Here we explore how college students across different courses may conceptualize symbolic algebrai... more Here we explore how college students across different courses may conceptualize symbolic algebraic properties. This work draws on the theory of Grundvorstellungen (GVs) as a tool to analyse how learner conceptions do or do not align with some desired goals of instruction. In analysing interviews, several categories of conceptions, or descriptive GVs, emerged, which may be a helpful first step in understanding learners' thinking and improving instruction on algebraic properties.
We prove that seesaw words exist in Thompson's group F (N) for N = 2, 3, 4,. .. with respect to t... more We prove that seesaw words exist in Thompson's group F (N) for N = 2, 3, 4,. .. with respect to the standard finite generating set X. A seesaw word w with swing k has only geodesic representatives ending in g k or g −k (for given g ∈ X) and at least one geodesic representative of each type. The existence of seesaw words with arbitrarily large swing guarantees that F (N) is neither synchronously combable nor has a regular language of geodesics. Additionally, we prove that dead ends (or k-pockets) exist in F (N) with respect to X and all have depth 2. A dead end w is a word for which no geodesic path in the Cayley graph Γ which passes through w can continue past w, and the depth of w is the minimal m ∈ N such that a path of length m + 1 exists beginning at w and leaving B |w|. We represent elements of F (N) by tree-pair diagrams so that we can use Fordham's method for computing word length. This paper generalizes results by Cleary and Taback, who proved the case N = 2.
This session focused on research being conducted at non-university tertiary institutions. Instruc... more This session focused on research being conducted at non-university tertiary institutions. Instructors in adult education as well as other non-university postsecondary institutions participated. The participants of the session shared concerns about wanting to know more about the non-university mathematics classroom through classroom-based research by practitioners. Examples of research presently being conducted were shared (Mesa, Wladis, & Watkins, 2014; Sitomer et al., 2012; Wladis, Conway, & Hachey, 2016) and are cited below. The goal for developing collaborative research and/or grant proposals among researchers of similar interest continues. A presentation of the National Science Foundation grant Algebra instruction at community colleges: An exploration of its relationship with student success (Watkins, Strom, Mesa, Kohli, & Duranczyk, 2015) design was shared. The researchers are exploring the impact of students' and instructors' pre-existing and moderating variable on the relationship between student-instructor interaction with mathematics in the classroom and students' performance outcomes. Research on how individual and institutional characteristics factor into failure rates and performance measures exists, but there is little information about the fundamental work of teachers in the classroom, and the interaction that occur between instructors, students, and the mathematical content. A qualitative study (Smith, 2016), provided the student point of view. It explored students' reflections on their experiences (unsuccessful and successful) in mathematics at the community college. Student voices provided essential insights as to how postsecondary educators might foster positive learning transformations, and avoid being the source of needless obstacles to degree attainment. Then the
International Journal of Algebra and Computation, Feb 1, 2011
We give criteria for determining the approximate length of elements in any given cyclic subgroup ... more We give criteria for determining the approximate length of elements in any given cyclic subgroup of the Thompson-Stein groups F (n 1 , ..., n k) such that n 1 − 1|n i − 1 ∀i ∈ {1, ..., k} in terms of the number of leaves in the minimal tree-pair diagram representative. This leads directly to the result that cyclic subgroups are quasi-isometrically embedded in the Thompson-Stein groups. This result also leads to the corollaries that Z n is also quasiisometrically embedded in the Thompson-Stein groups for all n ∈ N and that the Thompson-Stein groups have infinite dimensional asymptotic cone. 1991 Mathematics Subject Classification. 20F65. Key words and phrases. Thompson's group,piecewise-linear homeomorphisms,Stein groups, Higman groups, quasi-isometrically embedded subgroups, cyclic subgroups, asymptotic cone. The author would like to thank Martin Kassabov for interesting and relevant discussion during the preparation of this article. The author acknowledges support from the CUNY Scholar Incentive Award, and would like to thank the Technische Universität Berlin for its hospitality during the writing of this paper. She would also like to thank the anonymous referee for their helpful comments during the revision of this paper.
