Zombie already?

Ugh sorry this blog is fast becoming a "zombie blog" ! I am working on a blog about counting games and shall post soon!

PME-NA Part II: Tale of Three Great Tasks

One reason I enjoy going to PME-NA is seeing the work of my colleagues on children’s thinking. Inevitably what arises out of these studies are incredibly rich tasks that teachers can use in their teaching. I might add that these tasks are also good for teachers and prospective teachers to explore as they are learning about the ways children think.

Higinio Dominguez at Michigan State is an excellent interviewer of children-- :)Somewhere between Herb Ginsburg and Mister Rogers. He just gets those kids to talk, and does it with very little interference so what the kids say is remarkably close to what they really think. One thing he routinely does is leave off the question that usually ends the typical work problem. So for instance, when introducing this problem: “Joey has 14 cars. He lost some, so now he has 5 cars," and then asking, "How many cars did he lose?” Higinio says he simply asks, "So what so the problem?" The kids say all sorts of things such as, “Joey should really keep better track of his cars,” or “Probably his brother took them,” or “They are probably just under the bed.” Higinio’s point is that is you want kids to bring in their own personal connections (or resources--this is a much deeper construct) and this approach is one way to give them the opportunity to do that. Eventually the child or children will ask, "How many cars did he lose?" Or the teacher/interviewer can ask, "So how many are under the bed?”

Steven Greenstein, from Montclair State University has been investigating young children’s understanding of geometry, specifically the geometric reasoning involved in topology. He has noted that young children can identify important properties of sameness between 3-d objects. His research methodology included the use of a microworld, Configure, he developed to support children’s exploration on 3-D shapes. It is free and it looks like such a great way to get children to talk about their ways of thinking in geometry. The website includes suggestions for teachers who wish to use the program with their students. Here is the link: PLAYWITH SHAPES.com

Last, this idea comes from some researchers at Virginia Tech, Andrew Norton and Steven Boyce. The task is about unitizing, and children use their thinking about multiplication, and units to solve these problems mentally. I was familiar with the candy roll problem used to develop base ten understanding,

(see an explanation here: http://www-rohan.sdsu.edu/faculty/jbowers/macpics/intro.htm)

but was unfamiliar with this packing scenario:

A chip is worth 2 cents. When they are packed there are 3 chips to a cup, and 5 cups to a box. How many cents is the box worth? OR If you have 12 cents, how many more cents do you need to make a box? With this scenario, you can change the numbers around, and also you can begin the problem with actual chips, cups, and boxes. Eventually, kids begin to re-unitize and can solve them mentally. The problems can also extend to boxes per crate. I can see a relationship with this task to factoring and even prime numbers as well as fractions and folding problems!

PMENA 2013 in Chicago Part I: Lacking Legos™

I attended the North American Chapter of the international Group for Psychology of Mathematics Education. Thank goodness we just say "PME-NA" for short!

Anyhow, I attended a session about the parents' role in math education. This investigation focused on activities children do at home that could be resources for the development of young children's learning of math. To introduce herself to the parents the researcher planned a home visit to just talk to them in a more casual way before interviewing about what they do with their kids. What she realized after her home visits was that all but one of the 8 families had lots of access to computers, video games, smart phones, and TV (of course). Electronic access was not an issue at all. 

What she did notice was a lack of blocks and other such building and counting toys. When parents have a limited budget, they do not want to spend lots of money on blocks. She said that when parents have choice between spending dollars on these constructing type of toys (which are expensive) or electronics, they invest in electronics because those will be appealing to their kids in a variety of ways and will still be used as the child grows. The presenter of this research made this comment, “Schools should be interested in providing check out materials of these resources. Also, during school time, kids don’t actually need more time on screens.” She added that since her investigation seems to indicate that blocks and other physical materials are what is lacking in these homes, schools need to protect those spaces where children can be actively engaged in working and playing with physical materials. 

I really enjoyed this insight. It makes sense. Better off kids have lots of Legos ™ and K’nex ™, and tinker toys and building stuff, for the simple fact that these materials are expensive. The kids that have these toys also have the electronic devices and may not play with them much either.  Maybe Lego needs to start a campaign to get these things into classrooms to counteract the heavy marketing of tablets and smart devices that Gates and Apple seem to be pushing in these early childhood classrooms.

Summer

After yet another grueling, yet deeply satisfying day in the embedded class room, I reflected on my day. I thought about Summer, a second grader, who really struggled with some of the tasks we gave her. For Summer, each problem had to be counted one by one. So for 3+3 she needed to put out three fingers on each hand. Then carefully touched each finger to her cheek to count them all. She proudly said 6!

When we got to 5 + 6, her finger counting became more difficult, as she realized she did not have enough fingers to use her finger counting strategies. She counted several times slowly and carefully but she did not know what to to with the 6th finger on one hand when she needed it for the 5th finger too. She just kept trying! Finally, she looked up and said 11! She seemed to see an extra finger in her imagination. The point is, even though adding 5 + 6 was easy for most of our second graders that day, it was hard for Summer. But her persistence was more than awesome.

