# Why do children fail at physics and mathematics

__Understand numeracy problems__**Information paper on the phenomenon of arithmetic weaknesses / dyscalculia**

**© IML-Essen and RESI-Volxheim. All rights reserved. Reprinting, even in part, is only permitted with express permission. - Essen / Volxheim 1998/2001**

**Editor - collective authors of the numeracy therapists:**Boerner, Gabriele - state-certified teacher of mathematics / physics, Boerner, Klaus - Dipl. Psych., Supervisor BdP, Brettschneider, Jutta - Dipl. Päd., Spagl (Czerwinski), Carmen - Psychologist, Steeg, Friedrich H. - Dr . Dipl. Psych., Vogel, Jacqueline - Dipl. Päd.

**1. My child has a "numeracy weakness"!? - symptoms**

**2. Examples - individual error system**

**3. Learning - assessment - comparison - self-image**

**4. Mathematical weakness - what is it? - a question of definition?**

**5. Dyscalculia diagnostics - what does it have to do?**

**6. Dyscalculia therapy - what does it have to do**

**7. Plea for teacher training - what must it achieve**

**8. Our literature recommendations**

**1. My child has a "numeracy weakness" !? - symptoms**

Perhaps you have reasons and have registered indications that lead you to the assumption that “your problem child” suffers from a “weakness in arithmetic”. Perhaps you are also interested in this special learning process disorder in general and would like to find out more about this area - also so that you can become more aware of such children in the future.

The fact is: There are observations through which a child stands out and whose frequency, persistence and interaction should be taken seriously as an indication of the presence of a "weakness in numeracy". We would therefore like to first introduce you to a catalog of what we consider to be the most important observations:

**Observations on behavior in school and in mathematics**

1. Fear of school

2. Afraid of mathematics

3. Afraid of the math classwork

4. Fear of the teacher in mathematics

5. Failure in the subject of mathematics, although previously "successfully" practiced at home

6. Compared to classmates, it takes a lot of time to do math homework

7. Compared to other subjects, it takes a lot of time to spend on homework in mathematics

8. Frequent impression of total forgetfulness

9. Answers or questions often show a complete lack of understanding for the task at hand

10. Annoying to defensive reactions to help at home

11. Appeal to authorities to justify answers: "the teacher said - grandpa said - mom said!"

**Orientation problems and language problems**

12. left-handed or ambidextrous, or right-handed

13. Problems with positioning like top, bottom, left, right, between

14. One and tens are often swapped

15. Similar digits such as 9 and 6, 7 and 1 are often confused

16. Problems due to dialect, other mother tongue, poor vocabulary

17. There are visual defects, hearing defects, other delusional problems

**Observations in dealing with numbers and arithmetic**

18. Tasks are mostly dealt with counting

19. in the event of uncertainty, the counting starts again from the beginning

20. Problems counting backwards

21. Difficulties just counting from the imagination, e.g. "There are ... 6 chairs in our living room!"

22. Memorization as a compensation strategy

23. Need for donkey bridges or rhymes

24. Remember "rules" as a compensation strategy

25. Tasks such as 15 + 3 or 23 + 2 are also dealt with in writing

26. Difficulty distinguishing the quantity aspect and the number aspect of numbers

27. Interchanging calculation types - minus with plus, sometimes with plus

28. Difficulties in analyzing verbally or in writing given factual tasks and in translating the problems to be solved into mathematical operations

29. the meaning of the equal sign is not understood, formulations like: "then I write the result - then I write the larger number - then I write the smaller one - then I'm done!"

30. Problems with job transitions

31. Errors occur more frequently in connection with zero

If you have the impression that some of the problems listed above can be observed in your child, we recommend a diagnostic examination for the presence of a "mathematical weakness".

**2. Examples - individual error system**

In our experience, “mathematically weak” children do not simply produce nonsense, but their mistakes have a method and an “inner logic” that can be understood. Using a few examples from our practice, we want to demonstrate what we mean by that.

**Example 1:**

For example, we give a child the following task: 230 - 15. It delivers the following result:

230__- 15__

225

We ask ourselves: How does this - obviously wrong - result come about?

