Beginning a mathematics lesson involving a challenging task with a carefully chosen preliminary experience is an effective means of activating student cognition. In this article, the authors highlight a variety of preliminary experiences, each with a different structure and form, all designed to support students to more successfully engage with…

Seeds of Algebraic Thinking comes from the Knowledge in Pieces (KiP) perspective of learning. KiP is a systems approach to learning that stems from the constructivist idea that people learn by building on prior knowledge. As people experience the world, they acquire small, sub-conceptual knowledge elements. When people engage in a particular…

This paper describes a course designed to introduce students to mathematical thinking and a variety of lower level mathematics topics using baseball while satisfying the goals of quantitative reasoning. We give suggestions for sources, topics, techniques, and examples so any mathematics teacher can design such a course to fit their needs. The…

In this reflection of teaching, we describe a series of activities that introduce the Taylor series through dynamic visual representations with explicit connections to students' prior learning. Over the past several decades, educators have noted that curricular materials tend to present the Taylor series in a way that students often interpret as…

Every student has mathematical strengths beyond knowing basic facts, solving problems quickly, or showing work clearly. In this article, the author presents Smiles as an "on-ramp" task that supports students working together by unveiling and leveraging mathematical strengths. Nielsen describes on-ramp mathematics tasks as scaffolds that…

The complexity in understanding the [epsilon-delta] definition has motivated research into the teaching and learning of the topic. In this paper I share my design of an instructional analogy called the Pancake Story and four different questions to explore the logical relationship between [epsilon] and [delta] that structures the definition. I…

The transition to tertiary mathematics requires students to use definitions of mathematical objects instead of intuitions. However, routines of defining and of proving with definitions are difficult to engage in, as they are not familiar to students who come from secondary school mathematics. Defining is highly complex because of its underlying…

Children's own spontaneous mathematical activities are crucial for their mathematical development. Mathematical thinking and learning does not only occur in explicitly mathematical situations, such as the classroom. Those children with higher tendencies to recognize and use mathematical aspects of their everyday surroundings, both within the…

Why are math modeling problems the source of such frustration for students and teachers? The conceptual understanding that students have when engaging with a math modeling problem varies greatly. They need opportunities to make their own assumptions and design the mathematics to fit these assumptions (CCSSI 2010). Making these assumptions is part…

The Puppy Love problem asked fifth and sixth grade students to use their prior knowledge of measures of central tendency to determine a data set when given the mean, mode, median, and range of the set. The problem discussed in this article is a task with a higher level of cognitive demand because it requires that students (1) explore and…

Teachers have found that engaging students in justification can help students deepen and retain mathematical knowledge, gain a greater sense of ownership over the material, and improve communication and representation skills (Staples, Bartlo, and Thanheiser 2012). Student engagement in a justification activity can also lead to more equitable…

The Leading for Mathematical Proficiency (LMP) Framework (Bay-Williams et al.) has three components: (1) The Standards for Mathematical Practice; (2) Shifts in classroom practice; and (3) Teaching skills. This article briefly describes each component of the LMP framework and then focuses more in depth on the second component, the shifts in…

Teaching students how to multiply fractions is challenging, not so much from a computational point of view but from a conceptual one. The algorithm for multiplying fractions is much easier to learn than many other algorithms, such as subtraction with regrouping, long division, and certainly addition of fractions with unlike denominators. However,…

Using a nonroutine problem can be an effective way to encourage students to draw on prior knowledge, work together, and reach important conclusions about the mathematics they are learning. This article discusses a problem on the mathematical preparation of chocolate milk which was adapted from an old book of puzzles (Linn 1969) and has been used…

The author recently read a research paper by Padberg (2002), in which the development of understanding associated with decimal fractions was studied. Padberg (2002) outlined the situation that existed in Germany, where students were introduced to decimal fractions in the sixth year of school. He claimed that it was assumed students would have a…