"Italian psychologist Geatano Kanizsa first described this optical illusion in 1955 as a subjective or illusory contour illusion. The study of such optical illusions has led to an understanding of how the brain and eyes perceive optical information and has been used considerably by artists and designers alike. They show the power of human imagination in filling in the gaps to make implied constructions in our own minds.
Kanizsa figures and similar illusions are a really useful way to encourage learners to 'say what they see' and to explain how they see it. It offers a chance for others to become aware of the different views available in a diagram and share their own thoughts without the 'danger' of being wrong; many people see different things."
A large maths worksheet site where you can create sheets for a range of primary and secondary topics. [Be aware - site contains advertising and pop-ups]
http://ictmagic.wikispaces.com/Maths
LOG IN February 22, 2012 at 2pm Eastern US time: http://tinyurl.com/math20event
During the event, John Mason will lead a conversation about multiplication as scaling, and answer questions about his books, projects and communities.
All events in the Math Future weekly series: http://mathfuture.wikispaces.com/events
The recording will be at: http://mathfuture.wikispaces.com/JohnMason
Your time zone: http://bit.ly/wQYN1Y
Event challenge!
What good multiplication tasks about scaling do you know?
Share links and thoughts!
John writes about elastic multiplication: "It is often said that 'multiplication is repeated addition' when what is meant is that 'repeated addition is an instance of multiplication'. I have been developing some tasks which present 'scaling as multiplication' based around familiarity with elastic bands. Participants would benefit from having an elastic (rubber) band to hand which they have cut so as to make a strip; wider is better than thinner if you have a choice."
About John Mason
John Mason has been teaching mathematics ever since he was asked to tutor a fellow student when he was fifteen. In college he was at first unofficial tutor, then later an official tutor for mathematics students in the years behind him, while tutoring school students as well. After a BSc at Trinity College, Toronto in Mathematics, and an MSc at Massey College, Toronto, he went to Madison Wisconsin where he encountered Polya's film 'Let Us Teach Guessing', and completed a PhD in Combinatorial Geometry. The film released a style of teaching he had experienced at high school from his mathematics teacher Geoff Steel, and his teaching changed overnight.
His first appointment was at the Open University, which involved among other things the design and implementation of the first mathematics summer school (5000 students over 11 weeks on three sites in parallel). He called upon his experience of being taught, to institute active-problem-solving sessions, w
Our activities during Mathematics Week focus attention on Mathematics and Mathematics Education with the aim of making Mathematics more interesting, attractive, relevant, challenging, rewarding and engaging to learners and the community at large, e.g. by highlighting the impact of Mathematics on our daily lives and stressing the importance of Mathematics as a foundation for careers in science, technology and managerial jobs. Mathematics Week is a vehicle to popularise Mathematics and to increase public awareness, understanding and appreciation of Mathematics.
" The use of logarithms, an important tool for calculus and beyond, has been reduced to symbol manipulation without understanding in most entry-level college algebra courses. The primary aim of this research, therefore, was to investigate college students' understanding of logarithmic concepts through the use of a series of instructional tasks designed to observe what students do as they construct meaning. APOS Theory was used as a framework for analysis of growth.
APOS Theory is a useful theoretical framework for studying and explaining conceptual development. Closely linked to Piaget's notions of reflective abstraction, it begins with the hypothesis that mathematical activity develops as students perform actions that become interiorized to form a process understanding of the concept, which eventually leads students to a heightened awareness or object understanding of the concept. Prior to any investigation, the researcher must provide an analysis of the concept development in terms of the essential components of this theory: actions, process, objects, and schemas. This is referred to as the genetic decomposition. The results of this study suggest a framework that a learner may use to construct meaning for logarithmic concepts. Using tasks aligned with the initial genetic decomposition, the researcher made revisions to the proposed genetic decomposition in the process of analyzing the data. The results indicated that historical accounts of the development of this concept might be useful to promote insightful learning. Based on this new set of data, iterations should continue to produce a better understanding of the student's constructions. " (from the abstract)
The survey involved more than 600 "randomly selected mathematicians worldwide" and revealed that many mathematicians publish on arxiv.org, some on their personal websites, some publish in open access journals but disapprove of publishing fees, tenure and promotion influence publication, high awareness of publishing rights, little use of online collaboration tools
Kristine Fowler, mathematics librarian at the University of Minnesota, shares results from a survey of mathematicians concerning how they want to be able to reuse their work, discusses alternatives to standard copyright contracts and urges scholars and institutions to be aware of not only their options but the rights they may cede in entering into publishing agreements.
Six math projects that integrate real-world math problems are presented as a teaching strategy for helping students develop a greater understanding of math.
Is this the most important question to ask when solving the problem?
What questions need to be answered before answering this question?
What does this presume?
When you ask these and similar
questions, you are encouraging your students to move from passive to active
learning.
Avoiding Questions Easily Answered on the
Internet
The following examples are referred to “Google-Proofing” in some circles.
the frequency of questions is not as important as the quality of questions.
the following are factors to consider when asking students questions.
The average level of questions asked by teachers are 60 percent lower
cognitive, 20 percent procedural, and 20 percent higher cognitive.
Increasing the frequency of higher cognitive questions to the 50
With predominate use of lower cognitive questions; students tend toward lower
achievement
The use of higher cognitive questions tends to elicit longer student answers in
complete sentences, quality inference and conjecture by students, and the
forming of higher level questions.
Encouraging students to use critical thinking is more than an extension activity in science and math lessons, it is the basis of true learning.
Teaching students how to think critically helps them move beyond basic comprehension and rote memorization. They shift to a new level of increased awareness when calculating, analyzing, problem solving, and evaluating.
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