I chose to work with the binary circles and ended up rather far afield by the end of my exploration. I had curricular integration with an introductory unit on exponents in mind when contemplating the following extensions. In the introduction video, Susan spoke of inquiry and variations as integral components to the activities so I envision the first extensions as a "What do you notice, what do you wonder?" assuming students have done the original activity of addition with binary or base three numbers.
Extension 1: Using colours to represent digits as Ali and Colin did, the first extension compares the patterns made by two different bases. I used binary and base four as an example, but students could choose any two bases and their own colour schemes.
I envision guiding questions such as:
What do you notice about the patterns made?
How would you arrange the circles to make patterns more obvious?
Do circles that are coloured the same from the sets with different bases represent the same number (in decimal)?
Can you find a relationship between them beyond looking the same?
Extension 2: Using repeated addition with the same bases as before, can we figure out how to multiply?
Guiding questions could include:
What do you notice about the patterns made when you add the same number?
What "rules" can you find for colour changes?
How are the rules for different bases the same or different?
What about adding or multiplying numbers with different bases?
Extension 3: At this point I went off trail and asked myself, "What if we used shapes to indicate the base? Can fractals be used to illustrate numbers in exponential form?"
Possible guiding questions:
How can we represent different bases?
How do components of the fractal images correspond to parts of the notation for exponential form?
How can a number like 32 be represented in different artistic ways?
Brainstorming Sketch - Math 9 Exponents and Exponent Laws
- What do we know about binary, bits, & bytes?
- Kandinsky circles inspired binary numbers and addition activity.
- Binary expressed in exponential form with base of 2. Notation, coefficients, bases, exponents.
- Converting larger binary numbers to base 10 using exponential form. How many bits in a megabyte? Ex: 00110101 = 0*2^7 + 0*2^6 + 1*2^5 + 1*2^4+ 0*2^3 + 1*2^2 + 0*2^1 + 1*2^0 = 53
- What about other bases? Hexadecimal, duodecimal, e?
- Extension 1: Comparing different bases
- Can you use the principles from binary to decimal conversion to find a formula for converting base ___ to decimal? What about bases that are multiples of one another?
- What about multiplication?
- Extension 2: Multiplication by repeated addition
- What are the rules? How would you express the rules in mathematical notation?
- Extension 3: Exponential form and fractal art
- Move on to Exponent Laws
Reading and Reflection
It probably would have made more sense to choose the embodied activity rather than the art activity since my reading from Riley et al. reported results from a study integrating physical activity with mathematics through a program called EASY minds in grade 5/6 classes in New South Wales, Australia. Yet, I feel a little more comfortable finding ways to integrate outdoors and activity so I chose to challenge myself with Kandinsky in Binary.
The authors reported positive results in both mathematical attitudes of students and teachers as well as increased mathematics learning after a six week intervention to embed physical activity within the math curriculum. The authors specifically noted that math became a subject students looked forward to. The activities were either practice for procedural fluency, such as jumping rope while reciting multiplication tables, or finding math in their surroundings, such as estimating distances.
Some interesting thoughts from the teachers were "that the preparation took more time and effort because it involved teaching concepts in a new and practical way" (p. 1666). Time and effort seem to always come up as barriers to being more innovative in our classrooms. I know it is something we have discussed previously and I don't see any way around it. It is a matter of being content to take baby steps and making slow but steady progress towards the goal.
"The teachers also talked of EASY Minds having forced them to be more creative and forward thinking as well as having to structure their planning to get EASY Minds aligned with the scope and sequence of the existing curriculum" (p. 1666). This is something I have found as well. Since there are limited ready-made resources at the high school level it does require creativity and thinking outside the box to develop activities. This seems especially true if the goal is curricular integration and continuity rather than the orphaned or frivolous activities Susan cautioned against in her introduction. I am not sure how long it took me to create three somewhat cohesive extensions aligned with curriculum, but it was significant and included several thinking walks. I know the shift is towards competencies, but the content is also important.
The authors also suggested that a teacher's attitude towards mathematics and their pedagogical repertoire strongly influenced student's attitudes. I can certainly see how this could be true. If someone is passionate about a topic they engage with it in a very different way. Though, I do feel there are limits to the second-hand enthusiasm that can be generated. Just look at the way parents and children engage with Minecraft!
