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1. Potential conditional mutual information: Estimators and properties

   https://doi.org/10.1109/ALLERTON.2017.8262877  

   Rahimzamani, Arman; Kannan, Sreeram (October 2017, Communication, Control, and Computing (Allerton), 2017 55th Annual Allerton Conference on)

   The conditional mutual information I(X; Y|Z) measures the average information that X and Y contain about each other given Z. This is an important primitive in many learning problems including conditional independence testing, graphical model inference, causal strength estimation and time-series problems. In several applications, it is desirable to have a functional purely of the conditional distribution py|x, z rather than of the joint distribution pX, Y, Z. We define the potential conditional mutual information as the conditional mutual information calculated with a modified joint distribution pY|X, ZqX, Z, where qX, Z is a potential distribution, fixed airport. We develop K nearest neighbor based estimators for this functional, employing importance sampling, and a coupling trick, and prove the finite k consistency of such an estimator. We demonstrate that the estimator has excellent practical performance and show an application in dynamical system inference. 

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2. Design Process Geometries: Shapes and Learning Trajectories of Engineering Students’ Design Process Concept Maps

   Lande, Micah; Liu, Yue (June 2019, ASEE Annual Conference proceedings)

   The research objective of this NSF-funded study is to explore and understand how open-ended, hands-on making work and activities can reflect student learning trajectories and learning gains in the product-based learning, undergraduate engineering classroom. The aim is to expand understanding of what making learning in the context of engineering design education might be and to illustrate educational pathways within the engineering education curriculum. Making is rooted in constructionism – learning by doing and constructing knowledge through that doing. Aspects of making work and activities that are unique to making that could appear in the engineering classroom or curriculum include: sharing, practical ingenuity, personal investment, playful invention, risk taking, community building and self-directed learning. The main research questions of this work is: How do engineering students learn and apply making? What are the attributes of making in the engineering classroom? Empirical evidence of what making in the engineering classroom looks like, and how it changes over time, and how students conceptualize making through making, designerly, and engineering ways of knowing-doing-acting will come from revisiting and additional qualitative analysis of student project data collected during a product-based learning course engineering design course. To best address the research question, this proposed study proposes multiple qualitative methods to collect and analyze data on engineering students learning making. We aim to triangulate what students think they are learning, what they are being taught, and what students are demonstrating. This work is exploratory in nature. In our approach to understanding making outside of formal engineering education, at events like Maker Faires in the Maker Community, it does seem evident that there is a lot of overlap between a making mindset and a designerly way of knowing or engineering way of knowing. In the sphere of formal engineering education however, making is regularly viewed as lesser than engineering, engineering design without the engineering science or analysis. Making is not yet valued as part of formal engineering education efforts. If making is something that can be connected to beneficial student learning and is additive to the required technical content and provides a means for students to figure out what area of problems they want to tackle in the studies and beyond, it would make for a student-centered making revolution. This study advances the knowledge of the learning pathways of making by capturing empirical evidence of such learning trajectories. This study will advance the currently limited knowledge of learning in the making community and making in the engineering classroom. Initial findings generated during this study describe the learning trajectories of engineers learning making. By examining the engineering student making learning experience through the lens of cognitive science and illustrating empirical making learning trajectories, this work may impact the quality of engineering design teaching. By sharing learning trajectories across multiple communities, we seek to change the conversation by illuminating pathways for a wider array of student makers to become the makers and engineers of the future. 

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3. Design Process Geometries: Shapes and Learning Trajectories of Engineering Students’ Design Process Concept Maps

   Lande, Micah; Liu, Yue (June 2019, ASEE Annual Conference proceedings)

