Student Understanding of Linear Scale in Mathematics: Exploring What Year 7 and 8 Students Know

<p>This thesis set out to undertake a curriculum review. Scale was chosen as the focus of the review because the Mathematics and Science Taskforce (1997) indicated that measurement was an area of the curriculum that needed priority attention, however comparatively little had been done to provide this by 2005. Scales are a common mathematical object. A search of many (Western) homes is likely to find a variety of scales being used for different purposes. They can be found on car dashboards, kitchen stoves, tape measures, and clocks. They will be found in graphs, newspapers, and magazines. Scales also underpin important mathematical learning over and above their everyday application to measurement and graphing. For example, concepts like gradient, rate of change, functions, and limits all rely on an understanding of scale. But how much do we know about how students learn to use scales? What does learning about scale involve? The thesis approaches the review through an exploration of student understanding of scale in the context of mathematics. It focuses on answering the question "what understanding of scales do students in Year 7 and 8 at the case study schools have?" A definition of scale is developed and is based on the mathematics to which the curriculum document indicates Year 7 and 8 students should have been exposed. This could be identified as the notion of a linear scale. Students in Years 7 and 8 were chosen because by that age the mathematics curriculum document implies students should have a good understanding of scale; at the same time, their errors and misconceptions are likely to indicate learning barriers that need to be addressed. The literature and the New Zealand mathematics curriculum were used to define a construct of scale appropriate to explore with Year 7 and 8 students. Two tests were then developed to measure understanding of that construct. Where possible, items were initially developed or adapted from the literature then, when early findings suggested new avenues for exploration, new items were developed to investigate further issues of understanding. Both tests were used at different schools in the format of a cognitive interview; Test 1 was also used at another school as a written test. Additional items were developed to use with groups of teachers in an attempt to challenge early findings; these teacher trials used a third assessment format requiring both a written answer and a written explanation of method. In Test 1 students were assessed with pairs of items. In each pair one item used a decontextualised number line, the other a measurement or graphing context promoted by the curriculum. During the cognitive interviews, the verbal responses of students were recorded on audiotape, while field notes were used to capture non-verbal data. Follow-up probe questions were used to clarify solution strategies and the understanding underlying these strategies. The written test was then used to identify if the interpretations made could be transferred. Test 2 repeated the data collection methodology from Test 1 but used a different structure within the test. In total, 45 cognitive interviews and 81 written tests were undertaken with students, while 32 teachers participated in the teacher trials. Facilities, point bi-serial correlation coefficients, and levels of significance were used to ascertain the fitness for purpose of the developed test items. Data collected during the cognitive interviews were also analysed using both qualitative and quantitative methods to ascertain patterns of response. The mathematics curriculum in New Zealand had assumed that students would develop understanding about scale from exposure to scales in measurement and graphing situations. This approach might have been appropriate if a scale was a simple (or single) object to be mastered but it is not. A scale is a mathematical tool of vast flexibility that can be applied in numerous forms to a wide range of situations. The results suggest that teaching of scale needs to be more deliberate, and also needs to be considered when curricula are designed. A high proportion of the students in the study had not developed the expected understanding of scale by Years 7 and 8. A complex series of factors were identified that impacted on how the students worked with scale. These included: their understanding of number and number symbols; their understanding of the measurement conventions that are foundational to scale; the strategies they had developed to partition unmarked intervals; their strategies to decide on the value of a partition in marked intervals; their understanding of the role of the marks and spaces on the scale; and their ability to iterate a unit. These different bodies of understanding needed to be integrated and used in a coordinated manner for the students to become effective users of scale.</p>

Student Understanding of Linear Scale in Mathematics: Exploring What Year 7 and 8 Students Know

