As new technological developments continue to change the educational landscape, it is not an exception in the area of proof and proving. This classroom note introduces the use of one of the trending proofs assistants -- the Lean theorem prover. We first provide a technical account of Lean, then exemplify Lean proofs in propositional logic, number…

This paper presents a novel figure for teaching multiple geometric proofs of the Pythagorean theorem. Because it consists only of congruent given right triangles, the figure can be constructed using a template of the given right triangle or, if available, a computer program. Within the figure, called a Pythagorean multi-proof square, there are…

This book addresses the validity of think-aloud protocols (TAPs) in L2 writing research through a mixed methods study and proposes effective approaches for their valid implementation. The book uncovers the reactive effects that TAPs have on L2 writing performance and processes, and examines how individual factors moderate this reactivity. It…

We have revamped a classic analytical chemistry laboratory experiment, "Determination of an Unknown Acid", to provide students an opportunity to learn about new advances in the material sciences and their applications in analytical chemistry in an effort to reinforce the key concepts of chemical analysis. Specifically, students were…

The textbook used for fifth grade elementary students in an elementary school in Kuningan, West Java claims to offer Higher Order Thinking Skill (HOTS) questions on the cover of the textbook and some questions have been labeled as HOTS questions. This research investigated the validity of this claim, analyzing the levels of cognitive domain of…

Valid proofs need not be in the traditional two-column format. This classroom activity allows students to explore, discuss, and use specialized facts to create a general statement of truth.

Vocabulary learning is an incremental process. Vocabulary knowledge, especially for second-language learners, may develop across a lifetime. Teachers with experience in providing feedback on their students' vocabulary use in writing or speech might have noticed that it is sometimes difficult to pinpoint one aspect of word knowledge. The reason is…

The debate on the validity of teaching students how to identify their own preferred learning style is not going to go away anytime soon as evidenced by the number of articles still being written about the topic. One can find numerous research studies that say students have different strengths and weaknesses when it comes to learning and retaining…

Since the 1970s, the Mathematical Association of America's (MAA) journals "Mathematics Magazine" and "College Mathematics Journal" have published "Proofs without Words" (PWWs) (Nelsen 1993). "PWWs are pictures or diagrams that help the reader see why a particular mathematical statement may be true and how one…

How can we teach inquiry? In this paper, I offer practical techniques for teaching inquiry effectively using activities built from routine textbook exercises with minimal advanced preparation, including rephrasing exercises as questions, creating activities that inspire students to make conjectures, and asking for counterexamples to reasonable,…

Constructing viable arguments and reasoning abstractly is an essential part of the Common Core State Standards for Mathematics (CCSSI 2010). This article discusses the scenarios in which a mathematical task is impossible to accomplish, as well as how to approach impossible scenarios in the classroom. The concept of proof is introduced as the…

Students often have difficulty understanding algebraic proofs of statistics theorems. However, it sometimes is possible to prove statistical theorems with pictures in which case students can gain understanding more easily. I provide examples for two versions of Bayes' theorem.

The world of man-made design is all around, in everyday objects and appliances people use without a second thought. In this exercise, students have an opportunity to challenge the common refrigerator's design--and improve it. This approach can be used with many other appliances.

This article reports findings from a multi-year study of teachers' conceptual change coincident with the development of instructional expertise among teachers involved in educational reform efforts in schools in Santiago, Chile. Conceptual change in teachers is important because recent research indicates that students of teachers who function at…

In this article, we suggest an instructional intervention to help students understand statements involving multiple quantifiers in logical contexts. We analyze students' misinterpretations of multiple quantifiers related to the epsilon-N definition of convergence and point out that they result from a lack of understanding of the significance of…