Developing Preservice Teachers’ Mathematical and Pedagogical Knowledge Using an Integrated Approach

This paper describes how an integrated mathematics content and early field-experience course provides opportunities for preservice elementary teachers to develop understanding of mathematics and mathematics teaching. Engaging preservice teachers in solving and discussing mathematical tasks and providing opportunities to implement these tasks with elementary students creates an authentic context for the future teachers to reflect on their own understanding of mathematics, mathematics teaching, and students’ mathematical thinking. Essential elements of the cycle of events in the integrated model of instruction are discussed: preservice students’ acquisition of mathematical concepts in the context of selected tasks in the content course; subsequent posing of mathematical tasks in early field experiences; reflection on work with students; and response to instructors’ feedback

Pre-service Middle School Teachers’ Knowledge of Algebraic Thinking

In this study we examined the relationship between 18 pre-service middle school teachers’ own ability to use algebraic thinking to solve problems and their ability to recognize and interpret the algebraic thinking of middle school students. We assessed the pre-service teachers’ own algebraic thinking by examining their solutions and explanations to multiple algebra-based tasks posed during a semester-long mathematics content course. We assessed their ability to recognize and interpret the algebraic thinking of students in two ways. The first was by analyzing the preservice teachers’ ability to interpret students’ written solutions to open-ended algebra-based tasks. The second was by analyzing their ability to plan, conduct, and analyze algebraic thinking (AT) interviews of middle school students during a concurrent semester-long, field-based education class. We used algebraic habits of mind as a framework to identify the algebraic thinking that pre-service teachers exhibited in their own problem solving, and we asked students to use them to analyze the algebraic thinking of middle school students. The data revealed that pre-service teachers’ AT abilities varied across different features of algebraic thinking. In particular, their ability to justify a rule was the weakest of seven AT features. The ability to recognize and interpret the algebraic thinking of students was strongly correlated with the strength of the pre-service teachers’ own algebraic thinking. Implications for mathematics teacher education are discussed

Prospective K-8 Teachers’ Knowledge of Relational Thinking

The goal of this study was to examine two issues: First, pre-service teachers’ ability and inclination to think relationally prior to instruction about the role relational thinking plays in the K-8 mathematics curriculum. Second, to examine task specific variables possibly associated with pre-service teachers’ inclination to engage in relational thinking. The results revealed that preservice teachers engage in relational thinking about equality, however, their inclination to do so is rather limited. Furthermore, they tend to engage in relational thinking more frequently in the context of arithmetic than algebra-related tasks. Pre-service teachers’ inclination to engage in relational thinking appeared to also relate to the overall task complexity and the use of variables. Implications of these findings for pre-service teacher education are provided

Pre-Service Teachers’ Knowledge of Algebraic Thinking and the Characteristics of the Questions Posed for Students

In this study, we explored the relationship between the strength of pre-service teachers’ algebraic thinking and the characteristics of the questions they posed during cognitive interviews that focused on probing the algebraic thinking of middle school students. We developed a performance rubric to evaluate the strength of pre-service teachers’ algebraic thinking across 130 algebra-based tasks. We used an existing coding scheme found in the literature to analyze the characteristics of the questions pre-service teachers posed during clinical interviews. We found that pre-service teachers with higher algebraic thinking abilities were able to pose probing questions that uncovered student thinking through the use of follow up questions. In comparison, pre-service teachers with lower algebraic thinking abilities asked factual questions; moving from one question to the next without posing follow up questions to probe student thinking

Exploring the Relationship between K-8 Prospective Teachers’ Algebraic Thinking Proficiency and the Questions They Pose during Diagnostic Algebraic Thinking Interviews

In this study, we explored the relationship between prospective teachers’ algebraic thinking and the questions they posed during one-on-one diagnostic interviews that focused on investigating the algebraic thinking of middle school students. To do so, we evaluated prospective teachers’ algebraic thinking proficiency across 125 algebra-based tasks and we analyzed the characteristics of questions they posed during the interviews. We found that prospective teachers with lower algebraic thinking proficiency did not ask any probing questions. Instead, they either posed questions that simply accepted and affirmed student responses or posed questions that guided the students toward an answer without probing student thinking. In contrast, prospective teachers with higher algebraic thinking proficiency were able to pose probing questions to investigate student thinking or help students clarify their thinking. However, less than half of their questions were of this probing type. These results suggest that prospective teachers’ algebraic thinking proficiency is related to the types of questions they ask to explore the algebraic thinking of students. Implications for mathematics teacher education are discussed