- Did the child forget to carry over to the tens?
- Does the child mix plus and minus by the technique of "adding up"? - "5 and 0 is 5; 1 and 2 is 3"?
- Does the child have the problem of not being able to subtract the supposedly larger number from the smaller one? - "0 - 5 does not work - so I calculate 5 - 0 is 5" !?
- Does the child think: "There's a 0 there, so I don't have to calculate anything" !?

In order to distinguish these different interpretations from one another, we continue to follow the child's ideas. Therefore we give him two more tasks:

203 2003__- 51__ __- 511__

252 2512

In the course of the dialogue about the processing of the task, it emerges that the child is thoroughly satisfied with his method and the results. We still cannot clearly answer the questions posed above. It differs from the solution to the first task in that the results of these subtractions are greater than the number from which they are subtracted.

In the further dialogue, the child explains that "when calculating with minus values, the numbers get smaller". For example, it calculates the task: 2003-511 again and draws the conclusion that it has calculated correctly the first time: "1 and 2 is 3; 1 and 0 is 1; 5 and 0 is 5". The child works with the supplementary procedure ("from ... to"). "1 and 0 is 1; 5 and 0 is 5" explains it with the fact that because of the 0 there would be nothing to calculate. It again receives the result 2512.

However, we can at least deduce that the child has - in the best case - very vague concept of the number zero: For example, it reads "203" as a number. In terms of arithmetic, however, as the interview showed, it treats such a multi-digit number - such as "200" and "3" - like two numbers.

**Example 2:**

During the diagnostic session with another child, the following problem arises with regard to zero:

5 + 0 = 5 5 • 0 = 5

Carelessness? - Or: what was the thought?

- Is there no distinction between addition and multiplication ?:

2 + 3 = 5

2 • 3 = 5

and is that why the zero is treated equally in both types of calculation? - Can't the multiplication be reduced to adding the same summands?

(5 • 0 = 0 + 0 + 0 + 0 + 0) - and does it fail as a control instrument? - Does the child know that the 0 is the neutral element of addition and subtraction and wrongly concludes from this: "The 0 is in principle neutral." ?

In order to investigate the question of whether there has been a careless mistake, the child is given an additional task to work on: 2 • 3 • 0 • 4; the child arrives at the following solution:

2 • 3 • 0 • 4 = 24

It therefore differentiates between the types of calculation and multiplies in this case. Arbitrary handling of the arithmetic symbols is therefore excluded. A systematic error with regard to zero is now much more obvious, which is confirmed in the following:

2 • 3 • 0 + 4 = 10

Again, the child differentiates between the types of calculation and works through the corresponding tasks correctly in terms of numbers. However, it ascribes the property of being a neutral element to all types of arithmetic to zero.

In order to further clarify the child's - apparently wrong - conception of zero, we ask him to trace various multiplications back to the corresponding additions. His solutions:

2 • 3 = 3 + 3

2 • 5 = 5 + 5

3 • 5 = 10 + 5

4 • 5 = 10 + 10

5 • 0 = 2 + 3

The child can evidently also use the information taken from the lesson that the additive decomposition of numbers does not change their value. However, since it did not understand the mathematical context of the calculation types - multiplication includes the special cases of addition: all occurring summands are of the same size - it does not have the insight into the following identity:

4 • 5 = 5 + 5 + 5 + 5

so also not about this:

5 • 0 = 0 + 0 + 0 + 0 + 0

During the entire diagnosis, the child is purely results and technology-oriented. It considers this way of dealing with matter to be adequate; It does not see any logical connections. So answering the question about division by zero poses no problem at all: The rule: “Dividing by zero is prohibited!” Has been well remembered. After all, it's written: in his notebook, on a worksheet or in a book. Besides, he's been told it often enough.

When performing divisions in writing, his misunderstanding regarding zero becomes apparent:

453750 : 15 = 325__- 45__

0037

__- 30__

075

__- 75__

00

Or what do you think about that?