Claire Wladis is a professor at the Borough of Manhattan Community College and the Graduate Cente... more Claire Wladis is a professor at the Borough of Manhattan Community College and the Graduate Center at the City University of New York. She is particularly interested in research questions that address issues of equity and access to higher education and STEM fields. Currently she is investigating how access to and enrollment in online courses may relate to college access and persistence, particularly for students who have be historically underrepresented in college and STEM fields. She is also developing and validating a series of learning progressions and a concept inventory to measure the algebraic conceptual understanding of college students.
Journal of the London Mathematical Society, Nov 16, 2011
We consider the Thompson-Stein group F (n 1 , ..., n k) where n 1 , ..., n k ∈ {2, 3, 4, ...}, k ... more We consider the Thompson-Stein group F (n 1 , ..., n k) where n 1 , ..., n k ∈ {2, 3, 4, ...}, k ∈ N. We highlight several differences between the cases k = 1 and k > 1, including the fact that minimal tree-pair diagram representatives of elements may not be unique when k > 1. We establish how to find minimal tree-pair diagram representatives of elements of F (n 1 , ..., n k), and we prove several theorems describing the equivalence of trees and tree-pair diagrams. We introduce a unique normal form for elements of F (n 1 , ..., n k) (with respect to the standard infinite generating set developed by Melanie Stein) which provides a solution to the word problem, and we give sharp upper and lower bounds on the metric with respect to the standard finite generating set, showing that in the case k > 1, the metric is not quasi-isometric to the number of leaves or caret in the minimal tree-pair diagram, as is the case when k = 1. Contents 1. Introduction 2. Representing elements using tree-pair diagrams 2.1. Tree-pair diagrams as representatives of elements of F (n 1 , ..., n k) 2.2. Equivalence of trees: Basic results 2.3. Minimal tree-pair diagrams 2.4. Tree-pair diagram composition 2.5. Equivalence of trees: Further results 3. Equivalence of tree-pair diagrams 4. Normal Form and Solution to the Word Problem 4.1. Infinite Group Presentation 4.2. Unique Minimal Tree-pair Diagram Representatives 4.3. The Normal Form 4.4. The Normal Form of Positive Words 5. The Metric on F (n 1 , ..., n k) 5.1. Standard Finite Presentation 5.2. The Metric References 1991 Mathematics Subject Classification. 20F65. The author would like to thank Sean Cleary for his support and advice during the preparation of this article. The author acknowledges support from the CUNY Scholar Incentive Award and would like to thank the Techniche Universität Berlin for its hospitality during the writing of this paper.
We show that the inclusion map of the generalized Thompson groups F (n i) is exponentially distor... more We show that the inclusion map of the generalized Thompson groups F (n i) is exponentially distorted in the Thompson-Stein groups F (n 1 , ..., n k) whenever k > 1. One consequence of this is that F is exponentially distorted in F (n 1 , ..., n k) for k > 1 whenever n i = 2 m for some m (whenever no i, m exist such that n i = 2 m , there is no obviously "natural" inclusion map of F into F (n 1 , ..., n k)). This is the first known example in which the natural embedding of one of the Thompson-type groups into another is not quasi-isometric. 1991 Mathematics Subject Classification. 20F65.
We prove that Thompson's group F (n) is not minimally almost convex with respect to the standard ... more We prove that Thompson's group F (n) is not minimally almost convex with respect to the standard finite generating set. A group G with Cayley graph Γ is not minimally almost convex if for arbitrarily large values of m there exist elements g, h ∈ Bm such that d Γ (g, h) = 2 and d Bm (g, h) = 2m. (Here Bm is the ball of radius m centered at the identity.) We use tree-pair diagrams to represent elements of F (n) and then use Fordham's metric to calculate geodesic length of elements of F (n). Cleary and Taback have shown that F (2) is not almost convex and Belk and Bux have shown that F (2) is not minimally almost convex; we generalize these results to show that F (n) is not minimally almost convex for all n ∈ {2, 3, 4,. .. }.
HAL (Le Centre pour la Communication Scientifique Directe), Feb 2, 2022
This empirical paper explores students' conceptions of transformation as substitution equivalence... more This empirical paper explores students' conceptions of transformation as substitution equivalence by linking it to their definitions of substitution and equivalence. This work draws on Sfard's (1995) framework to conceptualize conceptions of substitution equivalence and its components, equivalence and substitution, each on a spectrum from computational to structural. We provide examples of student work to illustrate how students' understandings of substitution, equivalence, and substitution equivalence as an approach to justifying transformation may relate to one another.
