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Process related competencies: Consecutive integers

To train content and process related competencies it is useful to deal with the issue of “Sums of consecutive integers”. On this site you find the possibility to analyse the structure of tasks on this topic and to analyse videos that are focused on process related competencies.

1. Laura’s strategy of finding all solutions

Laura who is in grade three is looking for all sums of consecutive integers that are lower than twenty:

Do you understand how Laura solves her task? Describe Laura’s strategy of finding all solutions.

 

2. Consecutive integers - your explorations and typical procedures children use

Sums of consecutive integers are for example 2+3, 14+15+16 or 78+79+80+81. They are not 2+4+6 or 0+1+2. There can be tasks developed based on sums like that for all the different grades. We interviewed children of grade three and four. We asked them the following questions:

(1) Find all sums of consecutive integers that are lower than twenty.
(2) Why are these all?


To understand the procedures the children use and to see why these tasks are not only solving summation tasks, try to solve these two tasks yourself.

Try to find out how many sums of consecutive integers there are if the sum isn’t higher than 50 (100).

In case you are not sure that you found all solutions or that your explanations are correct you can read the following texts to find some advice.

Smart summation:
Steinbring, H. & Scherer, P. (2004): Zahlen geschickt addieren. In G. Müller, H. Steinbring & E. Wittmann (Hrsg.): Arithmetik als Prozess. Seelze: Kallmeyer. S. 55-69.

Summation formula:
Steinbring, H. & Scherer, P. (2004): Summenformeln. In: G. Müller, H. Steinbring & E. Wittmann (Hrsg.): Arithmetik als Prozess. Seelze: Kallmeyer. S. 237-254.

Selter and Schwätzer (2000) found out that many pupils start solving their task by writing down all the solutions that come to their mind unsystematically. After a little while most of the children start to work more systematically and use some strategies for finding more solutions. These strategies are usually not used consequently; more often the children are changing strategies quite often. You can see this very well in the example from the beginning where Laura is trying to find all solutions: Looking at the sums of two consecutive integers you can see that she increases each summand by one (2+3, 3+4, 4+5, 5+6 etc.), where else by looking at the sums of three consecutive integers you can see that she finds new solutions by taking the last summand of the first solution as the first summand of the following solutions (1+2+3, 3+4+5, 5+6+7).

To get an overview of the described strategies that were found out by Selter & Schwätzer (2000) have a look here.

 

3. Process related competencies in the context of „Sums of consecutive integers”

Children have acquired process related competencies if they are able to understand and handle mathematical methods so that they can transfer their knowledge (definition by the Kultusministerkonferenz, short KMK, conference of ministers of education). According to the standards for school mathematics the process related competencies include problem solving, communication, argumentation, mathematical modelling and demonstration.

To achieve these process related competencies and also the content related competencies are one main intention in the mathematic classes. The teacher has to provide material and tasks for the children that make it possible to acquire knowledge and calculation skills but still develop the children’s process related competencies further. This means that the teacher has to be able to find out how far the competencies are developed by watching the children, analysing their verbal comments and written documents. To know how far the competencies are developed makes it possible to support the children where they need it and to challenge them where they are already very good at.

You would like to know more about process related competencies? Have a look here.

At this point there will be the chance for you to analyse which process related competencies are needed to solve tasks based on sums of consecutive integers like above.

  • Have a look at the expected competences (especially the process related competencies) named in the mathematics curriculum of North Rhine-Westphalia, that the children should have acquired by the end of the fourth grade.
  • Think about which of the named process related competencies are related to the task from before: “Find all sums of consecutive integers that are lower than twenty. Explain what you found out.”

 

4. Analysing videos

Of course the differentiation between the process related competencies is not always possible. You can already see that by looking at the different definitions in the different curriculums and standards for school mathematics (see Process related competencies – An introduction). Sometimes it gets necessary to focus on one selected competence and to “ignore” other shown competencies for a while in order to better understand the children. Analysing the following videos you will have to focus on one competence.

Problem solving

The following videos show how Theresa, Nick and Sonja solve their task to find all sums of consecutive integers lower than twenty.