**Material tasks**

In the case of factual tasks, because of their clear content, viewers such as parents and teachers often ponder whether something is wrong with the child's intelligence, talent or state of mind. Produced results often appear “totally off the mark”, as if the child did not take note of the content of the text or had only inadequately taken note of it.

This judgment is only owed to the point of view of the adult, who may not discover any difficulty in such questions, since the matter to be dealt with seems to be obvious - if one only reads the text carefully. He can only wonder. The assessment criteria are not the comprehension and the content assessment of the child's thought processes, but the comparison of the child's “absurd” result with the child's own opinion. If such events accumulate, the child appears at best as inattentive or lazy to think. Most of the time, however, he is denied even the ability to think logically!

Here is a small excerpt of "mathematical products". All examples have one thing in common. The producers of results agree that there is something to be calculated, because after all it is about factual problems in mathematics - examples:

a) Read on the surface?

Beate buys an ice cream for 3 DM. The ice cream that her friend buys is a mark cheaper. How much do both cost together?

Calculation: 3 + 1 = 4

Answer: It costs 4 DM.

The child does not like to discover a contradiction. It does not see that the price for the second ice cream has to be calculated first. Comment of the child: "One mark is cheaper than three marks!" During the further discussion of the content of this task, the child explains the following:

1 is less than 3; 3 is greater than 1 - How much is 3 greater than 1? - At 3! - How much is 1 less than 3? - At 1!

b) Totally wrong?

A comic costs 3 DM. Fritz buys 2. He pays with a ten-mark note. How much money does he get back?

Calculation: 3 + 2 = 5

5 + 5 = 10

Answer: He gets 5 DM back.

Here the numbers are interpreted from the question - it's about money - as follows: A comic for 3 DM and one for 2 DM are 5 DM. 5 and again 5 are 10, so I get 5 DM back. The student said no to the question of whether this task could be calculated differently. “10 - 5 cannot be calculated because 10 is at the very bottom of the task! I have to calculate 3 + 2 first and you can't subtract 10 from 5, it's way too big! "

c) Numbers are there for arithmetic!

5. Lilo buys 30 eggs. They are packed in 3 boxes of the same size. How many eggs are in a carton?

Invoice:

5 + 30 = 35

35 + 3 = 38

Answer: There are 3 eggs.

Comment: "I have to calculate all the numbers. I have to start at the beginning. If I work in steps, it's easier. That's how we do it at school!" When prompted, the child does the math again aloud and confirms his result. "I think that's correct, there are also 3 numbers!" When the result is clearly checked, the child reacts with helpless amazement: "Something is wrong, but the numbers are in the story!"

d) Taken literally!

Twelve out of 36 students take part in the English class. The others have free work. How many students have free work?

Calculation: 36: 12 = 3

Answer: Three students have free work.

The “stimulus” word “part” tells the task that it has to be a division. The fact that everything “works out” fine is taken as confirmation.

e) Also totally wrong?

Anja goes on a trip to the farm with her friend Sonja and her parents. Sonja discovers 3 stray cats. Anja is playing with a dog. Create a task and solve it!

Question: Sonja asks her mother if she can take a cat home with her.

Calculation: 3 - 1 = 2

Answer: Sonja now has a cat.

In the interview, the child says that he would like to have a pet. Your friend has a hamster who is so cuddly. It imagined that if Sonja was allowed to take one animal with her, there would be one less in the yard. So the result is that Sonja now has an animal, that's great.

The child projects his wish into the task given to him and solves it accordingly. It has taken note of the text!

This is confirmed by the attempt to offer him the following solution to the task: How many animals are there on the farm? - 3 cats + 1 dog = 4 animals - comments as follows: “You can calculate that too, and you did the math correctly; but I like my job better: Sonja has a cat now! "

We can continue the list of similar tasks and their processing almost endlessly, even at higher requirement levels. At the latest when working on technical tasks that require the application of mathematical insights in the area covered, pure rule-based thinking and additional linguistic problems lead to chaos:

The handling of questions on this higher (= application) level requires a solid foundation - tools that can be used safely:

After gaining insight into the context of a task, the mathematical questions must be recognized and appropriate solutions found.