Springer Proceedings in Mathematics & Statistics, 2021
Multiple Answer Multiple Choice items (MAMC) have been used primarily for high-stakes assessments... more Multiple Answer Multiple Choice items (MAMC) have been used primarily for high-stakes assessments but are being used in broader circumstances. Unfortunately, they are a complex format for examinees, item writers, and scorers. This article features two novel approaches to scoring these items while comparing a low ability and a high ability examinee group. One of these approaches is the use of Latent Class Analysis (LCA) to better understand how examinees respond to these item formats related to a “credit earned approach” to scoring MAMC items. In addition, Jaccard’s distance is proposed as an alternative partial credit scoring method for assigning credit to students on these items. Both have their benefits and drawbacks, which are discussed in the article; however, both exploratory approaches reveal more about examinee knowledge than simple dichotomous scoring.
This session focused on research being conducted at non-university tertiary institutions. Instruc... more This session focused on research being conducted at non-university tertiary institutions. Instructors in adult education as well as other non-university postsecondary institutions participated. The participants of the session shared concerns about wanting to know more about the non-university mathematics classroom through classroom-based research by practitioners. Examples of research presently being conducted were shared (Mesa, Wladis, & Watkins, 2014; Sitomer et al., 2012; Wladis, Conway, & Hachey, 2016) and are cited below. The goal for developing collaborative research and/or grant proposals among researchers of similar interest continues. A presentation of the National Science Foundation grant Algebra instruction at community colleges: An exploration of its relationship with student success (Watkins, Strom, Mesa, Kohli, & Duranczyk, 2015) design was shared. The researchers are exploring the impact of students' and instructors' pre-existing and moderating variable on the relationship between student-instructor interaction with mathematics in the classroom and students' performance outcomes. Research on how individual and institutional characteristics factor into failure rates and performance measures exists, but there is little information about the fundamental work of teachers in the classroom, and the interaction that occur between instructors, students, and the mathematical content. A qualitative study (Smith, 2016), provided the student point of view. It explored students' reflections on their experiences (unsuccessful and successful) in mathematics at the community college. Student voices provided essential insights as to how postsecondary educators might foster positive learning transformations, and avoid being the source of needless obstacles to degree attainment. Then the
Claire Wladis is a professor at the Borough of Manhattan Community College and the Graduate Cente... more Claire Wladis is a professor at the Borough of Manhattan Community College and the Graduate Center at the City University of New York. She is particularly interested in research questions that address issues of equity and access to higher education and STEM fields. Currently she is investigating how access to and enrollment in online courses may relate to college access and persistence, particularly for students who have be historically underrepresented in college and STEM fields. She is also developing and validating a series of learning progressions and a concept inventory to measure the algebraic conceptual understanding of college students.
We prove that seesaw words exist in Thompson's group F (N) for N = 2, 3, 4,. .. with respect to t... more We prove that seesaw words exist in Thompson's group F (N) for N = 2, 3, 4,. .. with respect to the standard finite generating set X. A seesaw word w with swing k has only geodesic representatives ending in g k or g −k (for given g ∈ X) and at least one geodesic representative of each type. The existence of seesaw words with arbitrarily large swing guarantees that F (N) is neither synchronously combable nor has a regular language of geodesics. Additionally, we prove that dead ends (or k-pockets) exist in F (N) with respect to X and all have depth 2. A dead end w is a word for which no geodesic path in the Cayley graph Γ which passes through w can continue past w, and the depth of w is the minimal m ∈ N such that a path of length m + 1 exists beginning at w and leaving B |w|. We represent elements of F (N) by tree-pair diagrams so that we can use Fordham's method for computing word length. This paper generalizes results by Cleary and Taback, who proved the case N = 2.
