Summary: Quantitative reasoning skills are a fundamental tool in many part of Society, ranging from Mathematics and Science to Engineering and from Law and Business to Rhetoric. In Middle School and High School quantitative reasoning is almost never taught, except in the context of other disciplines, such as Mathematics or Physics. Recently a Middle School course in reasoning was developed and field tested . The course introduces basic elements of reasoning, such as the definition of a concept and the definition of a strategy. The course concepts are applied to algebraic proofing. We find that a diverse population of female middle school students readily accepts this approach and achieves proofing skills on a level which is comparable to university freshmen. We have established collaborations with educators in Jordan, Saudi Arabia, Egypt, and Palestine and find that approach is readily accepted there.  A. Hübler, A. Vlasic, E. Stiegler, L. Bievenue, D. Raineri, Interactive Middle School Courseware on Abstract Reasoning Skills, in C. Crawford et al. (Eds.), Proceedings of Society for Information Technology and Teacher Education International Conference 2006 (AACE: Chesapeake, VA, 2006) pp 389-394: http://server10.how-why.com/publications/2006/Site.pdf This work was funded in part by National Science Foundation Grant No. NSF DMS 03-25939 ITR, and NSF DGE 03-38215.
High quality course ware Why is Wikipedia so popular among researchers and students? - Its free. -Deeper reason: Many concepts are explained well, in contrast to standard textbooks. Standard textbooks give little intuition: They do not explain the range of applicability of Newton’s law, E&M, band structure, …. Example: Light bulb
Context of the concept High quality course ware
Definition of the concept High quality course ware
Discussion of the concept More abstract versions of the concept
Range of applicability, and examples High quality course ware: Discussion of the joke
Relations to other concepts High quality course ware: Discussion of the joke.
Definition of a Concept 1. A descriptive name, such as “Light bulb joke”; 2. An identification number, such as “Law 7” and a logo; 3. A description of the context in which the concept is typically used, such “The punch line of the joke is generally derogatory to a group.”; 4. A definition of the concept in terms of sub-concepts, prerequisite concepts, and common knowledge. 5. Discussion: -A range of applicability identifies when the concept is true and useful. Further synonyms and opposites are introduced. Examples: “The weight force is W=mg for an object on the surface of the earth.”, or “For any real number x …”; - Several single concept examples use the concept all by itself, such as “What is the weight force on a 2kg object?” - Several multiple concept examples and exercises use the concept plus one or several substitutions. They can also include translations to other fields and common knowledge examples (50% of content). High quality course ware
The Definition of the concept “Prime Factoring” in the Mathematics handbook “Math on Call” (Kaplan 1998). “Prime Factoring” is the name. “061” is the identification number and the graph of the “factor tree” serves as a logo. The first sentence describes the use context. The “One Way” symbol indicates the definition of the concept in terms of sub-concepts. Further a single concept example is given. There are no multiple concept examples. The authors omitted multiple concept examples since this book is considered a companion to a regular text book. Still, multiple concept examples would improve the definition of the concept “Prime Factoring”. High quality course ware
Conceptualization of Strategies: Reasoning Some facts about reasoning: -Fact 1: Most reasoning is chaotic, -i.e. small uncertainties in the assumptions lead to larger and larger differences in the conclusions, in a long chains of arguments. There are almost always small uncertainties in the assumptions. => inverse problem: only short chains of arguments can be inferred from knowing the assumptions and conclusions. - Fact 2: A large fraction of reasoning is in English, and not in terms of symbols. Only basic concepts are in symbols: +, -, *, …, in contrast to sin(), “is smooth”, “Solve for the unknown”,… High quality course ware: facts about reasoning
Edwards and Penney, Calculus with Analytical Geometry, Fourth Edition (Prentice Hall, Englewood Cliffs, New Jersey, 1994), p399 High quality course ware: facts about reasoning: abstract concepts are worded in English Most of the reasoning is in English, only basic concepts are phrased in symbols. Example :
Edwards and Penney, Calculus with Analytical Geometry, Fourth Edition (Prentice Hall, Englewood Cliffs, New Jersey, 1994), p399 High quality course ware: facts about reasoning Fact 3: We have to learn to reason and to communicate reasoning…it’s a language… … and the main reason why we learn it is: We do not want to disappoint our teacher, -i.e. a one-on-one relation with the teacher Fact 4: Teachers in public schools are often overworked (150students/day) and underpaid, -i.e. are not the primary bread winner of a family => teachers are seen as distant and cold Conclusion: Quality education for a large fraction of the population is only achievable with virtual tutors, that can reason and show emotions, humor, and have high expectations.