Arithmetically weak children who have never found this approach to the question deal with factual tasks accordingly "irrelevant": They see their task in making use of the "numbers" and "operations" that are visible to them - in whatever way.

Problems can arise on the linguistic, analytical level, on the mathematical level, as well as their connection.

The arithmetic operation selected is not due to the analysis of the facts in question, but to what the child thinks is feasible for himself: “I can do plus arithmetic!” Or what he has to hand because it is currently being dealt with in class (has been).

In this respect, however, hit rates that are acceptable in terms of grades are not that unusual, even for material tasks.

**3. Learning - assessment - comparison - self-image**

The beginning of school attendance now complements this childlike spontaneous and intuitive learning with systematic lessons and steers it into predetermined paths. In substance, this "forced" change in the activity of the child's interest has advantages and disadvantages:

On the one hand, the coincidences of childhood learning are translated into consistent work on a thing. On the other hand, learning is then no longer simply part of the child's everyday life and is mainly oriented towards his or her interests. It becomes a requirement of the child, owed to foreign standards set by the adults.

Even in preschool, the child's learning and its results are judged by the environment - the child receives praise and criticism.With the start of school attendance, the importance of assessment changes. The assessed learning is the decisive condition for the further course of the individual school career. The child notices this "new" standard in a purely practical way, both inside and outside of the classroom, and learns to work off it: Failures in mastering mathematical questions no longer mean simply not having understood something, but "failed in mathematics" to have.

In school, assessments are related to correct or incorrect results. In order to deal with the subject of mathematics in this way, it is appropriate that all students have one or more times the opportunity to "somehow" process a thought dealt with in class. However, all students have to present results at the specified time - regardless of whether they have come to a correct understanding with their processing at this point in time.

This is followed by the evaluation of their respective individual results = general grading.

The consequence of dealing with learning content in this way: The child's interest in the learning object is put into perspective. It shifts to evaluating the results - to grading. What the children now strive to do is to learn methods and strategies for achieving school success or avoiding failure. The good or bad grade gives students a “personal value”. Based on the grade, the logical conclusion of the student who was graded as poorly means at least: "I have failed!"

In school lessons, the learning child is subjected to a performance comparison with other children and thus also evaluated as a personality at the same time.All children have been given the same task - but: "You did it badly or not at all!" So which conclusion is closer than: "It's up to me!" develop mental problems.

The abstract assignment of grades to the students gets its psychological impact from the comparison within the class, through which this type of assessment alone makes "sense".

The “failure” in mathematics major weighs heavily on the child's self-assessment as a developing personality. Ultimately, the student does not consider what he has not yet correctly understood in mathematics to be his problem, but himself - his "motivation", his "skills", his "talent", his whole personality.This can go so far that his psychological consideration of the problem not only completely prevents the analysis of actual mathematical difficulties, but replaces it with dealing with other "problems". The result is that any further learning in the subject fails.

The more such a self-assessment becomes entrenched, the more likely the child will develop negative psychological sequelae with which the child, parents and teachers also have to struggle.

Praise and blame are given for good or bad performance, good or bad behavior. The objects of learning are subordinate to these points of view! The decisive offer of orientation for the child for the development of his or her personality lies in praise or criticism:

It has to prove itself against the given learning objectives in such a way that it adopts the desired evaluation criteria. The vicious circle begins: The child faces this probationary task and two options arise: "Success" or "Not success"!

It can usually be observed that the child's achievements, regardless of how they are to be assessed in terms of content,

Case 1: used to gain the pleasure of parents and teachers - those around him

Case 2: tries to shift to areas outside of school and pedagogically undesirable, to resist learning in school.

In case 2, learning becomes a struggle for survival or a battle of defense. The child tries to evade, to pretend performance, to gain recognition. "Self-worth" is largely determined by assertiveness in the educationally negative sense. Common judgments about such children are, for example: refusal to perform, behavioral disorder, ...

