Comments and answers for "Lack of ease?"
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The latest comments and answers for the question "Lack of ease?"Comment by Maria Droujkova on Maria Droujkova's answer
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Whose math? Yours - your friends' - your enemies' - mathematicians' - engineers' - ed testers' - ...? It's all different! Texting friends and publishing a science article means different writing. Likewise, people need to do math differently in different situations.
It is an excellent thing for students to make up their own terminology, for THEIR math. I always encourage it. This is similar to "baby signs" - studies show it promotes earlier, more advanced talking. Likewise, making up terminology promotes more robust math. But we need good scaffolding tools, including talking to others.Sun, 04 Aug 2013 08:45:42 GMTMaria DroujkovaComment by ccross on ccross's answer
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Certainly, being open to a different way of solving the problem, like Friend did with Problem 9, is great. But I don't know that it is a good thing for students to make up their own terminology or their own solutions to things like 0 factorial. He can re-define square number, but it just causes confusion when he is talking to other people who know the standard definition of the word. And if it comes up on a SAT test or a college math class or something, he is just going to be marked wrong, not celebrated for his individualized approach to math.Sun, 04 Aug 2013 08:28:22 GMTccrossComment by ccross on ccross's answer
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He isn't complaining about having to learn it, and he understands the explanation. He just doesn't agree with it and thinks his approach is better than the standard one.Sun, 04 Aug 2013 08:22:25 GMTccrossComment by Maria Droujkova on Maria Droujkova's answer
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Chris, spot on! WHY is the key question here.
A few variations:
- Why did you think the inventor decide on that definition? (check some math history, maybe)
- Why did this definition become popular? Why do people like it?
- What other definitions are possible? Which do you like? WHY?Sun, 04 Aug 2013 08:01:54 GMTMaria DroujkovaAnswer by ChrisYu
https://naturalmath.com/community/answers/712/view.html
Carol, my response to people who complain about all the rules and facts. Too many to memorize, doesn't make sense.
That is the tragedy of math education where the focus on memorizing formulas and procedures away from understanding. Once we understand, there's no effort in 'memorizing' how things work.
Instead of hating the 'fact' that 0! = 1, we should look at that as a challenge and ask why? When I have something that just bugs me, it's a clue that my understanding is not complete, that is where fruitful inquiry lies.Sun, 04 Aug 2013 07:57:20 GMTChrisYuComment by Maria Droujkova on Maria Droujkova's answer
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Carol, I'm thinking about solutions, vocabulary, and other "prescriptive" support. The goal of this course, and of all our efforts, is for kids and parents to MAKE THEIR OWN MATH. Your son's definition of square numbers is an excellent example of something we very much want to support.
We need a good balance - a flow channel between the extremes. On the one hand, we need to provide enough vocabulary, pictures, solutions, etc. to get all the kids and parents going. On the other hand, we need to provide LITTLE ENOUGH stuff to leave plenty of OPEN SPACE for THEIR MATH.
What do you think?Sun, 04 Aug 2013 07:11:01 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
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Carol, thank you for the detailed summary! "Get lucky with the students" made me chuckle! You CAN increase the chances, if you try math with different people, until your find good matches. Leaders of large Math Circles sometime have "math dating" (not called that) mixer activities, to help members find good partners for smaller problem-solving groups.Sun, 04 Aug 2013 07:02:53 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
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@nikkilinn, you write: "The course definitely stretched me, and I learned quite a bit, just with the four problems we did complete successfully." Making your own math has a very strong effect, and you are right - you can feel the difference even from one or two sessions! We'll organize more online discussion, chat and help for participants in the future courses.Sun, 04 Aug 2013 06:55:58 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
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Chris, "How do we reach the people that are "math challenged"?" is an excellent question! My main hope are local, friendly Math Circles, in combination with global online support. That's what this course is about. Several math-challenged people can help one another to feel better if they run a Circle together.
