Elementary math in elementary school: To learn the multiplication table, avoid proactive interference

A major challenge for elementary school students is memorizing the multiplication table. This is difficult because there are many facts to learn and they are similar to each other, which creates proactive interference in memory. Here, we examined whether reducing interference would improve the memorization of the multiplication table by first graders. In a series of 16 short training sessions over a period of 4 weeks, each child learned 16 multiplication facts – 4 facts per week. Learning was better when the 4 facts in a given week were dissimilar from each other, a situation that reduces the proactive interference among them. Critically, this similarity effect originated in the specific learning context, i.e., the grouping of facts to weeks, and could not be explained as an intrinsic advantage of some facts over others. The similarity effect persisted 5 weeks after the end of the training period, i.e., proactive interference affected the long-term memory. Furthermore, during training, the similarity effect was not observed immediately but only in later training sessions, and only when examined in the beginning of a session. This indicates that proactive interference affected the long-term memory directly – it did not originate in short-term memory processes and then “leak” to long-term memory. We propose that the effectivity of this low-interference training method, which is dramatically different from currently-used pedagogical methods, calls for a serious reconsideration of the way we teach the multiplication table in school.

Genetic code research: a precognition result (II)

In this second part of the short communication (Ref. 2), we give an argument more in favor of the validity of the precognition status of the final result of my 40 years of genetic code researches. It is shown that the changes in the number of atoms in the system-arrangements of protein amino acids, in relation to the Gaussian number (51) and the Dürer number (34 and 68, respectively), correspond to the changes in the products of number 5 in the Multiplication Table of the decimal number system. On the other hand, with the same changes (in the products of number 5), two unconscious narrations, said in the first part of this communication, correspond one hundred percent.

Orthogonality graphs of real Cayley–Dickson algebras. Part I: Doubly alternative zero divisors and their hexagons

We study zero divisors whose components alternate strongly pairwise and construct oriented hexagons in the zero divisor graph of an arbitrary real Cayley–Dickson algebra. In case of the algebras of the main sequence, the zero divisor graph coincides with the orthogonality graph, and any hexagon can be extended to a double hexagon. We determine the multiplication table of the vertices of a double hexagon. Then we find a sufficient condition for three elements to generate an alternative subalgebra of an arbitrary Cayley–Dickson algebra. Finally, we consider those zero divisors whose components are both standard basis elements up to sign. We classify them and determine necessary and sufficient conditions under which two such elements are orthogonal.

Teaching Aids and Manipulative Teaching Means

The subject matter of this essay concerns the use of manipulatives and teaching aids in the teaching of mathematics for the 1st and 2nd grade of primary school, combined with educational robotics applications. In particular, the use of the programmable bee-bot floor robot combines procedures that involve the use of manipulative means (creating a track as a cardboard model, painting, assembling), teaching aid tools (demonstration of the programmable bee-bot floor robot), and finally, the comprehension of simple programming and mathematical concepts. Through the implementation of an educational scenario aiming to familiarize students with the basic geometric concepts, mathematical operations (multiplication table), basic algorithmic structures (simple problems and step solving), there will be involvement with the cognitive areas of informatics (basic programming concepts)) and mathematics (geometry, calculations).

The Teaching of Multidigit Multiplication in the Japanese Approach

Multidigit multiplication in vertical form uses the idea of the distributive law such as 27 × 3 = (20 + 7) × 3 = 20 × 3 + 7 × 3 for using a multiplication table under the base ten place value system. Multiplication in vertical form is not simply repeated addition such as 27 + 27 + 27. In this meaning, through the extension of multiplication from single digit to multidigit by use of vertical form with a multiplication table, students have to integrate their knowledge on the base ten system with the definition of multiplication by measurement (a group of groups; see Chaps. 10.1007/978-3-030-28561-6_3, 10.1007/978-3-030-28561-6_4, 10.1007/978-3-030-28561-6_5, and 10.1007/978-3-030-28561-6_6 of this book) and so on. How does the Japanese approach enable students to develop multiplication in vertical form by and for themselves based on their learned knowledge?This chapter illustrates this process as follows. Firstly, the diversity of multiplication in vertical form is explained in relation to the multiplier and multiplicand, and the Japanese approach in comparison with other countries such as Chile and the Netherlands is clearly illustrated. Secondly, how a Japanese teacher enables students to develop multiplication in vertical form beyond repeated addition is explained with an exemplar of lesson study. Thirdly, the exemplar illustrates a full-speck lesson plan under school-based lesson study which demonstrates how Japanese teachers try to develop students who learn mathematics by and for themselves including learning how to learn (see Chap. 1). Fourthly, it explains the process to extend multiplication in vertical form to multidigit numbers by referring to Gakko Tosho textbooks.

Teaching the Multiplication Table and Its Properties for Learning How to Learn

Why do the Japanese traditionally introduce multiplication up to the multiplication table in the second grade? There are four possible reasons. The first reason is that it is possible to teach. The second reason is that Japanese teachers plan the teaching sequence to teach the multiplication table as an opportunity to teach learning how to learn. The third reason is that memorizing the table itself has been recognized as a cultural practice. The fourth reason is to develop the sense of wonder with appreciation of its reasonableness. The second and the fourth reasons are discussed in Chap. 10.1007/978-3-030-28561-6_1 of this book as “learning how to learn” and “developing students who learn mathematics by and for themselves in relation to mathematical values, attitudes, ways of thinking, and ideas.” This chapter describes these four reasons in this order to illustrate the Japanese meaning of teaching content by explaining how the multiplication table and its properties are taught under the aims of mathematics education. In Chap. 10.1007/978-3-030-28561-6_1, these were described by the three pillars: human character formation for mathematical values and attitudes, mathematical thinking and ideas, and mathematical knowledge and skills.