George Judd, Mathematics
Assessing Mathematics: Counting
Watch these video excerpts, which demonstrate fun, hands-on ways to teach mathematics. The original footage was created as part of a Webcast series, funded by the California Department of Education's Special Education Division, which offered professional development opportunities for teachers.
First activity, Follow the Money; so, that's not a political statement. Please get out your money now and arrange it in this little pattern like this. If you can just count, and if you start with one penny and you say 'okay I've got a row of one penny'. You can tell kids about rows or you can just count them. The idea is to simply start some information about what we can do with patterns of money and pennies.
So if I have one row with one penny, if I have two rows how many pennies do I have now? And get kids to count. What if I add another row? How many pennies am I going to have in that next row? Okay get your pennies out and start moving them. If you're doing kindergarten kids where they are just counting just have children recount: one, two, three, four, five, six. So in the first row we just have one penny. Second row – two pennies. What are we going to have in the third row? If you are going to ask children to predict what is going on they need a pattern to start predicting from. If students are just counting then just get them to count.
What I'm going to ask you to do now is set up a chart in which you start adding rows. So at this point you should have a handout somewhere that says…let me move this down just a little bit…you've got a problem and I'm going to give you one. Here's your problem: How many pennies are you going to have in the 10th row? When I work with students what I want those students to do is I would like those students to take this as a problem and then say 'how are you going to find out how many pennies there are if I gave you a thousand pennies?'
So if they are doing mathematical reasoning, setting up a problem is part of what is investigating about math. So if we look down over here I've set up a little problem, but I would ask students to say 'alright in row one you have two pennies, in row two you have two, row one is one, two is two, three is three, four is four.' If you added up all the pennies how many pennies would you have? How many pennies would you have if you added all the pennies in row one? Well in row one if you add these up you get one penny. In row two if you add them up what's the total of the pennies? Three pennies. I'm writing backwards and not too bad. So in row three how many pennies are there? Six pennies. Is that a six? Excuse me…six pennies, six pennies. Okay. Thank you. Six pennies. So how many do you have in row four? The total sum of the pennies is 10 pennies. I can do that one. Okay, at this point what are you going to have here? What are you going to have here? What are you going to have here?
So as you start taking this one simple object and you move it around and you start adding more pennies the idea is kids want to finish a pattern and if you're going to get to a point where mathematics is seen as patterns, give kids the chance to investigate patterns. What's great about manipulatives is they can see them, they can count them, they can predict them and then they can start putting them in some sort of arrangement.
Okay, make a reasonable prediction with 100 pennies. In the 100th row, how many pennies are there going to be? Now there is a procedure. How many more pennies grow? A helping column would say down here 'well how many pennies did you grow, you went with three, you increased by that many, gee what's the relationship between the growth here and the growth there?'
Setting up a problematic situation, somewhere there is a handout and in a minute I'm going to find out where it is, and it looks like this one. Hopefully you have this handout. If you don't have this handout, make one. It's called Triangular Numbers and what you're actually doing is taking the pattern and saying 'what's going to happen?' So I would like you to work on that as part of the things that you are doing and make a reasonable guess.
I'm going to come back to that a little later. I would also like to encourage you to go out and get some math materials that basically do stuff. Get some stuff. Go to, this is Teaching Mathematics to Children. There are other publications in CTM, National Council of Teachers of Mathematics can teach you stuff to do.
Also, be a mathematics investigator. Get involved in doing something. I'm going to back up a little bit about books. If you're going to access kids to books I would ask that you try to bring in more than just math books. I didn't bring any math textbooks or I brought only math textbooks. It's hard to decide what a math textbook is. If you're going to start with simple things you need to start engaging kids with a fun way; hands on is fun stuff.
So I'm not going to read to you very much, I'm just going to show you Snappy's little numbers. Snappy's little frog can catch one fly. I don't want to tease you too much but Snappy's little elephant has two tusks and hang around at lunch time, we will finish the rest of the story. What I would like you to do is get books, engage kids, get them to have fun.
One of the things I'd like to do is I'd like to encourage you to play around with this. When I get dull and boring try to find out what these patterns are. The idea is if you are going to become a math teacher, become a math investigator. Investigate this stuff. Play around and look for them. Now, if you'd like the answers of course I'm going to show them to you. Okay. Now, there are answers. I will tell you that number 17 is a tease; take out the commas. I hope I've made you just a little curious. Make kids curious. Make them find out what's going on. Get an idea of what is curiosity. I would like you to find out curiosity in other ways.
I've done this activity over and over again with students in the classroom with pumpkins, with squash, with whatever you have. And if you start with how many things are in it and say how are we going to find out, well sure, you're going to cut it open. I don't recommend you have kids cut it open. It's a basic safety issue. I've basically found you have a pretty good day if you don't have to all an attorney or an ambulance. So other than that, get prepared ahead of time. There are ways to get the seeds out and there are ways to start getting people involved in what's going on in these seeds. Take them out, scoop them out with a big spoon, put them out. If you have more than one group of children, it's nice to have kids do them in groups, but what you want people to do is to start counting and start pushing this activity up to greater than just counting the seeds. Now I found with lots of students that I just want to put a big circle out here and say 'just start pushing these seeds out'. If you want to dry the seeds out really quick simply put them between a paper towel and rub the little things off.
So kids are going to push these out and I'm sure that you would be substantially bored if I put them all out here. Once they get them out in a circle, we are going to pretend that we actually did that, if I could have glued them on that would have been fine, but okay it's a cooking show. Once you're done they're in a line. So now there is a linear relationship between the object and the number. And I'm only going to just start along this part and if they start counting, you're just doing one to one correspondents, one object - one count. At some point you can simply start upgrading the lesson and say 'ah, okay, I want you to put 10 seeds between each of the marks'.
So you can start children understanding the concept of base 10 by counting out 10 seeds, I don't have 10 seeds; one, two, three, four, five, six, seven, eight, nine, ten, and do the same thing in each section. You can spread out the seeds so that all of your students only are doing one squash or one pumpkin if it's October. Get them in, count them out. You may have six groups. Once you start collecting information now you're doing a data and statistics activity. You're finding out; well so if that group has 10 seeds and this group has 10 seeds and that group has 10 seeds, or 5 x 10, you can begin to write down the numbers, you can collect the data here and then begin to accumulate the data in another place. You can make a chart in which every group comes up and records their data and you start to collect the data and look at that as 'so how many seeds are there in a squash?' If you do another pumpkin on another day you get to repeat the activity and then you can choose to compare from one information to the other.
I think one of the great things about kids is they love to make guesses but guesses aren't estimates. A guess is just a number that comes to my head. An estimate is 'you know I've counted a squash before and the last one had 400 seeds, I don't think this one is going to have a million.' That's the difference between teaching kids to guess and teaching kids to make an estimate.
So get some hands on materials. Get some manipulative and let them go.
Produced by the Sacramento County Office of Education