strengthening skills - in the entry level (grades 1 and 2) (Vol. 134, volume 4). Frankfurt am Main: Primary School Association.

Schipper, Wilhelm (2005): Module G4: Recognizing learning difficulties - promoting understanding learning. Kiel.

Wartha, Sebastian / Schulz, Axel (2013): Preventing arithmetic problems. 2nd edition, Berlin: Cornelsen.

This funding material focuses on the change between the different forms of representation (task - mathematical action on the material - image - arithmetic history).

In the beginning lessons, addition and subtraction are the central operations that children need to develop ideas about. Even before they start school, many children can add and subtract and solve dynamic problems with materials or with the help of their fingers. In addition to dynamic, static ideas about addition and subtraction are required for the development of a deep mathematical understanding. In school, the previous knowledge is further developed and the calculation with materials is supplemented by the abstract numbers and mathematical symbols for plus and minus in order to solve the same factual situations with them. (cf. Gaidoschik 2007, pp. 70f.)
It is problematic if addition and subtraction are understood exclusively as counting up or down, because it is difficult to develop a comprehensive, diverse understanding of the operations. This also becomes clear in factual situations in which arithmetic operations should not only be activated on one side, but must be used flexibly. (cf. Wartha / Schulz 2013, p. 48)

The addition has different basic ideas. On the one hand, a set can be “added” dynamically or statically “combined” with a set (see Fig. 4).
The training of the operation concept can be done by translating arithmetic tasks into actions, representations and verbalization and vice versa. There is a multitude of possible representations and actions for every arithmetic problem. In contrast, there is no “one” representation of an addition problem. (see Häsel-Weide / Nührenbörger 2012. In: Bartnitzky / Hecker / Lassek, p. 29f.)
The “simple” addition tasks are initially a support that children can use to stimulate the operation ideas for addition. This includes tasks that they can master, such as doubling problems, adding up to 10, or adding an even ten. Since these tasks are often mastered memory-wise in the beginning of the lesson, you can build on them. If these tasks are used, further tasks can be opened with “1 away” or “1 in addition”. Because the actions are presented mentally, the children's operational ideas are further developed. (see Häsel-Weide / Nührenbörger 2012. In: Bartnitzky / Hecker / Lassek, p. 30f.)

Illustration 1

Basic concept

Illustration 2

Andrea has 5 tiles. She gets 3 in addition. How many tiles does she have now?

5 + 3

Sonja has 5 tiles. Pascal has 3. How many tiles do you have together?

+ as a summary

5 + 3

Michael receives 5 tiles from Axel and 3 from Claus. How many did he get in total?

+ as a summary of two changes

5 + 3 - as taking away

11 - 4

Alex has 11 tiles. Tom has 9. How many tokens must Tom get so that he has as many tokens as Alex.

- as a supplement

11 - 9 - as comparisons

9 - 4

Together, Helena and Melanie have 11 tiles. Melanie has 6. How many does Helena have?

- as the inverse of +

11 - 4

Fig. 4: Basic concepts of operations addition and subtraction.
(Based on: Wartha / Schulz 2013, p. 31)

### Basic concepts of subtraction

Subtraction offers three different concepts of operation. When "taking away", the subtrahend is dynamically deducted from the minuend, while when "comparing" the difference between the minuend and subtrahend is considered statically (see Fig. 4). “Complement” takes up an additive notion. Although this basic idea is used less often, it is adept at some tasks. For example, with 65 - 63 the result can be calculated via an addition, which would be much more time-consuming if it were removed. To illustrate the addition, for example, the calculation line can be used, which illustrates the step of minuend and subtrahend. (see Häsel-Weide / Nührenbörger 2012. In: Bartnitzky / Hecker / Lassek, p. 35f.)

