Many had tried the human knot. Bertine found that his mixed gender group was reluctant, so he went to single-gender tangles. Lynne tells of using this same Tangle for UW freshman during Summit 2009, having talked about touching BEFORE doing the tangle. Dave's 5th graders held onto two opposite ends of a shoestring, not hands.
Lynne tells a little of a talk at UW’s Center for Learning, e-volution with Gardner Campbell, "Narrate, Curate, Share: Integrative Learning and Web 2.0".
Steve is shared his “lock boxes” teaching strategy. (Steve has 56 physical lock boxes, and about 130 questions organized by the answers.) He tells his 7th graders that in a problem solving situation, rather than looking at looking for a rule, that they should look all around to figure things out. Steve is working with kids who have difficulty with even basic concepts and words can go out, discover relationships, etc. His example is to find a number that is 6 less than the product of the 2nd and 6th primes. Kids don’t immediately start asking questions --- learned helplessness of 'I don’t get it' --- now they read, investigate, etc. Mary adds that last year with her SPED kids would ask to do a lock box, and helped each other.
Ariane did number bracelets with all of her HS classes. Her point was that the problem of number bracelets engaged high performing and low performing students. One class was having a test under a sub, and the class wanted permission to make number bracelets after their test. A young man said that there were only 100 combos, so he would go home over the weekend, work out all 100 and let the class know what happens.
Lynne posted an NCTM site where they claim they “combine understanding with skill”. We had a small attendance (6), but the conversations were animated as we all face the challenge of implementing richer problems.
Kendall posed "Math Magic": pick 2 positive integers in a vertical fashion.
Then add the sum to the 2nd addend to get a 4th number. Now add the 3rd and 4th to get a 5th, keep going until you have 10 numbers. How can you quickly get the sum of all 10? Trick: 11 times 7th number = sum of all 10. Kendall’s problem is a challenge for his 13-year-old son Benjamin. Kendall also asks for help to decide favorites from Jackson to take to ND where he’ll present about TCs, and for suggests about other patterns that are like this, or ways to extend this. Maybe there is a similar pattern with 3 numbers to start.
Lynne & Kendall have a brief exchange about how people, even people who don’t think of selves as good in math, are drawn to this sort of problem.
Steve has us view a YouTube clip of Dr Who where we need to enter a Happy Prime:. Had his students exploring digit squares and looking for happy primes and happy numbers. Square the digits of a number, sum and then repeat. If this process reduces to 1, the original number is a happy number. Steve likes this for the training it gives for the 2-step process of squaring and adding. Examples of happy numbers are 13 and 19. Note that permutations of the digits will give another happy number, e.g. 91 or 31.
Ariane asks for more information about Kendall’s pattern, and Lynne suggests understanding the pattern using letters instead of specific numbers.
Steve relates that there are things we know about finding happy numbers. For example, every number along the path of converging to a happy number is in turn a happy number. For example, in expanding 19 we generated 82, 68 and 100.
Lynne suggests we return to our rubric for judging problems. Steve suggests that last two problems are not relevant to general population, but that it would be good for grades 4-5. Seem to rank well on all other points. Steve suggests that perhaps #8 is not perfect, and maybe we need a point about engagement.
Ariane suggests that there is a challenge in defending problems that don’t directly relate to her standards. Struggle will be to align and then get back to the book = boring things saying that the other part can be fun too. Steve agrees about the importance of some routines.