Online readiness surveys are commonly administered to students who wish to enroll in online cours... more Online readiness surveys are commonly administered to students who wish to enroll in online courses in college. However, there have been no well-controlled studies to confirm whether these instruments predict online outcomes specifically (as opposed to predicting course outcomes more generally). This study used a sample of 24,006 students to test the validity and reliability of an online readiness survey similar to those used in practice at a majority of U.S. colleges. Multilevel models were used to determine if it was a valid predictor of differential online versus face-toface course outcomes while controlling for unobserved heterogeneity among courses taken by the same student. Student self-selection into online courses was also controlled using studentlevel covariates. The study also tested the extent to which survey score correlated with subsequent decisions to enroll in an online course. No aspect of the survey was a significant predictor of differential online versus face-to-face performance. In fact, student characteristics commonly collected by institutional research departments were better predictors of differential online versus face-to-face course outcomes than the survey. Furthermore, survey score was inversely related to subsequent online enrollment rates, suggesting that the use of online readiness surveys may discourage some students from enrolling in online courses even when they are not at elevated risk online. This suggests that institutions should be extremely cautious about implementing online readiness surveys before they have been rigorously tested for validity in predicting differential online versus face-to-face outcomes.
We prove that Thompson's group F (n) is not minimally almost convex with respect to the standard ... more We prove that Thompson's group F (n) is not minimally almost convex with respect to the standard finite generating set. A group G with Cayley graph Γ is not minimally almost convex if for arbitrarily large values of m there exist elements g, h ∈ Bm such that d Γ (g, h) = 2 and d Bm (g, h) = 2m. (Here Bm is the ball of radius m centered at the identity.) We use tree-pair diagrams to represent elements of F (n) and then use Fordham's metric to calculate geodesic length of elements of F (n). Cleary and Taback have shown that F (2) is not almost convex and Belk and Bux have shown that F (2) is not minimally almost convex; we generalize these results to show that F (n) is not minimally almost convex for all n ∈ {2, 3, 4,. .. }.
We show that the inclusion map of the generalized Thompson groups F (n i) is exponentially distor... more We show that the inclusion map of the generalized Thompson groups F (n i) is exponentially distorted in the Thompson-Stein groups F (n 1 , ..., n k) whenever k > 1. One consequence of this is that F is exponentially distorted in F (n 1 , ..., n k) for k > 1 whenever n i = 2 m for some m (whenever no i, m exist such that n i = 2 m , there is no obviously "natural" inclusion map of F into F (n 1 , ..., n k)). This is the first known example in which the natural embedding of one of the Thompson-type groups into another is not quasi-isometric. 1991 Mathematics Subject Classification. 20F65.
Here we explore how college students across different courses may conceptualize symbolic algebrai... more Here we explore how college students across different courses may conceptualize symbolic algebraic properties. This work draws on the theory of Grundvorstellungen (GVs) as a tool to analyse how learner conceptions do or do not align with some desired goals of instruction. In analysing interviews, several categories of conceptions, or descriptive GVs, emerged, which may be a helpful first step in understanding learners' thinking and improving instruction on algebraic properties.
We prove that seesaw words exist in Thompson's group F (N) for N = 2, 3, 4,. .. with respect to t... more We prove that seesaw words exist in Thompson's group F (N) for N = 2, 3, 4,. .. with respect to the standard finite generating set X. A seesaw word w with swing k has only geodesic representatives ending in g k or g −k (for given g ∈ X) and at least one geodesic representative of each type. The existence of seesaw words with arbitrarily large swing guarantees that F (N) is neither synchronously combable nor has a regular language of geodesics. Additionally, we prove that dead ends (or k-pockets) exist in F (N) with respect to X and all have depth 2. A dead end w is a word for which no geodesic path in the Cayley graph Γ which passes through w can continue past w, and the depth of w is the minimal m ∈ N such that a path of length m + 1 exists beginning at w and leaving B |w|. We represent elements of F (N) by tree-pair diagrams so that we can use Fordham's method for computing word length. This paper generalizes results by Cleary and Taback, who proved the case N = 2.