### Illustrate subtraction

When depicting the basic idea of ​​“taking away”, the difficulty lies in depicting all three subsets in such a way that they remain visible. However, if something is removed from the initial quantity, for example a turning plate, then the initial quantity can no longer be recognized when the operation has been carried out. While with the addition (e.g. 4 + 3) both quantities can be placed next to each other, with 4 - 3 the latter 3 are no longer there. The mapping is much more difficult to relate to the task. In representations, an implied removal, crossed-out platelets or slightly transparent foils over platelets (see Fig. 5) are often used. This covers the subtrahend without breaking the minuend. Even with these representations, which a trained eye can clearly see as a subtraction, children must first learn to “read” the representation. This can be achieved with actions and speaking. (see Häsel-Weide / Nührenbörger 2012. In: Bartnitzky / Hecker / Lassek, p. 32f.) Fig. 5: Different representation of the subtraction problem 14 - 5 = 9 through transparent foils over plates.
(Based on: Häsel-Weide / Nührenbörger 2012 In: Bartnitzky / Hecker / Lassek, p. 33)

After basic concepts of addition and subtraction have been developed, doubling and halving are necessary as pioneers of multiplication and division for a deep mathematical understanding. (see ibid., p. 29)

### Literature:

Häsel-Weide, Uta / Nührenbörger, Marcus (2012): Support in mathematics lessons. In: Bartnitzky, Horst / Hecker, Ulrich / Lassek, Maresi (eds.): Individual support - strengthening skills - in the entry level (grades 1 and 2) (Vol. 134, volume 4). Frankfurt am Main: Primary School Association.

Wartha, Sebastian / Schulz, Axel (2013): Prevent arithmetic problems, 2nd edition, Berlin: Cornelsen.

The material “Maths safely. Handout for a diagnosis and support concept to secure basic mathematical skills “possible tasks to diagnose and promote surgical concepts.

Link tip:Math sure: "Material on the content area" Natural numbers ""

### Ideas about multiplication & division

Even before multiplication and division were discussed with children in the classroom, they can already solve specific problems with their prior knowledge. Without a viable idea of ​​the operation, they often come up with solutions by modeling directly with materials and then counting them. With other strategies - e.g. rhythmic counting, learned sequences of numbers as well as links for addition in multiplication tasks or repeated subtractions and links for multiplication in division tasks - some children, on the other hand, solve the tasks with previous knowledge that indicate an understanding of operations. (cf. Padberg 2009, p. 114ff .; ibid., p. 138ff.)
Sustainable and diverse ideas about multiplication and division have to be developed in the classroom and are often a challenge for children even at the end of primary school and internalize. (see Gaidoschik 2009, p. 50)
To diagnose the children's level of knowledge, tasks with contexts of the children's realities and open tasks that allow different processing methods and results and can thus be solved in a variety of ways are suitable. The development or further development of the basic ideas can be based on the solution strategies used by the children.

Link tip:KIRA: "Multiplication and division: learning levels and developments"

### Operation presentation for multiplication

In school books, multiplication tasks are often depicted both as a “temporal-successive action” (see Fig. 6) and as a “spatially simultaneous arrangement” (see Fig. 7).  Fig.6: Example of multiplication as "temporal successive action" Fig. 7: Example of multiplication as "spatially simultaneous arrangement"

In addition, it is also possible to imagine multiplication in a combinatorial context.
In the dynamic presentation of a successive action for multiplication over time, the total amount or the product is created by repeating the same action several times, in which an equal amount is added, e.g. by fetching 5 apples three times. Here the connection between multiple addition and multiplication becomes particularly clear. Due to the everyday use of e.g. "three times", this idea is often very close to everyday life for children. (cf. Padberg 2009, p. 117f.)
Spatial-simultaneous arrangements, on the other hand, statically depict a total quantity as several equally powerful quantities. Here, the quantities are statically combined, which can also be associated with the basic static concept of addition ("joining"). The spatial-simultaneous arrangement can also be seen as the result of the temporal-successive action and establishes the close connection between the two ideas. It is important that children can recognize multiplication tasks in both situations. (see ibid., pp. 118f.)
Combinatorics addresses another idea of ​​multiplication. If, for example, you are looking for all options for two-course menus where you can choose from 3 starters and 5 main courses, you can use 3 x 5 to determine the number of combinations. Here, 5 main courses are possible for each of the two starters. This notion does not make sense for the introduction of multiplication, as it is more difficult to work with materials and representations. However, the combinatorial aspect of multiplication is indispensable for technical problems. (cf. ibid., p. 120f.)

