Tuesday, October 28, 2014

Patty Paper

Patty Paper can be used in elementary, middle school of high school settings. It is a paper that was used to separate hamburger patties and is sort of like parchment paper, but really thin. You can do more with it than folding origami or regular paper for two reasons. It is cheap, and the lines show really well for the folds. You can use it for teaching many mathematical ideas including fractions, multiplication, algebra and geometry.
LINKS
This You-Tube video demonstrates how to find the line that bisects another line. This is fairly simple and direct teaching of the activity but there are lots of other things you can do that are more open ended. Copy and paste this link:  https://www.youtube.com/watch?v=WvgAvVKUISQ
EDUCATIONAL VALUE
Once you get to know the uses of patty paper you can actually use it for developing geometric proof and other geometric properties. We used patty paper for a study of young children's multiplicative thinking is a study that was later published as an article in Teaching Children Mathematics (Turner, E., Junk, D. & Empson, S. (2007) The Power of Paper Folding Tasks to Support Multiplicative Thinking and Rich Mathematical Discussion Teaching Children Mathematics, 13, 6, 322-329.). A side note to the study is that patty paper serves as an instant feedback mechanism for the learner. We had two basic tasks: predicting what would happen given a particular sequence of folds, and propose what sequence of folds would be needed to make a certain number of sections. (approximate examples of each type: "If you fold paper into 2 equal parts and then 3 equal parts how many equal parts will you see when you open the paper back up?" and "If you want the paper to show exactly 8 folds, what are the steps to folding it?") Students make predictions or make the folds then reflect on what happened and make appropriate adjustments. Class time can be spent talking about why the sequences of folds worked or did not work. Also records can be made of the folds that will eventually reflect the relationship of multiplication, division and fractions.

Monday, October 27, 2014

Supporting Struggling Students

I have been participating in an online course, and one of the assignments was to outline an issue in my work that needs attention, and that the attention could be in the form of an innovative technology or use of a current technology in an innovative way. Here is my post about supporting teachers to learn how to support students who are struggling:


Helping teachers to learn how to effectively responding to students who are struggling is a difficult thing to do in a teacher workshop setting. Teachers need lots of practice in doing this and also need to do it in a supportive non-stress environment. In a workshop setting there is just not enough time to help 20 to 25 teachers learn this important skill. Also just saying "they will get it with practicing with their own students" is only a partial solution, since, those particular students NEED competent responses NOW, and teachers who are actively teaching need time to reflect and perhaps some real time coaching at those very moments which are often not predictable. One way might be to develop a computer program that allows teachers to view student teacher interactions, choose how they would respond, then try it out virtually, and see the results.
Here is a description of the issue:
Teachers typically respond to struggling students in math class by providing instructions on how to get the answer. These instructions often remove the challenge of the problem in such a way that a student avoids actually learning the thing he or she needs to be able to solve future problems. Complicating the issue is teachers’ perceptions of struggle and their role as teachers. For example, when a child is attempting solve word problem, it is natural for a student to model the problem as it is stated by physically representing the objects and then counting them. Sometimes the child struggles to understand in what order to do things. For example this problem can be challenging for young children:
Jeff has some toy animals in a box. He gives 5 of them to his little brother to play with. Now he has 4 little animals in his box. How many did he have to start with?
If a teacher intervenes and tells the child the order (e.g. “first put out 5 blocks… then put out 4 blocks… now count them all”), the very thing the activity was supposed motivate the child to do has now been done by the teacher. All the child is doing now is following the teacher (which can be cognitively challenging), but not thinking about the problem’s meaning and how that might used to make the decision about what to do first. Teachers need to know several things about teaching and learning mathematics and employ them as they are interacting with students. First they need to understand that struggle isn’t necessarily a bad thing, and part of the job while intervening is to maintain the present cognitive demand of the problem, but possibly create a more accessible entry point. Often that entails a subtle suggesting of a problem solving tool (“yesterday I saw you using blocks to figure a problem do you think that might help you now?”); supporting the child to understand what the story in the problem is about (“so have you ever collected little toy animals like the boy in this story? can you tell me what is happening in this story?”); understanding why this problem might be hard—it is structurally a subtraction problem, but to solve it the two given amounts need to be added, so going back to an slightly easier problem with the same context might be helpful (“jeff has 8 toys and gives 3 to his brother, how many does he have left?”)
What teachers need to be able to do:
Teaching teachers how to respond to students without taking away the challenges in the task that are designed to help kids learn the concept is challenging.  Each teacher-student interaction can feel different and teachers feel that to be responsive they need to experience every situation or be provided with endless sets of prompts and questions that will solve all of the issues with struggling kids. OR (and this is sort of worse) teachers see all struggles as virtually the same and responses become all of one kind (usually just provide the child with the appropriate steps to solve the problem). Neither way of responding is ideal. Being responsive means that teachers can customize their responses, but also be able to see a set of general ways children behave when offered particular problems, understand the goals of the task the child is having difficulty with, and then provide the most minimum “intervention” so that the child learns the hard stuff through thinking, and doing rather than following.