This session focused on research being conducted at non-university tertiary institutions. Instruc... more This session focused on research being conducted at non-university tertiary institutions. Instructors in adult education as well as other non-university postsecondary institutions participated. The participants of the session shared concerns about wanting to know more about the non-university mathematics classroom through classroom-based research by practitioners. Examples of research presently being conducted were shared (Mesa, Wladis, & Watkins, 2014; Sitomer et al., 2012; Wladis, Conway, & Hachey, 2016) and are cited below. The goal for developing collaborative research and/or grant proposals among researchers of similar interest continues. A presentation of the National Science Foundation grant Algebra instruction at community colleges: An exploration of its relationship with student success (Watkins, Strom, Mesa, Kohli, & Duranczyk, 2015) design was shared. The researchers are exploring the impact of students' and instructors' pre-existing and moderating variable on the relationship between student-instructor interaction with mathematics in the classroom and students' performance outcomes. Research on how individual and institutional characteristics factor into failure rates and performance measures exists, but there is little information about the fundamental work of teachers in the classroom, and the interaction that occur between instructors, students, and the mathematical content. A qualitative study (Smith, 2016), provided the student point of view. It explored students' reflections on their experiences (unsuccessful and successful) in mathematics at the community college. Student voices provided essential insights as to how postsecondary educators might foster positive learning transformations, and avoid being the source of needless obstacles to degree attainment. Then the
International Journal of Algebra and Computation, Feb 1, 2011
We give criteria for determining the approximate length of elements in any given cyclic subgroup ... more We give criteria for determining the approximate length of elements in any given cyclic subgroup of the Thompson-Stein groups F (n 1 , ..., n k) such that n 1 − 1|n i − 1 ∀i ∈ {1, ..., k} in terms of the number of leaves in the minimal tree-pair diagram representative. This leads directly to the result that cyclic subgroups are quasi-isometrically embedded in the Thompson-Stein groups. This result also leads to the corollaries that Z n is also quasiisometrically embedded in the Thompson-Stein groups for all n ∈ N and that the Thompson-Stein groups have infinite dimensional asymptotic cone. 1991 Mathematics Subject Classification. 20F65. Key words and phrases. Thompson's group,piecewise-linear homeomorphisms,Stein groups, Higman groups, quasi-isometrically embedded subgroups, cyclic subgroups, asymptotic cone. The author would like to thank Martin Kassabov for interesting and relevant discussion during the preparation of this article. The author acknowledges support from the CUNY Scholar Incentive Award, and would like to thank the Technische Universität Berlin for its hospitality during the writing of this paper. She would also like to thank the anonymous referee for their helpful comments during the revision of this paper.
Claire Wladis is a professor at the Borough of Manhattan Community College and the Graduate Cente... more Claire Wladis is a professor at the Borough of Manhattan Community College and the Graduate Center at the City University of New York. She is particularly interested in research questions that address issues of equity and access to higher education and STEM fields. Currently she is investigating how access to and enrollment in online courses may relate to college access and persistence, particularly for students who have be historically underrepresented in college and STEM fields. She is also developing and validating a series of learning progressions and a concept inventory to measure the algebraic conceptual understanding of college students.
Journal of the London Mathematical Society, Nov 16, 2011
We consider the Thompson-Stein group F (n 1 , ..., n k) where n 1 , ..., n k ∈ {2, 3, 4, ...}, k ... more We consider the Thompson-Stein group F (n 1 , ..., n k) where n 1 , ..., n k ∈ {2, 3, 4, ...}, k ∈ N. We highlight several differences between the cases k = 1 and k > 1, including the fact that minimal tree-pair diagram representatives of elements may not be unique when k > 1. We establish how to find minimal tree-pair diagram representatives of elements of F (n 1 , ..., n k), and we prove several theorems describing the equivalence of trees and tree-pair diagrams. We introduce a unique normal form for elements of F (n 1 , ..., n k) (with respect to the standard infinite generating set developed by Melanie Stein) which provides a solution to the word problem, and we give sharp upper and lower bounds on the metric with respect to the standard finite generating set, showing that in the case k > 1, the metric is not quasi-isometric to the number of leaves or caret in the minimal tree-pair diagram, as is the case when k = 1. Contents 1. Introduction 2. Representing elements using tree-pair diagrams 2.1. Tree-pair diagrams as representatives of elements of F (n 1 , ..., n k) 2.2. Equivalence of trees: Basic results 2.3. Minimal tree-pair diagrams 2.4. Tree-pair diagram composition 2.5. Equivalence of trees: Further results 3. Equivalence of tree-pair diagrams 4. Normal Form and Solution to the Word Problem 4.1. Infinite Group Presentation 4.2. Unique Minimal Tree-pair Diagram Representatives 4.3. The Normal Form 4.4. The Normal Form of Positive Words 5. The Metric on F (n 1 , ..., n k) 5.1. Standard Finite Presentation 5.2. The Metric References 1991 Mathematics Subject Classification. 20F65. The author would like to thank Sean Cleary for his support and advice during the preparation of this article. The author acknowledges support from the CUNY Scholar Incentive Award and would like to thank the Techniche Universität Berlin for its hospitality during the writing of this paper.