According to researchers who look at how people think of themselves as learners (Carol Dweck is one), it is exactly this kind of persistence, a belief that a person isn't stuck in his or her knowledge state (e.g. "smart" or "dumb") but that we are malleable--our intelligence is malleable and can be improved by hard work. Summer is just that kind of kid! Hopefully, her teacher will see that and encourage her in this way.

Later, in whole group, I offered the entire class a problem that turned out to be pretty hard for most of them. I spied Summer at the back of the room counting her fingers so very carefully. The problem was: Abby has 6 candy bars, and she gets some more while she is trick o treating. Now she has 13. How many did she get when she was out trick o treating?  About half the class was convinced it was 19. After class she came quietly up to me to show me what she had been doing. She counted up from 6 to 13 using her fingers. She told me the answer was 6. Okay, so she made a counting error (not surprising since I knew was still working on counting). But, she was proud, and she had a valid strategy for that problem. Go Summer!

Small things are Big things!

I am in Arkansas this week working with elementary teachers. They are in the their second year of Cognitively Guided Instruction. During this year we do this workshop format that is very ambitious to say the least. We conduct embedded class lessons. This entails LOTS of planning by myself, the hosting teacher and the CGI participants. On the first day on this follow-up (we spent 4 days together this summer, this is Day 5 of the PD), we interview all of the kids in one 2nd grade class. Each teacher pair interviews two children (one at a time). Then we all gather back together to analyze what the kids did, how they were thinking and try to create a sort of class profile. Then we decide what what we want as our focus group of kids, and what our goals for the class and this focus group are in particular. Then we choose/design problems and or tasks to present to the kids and anticipate what they might do and how the teacher might respond. Next we go the classroom. I taught the lesson. What a hoot. All 24 teachers crowded into the room watching the lesson. 24 kids learning math. One teacher. After the lesson we return the training room to debrief and talk about what we learned and what we would plan for the next lesson. A lot to do in one day and so so so exhausting, but well worth it.

The next day(Day 6, or second day of the follow-up) we do the same thing all over again, only with kindergarten! Kindergarteners solve math story problems and explain their strategies to the class, while being observed by 24 teachers. WOW. Lots of mathematics and teaching talk happened during these two days, too much to say here. Along the way we also learn some interesting things about kids ways of thinking about numbers. We learned that one Kindergartner had invented a new number! All this time his teacher did not know why his counts were one off, but we discovered he counted 11 cubes like this" one, two, three, four, five, six, seven, eight, nine,teneleven, twelve. HA. Only in Kindergarten classes do you learn how new everything is for them and how small things are big things for new learners.

Why Math Nerd?

Lots of times I am confronted with negative comments about what I do for a living. My sister calls me a math nerd. People automatically believe I love Sudoku. If I make a calculation error, someone will say-and you are a mathematician? Ha!  The other night while at a dinner with professionals including two lawyers, an engineer and a realtor, I mentioned the word quadratic and I was immediately hit with a barrage of statements including, "oh come on! math who wants to talk math?".  Huh? I did not have that reaction when you said " latest house bill" or when you say "Selling trend"? What? I thought you guys were educated, and most of all my friends?

I am over it (sort of), but I realize that I AM a math nerd, just like someone is a Star Wars nerd, or a chess nerd or even a football nerd. I am interested in math. I like talking and hearing about things that involve interesting math ideas, like "I wonder if the number of people have died in the world is more or less than the number of people living right now?" No, I am not GREAT at math. I do not use calculus at all, for anything, but I am interested in how calculus works and how it was developed and how the graphs created using calculus change when something about the equation changes. Anyway, I like math in a way that some people enjoy lots of things but aren't experts at it. Talking about math stuff (and actually science too) is enjoyable social activity for me.

This blog will give me a chance to explain this point of view, and also describe what I do that makes me a math educator with a PhD. I get to do research on how people learn and think about math. How people feel about math and how that effects their ability to do math. I get to teach. In the past two months I've taught a bunch of second graders, some pre-service teachers, some in-service teachers and some teachers of teachers. I have also been researching the relationship between the quality of a professional development experience and teachers' content knowledge. It is fun and promises to be "funner" all the time. Yup I am a math nerd.

Other Cool Blogs

Stinky's Cool Math Blog

I really like this blog at first glance because it seems honest and also informed. Plus, the author highlights Diane Ravitch, who is fast becoming one of my most favorite school policy persons.

Learning Math, Teaching Math

This is an excellent blog (the link is actually to the archived version) about many issues in math education. Susan Empson, my friend, advisor (and overall guru or sensei), and author of  this blog is in the process of publishing much of the work you can see here that was done with struggling 2nd graders.

Enjoy!