Several students claimed the multi-tasking required to combine math with physical activity was beneficial. I wonder about this. Did the multi-tasking help with learning process or did it facilitate higher engagement? Is this related to shortened attention spans (apparently now legitimately shorter than a goldfish)? Do the activities and math concepts use the same neural pathways? The researchers noted the physical activity before the learning was most beneficial. Is it partly burning off some of the extra energy of 11-year-olds? Related to this, are there gender differences in attitudes and learning combined with physical activity? Lots of questions, lots to think about...
Hi Danielle,
ReplyDeleteI LOVED your extensions using exponents! It's fascinating the way you showed the bases and patterns. I would never have thought of those patterns in the context of exponents! Amazing work!
Your use of fractals to show the exponents completely spoke to me. The use of exponents in the fractals describes my class in FOM12 from Wednesday to Friday. Students were working on a project on fractals and have to relate them to real-life (easy, google), build a 3D model (like, how hard could that be?), and show the mathematical equations for the construction. I had a few very frustrated students who said it just wouldn't work! Impossible! I said one word to help them figure it out: "exponents" and it made complete sense to many of them. One student who is away a lot still couldn't get it, but eventually did. He built the fractal before measuring and that was the problem. Sometimes I think because we do project based learning in my FOM12 class, they think its easy. Actually, the math is more complicated because there aren't any ready answers in the back of the book for projects that students create.
We seem to by in synch this week. "Time and effort seem to always come up as barriers to being more innovative in our classrooms." Agree. I mentioned it too in my post this week.
"The authors also suggested that a teacher's attitude towards mathematics and their pedagogical repertoire strongly influenced student's attitudes." Completely agree too. Again, something that I also mentioned in this week's blog.
Thanks for posting a very thought provoking post with innovating activities for me to think about,
I agree with Maria. I just completed exponents with my grade 9 class and as I was completing the embodied multiplication activity with my daughter I wondered how I could do exponents through dance movements. I could not develop it soon enough last week to incorporate it but I could "practice" for next year.
DeleteThanks for checking out my activities and slogging through my overly long post. I think I overcompensated for being late last week.
DeleteMaria, I am happy to hear that the fractal idea could actually work!
Oh dear. A goldfish. I am pretty sure if I told my students that fact, they wouldn't believe that the technology in their hands is contributing to the reason that they cannot function in classroom activities. Cell phones, ear buds, and even laptops in class are a constant issue and a battle that every educator is losing. No solution here. Just a rant.
ReplyDeleteI find that my fixed last class of the day has the hardest time learning. Last class of the day, tired, have not consumed proper calories all day (see Maria's blog), and then add on the tech issue. However, you would think that they just had some physical activity and were up and moving the lunch hour before class that they would be more inclined to learn than the classes in the morning. I find that movement before class actually makes it harder for students to regain their focus sometimes. I wonder why.
I actually read this article before I read the Kelton and Ma article for the week. I was intrigued by the title and abstract and just decided to read the whole thing. Many of the comments made by the participants resonated with me and I agree that planning time constraints and individual levels of comfort with the embodied pedagogies are significant and important considerations. I agree that picking one small aspect and trying something is all we can do and we often put pressure to make huge changes that end up challenging our comfort levels too much.
"I find that movement before class actually makes it harder for students to regain their focus sometimes. I wonder why." Exactly what I have noticed too. I can get no work after that.
DeleteI find that last class of the day most challenging and least productive for the same reasons you mentioned. It is manageable with a course like Biology where you can do nature walks and labs, but courses like Calculus are pretty rough! It does give me more motivation to find ways of integrating embodiment into the upper-level courses even though it is a bit more challenging with the more specific concepts in those courses as you mentioned in your post.
ReplyDeleteAs an aside, I think second block is ideal. The students are not half asleep or straggling in late like first block, they aren't full of beans like after lunch, and not yet mentally done for the day like last block.
Danielle, I enjoyed the ways in which words like ‘What do you notice? What do you wonder?’ open activities in unexpected ways.
ReplyDeleteThe three workable parts of your activity give room for students to take responsibility for doing math in many and varying ways. Extension 3 made me wonder how this activity might unfold in different places. Bringing together your outdoor embodied teaching experiences with arts activities? Beaches as shapes? Fractals in waves?
Well done for challenging yourself to take time and effort with Kandinsky in Binary and for an encouraging discussion of Riley et al’s writing.
I hope that this week’s response and conversations help towards considering topics and approaches for the course assignment.