We show that the inclusion map of the generalized Thompson groups F (n i) is exponentially distor... more We show that the inclusion map of the generalized Thompson groups F (n i) is exponentially distorted in the Thompson-Stein groups F (n 1 , ..., n k) whenever k > 1. One consequence of this is that F is exponentially distorted in F (n 1 , ..., n k) for k > 1 whenever n i = 2 m for some m (whenever no i, m exist such that n i = 2 m , there is no obviously "natural" inclusion map of F into F (n 1 , ..., n k)). This is the first known example in which the natural embedding of one of the Thompson-type groups into another is not quasi-isometric. 1991 Mathematics Subject Classification. 20F65.
We prove that Thompson's group F (n) is not minimally almost convex with respect to the standard ... more We prove that Thompson's group F (n) is not minimally almost convex with respect to the standard finite generating set. A group G with Cayley graph Γ is not minimally almost convex if for arbitrarily large values of m there exist elements g, h ∈ Bm such that d Γ (g, h) = 2 and d Bm (g, h) = 2m. (Here Bm is the ball of radius m centered at the identity.) We use tree-pair diagrams to represent elements of F (n) and then use Fordham's metric to calculate geodesic length of elements of F (n). Cleary and Taback have shown that F (2) is not almost convex and Belk and Bux have shown that F (2) is not minimally almost convex; we generalize these results to show that F (n) is not minimally almost convex for all n ∈ {2, 3, 4,. .. }.
HAL (Le Centre pour la Communication Scientifique Directe), Feb 2, 2022
This empirical paper explores students' conceptions of transformation as substitution equivalence... more This empirical paper explores students' conceptions of transformation as substitution equivalence by linking it to their definitions of substitution and equivalence. This work draws on Sfard's (1995) framework to conceptualize conceptions of substitution equivalence and its components, equivalence and substitution, each on a spectrum from computational to structural. We provide examples of student work to illustrate how students' understandings of substitution, equivalence, and substitution equivalence as an approach to justifying transformation may relate to one another.
Springer Proceedings in Mathematics & Statistics, 2021
Multiple Answer Multiple Choice items (MAMC) have been used primarily for high-stakes assessments... more Multiple Answer Multiple Choice items (MAMC) have been used primarily for high-stakes assessments but are being used in broader circumstances. Unfortunately, they are a complex format for examinees, item writers, and scorers. This article features two novel approaches to scoring these items while comparing a low ability and a high ability examinee group. One of these approaches is the use of Latent Class Analysis (LCA) to better understand how examinees respond to these item formats related to a “credit earned approach” to scoring MAMC items. In addition, Jaccard’s distance is proposed as an alternative partial credit scoring method for assigning credit to students on these items. Both have their benefits and drawbacks, which are discussed in the article; however, both exploratory approaches reveal more about examinee knowledge than simple dichotomous scoring.
This session focused on research being conducted at non-university tertiary institutions. Instruc... more This session focused on research being conducted at non-university tertiary institutions. Instructors in adult education as well as other non-university postsecondary institutions participated. The participants of the session shared concerns about wanting to know more about the non-university mathematics classroom through classroom-based research by practitioners. Examples of research presently being conducted were shared (Mesa, Wladis, & Watkins, 2014; Sitomer et al., 2012; Wladis, Conway, & Hachey, 2016) and are cited below. The goal for developing collaborative research and/or grant proposals among researchers of similar interest continues. A presentation of the National Science Foundation grant Algebra instruction at community colleges: An exploration of its relationship with student success (Watkins, Strom, Mesa, Kohli, & Duranczyk, 2015) design was shared. The researchers are exploring the impact of students' and instructors' pre-existing and moderating variable on the relationship between student-instructor interaction with mathematics in the classroom and students' performance outcomes. Research on how individual and institutional characteristics factor into failure rates and performance measures exists, but there is little information about the fundamental work of teachers in the classroom, and the interaction that occur between instructors, students, and the mathematical content. A qualitative study (Smith, 2016), provided the student point of view. It explored students' reflections on their experiences (unsuccessful and successful) in mathematics at the community college. Student voices provided essential insights as to how postsecondary educators might foster positive learning transformations, and avoid being the source of needless obstacles to degree attainment. Then the
Claire Wladis is a professor at the Borough of Manhattan Community College and the Graduate Cente... more Claire Wladis is a professor at the Borough of Manhattan Community College and the Graduate Center at the City University of New York. She is particularly interested in research questions that address issues of equity and access to higher education and STEM fields. Currently she is investigating how access to and enrollment in online courses may relate to college access and persistence, particularly for students who have be historically underrepresented in college and STEM fields. She is also developing and validating a series of learning progressions and a concept inventory to measure the algebraic conceptual understanding of college students.