In case 1, the child usually has an even bigger problem: it tries to please "the other". It makes one's own “self-worth”, beyond the ability to achieve one's own success, dependent on recognition from parents, teachers and classmates. If success does not materialize, this result stands for "personal failure", i.e. worthlessness of oneself. If success is formal - e.g. a good class test with more or less randomly correct results, the praise received for it can raise self-esteem. The next failure, the rebuke is felt all the harder: “You could do it last time!” The feeling of failure intensifies: “I should have been able to do it and I seriously disappointed“ the others ”in their expectations.” That weighs twice as much and drives up the spiral of guilt and self-doubt!

Such connections can lead to neurotic undesirable developments which, once they have become an integral part of the child's personality, later have to be absorbed again in lengthy psychotherapeutic learning processes.

We hope that with our explanations we have contributed to the fact that you can now better understand that "arithmetic" children

- ... are not stupid ...
- are not lazy ...
- don't practice too little ...
- ... too much practice the wrong thing ...
- memorize too much to survive without understanding ...
- often do not dare to ask when they do not know what to do ...
- often not finding help because nobody understands what they don't understand ...
- who are unable to discuss many of their own ideas and theories ...
- often wonder why they are wrong again ...
- at some point think: something is wrong with me ...
- therefore almost despair ...
- ... and at the end draw the conclusion: I am just like that!

**4. Poor numeracy - what is it? - a question of definition?**

“Computational thinking” is an important part of intelligence measurement. The main subject of mathematics is groundbreaking for school and professional careers. Failure in this subject torpedoes the entire life plan.

The phenomenon of “weaknesses in numeracy” has been known to science for decades. It is also known - at least to educational advisors, school psychologists, paediatricians, child and adolescent psychiatrists - the high probability that these children will develop psychoneurotic secondary problems in individual cases. There are many theoretical approaches and definitions. Almost all of them have one thing in common - the characterization of the phenomenon as "partial performance weakness". The reference is the comparison of performance with other subjects.

Two exemplary definitions:

“Arithmetic disorder: impairment of basic arithmetic skills. This disorder includes a specific impairment of numeracy skills that cannot be explained solely by a general intellectual disability or clearly inadequate schooling. The deficit concerns the mastery of basic arithmetic skills such as addition, subtraction, multiplication and division, less the higher mathematical skills that are required for algebra, trigonometry, geometry and differential and integral calculus. "

(WHO / ICD 10 - International Classification of Mental Disorders 1995, p. 277 under F8 Developmental disorders, F81 circumscribed developmental disorders of school skills) (ICD 10 F81.2)

“If a child with a normal level of intelligence is consistently weak at arithmetic or fails at all, it can be justified to suspect a weakness in arithmetic. Not every child who calculates badly has a numeracy weakness. (...)

There is also no arithmetic weakness, but as many different arithmetic weaknesses as there are arithmetic weak children. No one is exactly the same as the other. The arithmetic weakness is an abstract collective term. In the concrete case we are dealing with the individual arithmetic weakness of a certain student. "(Wolfensberger, 1981)

If mathematicians are allowed to grasp logical connections in non-mathematical fields, one must look for the reason for the failure of mathematics in the matter itself, its presentation and the handling of it.

Beyond all intelligence tests, grades and “hereditary traits”, the many years of experience of numeracy therapists in dealing with the thoughts of their clients shows that they are also capable of learning in the mathematical field.

A concrete assessment of poor computational performance shows that the clients did not understand mathematical facts at all or did not understand them correctly. However, most of their flawed math solutions can be traced back to justifiable strategies. If a “mathematically weak” child has not understood a subject presented in the classroom, it makes its own “verse” on it. For the mathematically versed “spectator” this may take on absurd traits. The content of the thoughts of the “mathematically weak” children is nonetheless oriented towards providing the “little thinker” with a support that enables him to “somehow carry on” - in a world in which it is a matter of achievements that are absolutely necessary are to be provided!

The children develop the most imaginative, intelligent techniques and strategies. These own, flawed procedures are called subjective algorithms in science. In a more or less systematic way, they show what notions “poor math” children have of the “world of numbers and arithmetic”. The starting point for understanding the “arithmetic weakness” of the individual child, its assessment and appropriate treatment must be found in the characteristics of this individual world of thought.