   The research objective of this NSF-funded study is to explore and understand how open-ended, hands-on making work and activities can reflect student learning trajectories and learning gains in the product-based learning, undergraduate engineering classroom. The aim is to expand understanding of what making learning in the context of engineering design education might be and to illustrate educational pathways within the engineering education curriculum. Making is rooted in constructionism – learning by doing and constructing knowledge through that doing. Aspects of making work and activities that are unique to making that could appear in the engineering classroom or curriculum include: sharing, practical ingenuity, personal investment, playful invention, risk taking, community building and self-directed learning. The main research questions of this work is: How do engineering students learn and apply making? What are the attributes of making in the engineering classroom? Empirical evidence of what making in the engineering classroom looks like, and how it changes over time, and how students conceptualize making through making, designerly, and engineering ways of knowing-doing-acting will come from revisiting and additional qualitative analysis of student project data collected during a product-based learning course engineering design course. To best address the research question, this proposed study proposes multiple qualitative methods to collect and analyze data on engineering students learning making. We aim to triangulate what students think they are learning, what they are being taught, and what students are demonstrating. This work is exploratory in nature. In our approach to understanding making outside of formal engineering education, at events like Maker Faires in the Maker Community, it does seem evident that there is a lot of overlap between a making mindset and a designerly way of knowing or engineering way of knowing. In the sphere of formal engineering education however, making is regularly viewed as lesser than engineering, engineering design without the engineering science or analysis. Making is not yet valued as part of formal engineering education efforts. If making is something that can be connected to beneficial student learning and is additive to the required technical content and provides a means for students to figure out what area of problems they want to tackle in the studies and beyond, it would make for a student-centered making revolution. This study advances the knowledge of the learning pathways of making by capturing empirical evidence of such learning trajectories. This study will advance the currently limited knowledge of learning in the making community and making in the engineering classroom. Initial findings generated during this study describe the learning trajectories of engineers learning making. By examining the engineering student making learning experience through the lens of cognitive science and illustrating empirical making learning trajectories, this work may impact the quality of engineering design teaching. By sharing learning trajectories across multiple communities, we seek to change the conversation by illuminating pathways for a wider array of student makers to become the makers and engineers of the future. 

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4. Triforce and corners

   https://doi.org/10.1017/s0305004119000173  

   FOX, JACOB; SAH, ASHWIN; SAWHNEY, MEHTAAB; STONER, DAVID; ZHAO, YUFEI (July 2020, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract May the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ 4– o (1) but not O ( δ 4 ). Let M ( δ ) be the maximum number such that the following holds: for every ∊ > 0 and $$G = {\mathbb{F}}_2^n$$ with n sufficiently large, if A ⊆ G × G with A ≥ δ | G | 2 , then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” ( x , y ), ( x + d , y ), ( x , y + d ) ∈ A is at least ( M ( δ )–∊)| G | 2 . As a corollary via a recent result of Mandache, we conclude that M ( δ ) = δ 4– o (1) and M ( δ ) = ω ( δ 4 ). On the other hand, for 0 < δ < 1/2 and sufficiently large N , there exists A ⊆ [ N ] 3 with | A | ≥ δN 3 such that for every d ≠ 0, the number of corners ( x , y , z ), ( x + d , y , z ), ( x , y + d , z ), ( x , y , z + d ) ∈ A is at most δ c log(1/ δ ) N 3 . A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3. 

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5. Student Learning Trajectories from Making and Engineering Activities

   Lande, Micah (January 2018, Review & directory - American Society for Engineering Education)

   The research objective of this NSF-funded study is to explore and understand how open-ended, hands-on making work and activities can reflect student learning trajectories and learning gains in the product-based learning, undergraduate engineering classroom. The aim is to expand understanding of what making learning in the context of engineering design education might be and to illustrate educational pathways within the engineering education curriculum. Making is rooted in constructionism – learning by doing and constructing knowledge through that doing. Aspects of making work and activities that are unique to making that could appear in the engineering classroom or curriculum include: sharing, practical ingenuity, personal investment, playful invention, risk taking, community building and self-directed learning. The main research questions of this work is: How do engineering students learn and apply making? What are the attributes of making in the engineering classroom? Empirical evidence of what making in the engineering classroom looks like, and how it changes over time, and how students conceptualize making through making, designerly, and engineering ways of knowing-doing-acting will come from revisiting and additional qualitative analysis of student project data collected during a product-based learning course engineering design course. To best address the research question, this proposed study proposes multiple qualitative methods to collect and analyze data on engineering students learning making. We aim to triangulate what students think they are learning, what they are being taught, and what students are demonstrating. This work is exploratory in nature. In our approach to understanding making outside of formal engineering education, at events like Maker Faires in the Maker Community, it does seem evident that there is a lot of overlap between a making mindset and a designerly way of knowing or engineering way of knowing. In the sphere of formal engineering education however, making is regularly viewed as lesser than engineering, engineering design without the engineering science or analysis. Making is not yet valued as part of formal engineering education efforts. If making is something that can be connected to beneficial student learning and is additive to the required technical content and provides a means for students to figure out what area of problems they want to tackle in the studies and beyond, it would make for a student-centered making revolution. This study advances the knowledge of the learning pathways of making by capturing empirical evidence of such learning trajectories. This study will advance the currently limited knowledge of learning in the making community and making in the engineering classroom. Initial findings generated during this study describe the learning trajectories of engineers learning making. By examining the engineering student making learning experience through the lens of cognitive science and illustrating empirical making learning trajectories, this work may impact the quality of engineering design teaching. By sharing learning trajectories across multiple communities, we seek to change the conversation by illuminating pathways for a wider array of student makers to become the makers and engineers of the future. 

   more » « less