In this article, the authors share frameworks for problem types and students' reasoning about integers. The authors found that all ways of reasoning (WoRs) were used across grade levels and that specific problem types tended to evoke particular WoRs. Specifically, students were more likely to use analogy-based reasoning on all-negatives problems…

Duncan Symons and Derek Holton discuss the different types of mathematical reasoning and what each of these might look like in the classroom. By suggesting language that can be used to describe the different methods of reasoning, they hope to provide teachers with the tools they need to better recognise and assess student reasoning.

The hammer-and-nail phenomenon highlights human tendency to approach a problem using a tool with which one is familiar instead of analyzing the problem. Pedagogical suggestions are offered to help students minimize their mathematical impulsivity, cultivate an analytic disposition, and develop conceptual understanding.

This article examines a common proportional reasoning problem used in schools, often referred to as the Orange Juice task. The authors show how these six strategies described by Nikula (e.g., Unitizing, Norming, etc.) and one additional strategy can be used to either solve or make progress in the Orange Juice task. The article presents work from…

This groundbreaking book looks at the development of mathematical thinking in infants and toddlers, with an emphasis on the earliest stage, from zero to three, when mathematical thinking and problem solving first emerge as natural instincts. The text explores the four precursor math concepts--attribute, comparison, change, and pattern--with an…

Students become attentive, curious, and passionate about learning when they can see its relevance to their lives and when they're empowered to use that learning to solve problems that matter. Regardless of the subject or grade level you teach, you can infuse your instruction with the meaning students crave by implementing design thinking. Design…

Today, schools are introducing STEM education and robotics to children in ever-lower grades. In "Beyond Coding," Marina Umaschi Bers lays out a pedagogical roadmap for teaching code that encompasses the cultivation of character along with technical knowledge and skills. Presenting code as a universal language, she shows how children…

Working with language-independent logic structures can help students develop both inductive and deductive reasoning skills. The Japanese publisher Nikoli (with resources available both in print and online) produces a treasure trove of language-independent logic puzzles. The Nikoli print resources are mostly in Japanese, creating the extra…

Students who are adept in modeling with mathematics have the capability to use mathematics in situations that arise in everyday life. The German Tank problem described in this article created the expectation that student reasoning was rooted in logical deductions (NCTM 2000). By engaging in this problem, students grappled with challenging…

Knowing the facts is not enough. If we want students to develop intellectually, creatively problem-solve, and grapple with complexity, the key is in "conceptual understanding." A Concept-Based curriculum recaptures students' innate curiosity about the world and provides the thrilling feeling of engaging one's mind. This updated edition…

The Common Core State Standards (CCSSI 2010) for Mathematical Practice have relevance even for those not in CCSS states because they describe the habits of mind that mathematicians--professionals as well as proficient school-age learners--use when doing mathematics. They provide a language to discuss aspects of mathematical practice that are of…

In the authors' attempts to incorporate problem solving into their mathematics courses, they have found that student ambition and creativity are often hampered by feelings of risk, as many students are conditioned to value a produced solution over the actual process of building one. Eliminating risk is neither possible nor desired. The challenge,…

How do students think about an angle measure of ninety degrees? How do they think about ratios and values on the unit circle? How might angle measure be used to connect right-triangle trigonometry and circular functions? And why might asking these questions be important when introducing trigonometric functions to students? When teaching…

"Invention through Form and Function Analogy" is an invention book for teachers and other leaders working with youth who are involving students in the invention process. The book consists of an introduction and set of nine learning cycle formatted lessons for teaching the principles of invention through the science and engineering design…

Students' success in solving problems involving proportional reasoning is an indication that they have moved beyond additive thinking to multiplicative thinking. However, classroom work indicates that many students do not reason proportionally in many practical contexts. The authors discuss a particular task that reveals students'…