The conditions and reasons for the development of this world of thought lie in the world in which the children develop themselves and their ideas. The decisive factor is the world of experience at school, the experience of the behavior of teachers, parents and other important caregivers such as friends, as well as the television program, all kinds of leisure activities ... - in short: all experiences that a child has in his life and how it processes them.

The reasons for the individual “arithmetic weakness” can therefore only be found in conversation with the child - about his world of thoughts, his algorithms, his mistakes, his contradictions. And so the correct exploration of the phenomenon of the superficial "weakness" of the individual thinking of the child provides at the same time the connection point (s) for the development of his intellectual potentials and capacities - cornerstones for the project: Therapy with "mathematically weak" children!

It is no wonder that a “mathematically weak” child can achieve good or average results in other subjects, and neither is it an indication or proof of a certain “partial disorder” of learning. Conversely, it becomes an argument: children can achieve intelligent performance in all areas of thought and knowledge if they meet the intact minimum requirements. Whether the achievements in different areas correspond to the testable knowledge of the curriculum in the sense of objective knowledge depends essentially on the fact that a continuous learning process has been set in motion. In children with “poor numeracy”, the learning process breaks down or has never really started. The unaffected continuation of lessons in school turns math lessons into a nightmare for these children. They no longer understand what the class is trying to teach them. They look for hope and hit rates through their algorithms. Teachers can no longer individually track down and catch up with the exit points. Sometimes they even consider the child no longer capable of learning.

The project to fathom the “arithmetic weakness” of an individual is therefore not a question of quantitative comparative measurements, not a question of correct and incorrect solutions in a limited time, with whatever degree of difficulty and in whatever amount. The reasons for "arithmetic weakness", the content of the arithmetic weakness, is a question of the qualitative analysis of a momentary individual thinking in its "(in) mathematical" shades - in this respect also a question of diagnostic competence and care!

**5. Dyscalculia diagnostics - what does it have to do?**

Dyscalculia diagnostics is differential and support diagnostics. It examines the client's specific difficulties in the mathematical field as well as their extent and manifestations (individual algorithms). The exact location in the mathematics building is the foundation of the therapeutic concept. Signs of impending or unfavorable developments that have already occurred and which additionally impair performance are: fear of performance and failure, area-specific concentration disorders. They are important indicators in determining a "learning process disorder".

The central aspect of the diagnosis of dyscalculia is the review of the assumed basic knowledge. Incorrect and correct results are analyzed for the individual solution strategies of the client. Clinical interviews and behavioral observation are therefore the adequate means. Paths of thought are revealed and thus an objective assessment of the quality of the results obtained is possible.

The diagnostic procedure is questioning, explanatory and motivating. On this basis, the therapist creates an overall qualitative profile. Support diagnostics are therefore individual and not aimed at comparing children. Such a procedure cannot and does not want to be standardized.

**6. Dyscalculia therapy - what does it have to do**

Therapy for weaknesses in numeracy is always individual therapy. In terms of content, this means: Productive disputes are held - individual knowledge dialogues with a partner who is trained in mathematics and pedagogy and psychology and who understands and presents the basic mathematics in a differentiated manner.

Personal security and a stable sense of self-worth are based on independent intellectual performance and the certainty of underlying knowledge. Acquiring new knowledge about numbers and arithmetic, which one can dispose of beyond praise and blame, gives self-confidence. The therapeutic learning dialogue promotes mutual, objective criticism, productive contradiction and questions that are interested in content - this is how security in the mathematical fundamentals is gradually established.

In numerical weakness therapy, the course of all systematic learning steps is made dependent on the child's individual difficulties during the entire teaching / learning process. This means that the learning sequence is tailored to the particular problems and trained habits of the child. Their own calculation strategies are specifically addressed and processed so that no beliefs and ambiguities run along subliminally in the new knowledge structure. The child is made aware of the uselessness of his wrong strategies - otherwise he or she may not even make the transition to logical mathematical thinking or think that it is about "alternative tricks" for arithmetic - whatever the child imagines under "arithmetic" .

