Inquiry is the essence of social studies, and inquiry "looping" in social studies curricula means offering students opportunities to engage in inquiry in regular intervals and in a coherent fashion within and across grade levels. This can benefit the youngest students since they are filled with hope and want to make a difference in the…

A test method is described for determining the divisibility of non-negative integers by a prime number. The test uses an integer multiplying factor that is defined for each prime, designated as [beta], to reduce the non-negative integer that is being tested by an order of magnitude in each of a sequence of steps to obtain a series of new numbers.…

Explore the impact technology has on mathematical identity and agency when students use mathematical action technology to engage in cycles of proof and support case-based reasoning. This article showcases a mathematics task used in a fourth-grade class that allowed students to develop their conceptual understanding of division. The authors…

This article describes some of the essential mathematics that underpins the use of algorithms through a series of learning pathways. To begin, a graphic depicting the mathematical ideas and concepts that underpin the learning of algorithms for multiplication and division is provided. The understanding and use of algorithms is informed by two…

Productive struggle is an essential part of mathematics instruction that promotes learning with deep understanding. A video scenario is used to provide a glimpse of productive struggle in action and to showcase its characteristics for both students and teachers. Suggestions for supporting productive struggle are provided.

Students' difficulties understanding the meaning of logarithms could stem in part from differences between teachers' and students' views of them. The purpose of this article is to unpack some specialized content knowledge for teaching logarithms. The author discusses the history of logarithms to show why they can be understood as repeated…

Recently a team of third- and fourth-grade teachers tested instructional ideas on the topic of division with remainders (DWR) using lesson study. The Common Core State Standards for Mathematics give little attention to the concept of the remainder, specifying only that students are to solve problems for which remainders must be interpreted (NGA…

The importance of multiplicative thinking in supporting students' learning of key topics and success in further mathematics is widely and clearly stated in mathematics education literature. However, multiplicative thinking is not clearly stated in curriculum documents of many countries including Australia. In the "Australian Curriculum:…

This document includes instructional tips on: (1) Building on students' informal understanding of sharing and proportionality to develop initial fraction concepts; (2) Helping students recognize that fractions are numbers that expand the number system beyond whole numbers, and using number lines to teach this and other fraction concepts; (3)…

Developing multiplicative reasoning is an important milestone for elementary school students, which influences their learning of later mathematical concepts (Hackenberg and Tillema 2009). For children to conceptually understand multiplication, one should move beyond merely counting by ones to dealing with composites (twos, fives, etc.) and other…

Children learn to find answers when multiplying two whole numbers (e.g., 3 × 7 = 21). To this end, they may repeatedly add one number (e.g., 7 + 7 + 7 = 21). But what meanings do they have for multiplication? The authors address this issue while sharing an innovative, playful task called Please Go and Bring for Me (PGBM). Drawing on the…

This is the first of two articles on the use of a written multiplication algorithm and the mathematics that underpins it. In this first article, the authors present a brief overview of research by mathematics educators and then provide a small selection of some of the many student work samples they have collected during their research into…

The article describes Orthodox Christian education as a kind of religious education. It offers a definition of Orthodox Christian education and characterizes its structure in modern Russia. It proposes a model for the Orthodox Christian education system, including introductory, basic, professional, and advanced academic levels. The author…

During a time when students are accessing instruction virtually because of COVID-19, reflecting on the sixth of the Standards for Mathematical Practice (SMP 6) is a vital step for educators to take. Although the Common Core State Standards for Mathematics (NGA Center and CCSSO 2010) identifies SMP 6 as Attend to precision, associated expertise…

Civics education shouldn't be confined to dusty textbooks, as evidenced by this assortment of projects. In one unit, elementary students play out the presidential election--from campaigning to inauguration day--using the Storypath approach. In another project, 5th graders explore the controversy about Confederate monuments by studying a local…