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## Technology and the Yin & Yang of Teaching and Learning Mathematics

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**Technology and the Yin & Yang ofTeaching and Learning**Mathematics B Kutzler, Austria**Humans are ruled by two opposite forces:**hold on let go**Humans are ruled by two opposite forces:**hold on (yin) let go (yang)**In our context these show as**hold on = connect let go = automate**Automate**Automate = compensate + amplify = compensate + (solve + explore)**automate**compensate explore solve**Connect**Connect= = represent + document + communicate**Connect**Connect = model = represent + document + communicate model**Connect**Connect = model model = represent + document + communicate model human**Connect**Connect = model model human = represent + document + communicate model human human**connect**document communicate represent**connect**automate document communicate compensate represent explore solve**connect**automate document communicate compensate represent explore solve**Math-Pedagogy-Technology Space**technology mathematics pedagogy**Math-Pedagogy-Technology Space**technology mathematics**With a helicopter one can reach places that one cannot**reach otherwise.**With a helicopter one can reach places that one cannot**reach otherwise.B Waits: Tech makes some math possible**With a helicopter one can reach places that one cannot**reach otherwise.B Waits: Tech makes some math possible Tech makes some math pedagogy possible**Not the tool, but the use of the tool is or is not**pedagogical. (Vlasta Kokol-Voljc)**Teacher training:**Step 1: technology axis Step 2: math/technology plane Step 3: math/pedagogy/technology space Introducing teachers to technology needs as much (pedagogical) care as any other teaching**(1) Represent**Mathematical problem solving = the art of transforming representations (until the solution is visible)**(1) Represent**Mathematical problem solving = the art of transforming representations (until the solution is visible)such as transforming 5x-6=2x+15into x=7**(1) Represent**Algebraic/graphic/numeric 1+2 3 3a+4a 7a 24 2 6 24 4.89898**(1) Represent**Use technology to simulate a mathematician‘s mind New pedagogy (helps more students to understand)**(2) Document**= share your work with an anonymous person argue (= convince sb with certain level of knowledge) designdescribe**(2) Document**Describe/understand Newspaper articles Mathematical logic27 is divisible by 3 x(3x=27) every number is greater than its predecessor x(x>x-1) Use wordproblems to practice understanding**(2) Document**Argue - Examples such asExplain why one uses the first derivative to find the extreme values of a function If a curve is symmetric in both the x- and y-axes, is it then symmetric through the origin? Give a reason or find a counter-example.**(2) Document**With technology: Result loses importance Choosing keys/commands is mathematical work New pedagogy (shift focus fromexecuting algorithms to documenting mathematical work)**(3) Communicate**= impart or transmit idea, knowledge, information Documentation = „offline“ communication „Online“ communication**(3) Communicate**Student teacher: - front teaching - teacher as advisor and assistant Student student: - alone (cf traditional exam) - teamwork New pedagogy (enables new teaching/learning styles)