More frequently, a person interested in math invites local kids - AND THEIR PARENTS - into their Circle, and helps them feel better about math. I am thrilled about reports of grown-ups in this course helping their (grown-up) friends and family members engage with math!Sun, 04 Aug 2013 06:52:20 GMTMaria DroujkovaComment by Maria Droujkova on Maria Droujkova's answer
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Rodi, what an interesting piece of advice about researching anecdotes! Laughter boosts math success very significantly. First, it relaxes the body (which is essential for the focus). Second, it oxygenates the brain. Third, it feels good overall, calms you down, makes the group mingle...
You got me thinking of what other such extras I gather. I prepare cute, funny, or beautiful math pictures to share; and sometimes short videos to end the meeting; and "Eat Your Math" jokes or micro-activities for the snack time. It's a different sort of prep than people would expect!Sun, 04 Aug 2013 06:42:28 GMTMaria DroujkovaAnswer by ccross
https://naturalmath.com/community/answers/703/view.html
Oh, and I forgot one.
**Add Vocabulary Words**
We had to look up words several times in order to figure out the problem. The best example was the question about a square number with a last digit of 8. My son argued that 8 itself would work, because it is a square of something (2.8284 etc. according to our calculator). In his mind, every number is a square number, because they all have square roots. But I said I thought the term square number meant the square of a whole number. So we had to look it up, and it was defined as the square of an integer, which we also had to look up...
So it wasn't a big deal. But if this were a book, and I was using it in a class or other location where I didn't have access to a computer, then it would have been a problem.
(OT--Of course, my son still thinks he is right about the principle of the word usage, and doesn't like the term square number to be restricted to integers. So he has added that to math rules, definitions, and principles that he doesn't like, like the fact the 0 factorial is 1.)Sat, 03 Aug 2013 14:11:02 GMTccrossAnswer by ccross
https://naturalmath.com/community/answers/702/view.html
This is mostly a repeat of things I have said in other discussions or questions or whatever they are called in this system, but I'll pull them all together here.
I am also one of Chris Yu's math-challenged parent educators, so these are coming from that perspective.
**Give the solution and explanation to all problems.**
I understand the value of making us wrestle with the problems ourselves. And I have really enjoyed and gotten value from this course. However, I would NOT buy this as a book if there isn't some way that I can check the answers. Some of the problems I couldn't figure out; others I thought I could do, but didn't trust that was absolutely the right answer. Finally, there were one or two that I could do based on the hint, but I couldn't explain why they worked. I don't want to pass on my own math misconception or bad attitudes ("you don't need to know WHY it works--just do it this way and get the right answer") to the next generation. So a book that has problems where I can't check the answers (even if it was on a website or something) is a non-starter to me.
I also believe this approach of just giving hints instead of answers just exacerbates the negative feelings that we math-challenged folks have about math. I read the problem, I read the hint, and it gives me the message "this SHOULD be enough information for you to figure this out." But it isn't, for me, because I can't figure it out. So it just makes be feel bad, like I'm a math failure once more. And then, being human and not liking to feel bad about ourselves, I turn it against the course/book/authors--"This is obnoxious. Why can't they just give us the answers? This is badly done....blah blah blah."
I'm not saying I feel that way about the authors--I know you guys are all outstanding professionals in your fields, and are deeply and sincerely committed to helping people at all levels embrace math. But I noticed those sorts of thoughts coming up for me, although I just dismissed them. However, I've had the great benefit of hours of personalized counseling from Maria to help me get over my math hurdles. I can just imagine how my peers who haven't had someone work with them to reverse their negative math feelings might feel about this.
I know there has been a suggestion to have an online math help forum, and that might help. But a lot of people don't like to admit publicly to their math deficiencies.
So, working on problems that I didn't immediately "get" gave me, if not joy, at least an intriguing challenge. But not being able to check my answer was definitely a source of frustration and non-joy for me.
