Improving students’ understanding of algebra and multiplicative reasoning: did the ICCAMS intervention work?

In this paper we report on the intervention phase of an ESRC-funded project, Increasing Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS). The intervention was designed to enable teachers to use formative assessment in mathematics classrooms by evaluating what students already knew, then adapting their teaching to students’ learning needs. A key feature was the use of models and representations, such as the Cartesian graph, both to help students better understand mathematical ideas and to help teachers appreciate students’ difficulties. Twenty-two teachers and their Year 8 classes from 11 schools took part in the intervention during 2010/11. Pre! and post-tests in algebra, decimals and ratio were administered to the students of these classes, and compared to a control group of students matched from the ICCAMS national longitudinal survey (using propensity score matching). The students in the intervention group made greater progress than the matched control

Improving students’ understanding of algebra and multiplicative reasoning: did the ICCAMS intervention work?

In this paper we report on the intervention phase of an ESRC-funded project, Increasing Competence and Confidence in Algebra and Multiplicative Structures (ICCAMS). The intervention was designed to enable teachers to use formative assessment in mathematics classrooms by evaluating what students already knew, then adapting their teaching to students’ learning needs. A key feature was the use of models and representations, such as the Cartesian graph, both to help students better understand mathematical ideas and to help teachers appreciate students’ difficulties. Twenty-two teachers and their Year 8 classes from 11 schools took part in the intervention during 2010/11. Pre! and post-tests in algebra, decimals and ratio were administered to the students of these classes, and compared to a control group of students matched from the ICCAMS national longitudinal survey (using propensity score matching). The students in the intervention group made greater progress than the matched control

A Rigorous Free-form Lens Model of Abell 2744 to Meet the Hubble Frontier Fields Challenge

Hubble Frontier Fields (HFF) imaging of the most powerful lensing clusters provides access to the most magnified distant galaxies. The challenge is to construct lens models capable of describing these complex massive, merging clusters so that individual lensed systems can be reliably identified and their intrinsic properties accurately derived. We apply the free-form lensing method (WSLAP+) to A2744, providing a model independent map of the cluster mass, magnification, and geometric distance estimates to multiply-lensed sources. We solve simultaneously for a smooth cluster component on a pixel grid, together with local deflections by the cluster member galaxies. Combining model prediction with photometric redshift measurements, we correct and complete several systems recently claimed, and identify 4 new systems - totalling 65 images of 21 systems spanning a redshift range of 1.4<z<9.8. The reconstructed mass shows small enhancements in the directions where significant amounts of hot plasma can be seen in X-ray. We compare photometric redshifts with "geometric redshifts", finding a high level of self-consistency. We find excellent agreement between predicted and observed fluxes - with a best-fit slope of 0.999+-0.013 and an RMS of ~0.25 mag, demonstrating that our magnification correction of the lensed background galaxies is very reliable. Intriguingly, few multiply-lensed galaxies are detected beyond z~7.0, despite the high magnification and the limiting redshift of z~11.5 permitted by the HFF filters. With the additional HFF clusters we can better examine the plausibility of any pronounced high-z deficit, with potentially important implications for the reionization epoch and the nature of dark matter.Comment: Accepted for publication in ApJ with newly identified lensed images in complete HFF dat

A Free-Form Prediction for the Reappearance of Supernova Refsdal in the Hubble Frontier Fields Cluster MACSJ1149.5+2223

The massive cluster MACSJ1149.5+2223(z=0.544) displays five very large lensed images of a well resolved spiral galaxy at $z_{\rm spect}=1.491$. It is within one of these images that the first example of a multiply-lensed supernova has been detected recently as part of the Grism Lens-Amplified Survey from Space. The depth of this data also reveals many HII regions within the lensed spiral galaxy which we identify between the five counter-images. Here we expand the capability of our free-form method to incorporate these HII regions locally, with other reliable lensed galaxies added for a global solution. This improved accuracy allows us to estimate when the Refsdal supernova will appear within the other lensed images of the spiral galaxy to an accuracy of $\sim$ 7\%. We predict this supernova will reappear in one of the counter-images (RA=11:49:36.025, DEC=+22:23:48.11, J2000) and on November 1$^{st}$ 2015 (with an estimated error of $\pm$ 25 days) it will be at the same phase as it was when it was originally discovered, offering a unique opportunity to study the early phases of this supernova and to examine the consistency of the mass model and the cosmological model that have an impact on the time delay prediction.Comment: 12 images, 11 pages. Mateches accepted version in MNRAS. MNRAS in pres