We prove that seesaw words exist in Thompson's group F (N) for N = 2, 3, 4,. .. with respect to t... more We prove that seesaw words exist in Thompson's group F (N) for N = 2, 3, 4,. .. with respect to the standard finite generating set X. A seesaw word w with swing k has only geodesic representatives ending in g k or g −k (for given g ∈ X) and at least one geodesic representative of each type. The existence of seesaw words with arbitrarily large swing guarantees that F (N) is neither synchronously combable nor has a regular language of geodesics. Additionally, we prove that dead ends (or k-pockets) exist in F (N) with respect to X and all have depth 2. A dead end w is a word for which no geodesic path in the Cayley graph Γ which passes through w can continue past w, and the depth of w is the minimal m ∈ N such that a path of length m + 1 exists beginning at w and leaving B |w|. We represent elements of F (N) by tree-pair diagrams so that we can use Fordham's method for computing word length. This paper generalizes results by Cleary and Taback, who proved the case N = 2.
Online readiness surveys are commonly administered to students who wish to enroll in online cours... more Online readiness surveys are commonly administered to students who wish to enroll in online courses in college. However, there have been no well-controlled studies to confirm whether these instruments predict online outcomes specifically (as opposed to predicting course outcomes more generally). This study used a sample of 24,006 students to test the validity and reliability of an online readiness survey similar to those used in practice at a majority of U.S. colleges. Multilevel models were used to determine if it was a valid predictor of differential online versus face-toface course outcomes while controlling for unobserved heterogeneity among courses taken by the same student. Student self-selection into online courses was also controlled using studentlevel covariates. The study also tested the extent to which survey score correlated with subsequent decisions to enroll in an online course. No aspect of the survey was a significant predictor of differential online versus face-to-face performance. In fact, student characteristics commonly collected by institutional research departments were better predictors of differential online versus face-to-face course outcomes than the survey. Furthermore, survey score was inversely related to subsequent online enrollment rates, suggesting that the use of online readiness surveys may discourage some students from enrolling in online courses even when they are not at elevated risk online. This suggests that institutions should be extremely cautious about implementing online readiness surveys before they have been rigorously tested for validity in predicting differential online versus face-to-face outcomes.
We prove that Thompson's group F (n) is not minimally almost convex with respect to the standard ... more We prove that Thompson's group F (n) is not minimally almost convex with respect to the standard finite generating set. A group G with Cayley graph Γ is not minimally almost convex if for arbitrarily large values of m there exist elements g, h ∈ Bm such that d Γ (g, h) = 2 and d Bm (g, h) = 2m. (Here Bm is the ball of radius m centered at the identity.) We use tree-pair diagrams to represent elements of F (n) and then use Fordham's metric to calculate geodesic length of elements of F (n). Cleary and Taback have shown that F (2) is not almost convex and Belk and Bux have shown that F (2) is not minimally almost convex; we generalize these results to show that F (n) is not minimally almost convex for all n ∈ {2, 3, 4,. .. }.
We show that the inclusion map of the generalized Thompson groups F (n i) is exponentially distor... more We show that the inclusion map of the generalized Thompson groups F (n i) is exponentially distorted in the Thompson-Stein groups F (n 1 , ..., n k) whenever k > 1. One consequence of this is that F is exponentially distorted in F (n 1 , ..., n k) for k > 1 whenever n i = 2 m for some m (whenever no i, m exist such that n i = 2 m , there is no obviously "natural" inclusion map of F into F (n 1 , ..., n k)). This is the first known example in which the natural embedding of one of the Thompson-type groups into another is not quasi-isometric. 1991 Mathematics Subject Classification. 20F65.
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