## Check out Wildfogel’s fascinating talk on different sizes of infinity.

## Click on __Off-topic__ *All things off-topic *to read four posts about recent in-class explorations of complex and imaginary numbers.

## One Page Update on ImagineIT

We are 11 weeks into the 2015-16 school year, and my ImagineIT project on the Real Number Line (RNL) has led me in excitingly new directions. Class discussions in Algebra I about the infinite number of numbers between 0 and 1 has piqued students’ curiosity. When graphing inequalities, I, for the first time, required students to list at least one fraction in their solution set. They rose to the occasion, offering correct solutions at over an 80% success rate. In Honors Advanced Algebra, we’ve dug deeply into the world of imaginary numbers. Today, I blogged about four lessons over the past four weeks that have been inspired by this year’s ImagineIT project. While the project is on the RNL, my Advanced Algebra class has spent weeks exploring imaginary numbers and their relationship to real and complex numbers. You could argue that I haven’t focused enough on the RNL – complex numbers have been the focus over the past month – but I feel good about our exploration of this ‘new’ type of number (new to my students, and relatively new to mankind). Discussing how these numbers came to be, their applications and why they’re important have led to illuminating talks I think students enjoy, even if they’re bewildered at first.

I would not say that my ImagineIT project, at least to date, has been a complete success. While students seem to taken to imaginary and complex numbers, we haven’t focused enough on the topic of my ImagineIT – the RNL. Moreover, and more importantly, I haven’t had students

I would not say that my ImagineIT project, at least to date, has been a complete success. While students seem to taken to imaginary and complex numbers, we haven’t focused enough on the topic of my ImagineIT – the RNL. Moreover, and more importantly, I haven’t had students

*do*anything, outside of traditional pencil-and-paper exercises, with what we’ve learned. Presenting their findings, or representing their learning creatively, has been lacking. I need to find or create a project that affords them the opportunity for independent learning, and then have them present what they’ve learned in small groups or to the entire class. In other words, class discussions on number theory have been enjoyable and enlightening, but I need to have students apply what they’ve learned and what they have yet to discover. Finding time in the curriculum, and time to plan such a project, has not happened to date. I am thrilled at the idea of having students conduct more research, but I am anxious about how to execute such a project. Perhaps I need to start small: a two to three day assignment using manipulatives or technology to display such learning should not be daunting. And, after conducting my focus group, I am heartened that students want to explore different number sets/systems in greater detail. In Phase V, I reflected on the following:*The students in the focus group, many of them reluctant learners of math, were energized – as was I – by the idea of exploring number history and theory. They groaned when I mentioned fractions, and I sympathized with their emotional reaction. In spite of their lack of success with or negative feelings toward math, they were interested when I brought up different sizes of infinity, whether numbers were discovered or invented, and the idea that 1 = 0.9999... . Viewing numbers from a philosophical point of view sparked their interest. There was a consensus that carving out a few days to a week or more exploring numbers in this way, as opposed to following the curriculum only, would (a) shore up their understanding of numbers and (b) help them in future math classes.*I must do well by the students by allowing them the opportunity to explore mathematics from philosophical*and*mathematical viewpoints. I plan to do this no later than March. Meantime, we’ll add different ideas, such as the Tower of Hanoi, Two Balloons, Infinite Hotel Paradox and 1 = 0.9999…, to their mathematical toolbox.## Final ImagineIT Report

One of the most inspiring mornings of the year was spent with my focus group. In discussing with students my ImagineIT – the direction I wanted to take the class, the topics students wanted to home in on, the way in which we could cover the material – I felt that this helped me form a stronger professional identity (i.e. to trust my instincts and take calculated risks), shore up my student-teacher rapport and construct a meaningful investigation of material (Richert 17). The focus group in particular informed the dilemma of (a) what to cover and (b) how long to spend on the material. Indeed, ‘establishing caring relationships with students is central to teaching them well’ (Richert 57). Listening to their feedback and figuring out ways to implement their ideas helped to build rapport, which, in turn, has helped to teach them well.

My colleagues validated my primary concern of pacing with respect to the Real Number Line (RNL). They also offered ideas on how to implement my ImagineIT. While Algebra I was not going to be my focus with the RNL, a colleague suggested plotting fractions on the number line, something that at first didn’t seem ambitious. Upon further reflection, however, this is absolutely necessary given student struggles with rational numbers.

Since writing the implementation report, I have learned that my ImagineIT project will span the entire school year. Finding time and space for students to explore real (and imaginary) numbers, especially given missed time because of paternity leave, makes the pacing dilemma even greater. What is clear is this: students want to know more about the relationships among number sets and how this can benefit them in the world and in future math classes.

In thinking ahead to ImagineIT Round 2, I see opportunity to hook students into number and set theory with videos like Jeff Dekofsky’s ‘The Infinite Hotel Paradox’, the idea that you, Missy, suggested (how does 1 = 0.999…?), and projects like ‘The Tower of Hanoi’ and ‘Two Balloons’. These touch on the RNL and infinity, one of the first and most challenging concepts we covered this year. Students have expressed difficulty with infinity in different contexts, namely with domain and range and end behavior. Seeing this idea in these situations might clear up some of the ambiguity that exists with infinity, thus strengthening student understanding of the RNL. Having students

References

Richert, A. (2012).

My colleagues validated my primary concern of pacing with respect to the Real Number Line (RNL). They also offered ideas on how to implement my ImagineIT. While Algebra I was not going to be my focus with the RNL, a colleague suggested plotting fractions on the number line, something that at first didn’t seem ambitious. Upon further reflection, however, this is absolutely necessary given student struggles with rational numbers.

Since writing the implementation report, I have learned that my ImagineIT project will span the entire school year. Finding time and space for students to explore real (and imaginary) numbers, especially given missed time because of paternity leave, makes the pacing dilemma even greater. What is clear is this: students want to know more about the relationships among number sets and how this can benefit them in the world and in future math classes.

In thinking ahead to ImagineIT Round 2, I see opportunity to hook students into number and set theory with videos like Jeff Dekofsky’s ‘The Infinite Hotel Paradox’, the idea that you, Missy, suggested (how does 1 = 0.999…?), and projects like ‘The Tower of Hanoi’ and ‘Two Balloons’. These touch on the RNL and infinity, one of the first and most challenging concepts we covered this year. Students have expressed difficulty with infinity in different contexts, namely with domain and range and end behavior. Seeing this idea in these situations might clear up some of the ambiguity that exists with infinity, thus strengthening student understanding of the RNL. Having students

*do*more with the RNL is one change I’d like to make. Holding whole class discussions on the topic is important and interesting, but I think students, when they play with ideas in a hands-on fashion, will shore up their understanding of the RNL, its importance in math and their understanding of math in general.References

Richert, A. (2012).

*What Should I Do? Confronting Dilemmas of Teaching in Urban Schools.*New York, NY: Teachers College Press.