### Operation presentation for the division

After the multiplication has been discussed, the division should be introduced separately in the classroom. It can be interpreted as a division or distribution situation.
When splitting, the total amount is divided into subsets that have a known equal number. An example of this can be a group of children with a total of 6 children, which is divided into groups of three. The number of small groups (2) corresponds to the result. When dividing, the number of partial quantities is sought, while the total quantity (6) and the group size (3) are given (see Fig. 8). Fig. 8: Example of division as "splitting up"

This type of presentation is close to everyday life and vivid and at the same time shows a connection to multiplication if the groups were arranged spatially and simultaneously. (cf. ibid., pp. 141f.)
In contrast, when distributing, the number of subgroups is known and the number of elements of the subsets is searched for. An example of this could be distributing 16 marbles to 4 children. Tasks that address this idea offer actions with materials in which, for example, marbles are distributed individually or in steps (see Fig. 9). Fig. 9: Example of division as "distribute"

It is important that a fair division is required in subject contexts, which relates solely to the number and does not, for example, equate two small marbles with one large one.
Distributing is also useful to illustrate division with the remainder. If, for example, 22 marbles are distributed to 5 children, the action shows that this cannot be done completely. (cf. ibid., p. 143f.)

### Development of basic ideas

“You just have to memorize the multiplication table” (Gaidoschik 2009, p. 52), and one goal of the lesson is of course that children can automatically call up the multiplication tables. However, this should not be done by memorizing the individual rows in the manner of a poem. Instead, the retrieval should be at the end of an understanding-building lesson. Otherwise the individual tasks will only be mastered as long as the series are memorized and will be forgotten after a while. (see ibid., p. 52)
Memorization, in particular, often leads to children with difficulty learning math performing well when introducing multiplication. They can call up the tasks or count up the rows, but only a viable operation concept is sustainable for mathematical understanding. (see Häsel-Weide / Nührenbörger 2013 In: Bartnitzky / Hecker / Lassek, p. 35)

In order to learn the various basic ideas about multiplication and division, dealing with materials is indispensable for all children. The concrete multiplication of an action, e.g. if four times 5 marbles are fetched, let children experience the idea of ​​a time-successive multiplication. The translation of the action into a linguistic form ("four times five marbles") and finally into a calculation term (4 · 5) and the back-translations promote the understanding of multiplication. (see Gaidoschik 2009, p. 53)
The material used must be selected with regard to the higher number range and a structure. So that it does not have to be counted repeatedly, but still ambiguous and flexible representations can be found, special point stripes are suitable, for example, to represent a multiplication (see Fig. 10). Here, evenly-sized sets are dealt with (point strips with the same number of points in each case), these are used to carry out actions (adding and subtracting several point strips) and increasingly detached from the actions and instead concealing them or performing them mentally. The point stripes can be arranged and interpreted as a representation of a multiplication as rows of points or as point fields. For example, the subsequent geometric area calculation is prepared with flat point fields. (see Häsel-Weide / Nührenbörger 2013 In: Bartnitzky / Hecker / Lassek, p. 36) Fig. 10: Stripes of points with 4 points each can be used to represent multiplications, such as 5 · 4 here.

### Literature:

Häsel-Weide, Uta / Nührenbörger, Marcus (2013): Support in mathematics lessons. In: Bartnitzky, Horst / Hecker, Ulrich / Lassek, Maresi (eds.): Individual support - strengthening skills from grade 3 (Vol. 135, Book 2). Frankfurt am Main: Primary School Association.

Gaidoschik, Michael (2009): “You just have to remember that” ??? - Difficulties with the multiplication table: Some suggestions for prevention and remedial measures. Vienna. http://www.recheninstitut.at/mathematische-lernschwierbaren/endungips/einmaleins/ [07/05/2015]

Padberg, Friedhelm (2009). Didactics of arithmetic. For teacher training and teacher training (3rd expanded, completely revised edition). Heidelberg: Spectrum Academic Publishing House.