<p>This thesis set out to undertake a curriculum review. Scale was chosen as the focus of the review because the Mathematics and Science Taskforce (1997) indicated that measurement was an area of the curriculum that needed priority attention, however comparatively little had been done to provide this by 2005. Scales are a common mathematical object. A search of many (Western) homes is likely to find a variety of scales being used for different purposes. They can be found on car dashboards, kitchen stoves, tape measures, and clocks. They will be found in graphs, newspapers, and magazines. Scales also underpin important mathematical learning over and above their everyday application to measurement and graphing. For example, concepts like gradient, rate of change, functions, and limits all rely on an understanding of scale. But how much do we know about how students learn to use scales? What does learning about scale involve? The thesis approaches the review through an exploration of student understanding of scale in the context of mathematics. It focuses on answering the question "what understanding of scales do students in Year 7 and 8 at the case study schools have?" A definition of scale is developed and is based on the mathematics to which the curriculum document indicates Year 7 and 8 students should have been exposed. This could be identified as the notion of a linear scale. Students in Years 7 and 8 were chosen because by that age the mathematics curriculum document implies students should have a good understanding of scale; at the same time, their errors and misconceptions are likely to indicate learning barriers that need to be addressed. The literature and the New Zealand mathematics curriculum were used to define a construct of scale appropriate to explore with Year 7 and 8 students. Two tests were then developed to measure understanding of that construct. Where possible, items were initially developed or adapted from the literature then, when early findings suggested new avenues for exploration, new items were developed to investigate further issues of understanding. Both tests were used at different schools in the format of a cognitive interview; Test 1 was also used at another school as a written test. Additional items were developed to use with groups of teachers in an attempt to challenge early findings; these teacher trials used a third assessment format requiring both a written answer and a written explanation of method. In Test 1 students were assessed with pairs of items. In each pair one item used a decontextualised number line, the other a measurement or graphing context promoted by the curriculum. During the cognitive interviews, the verbal responses of students were recorded on audiotape, while field notes were used to capture non-verbal data. Follow-up probe questions were used to clarify solution strategies and the understanding underlying these strategies. The written test was then used to identify if the interpretations made could be transferred. Test 2 repeated the data collection methodology from Test 1 but used a different structure within the test. In total, 45 cognitive interviews and 81 written tests were undertaken with students, while 32 teachers participated in the teacher trials. Facilities, point bi-serial correlation coefficients, and levels of significance were used to ascertain the fitness for purpose of the developed test items. Data collected during the cognitive interviews were also analysed using both qualitative and quantitative methods to ascertain patterns of response. The mathematics curriculum in New Zealand had assumed that students would develop understanding about scale from exposure to scales in measurement and graphing situations. This approach might have been appropriate if a scale was a simple (or single) object to be mastered but it is not. A scale is a mathematical tool of vast flexibility that can be applied in numerous forms to a wide range of situations. The results suggest that teaching of scale needs to be more deliberate, and also needs to be considered when curricula are designed. A high proportion of the students in the study had not developed the expected understanding of scale by Years 7 and 8. A complex series of factors were identified that impacted on how the students worked with scale. These included: their understanding of number and number symbols; their understanding of the measurement conventions that are foundational to scale; the strategies they had developed to partition unmarked intervals; their strategies to decide on the value of a partition in marked intervals; their understanding of the role of the marks and spaces on the scale; and their ability to iterate a unit. These different bodies of understanding needed to be integrated and used in a coordinated manner for the students to become effective users of scale.</p>

Local Linear Scale Factors in Map Projections of an Ellipsoid

The main problem in cartography is that it is not possible to map/project/transform a spherical or ellipsoidal surface into a plane without distortions. The distortions of areas, angles, and/or distances are immanent to all maps. It is known that scale changes from point to point, and at certain points, the scale usually depends on the direction. The local linear scale factor c is one of the most important indicators of distortion distribution in the theory of map projections. It is not possible to find out the values of the local linear scale factor c in directions of coordinate axes x and y immediately from the definition of c. To solve this problem, in this paper, we derive new formulae for the calculation of c for a rotational ellipsoid. In addition, we derive the formula for computing c in any direction defined by dx and dy. We also considered the position and magnitude of the extreme values of c and derived new formulae for a rotational ellipsoid.

Elastic-viscous properties of acrylic acid 2-propenamide gel

The rheological properties of the gel-like material, the monomer of which is a crosslinked and modified 2-propenamide of acrylic acid, were determined by relaxation rheometry methods. The values of its elastic modulus and modulus of losses and complex viscosity depending on: deforming stress and its frequency are determined; relative deformation; temperature in the range (20-100) ° C and the regularities of these dependences are noted. It is established that: 1) the dependence of the modulus of elasticity (G'); modulus of loss (G'') and complex viscosity from: relative deformation; voltage; temperature; frequencies indicate that in the linear scale they change according to nonlinear dependencies, and in the transition to the logarithmic scale contain plateau-like areas; 2) analytical dependences of the above parameters on stress, strain rate and temperature are complex and difficult to establish; 3) in the range (20-80) ° C and relative deformations (10-100)% hydrogel has a virtually unchanged value of the modulus (G ') ten times greater than the modulus (G' '), whichdetermines the uniqueness of its rheological and biophysical properties ; 4) in the region (20-80) ° C hydrogel in terms of modulus of elasticity and tangent of the angle of loss is close to a completely elastic body; 5) when the frequency of the deforming voltage is more than 15.8 Hz and the relative deformation ≥100%, the gel is brittlely deformed; while the modulus of its elasticity decreases abruptly and the modulus of losses increases rapidly with increasing frequency of the deforming stress. 6) the dependence of the elastic-viscosity characteristics of the samples washed and unwashed in saline gel in the temperature range (20-80) ° C differ little and indicate that the equilibrium structure of the hydrogel 2-propenamide acrylic acid belongs to the typical colloidal dispersed structure of gelatinous substances.