**Add handouts, or at least the graphics attached separately as jpegs so people can download them and create their own handouts.**
This may be less important for those working with younger students. But in working with teenagers, I found that adding in manipulatives was more distracting than helpful, and they mostly just wanted to see the problem and work it out on paper. But to reproduce the graphics, I had to use Jing to cut out the graphics on a page. Not a huge thing, but it seemed unnecessary, since I'm sure they were all created as jpegs and inserted into the text, and it would have been easy to just attach them for our use. And I know some people don't know how to use Jing or such software.
**Maybe tag the responses by student age/grade range or something like that?**
I wouldn't necessarily want to have completely different discussions for the age ranges, because in the beginning, it was interesting to read about how people were adapting the problems for different ages. And for a few of the problems, I tried working with younger students as well. But later on, when our time got crunched because my son was doing a camp and I seemed to be spending much of my day transporting him to and from said camp, I wanted to skip the stuff that didn't relate to my age range of teenagers.
In reality, at least by the end, it seemed like only Denise and Maria were also working with teenagers, so I just checked out their posts. But if this were at a larger scale, that probably wouldn't work as well. So I don't know how exactly, but it would be helpful to search for the posts that relate to the age of student that I am working with.
**Make it clear where to go for assistance**
I know Moby Snoodles said that if people didn't understand how to solve the problem, the course authors thought they would ask in the "How are you going to adapt Problems ####" forum. But that hadn't occurred to me. I thought that if I didn't know how to do something, I just shouldn't say anything.
There is a suggestion about a math support forum, so that might have helped. Personally, I wouldn't call it a "newbie" forum, because most of us have been doing this, however badly we do it, for years and years now.
I took another MOOC this summer that had a forum called "I Need HELP!" I thought that was great! It made it really clear where you could go to ask about anything--how the MOOC worked, technical issues, all sorts of things. There was a lot of peer-to-peer responses in that forum as well, so it wasn't just the course developers responding, especially when it came to technical issues.
Personally, I'm adding that to any online class I've developing from now on. It is so easy to get overwhelmed in these courses, so making it really easy for people to know where to go when they are have problems is really important to improving the completion rates of these online experience, I think.
**I like reflecting through writing.**
I know the organizers asked about posting videos discussing your experience instead of writing these posts. That doesn't really excite me. I prefer writing to talking on a video. I'm way more coherent in text than when I speak, and the process of writing helps me figure things out.
But, that's just me. Probably younger people who are growing up with more use of video do better in that medium than I do.
**Having a weekly live session**
If there had been a weekly live discussion, I probably would have participated and probably would have benefitted from that, particularly if you GAVE US THE ANSWERS! However, based on my participation in my other MOOCs (this is my 4th this year), I would only do it if it were scheduled at a time that I could participate live. In my other courses, I would do the live sessions that fit my schedule, but I never went back and listened to any of the ones that I missed.
But, of course, scheduling live sessions at work for everyone in an international course is just impossible.
**Get lucky with the students you are teaching?**
So I know this isn't a very helpful suggestion. But the most joyful experience of this course for me was the case I described in my writeup of Problem 9, when one of my son's friends can up with an absolutely beautiful solution that was completely different from the one described in the book. It was joyful to see his facility with math, and it was joyful to see how he shared that with my son. It was joyful to see them doing math together without me.
So my issue isn't about control. I don't need to know the answer if I have a student who can explain a problem that I can't. But it is hard to guarantee I can always have someone around to do that for me!