Geometric Corroboration of the Earliest Lensed Galaxy at z ≃ 10.8 from Robust Free-form Modelling

A multiply lensed galaxy, MACS0647-JD, with a probable photometric redshift of z ≃ 10.7_(-0.4)^(+0.6) is claimed to constitute one of the very earliest known galaxies, formed well before reionization was completed. However, spectral evidence that MACS0647-JD lies at high redshift has proven infeasible and so here we seek an independent-lensing-based "geometric redshift" derived from the angles between the three lensed images of MACS0647-JD, using our free-form mass model (WSLAP+) for the lensing cluster MACSJ0647.7+7015 (at z = 0.591). Our lens model uses the nine sets of multiple images, including those of MACS0647-JD, identified by the CLASH survey toward this cluster. We convincingly exclude the low-redshift regime of z < 3, for which convoluted critical curves are generated by our method, as the solution bends to accommodate the wide angles of MACS0647-JD for this low redshift. Instead, a best fit to all sets of lensed galaxy positions and redshifts provides a geometric redshift of z≃ 10.8_(-0.4)^(+0.3) for MACS0647-JD, strongly supporting the higher photometric redshift solution. Importantly, we find a tight linear relation between the relative brightnesses of all nine sets of multiply lensed images and their relative magnifications as predicted by our model. This agreement provides a benchmark for the quality of the lens model, and establishes the robustness of our free-form lensing method for measuring model-independent geometric source distances and for deriving objective central cluster mass distributions. After correcting for its magnification the luminosity of MACS0647-JD remains relatively high at M UV = −19.4, which is within a factor of a few in flux of some surprisingly luminous z sime 10–11 candidates discovered recently in Hubble blank field surveys

Geometric Corroboration of the Earliest Lensed Galaxy at z ≃ 10.8 from Robust Free-form Modelling

A multiply lensed galaxy, MACS0647-JD, with a probable photometric redshift of z ≃ 10.7_(-0.4)^(+0.6) is claimed to constitute one of the very earliest known galaxies, formed well before reionization was completed. However, spectral evidence that MACS0647-JD lies at high redshift has proven infeasible and so here we seek an independent-lensing-based "geometric redshift" derived from the angles between the three lensed images of MACS0647-JD, using our free-form mass model (WSLAP+) for the lensing cluster MACSJ0647.7+7015 (at z = 0.591). Our lens model uses the nine sets of multiple images, including those of MACS0647-JD, identified by the CLASH survey toward this cluster. We convincingly exclude the low-redshift regime of z < 3, for which convoluted critical curves are generated by our method, as the solution bends to accommodate the wide angles of MACS0647-JD for this low redshift. Instead, a best fit to all sets of lensed galaxy positions and redshifts provides a geometric redshift of z≃ 10.8_(-0.4)^(+0.3) for MACS0647-JD, strongly supporting the higher photometric redshift solution. Importantly, we find a tight linear relation between the relative brightnesses of all nine sets of multiply lensed images and their relative magnifications as predicted by our model. This agreement provides a benchmark for the quality of the lens model, and establishes the robustness of our free-form lensing method for measuring model-independent geometric source distances and for deriving objective central cluster mass distributions. After correcting for its magnification the luminosity of MACS0647-JD remains relatively high at M UV = −19.4, which is within a factor of a few in flux of some surprisingly luminous z sime 10–11 candidates discovered recently in Hubble blank field surveys