A Self-Calibration Stitching Method for Pitch Deviation Evaluation of a Long-Range Linear Scale by Using a Fizeau Interferometer

An interferometric self-calibration method for the evaluation of the pitch deviation of scale grating has been extended to evaluate the pitch deviation of the long-range type linear scale by utilizing the stitching interferometry technique. Following the previous work, in which the interferometric self-calibration method was proposed to assess the pitch deviation of the scale grating by combing the first-order diffracted beams from the grating, a stitching calibration method is proposed to enlarge the measurement range. Theoretical analysis is performed to realize the X-directional pitch deviation calibration of the long-range linear scale while reducing the second-order accumulation effect by canceling the influence of the reference flat error in the sub-apertures’ measurements. In this paper, the stitching interferometry theory is briefly reviewed, and theoretical equations of the X-directional pitch deviation stitching are derived for evaluation of the pitch deviation of the long-range linear scale. Followed by the simulation verification, some experiments with a linear scale of 105 mm length from a commercial interferential scanning-type optical encoder are conducted to verify the feasibility of the self-calibration stitching method for the calibration of the X-directional pitch deviation of the linear scale over its whole area.

Critical behavior of a two-dimensional ferromagnetic film on a non-magnetic substrate

In this paper, we investigate the behavior of a ferromagnetic (FM) film on a nonmagnetic substrate near the Curie point by the computer simulation. The influence of the substrate is specified using the two-dimensional Frenkel-Kontorova (FK) potential. The study is carried out for a two-dimensional film described by the Ising model. At the first step, we calculate the positions of the substrate’s atoms in the ground state depending on the parameters. The parameters are (i) the ratio of the substrate periods and the crystal lattice of the film; and (ii) the ratio of the substrate potential amplitude to the elasticity coefficient of interatomic interaction. The period ratio determines the system coverage ratio. Minimization of the system’s total energy determines the ground state. Calculations show that the ground state has a periodic structure that differs from a square lattice with a coverage coefficient not equal to unity. We calculate the displacements of atoms from the equilibrium position for systems with a different linear scale.

Experimental Research on the Thresholds of Scale-Dependent Dispersivity of Solute Transport

The conventional advection-dispersion equation (ADE) has been widely used to describe the solute transport in porous media. However, it cannot interpret the phenomena of the early arrival and long tailing in breakthrough curves (BTCs). In this study, we aim to experimentally investigate the behaviors of the solute transport in both homogeneous and heterogeneous porous media. The linear-asymptotic model (LAF solution) with scale-dependent dispersivity was used to fit the BTCs, which was compared with the results of the ADE model and the conventional truncated power-law (TPL) model. Results indicate that (1) the LAF model with linear scale-dependent dispersivity could better capture the evolution of BTCs than the ADE model; (2) dispersivity initially increases linearly with the travel distance and is stable at some limited value over a large distance, and a threshold value of the travel distance is provided to reflect the constant dispersivity; and (3) compared with the TPL model, both the LAF and ADE models can capture the behavior of solute transport as a whole. For fitting the early arrival, the LAF model is less than the TPL; however, the LAF model is more concise in mathematics and its application will be studied in the future.

Perseus arm - a new perspective on star formation and spiral structure in our home galaxy

Expansion of (sub)millimetre capabilities to high angular resolution offered with interferometers allows to resolve giant molecular clouds (GMCs) in nearby galaxies. This enables us to place the Milky Way in the context of other galaxies to advance our understanding of star formation in our own Galaxy. We thus remap 12CO (1 - 0) data along the Perseus spiral arm in the outer Milky Way to a fixed physical resolution and present the first spiral arm data cube at a common distance as it would be seen by an observer outside the Milky Way. To achieve this goal we calibrated the longitude-velocity structure of 12CO gas of the outer Perseus arm based on trigonometric distances and maser velocities provided by the BeSSeL survey. The molecular gas data were convolved to the same spatial resolution along the whole spiral arm and regridded on to a linear scale map with the coordinate system transformed to the spiral arm reference frame. We determined the width of the Perseus spiral arm to be 7.8 ± 0.2 km s−1 around the kinematic arm centre. To study the large scale structure we derived the 12CO gas mass surface density distribution of velocities shifted to the kinematic arm centre and arm length. This yields a variation of the gas mass surface density along the arm length and a compression of molecular gas mass at linear scale. We determined a thickness of ∼63 pc on average for the Perseus spiral arm and a centroid of the molecular layer of 8.7 pc.

A NOVEL DESIGN CONCEPT FOR A THERMALLY STABLE LINEAR SCALE USING TWO DIFFERENT MATERIALS

Linear scale has significant impacts on the machine tool accuracy, since the positioning of the linear axes are controlled by its measurement. This paper presents a novel concept of linear scale design which can provide high thermal stability with low cost. This concept applies two different materials: a steel linear scale attached mechanically on a carbon fiber reinforced plastic (CFRP) tube. Attaching this two materials, the thermal behavior of the steel scale can be mechanically compensated by the CFRP tube when the temperature changes. The potential of the design concept is analyzed based on the experiment results.