But if a student like that wanders into your sphere and does something beautiful like that--it is a magical moment.Sat, 03 Aug 2013 10:18:35 GMTccrossComment by Rodi.Steinig on Rodi.Steinig's answer
https://naturalmath.com/community/comments/687/view.html
Here's one more thing I thought of after I wrote the above: A compelling problem (or what the Kaplans call and "accessible mystery") takes weeks and weeks to unravel, even with the youngest kids. What I have found is that over the weeks that I explore such a problem with my group, the prep time needed each week decreases.Thu, 01 Aug 2013 13:04:53 GMTRodi.SteinigComment by Rodi.Steinig on Rodi.Steinig's answer
https://naturalmath.com/community/comments/686/view.html
One thing that may be helpful is the database of Math Circle Problems and Lesson Plans that the NAMC (National Association of Math Circles) is putting together on the their website:
https://www.mathcircles.org/content/lesson-plansThu, 01 Aug 2013 13:03:12 GMTRodi.SteinigAnswer by nikkilinn
https://naturalmath.com/community/answers/677/view.html
As a home schooler who is not a math teacher, I found the course a little too much to keep up with after the first week. I also had a difficult time knowing exactly how to present the more complex problems. I really appreciated all of the ideas from the experienced math teachers, and the handouts that were shared. We will keep working through the problems that we have not yet completed. I am one of the "math challenged" that ChrisYu mentioned above :), and I felt that I could have used a little more instruction/help for my own understanding prior to presenting to the kids. The course definitely stretched me, and I learned quite a bit, just with the four problems we did complete successfully.Thu, 01 Aug 2013 09:23:04 GMTnikkilinnAnswer by ChrisYu
https://naturalmath.com/community/answers/675/view.html
After reading Bob Kaplan's book, he states that Kits are not needed. But I do think it would be nice to have some thing prepackaged because not everyone has the time to prepare. Like a crutch to help out people new to this.
It seems like many of the people who signed up for the course have some math interest or background. How do we reach the people that are 'math challenged'?Thu, 01 Aug 2013 02:41:44 GMTChrisYuComment by Rodi.Steinig on Rodi.Steinig's answer
https://naturalmath.com/community/comments/667/view.html
One other thought about giving up some of the math prep: I’ve internalized that fact that it’s okay, and even good, to not know everything. We need to know enough to (very) occasionally redirect students with pointed questions, but if we do our job as merely being secretaries well, the students can really discover the joy of math.Tue, 30 Jul 2013 16:45:37 GMTRodi.SteinigAnswer by Rodi.Steinig
https://naturalmath.com/community/answers/665/view.html
I used to spend a lot more time preparing for Math Circles. (I guess you would call mine “hardcore.”) I researched history, relevant contemporary cultural references, and of course every possible direction I could imagine that the math could go in. I brought this up at a Math Circle training I attended at Notre Dame with Bob and Ellen Kaplan, considered founders of Americanized math circles. Bob said in reply, “Most people over-prepare for math circles.” I’ve been thinking about that over the past year, and have given up the time spent on the math preparation. What has happened in my math circle, and this course this summer, is that I’ve seen that I don’t need to do that. So much interesting math that I already know about comes up that I don’t need all those questions in my back pocket. Of course, the initial question needs to be compelling.
I’ve found that, though, that researching anecdotes is well worth the time. Sometimes students need to get away from the math during a session. In fact, during my last session here (questions 7-10) I ended up telling the kids Dr. Tanton’s anecdote from the beginning of his books of how ended up a mathematician from studying patterns on his ceiling as a boy. Just a few sentences about that and the kids were ready to get back to a challenging question.
For this course, I don’t think a 1:1 ratio is possible if participants need to read the assignments, sample problems, essays, and then to reflect. A 2:1 ratio might be possible if doing things bare bones. Also, the ratio gets friendlier as the course goes on and we gain experience.
I agree that one problem per week and per session would work better (for the kids, too). I still haven't found the time to write up what happened when I did problems 5 and 6 with the kids, but I hope to do that this week.
To make the work more joyful, I think that a weekly online interactive voice session (via Elluminate or some program like that) with the course leaders would be fantastic.Tue, 30 Jul 2013 16:37:27 GMTRodi.SteinigComment by Rodi.Steinig on Rodi.Steinig's answer
https://naturalmath.com/community/comments/664/view.html
A reflection method that works for me is to have a camera in class and take a photo of the board (or other work) each time I'm about to erase. When students say things that are memorable, I write them on the board, and they're captured in the photos. I also keep running lists on the board entitled "Questions" "Conjectures" and "Assumptions." This helps me remember, but also helps the kids to focus on the problem.