K-8 Pre-service Teachers’ Algebraic Thinking: Exploring the Habit of Mind Building Rules to Represent Functions

In this study, through the lens of the algebraic habit of mind Building Rules to Represent Functions, we examined 18 pre-service middle school teachers\u27 ability to use algebraic thinking to solve problems. The data revealed that pre-service teachers\u27 ability to use different features of the habit of mind Building Rules to Represent Functions varied across the features. Significant correlations existed between 8 pairs of the features. The ability to justify a rule was the weakest of the seven features and it was correlated with the ability to chunk information. Implications for mathematics teacher education are discussed

Sheared Ising models in three dimensions

The nonequilibrium phase transition in sheared three-dimensional Ising models is investigated using Monte Carlo simulations in two different geometries corresponding to different shear normals. We demonstrate that in the high shear limit both systems undergo a strongly anisotropic phase transition at exactly known critical temperatures T_c which depend on the direction of the shear normal. Using dimensional analysis, we determine the anisotropy exponent theta=2 as well as the correlation length exponents nu_parallel=1 and nu_perp=1/2. These results are verified by simulations, though considerable corrections to scaling are found. The correlation functions perpendicular to the shear direction can be calculated exactly and show Ornstein-Zernike behavior.Comment: 6 pages, 3 figure

Study of the Λp\Lambda p Interaction Close to the Σ+n\Sigma^+n and Σ0p\Sigma^0p Thresholds

The $\Lambda p$ interaction close to the $\Sigma N$ threshold is considered. Specifically, the pronounced structure seen in production reactions like $K^-d \to \pi^- \Lambda p$ and $pp\to K^+ \Lambda p$ around the $\Sigma N$ threshold is analyzed. Modern interaction models of the coupled $\Lambda N - \Sigma N$ systems generate such a structure either due to the presence of a (deuteron-like) unstable bound state or of an inelastic virtual state. % A determination of the position of the prominent peak as observed in various experiments for the two aforementioned reactions leads to values that agree quite well with each other. Furthermore, the deduced mean value of $2128.7\pm 0.3$ MeV for the peak position coincides practically with the threshold energy of the $\Sigma^+ n$ channel. This supports the interpretation of the structure as a genuine cusp, signaling an inelastic virtual state in the $^3S_1-^3D_1$ partial wave of the $\Sigma N$ isospin 1/2 channel. % There is also evidence for a second peak (or shoulder) in the data sets considered which appears at roughly 10-15 MeV above the $\Sigma N$ threshold. However, its concrete position varies significantly from data set to data set and, thus, a theoretical interpretation is difficult.Comment: accepted for publication Nucl. Phys.

Average distance in growing trees

Two kinds of evolving trees are considered here: the exponential trees, where subsequent nodes are linked to old nodes without any preference, and the Barab\'asi--Albert scale-free networks, where the probability of linking to a node is proportional to the number of its pre-existing links. In both cases, new nodes are linked to $m=1$ nodes. Average node-node distance $d$ is calculated numerically in evolving trees as dependent on the number of nodes $N$. The results for $N$ not less than a thousand are averaged over a thousand of growing trees. The results on the mean node-node distance $d$ for large $N$ can be approximated by $d=2\ln(N)+c_1$ for the exponential trees, and $d=\ln(N)+c_2$ for the scale-free trees, where the $c_i$ are constant. We derive also iterative equations for $d$ and its dispersion for the exponential trees. The simulation and the analytical approach give the same results.Comment: 6 pages, 3 figures, Int. J. Mod. Phys. C14 (2003) - in prin

New algorithm for the computation of the partition function for the Ising model on a square lattice

A new and efficient algorithm is presented for the calculation of the partition function in the $S=\pm 1$ Ising model. As an example, we use the algorithm to obtain the thermal dependence of the magnetic spin susceptibility of an Ising antiferromagnet for a $8\times 8$ square lattice with open boundary conditions. The results agree qualitatively with the prediction of the Monte Carlo simulations and with experimental data and they are better than the mean field approach results. For the $8\times 8$ lattice, the algorithm reduces the computation time by nine orders of magnitude.Comment: 7 pages, 3 figures, to appear in Int. J. Mod. Phys.