1. Look for the moments where you can see that the children show the competencies named in the mathematics curriculum of North-Rhine Westphalia

  • The children try out things more systematically and goal-orientated
  • They use their knowledge of correlations to find the solution
  • They check their results

2. Discuss the competencies concerning problem solving by focussing on one of the videos. Explain your thoughts.

Theresa
Nick
Sonja

Argumentation

In the following interviews we asked the children to explain why there aren’t any more sums of consecutive integers that are twenty or lower than twenty.

Jakob
Dennis
Sonja

1. Look for the moments where you can see that the children show the competencies named in the mathematics curriculum of North-Rhine Westphalia

  • The children make educated guesses about mathematical correlations and specific characteristics.
  • The children question and check their guesses, solutions, statements, etc
  • They prove or disprove their guesses and develop first general statements (by using examples)

2. Describe how the children give reasons for having found all the solutions.

 

 5. Advanced task

Christoph, who is in grade three, chose a way of presenting his results that makes it possible to find out more than what was asked.

1. Look for the moments where you can see that the children show the competencies named in the mathematics curriculum of North-Rhine Westphalia

  • What does Christoph find out? Try to describe this in your own words.
  • How does he explain the pattern he found?
Christoph

 

6. Related issues

Process related competencies
packages with a pattern
number grid
combinatorics

You can find more material for information, teaching and advanced training on the topic of process and content related competencies on the homepage of our partner project PIK AS (House 1 – „discovery, description, reasoning“). In house 7 you will find more information on how to use ‘good tasks’ in classes and how to encourage process related competencies.

7. Material

Guideline for an interview: Consecutive integers

 

8. Bibliography

KMK (2005): Bildungsstandards im Fach Mathematik für den Primarbereich (Jahrgangsstufe 4). Beschluss der Kultusministerkonferenz vom 15.10.2004. Verfügbar unter: http://www.kmk.org/fileadmin/veroeffentlichungen_beschluesse/2004/2004_10_15-Bildungsstandards-Mathe-Primar.pdf (Abruf am: 13.07.2011)

Ministerium für Schule und Weiterbildung NRW (Hrsg.) (2008): Lehrplan Mathematik für die Grundschulen des Landes NRW. Verfügbar unter: http://www.schulentwicklung.nrw.de/lehrplaene/upload/lehrplaene_download/grundschule/grs_faecher.pdf (Abruf am: 13.07.2011)

Schwätzer, U. & Selter, Ch. (2000): Plusaufgaben mit Reihenfolgezahlen - eine Unterrichtsreihe für das 4. bis 6. Schuljahr. In: Mathematische Unterrichtspraxis. H. 2, S. 28- 37

Steinbring, H. & Scherer, P. (2004): Zahlen geschickt addieren. In G. Müller, H. Steinbring & E. Wittmann (Hrsg.): Arithmetik als Prozess. Seelze: Kallmeyer. S. 55-69.

Steinbring, H. & Scherer, P. (2004): Summenformeln. In: G. Müller, H. Steinbring & E. Wittmann (Hrsg.): Arithmetik als Prozess. Seelze: Kallmeyer. S. 237-254.

Walther, G. (2004): Gute Aufgaben. Basispapier zum Modul 1: Umgang mit Aufgaben im Mathematikunterricht. Verfügbar unter: http://www.sinus-an-grundschulen.de/fileadmin/uploads/Material_aus_STG/Mathe-Module/Mathe1.pdf (Abruf am: 05.07.2011)

 

9. More literature – more information

Schwätzer, U. & Selter, Ch. (1998): Summen von Reihenfolgezahlen - Vorgehensweisen von Viertklässlern bei einer arithmetisch substantiellen Aufgabenstellung. In: Journal für Mathematikdidaktik (JMD). H. 19 (98) 2/3, S. 123-148.

Selter, Ch. (2004): Mehr als Kenntnisse und Fertigkeiten. Basispapier zum Modul 2: Erforschen, entdecken und erklären im Mathematikunterricht der Grundschule. Verfügbar unter: http://www.sinus-grundschule.de/fileadmin/Materialien/Modul2.pdf (Abruf am: 05.07.2011)