I used to have kids in the class jot down notes in the car on the ride home, or even get a kid not in the class to take notes for me, but that's not ideal.Tue, 30 Jul 2013 16:24:02 GMTRodi.SteinigComment by Maria Droujkova on Maria Droujkova's answer
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Hmm, good points! Maybe the leader can do the quick "mind dump" recording, and a volunteer parent can type it up for reference?Wed, 24 Jul 2013 10:59:11 GMTMaria DroujkovaComment by Denise Gaskins on Denise Gaskins's answer
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Written notes are the easiest to collect, keep, and glance back over for future ideas. Recording might be a quick way to do a mind-dump right after a meeting, but it's not very practical as a reference.Wed, 24 Jul 2013 10:49:49 GMTDenise GaskinsComment by Maria Droujkova on Maria Droujkova's answer
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Is there an easier, quicker way to reflect? How about recording a short webcam video of yourself?Wed, 24 Jul 2013 10:31:34 GMTMaria DroujkovaAnswer by Denise Gaskins
https://naturalmath.com/community/answers/640/view.html
For my older groups in the past, I used to start with a minimal-preparation game or warm-up, like these:
[Hit Me! (A Math Game)][1]
[Math Warm-Up: Today Is February 4×3×2×1][2]
I would also keep on hand a second minimal-prep game or activity as a time-filler, if needed. As I collected a handful of games I liked, I knew I could always fall back on these to rescue a failed meeting. The kids never minded repeating these, as long as we didn't play the same thing every time.
Then I would plan one main project/investigation/activity that I figured would keep us busy for at least half an hour. That way, I would only have one thing that needed much in the way of research or preparation time, which kept my prep time much closer to 1:1 and kept me from burning out on the whole idea of math club. It also allowed us freedom to go deeper into the topic (like the MAA-AMC lessons) or follow rabbit trails as we wished.
I also neglected to write down much in the way of notes afterwards, which I now regret. Even though it takes time to write things out, it's much more helpful than just thinking about how the meeting went and telling myself, "Next time I'll remember [whatever] ..."
[1]: http://letsplaymath.net/2008/05/29/hit-me-math-game/
[2]: http://letsplaymath.net/2009/02/24/math-warm-up-today-is-february-4-x-3-x-2-x-1/Wed, 24 Jul 2013 10:28:17 GMTDenise GaskinsComment by MobySnoodles on MobySnoodles's answer
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You can use big flat edible leaves for the cake problem, too (lettuce or collard greens).
Maybe we can develop a collection of intro and exit/break tasks. I have "apple math" (snack break) tasks from my math clubs. For starters, we used to do a Show and Tell. Kids brought their favorite objects, and explained the math they found in them. If it was a new concept, they put it up on a poster board.
You can ask other parents for some help in the fall. For example, they can take photos, or help you search for tasks. 4-6 weeks (weekly) is easier than every next week: people forget which week!Wed, 24 Jul 2013 06:31:12 GMTMobySnoodlesAnswer by mirandamiranda
https://naturalmath.com/community/answers/633/view.html
I would say I have spent more than an hour on prep/reflection per hour of contact time. But I have enjoyed it! I have been considering continuing the 'math circle' we are doing into the autumn but am slightly put of by the amount of time it might take me. Maybe doing it less frequently than weekly would help. I would love any suggestions from experienced leaders, although my circle will probably be more softcore than hardcore!
In terms of suggestions for the future, I think having fewer problems per session could help. I would also really appreciate something that would support a little more structure; say, one short discussion/intro problem, one more open-ended exploratory problem, and then a 'math snacks' problem to round things up. Of course this might be asking you to do my job of adapting the problems...
Hmm, now I am thinking I should adapt the L-shaped cake problem with actual cake! Slices of bread maybe?Wed, 24 Jul 2013 00:28:05 GMTmirandamiranda