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Can Early Algebra lead non-proficient students to a better arithmetical understanding? - page 3 / 3





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Children of a 5th grade were given the equations L - R = U and N + M = B – J and asked to invent fitting word problems. Below we want to give some examples. The equation L – R = U resulted in the following word problems: Lena geht in den Laden und will 10 Buntstifte von Pelikan kaufen. 10 Stifte = R. Doch es gibt noch so viele schöne andere, dass Lena noch mehr kauft. Sie kauft 23. Was für ein Wert hat U? Wie viele Stifte kauft Lena mehr? (Angelina) Lena walks into a shop and wants to buy ten coloured crayons. 10 crayons = R. But there are so much other pretty ones, that is why Lena buys some more. She buys 23. What is the value of U? How much crayons more did Lena buy?(translation by the author) Although the children learned to use letter equations only in the context of geometric magnitudes like length, area and volume Angelina chose the context of money for their word problems. We assume they chose money because it plays a major role in their everyday life and therewith a much bigger role than geometric magnitudes. The details on the brand of the crayons and the reason why she bought more are evidence that here we see an episode that really has happened or could happen in her life. The word problem fits to the equation which is revealed by Angelina as she is relating some of the letters to the values. Other children only used numbers or only used letters: Kim hat 20 Blumen, sie verliert 5. Wie viele hat sie noch? (Axel) Kim has 20 flowers. She loses 5. How many are left over? (translation by the author) Horst hat L Boote geschnitzt. Ihm fallen R Boote ins Wasser. Wie viele hat er noch übrig? Horst carved L boats. R boats are falling into the water. How many are left over? (translation by the author) The above examples show that the children not only use the letters for magnitudes but also for numbers of objects. The equation N + M = B – J resulted in the following word problem: Lara geht zu Faberkastell und will einen Radiergummi von 2,00 kaufen und einen Bleistift von 3,00 . Sie hat aber nur 5,50 mit. Reicht das Geld und wenn ja, w ie viel bekommt sie zurück? (Lana) Lara walks to Faber-Castell and wants to buy an eraser of 2.00 and a pencil of 3.00 . She only has 5.50 with her. Is this enough money and if yes, how many money will she get back? (translation by the author) The word problem fits to the equation. Lana invents values for N and M (2.00 and 3.00 ) and B (5.50 ) and wants to know how big J is. She as well implicitly writes down, why it is important for her to know how big J is:

she wants to know, if she has enough money for her buying.


The next step is to explore if children will and can use their knowledge about abstract symbolic equations for solving word problems only containing numbers and no letters. First observations showed that low-achieving children who have not been able to solve a word problem directly came back to abstract symbolic equations. For example a low-achieving 3rd grader’s first reaction after reading the word problem “A street has length 845 m. Hans has already walked 220m. How far does he still have to go?” was “I want to do that with letters.”

Literature Auernheimer, G. (ed.) (2003): Schieflagen im Bildungssystem. Die Benachteiligung der Migrantenkinder. Opladen: Leske + Budrich. Davydov, V. V. (1975). The Psychological Characteristics of the “Prenumerical” Period of Mathematics Instruction. In: L. P. Steffe (Ed.), Children's Capacity for Learning Mathematics. Soviet Studies in the Psychology of Learning and Teaching Mathematics. Vol. VII. (pp. 109-205). Chicago: University Of Chicago. Dougherty, B. J. & Venenciano, L. C. H. (2007): MeasureUp for Understanding. In: Teaching children mathematics, 13(9), 452-456. El’konin, D. B. (1975). Primary Schoolchildren’s Intellectual Capabilities and the Content of Instruction. In: L. P. Steffe (Ed.), Children's Capacity for Learning Mathematics. Soviet Studies in the Psychology of Learning and Teaching Mathematics. Vol. VII. (pp. 13-54). Chicago: University Of Chicago Hasemann, K & Stern, E (2002): Die Förderung des mathematischen Verständnisses anhand von Textaufgaben - Ergebnisse einer Interventionsstudie in Klassen des 2. Schuljahres. Journal für Mathematik-Didaktik, 23/3, 222-242. McNeil, N. (2004), Don't teach me 2 + 2 equals 4: Knowledge of arithmetic operations hinders equation learning. In K. Forbus; D. Gentner & T. Regier, (eds.), Proceedings of the Twenty-sixth Annual Conference of the Cognitive Science Society, pp. 938-943. Sekretariat der Ständigen Konferenz der Kultusminister der Länder in der Bundesrepublik Deutschland (KMK) (eds.) (2005): Beschlüsse der Kultusministerkonferenz: Bildungsstandards für den Primarbereich im Fach Mathematik (Jahrgangsstufe 4). Beschluss vom 15.10.2004. München, Neuwied: Wolters Kluwer Deutschland GmbH Wittmann E.C. & Müller G.N. (2005). Das Zahlenbuch 4. Leipzig: Klett.


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