As an alternative to the misconception of mathematics, which has become more and more entrenched in children - it is a pure drum subject, you have to practice, practice and practice again - it is important to give the child the insight that you are who can understand math! This insight is not so natural and simple:

Can you explain offhand what the difference and the connection between the concept of quantity and concept of number is?

Why is a pear = 1 wrong? Because it doesn't fit?

What is it anyway, the one? 1 candy, 1 DM, 1 something ...?

Why is 11 not 1 and 1 because 11 is bigger and / or comes later?

Do you understand?

The worst idea that a “mathematically weak” child has at the beginning of therapy is that it should no longer use its routine methods for the production of calculation results. It is correct in this notion. But so that an insight emerges from this - and fear only takes this away - the problematic of these methods must be specifically addressed in order to bring the misleading and competing worlds of thought to one's own examination with deliberately correctly experienced resolutions.

The objective knowledge exit point is the starting point for further learning. It is not appropriate to subject “poor numeracy” children to an unspecific total repetition. They too have the basics. However, these must be checked for correctness and security (progress diagnosis is permanent support diagnosis here). Finding the most appropriate entry and focus points for the respective therapy and working step by step with the individual child in each therapy session means: permanent progress diagnosis, therapy dialogue and planning of the next learning steps.

In individual therapy, there is no comparison of performance or competition. The necessary, useful, topic-centered dialogue is not disturbed. However, the problems that arise from the ongoing comparison of performance and the competition in school must also be addressed in the context of therapy.

Therapy must have a protective function to the outside world: The requirements of the school should - in relation to mathematics - be put on hold. In counseling meetings with parents and teachers, agreements must be made and, as far as possible, mutually agreed. In this way, the child is given the freedom to learn without competition and without fear. It must be psychologically freed from the pressure to perform and evaluate in order to be able to build up correct, self-developed knowledge, supervised by the therapist.

The child, as it thinks and feels right now, is at the center of the therapeutic process. Mutual criticism and self-criticism are possible and desirable. Mathematical weakness therapy moves within the methodical framework of client-centered psychotherapeutic concepts aimed at positive behavioral changes - without being psychotherapy itself!

**7. Plea for teacher training - what must it achieve**

Teachers who consider it problematic and worth explaining that not all children achieve at least minimal learning success develop - ideally representable - two extreme cases of dealing with school reality in the sense of professional survival:

1. They accept the results of school selection, no matter how catastrophic they are for individual children, and calm down with the fact that not all people are the same - which is why not all could learn the same thing.

The alternative is:

2. You work endlessly - in the context of school lessons - to impart a minimum of basic knowledge to each of your students if possible. You pursue the claim: It should not have been my fault if some of my students did not understand the basics.

Both cases - ideally described - of dealing with school reality do not change the educational system-related dilemma of contradicting learning conditions. There is, however, the possibility of opening doors to a more expedient handling of phenomena such as "arithmetic weakness":

Further training courses that address the following task: Dealing with the phenomenon of arithmetic weaknesses, which enables the teaching staff to judge the difficulties of their students more competently: What systematic errors do the children produce in arithmetic - and: do they therefore fail in school? !

The teachers themselves should have the opportunity to combine their everyday teaching experience with knowledge of problems inherent in the education system for individual learning through further training in the field of learning disabilities.

The focus of further training courses on the phenomenon of "weaknesses in numeracy" should therefore primarily be to give the teaching staff a "diagnostic perspective" for their everyday school life:

The knowledge gained from scientifically based practice of diagnostic and therapeutic support for “mathematically weak” clients can help individual teachers understand and guide students better and protect them from being overwhelmed. Competent action in this context means: taking or encouraging well-founded preventive measures, encouraging pupils to enter into dialogue, educating parents or suggesting extracurricular therapeutic measures.

**8. Our constantly updated literature recommendations:**

__Annotated list of the literature recommended by BIB-Essen and RESI-Volxheim__

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