High quality course ware: virtual tutor: evaluating expressions The language of reasoning
High quality course ware: virtual tutor: evaluating expressions
Laws: Law 1 is "a+0=a". Law 2 is "a*0=0". Law 3 is "a*b=b*a". Law 4 is "a+b=b+a". Law 5 is "(a+b)*c=a*c+b*c". Law 6 is "(a+b)+c=a+(b+c)". Law 7 is "(a*b)*c=a*(b*c)". Law 8 is "a*1=a". More details More details Law 6 is the associative axiom for addition. It means "(a+b)+c=a+(b+c)". This law applies for any real number a, any real number b, and any real number c. Example: Write the given equation: x=(a+2)+3 Use law 10 to replace (a+2)+3: x=a+(2+3) High quality course ware: virtual tutor: proofing Computer assisted proofing: A typical set of given concepts for a Middle School level proof:
Commands: 1: Write the given equation. 2: Memorize this law. 3: Remove parenthesis. 4: Set parenthesis. 5: Do the arithmetic. 6: Add number. 7: Multiply by number. 8: Use the symmetric axiom of equality. 9: Use law index. 10: Replace name. 11: Replace name by other-name. 12: Replace name by other-name. 13: Replace name with law law. 14: Replace expression with law name. More details More details 8: Use the symmetric axiom of equality. Exchange the two members of an equation. Example: Use law 3. a*(b+c)=a*b+a*c Use the symmetric axiom of equality. a*b+a*c=a*(b+c) High quality course ware: virtual tutor: proofing
High quality course ware: virtual tutor: proofing Before e-learning: After e-learning: Students use reasononing skills even in hand written homework
High quality course ware: virtual tutor: results Computer assisted reasoning, a 3-year test at Campus Middle School: Results Students complete 15-20 homework problems/day
High quality course ware: virtual tutor: proofing A cumbersome evaluation ….
High quality course ware: virtual tutor: proofing … takes less than one minute
Summary: Why is Wikipedia so popular among students and scientist? - Its well conceptualized. A concept contains a name, icon, definition & discussion. Strategies are recipies, algorithms. Methods are short strategies. Why do students do their homework? - Because they do not want to disappoint the teacher. Why do students have to show their work? - Most reasoning is chaotic, -i.e. small uncertainties in the assumptions lead to larger and larger differences in the conclusions, in a long chains of arguments => inverse problem: only short chains of arguments can be inferred from knowing the assumptions and conclusions. Most multiple choice homework problems require very little reasoning, test methods, but don’t test strategies. -The students learn to communicate their reasoning. -The virtual teacher can learn new strategies from the students. Why does the virtual teacher have to speak and understand English (Arabic)? -Abstract concepts are phrased in English (Arabic). -Everybody who knows something about the subject area, can teach the virtual teacher, no computer programming knowledge is required. Students, teachers, researchers, everybody who speaks English (Arabic) can teach the virtual teacher. - The virtual teacher can rephrase an abstract concept in terms of everyday concepts, and group a set of everyday concepts into an abstract concepts and name it.
From: Dr. Zain Yamani Date: Fri, Nov 28, 2008 at 10:41 PM Subject: RE: draft report of the nano energy and electronic group To: Alfred Hubler Cc: Zain Yamani Dear Professor Hubler, This issue of teaching high school teachers is also one of my favorites. Since we do not expect many of them to be used to mathematical rigor, and even if they did the actual high schools students are not interested..hence.. it is a challenge for us (me and you..) to get the gut-feelings of physics into their souls. Regarding your kind offer that we might develop something together re- HTW.. I'd LOVE that!! However, I face "one" problem, my current Administrative responsibility as Center Director which has kept me away from my lab AND undergraduate students. We have holidays coming up next week. May it be a time for me to have some peace of mind, and to reflect on how can (better) balance my different wishes and responsibilities. So, I promise to get back to you after the holidays. Thank you for taking initiative. I really appreciate that. Take care, Zain
On Fri, Jan 16, 2009 at 12:48 AM, wrote: Dear Professor Hubler, Thank you very much for your email. I am willing to collaborate on creating an online interactive course on nanotechnology. In fact just last week we suggested to the vice president for academic affairs to create such a modular course for senior students of science, engineering, and IT students. Your collaboration in such an effort will be highly appreciated. I am currently working on an online course in introductory mechanics physics course for science and engineering students. This course will be offered and evaluated next semester. For the high and middle school material in science and math, I have been involved in such activity for a long time, and will be willing to collaborate in writing a proposal to enhance science and math education in Palestine. Thanks again, and looking forward to hearing from you. Aziz
Conclusion Alfred Hubler, email@example.com Center of Complex Systems Research, Physics, UIUC http://server10.how-why.com/blog http://server10.how-why.com/blog The greatest natural resource in the Middle East is the thoughtfulness and ingenuity of the students. Quantitative reasoning skills are a fundamental tool in many fields, ranging from Mathematics to Engineering and from Business to Rhetoric. Quantitative reasoning is almost never taught as a course, but in the context of other disciplines, such as Mathematics or Physics. We introduce basic elements of reasoning, such as the definition of a concept, a method, and a strategy and study the impact on the students.
We also conceptualize strategies. We teach understanding – teach translating real world problems to Algebra, Physics,… and back, - abstract concepts to practical concepts und vice versa – Biology problems to Algebra and back. We find that a diverse population of female middle school students readily accepts this approach and outperforms their peers, and we noticed a low drop out rate in Physics & Chemistry UG introductory courses. We are open for collaborations. Thank you. - D. Raineri, A. Hübler, B. Mertens, Cyberprof TM : An Intelligent Human-Computer Interface for Interactive Instruction on the World Wide Web, JALN 1, 20 (1997) - A. Hübler, A. Vlasic, E. Stiegler, L. Bievenue, D. Raineri, Interactive Middle School Courseware on Abstract Reasoning Skills, in C. Crawford et al. (Eds.), (AACE: Chesapeake, VA, 2006) pp 389-394.