Hey Lehrer:innen: wie streng geht ihr bei sowas mit der Bewertung um? Faktoren lassen sich tauschen - hier aber komplette Uneinsichtigkeit ala „auf dem Bild sind aber 4 Hände mit 5 Fingern“. pic.twitter.com/f1PGsCQBQV— Carsten Paul (@CarstenPaul1982) April 21, 2021
Voitteko kertoa miksi 2. Luokkalaisille lapsille opetetaan näin?— A. Rautio (@riisemcpiise) October 28, 2022
Tässähän oikeasti vain hämmennetään ja rangaistaan lasta oikeasta vastauksesta. @liandersson @okmfi @Opetushallitus pic.twitter.com/ciCFoN1xxx
- Why Was 5 x 3 = 5 + 5 + 5 Marked Wrong | by Brett Berry | Math Hacks | Medium
- Parents criticise US Common Core maths after third grade pupil told solution for 5+5+5=15 is incorrect | The Independent
- Verwirrende Matheaufgabe: Warum 5x3 nicht dasselbe ist wie 3x5 | STERN.de (Germany)
- Perkalian Galau “4×6″ atau 6×4” | Faith -- Hope -- Love (Indonesia)
- 하늘 비양구 길을 보면서 그리움을 느끼고 | 21. 곱셈의 교환 법칙 - Daum 카페 (Belgium; written in Korean)
- „Das muss man sich einfach merken“??? (Germany)
- Isoda, M. and Olfos, R. (2021). Teaching Multiplication with Lesson Study - Japanese and Ibero-American Theories for Mathematics Education. Springer. https://link.springer.com/book/10.1007%2F978-3-030-28561-6
- Daroczy, G., Wolska, M., Meurers, W. D. and Nuerk, H-C. (2015). Word problems: a review of linguistic and numerical factors contributing to their difficulty. Frontiers in Psychology, Vol.6, No.348. https://doi.org/10.3389/fpsyg.2015.00348
- Johansson, I.: Collections as One-and-Many on the Nature of Numbers, Grazer Philosophische Studien, Vol.91, pp.17-58 (2015). http://dx.doi.org/10.1163/9789004302273_003
- Lannin, J., Chval, K., and Jones, D. (2013). Putting Essential Understanding of Multiplication and Division into Practice in Grades 3-5. National Council of Teachers of Mathematics. [isbn:9780873537155] (sample: https://www.nctm.org/Handlers/AttachmentHandler.ashx?attachmentID=b8y3Aq4tIwM%3D)
- Chapin, S. H., O'Connor, C. and Anderson, N. C. (2009). Classroom Discussions - Using Math Talk to Help Students Learn, Grades K-6, Second Edition. Math Solutions. [isbn:1935099019] (preview: http://books.google.co.jp/books?id=2NX4I6mekq8C&pg=PA3)
- Isoda, M. and Olfos, R. (2009). La enseñanza de la multiplicación : el estudio de clases y las demandas curriculares. Ediciones Universitarias de Valparaíso. http://math-info.criced.tsukuba.ac.jp/upload/MultiplicationIsodaOlfos.pdf
- Yoshida, M. (2009). Is Multiplication Just Repeated Addition? - Insights from Japanese Mathematics Textbooks for Expanding the Multiplication Concept, 2009 NCTM Annual Conference. https://web.archive.org/web/20160303173651/http://www.globaledresources.com/resources/assets/042309_Multiplication_v2.pdf
- Greer, B. (1992). Multiplication and Division as Models of Situations. In Grouws D.A. (Ed.), Handbook of Research on Mathematics Teaching and Learning, National Council of Teachers of Mathematics, pp.276-295. [isbn:1593115989]
- Anghileri, J. and Johnson, D. C. (1988). Arithmetic Operations on Whole Numbers: Multiplication and Division. In Post, T.R. (Ed.), Teaching Mathematics in Grades K-8, Longman Higher Education, Allyn and Bacon, pp.146-189. [asin:0205110762]
- Vergnaud, G. (1988). Multiplicative Structures. In Hiebert, J. and Behr, M. (Eds.), Number Concepts and Operations in the Middle Grades, Vol.2, pp.141-161. [isbn:0873532651]
My opinion is ...
- A-1. Two factors are commutative. By the commutative law of multiplication, 5 x 3 = 3 x 5 holds.
- A-2. A "dealing-out" operation can swap the roles of multiplicand and multiplier; we can write "5 apples/round x 3 rounds = 15 apples".
- A-3. We can write "5 x 3" by arranging the apples in a rectangle shape.
- A-4. We can recognize the number of dishes as the multiplicand (or the basic quantity) and the number of apples per dish as the multiplier.
- A-5. With counters attached, "5 x 3 apples" and "3 apples x 5" are equivalent, and so are "5 dishes x 3 apples/dish" and "3 apples/dish x 5 dishes".
- A-6. In other countries, they write the multiplicand and the multiplier oppositely or do not think that the order matters.
- B-1. In this case, the number of apples per dish is the multiplicand while the number of dishes is the multiplier.
- B-2. "5 x 3" and "3 x 5" differ in meaning, although the products are the same.
- B-3. "5 x 3 = 15" shows the opposite situation regarding the numbers of apples and dishes.
- B-4. "5 x 3 = 15" leads to 15 dishes but not apples.
- B-5. With counters attached, "5 apples x 3" and "3 apples x 5" are different, and so are "5 apples/dish x 3 dishes" and "3 apples/dish x 5 dishes".
- B-6. Writing expressions in consideration of language and cultural differences is educationally valuable.
Towards Order-of-Multiplication Dispute (English Version)
- I disagree with all of A-1 to A-6.
- Just the same, I would not approve of all the reasons B-1 to B-6. I agree with the first two reasons under the condition that the apple problem and others are on the test after the pupils learn B-1 and B-2 properly in class. The condition seems to be met on the grounds of lesson plans developed by teachers as well as textbooks which are well-organized.
It may be useful to focus attention on where the commutative law appears in an argument. Those who complain about the incorrectness of the red-pencil seen in imgur's picture often think that although a x b and b x a are different literally, a x b = b x a should hold by the commutative law. On the other hand, people for it seem to understand that even if they accept the commutative law of multiplication, a x b and b x a represent different things.Multiplication in classes