Students need to achieve automaticity in learning mathematics without sacrificing conceptual understanding of the algorithms that are essential in being successful in algebra and problem solving, as well as in science. This research investigated the relationship between science-contextualized problems and computational fluency by testing an…

Since 2001, the authors have been working with groups of teachers to investigate students' early algebraic thinking--learning representations, connections, and generalizations in the elementary school grades. They began paying attention to students' explicit remarks about regularities in the number system or what students imply by their…

As part of individual interviews incorporating whole number and rational number tasks, 323 grade 6 children in Victoria, Australia were asked to nominate the larger of two fractions for eight pairs, giving reasons for their choice. All tasks were expected to be undertaken mentally. The relative difficulty of the pairs was found to be close to that…

The mathematics for finding and plotting the locations of stars and constellations are available in many books on astronomy, but the steps involve mystifying and fragmented equations, calculations, and terminology. This paper will introduce an entirely new unified and cohesive technique that is easy to understand by mathematicians, and simple…

The effectiveness of problem solving as a learning tool is often diminished because students typically use only an algorithmic approach to get to the answer. We discuss a way of encouraging students to reflect on the solution to their problem by requiring them--after they have arrived at their solution--to draw solution maps. A solution map…

The final report of the Williams committee (DCSF, 2008: 68) argues that the revised mathematics Framework (DfES, 2006) "should be reconsidered to achieve a more suitable, user-friendly form." It might also have added that there is not much help and support in it for early years teachers. A much more useful document is the "Practice guidance for…

For RCTs of education interventions, it is often of interest to estimate associations between student and mediating teacher practice outcomes, to examine the extent to which the study's conceptual model is supported by the data, and to identify specific mediators that are most associated with student learning. This paper develops statistical power…

339 children aged 6 and 7 at Oxford primary schools took part in a study of arithmetic. 204 of the children had been selected by their teachers as having mathematical difficulties and the other 135 children were unselected. They were assigned to an Addition Performance Level on the basis of a calculation pretest, and then given Dowker's (1998)…

The subject of uncertainties (sometimes called errors) is traditionally taught (to first-year science undergraduates) towards the end of a course on statistics that defines probability as the limit of many trials, and discusses probability distribution functions and the Gaussian distribution. We show how to introduce students to the concepts of…

It is estimated that 50% of beginning teachers leave the profession within the first five years on the job (Murnane, Singer, Willett, Kemple, & Olsen, 1991; Colbert & Wolff, 1992; Ingersoll, 2003b; Schlechty & Vance, 1981). When teachers depart, they take with them their knowledge of instructional techniques, students' learning styles, and…

Cantor's diagonal proof that the set of real numbers is uncountable is one of the most famous arguments in modern mathematics. Mathematics students usually see this proof somewhere in their undergraduate experience, but it is rarely a part of the mathematical curriculum of students of the fine arts or humanities. This note describes contexts that…

In earlier columns, the celestial coordinate system of hour circles of right ascension and degrees of declination was introduced along with the use of an equatorial star chart (see SFA Star Charts in Resources). This system shows the planets' motion relative to the ecliptic, the apparent path the Sun follows during the year. An alternate system,…

Three elementary students with autism were taught single-digit addition problem-solving skills using number and touch-point strategies. Prior to the study, all students were unable to correctly calculate single-digit addition problems. An alternating-treatments design was used to compare the acquisition performance of single-digit addition…

In this article, the authors discuss the importance of unexpected graphs as a vehicle for encouraging critical classroom dialogue. By examining such graphs more critically, teachers and their students can reexamine beliefs about the authority of technology in their classrooms. (Contains 15 figures.)

The speed of sound is measured in several gases using a pulsed laser to create a micro-spark on a carbon rod and a microphone connected to a digital oscilloscope to measure the time-of-flight of the resulting shockwave over a known distance. These data are used to calculate the heat capacity ratios (C[subscript p]/C[subscript V]) of the gases and…