Update from the Second Grade Landscape

So this year I have been working with second grade again. I am attempting to learn from what we did last year. I think that our understanding about how the students work with base ten ( and especially my host teacher's understanding) has grown. A recurring issue is the representation of base ten blocks with stick and ball drawings that almost immediately replaces the actual blocks after a few limited counting experiences with the blocks. This, apparently, has become common practice and is partly a result of teachers feeling pressured to increase the number sizes too fast, which keeps the students in the direct modeling stages longer since they have not developed a good understanding of the use of facts and related facts to solve the smaller number problems. Teachers also feel that the time spent in getting the blocks out, managing them and putting them back wastes precious time. Of course I am constantly pushing against this practice, but it is not my classroom and I am only there once a week.

Nevertheless the kids and their teacher are making progress, the conversations and discussions are pretty complex and meaningful and the kids love doing real math. No M&Ms needed.  Here is an example of the board work from one of our problem solving discussions. This year, we frequently write problems after reading a book. The photo is the board work from a problem written by a student after reading the book Porkenstein by Kathryn Laskey (the gist of the book is friendship). The problem was:

Porkenstein has 35 friends. Some don't like him anymore. Now he has 19 friends. How many friends don't like him anymore?

The strategies were presented from left to right. The sticks in the upper left were a problem for the girl who drew them, so as a class we talked about what she did instead (insert: the problem was that she did not believe that a stick had ten in it, and she was treating the problem as a separate result unknown: 35-19 = ?). When she explained her second drawing (5 by 7 circles below), she said she covered up 19 and counted how many needed to be taken away (now she was treating the problem as a separate change unknown: 35 - ? = 19). Two more students shared, one who subtracted 19 from 35 using the base ten representation and another who used the number line to count back 19, and then as a class we used the numberline to count from 19 to 35. My goal for this lesson was to help kids see that problem can be solved a number of ways, and I hoped that they would treat the problem like a join change unknown and use the anchor of 19 close to 20 ( e.g. 19 + 1 >>20 + 10 >>30+ 5 >> 35; so 1 + 10 + 5 = 16.) Instead we ended up talking more about base ten approaches and recording number sentences that matched their thinking.

                   

I am reposting this very nice explanation of the difference between teaching algorithms versus teaching conceptual understanding. The author is Sam Otten from The University of Missouri, and is responding to a viral post by a dad earlier this year (that I commented in in my May 16th post). The context is within the Common Core, but the issue is common to any math teacher no matter if you are in a Common Core state or not:

https://www.youtube.com/watch?v=dmybO35F_EI

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.

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.

Check out my new page under the Adventures in Second Grade tab!

Here is my response to yet another uninformed journalist/parent ranting about mathematics instruction:

Oh please. I am so tired of uninformed people posting stuff like this! We have all seen kids who create "another way" to solve a problem that is not as efficient as another way. Again the point of solving problems by decomposition is partly to become efficient at computing, but another reason to do it is to build a robust understanding of numbers. The only problem I have with this example is the judgement of the US Standard Algorithm as "old fashioned", as it communicates a negative attitude toward a pretty slick way of calculating. And, saying "new way" (these terms were probably introduced by a well-meaning teacher, but misguided) also seems mislabeled, since the goal is not to do something new, but something smart. Impressively (and probably not understood by the everyday person), the representation describes a strategy related to the number line, which can become more and more efficient over time.

One more thing: I have seen teachers (again, well-intentioned, but misguided) interpret "use a numberline", or "decompose numbers", or "use addition to solve subtraction", as an entry point for showing kids a specific way to solve problems rather than letting kids analyze the numbers and make computing decisions and then discuss those decisions as a mathematical activity.

Real World Photos and Counting Brownies with Second Graders

I have been fortunate to work with a bunch of really great second graders as a guest teacher this year. It has been super fun to see them grow in math. They love to be challenged with big numbers, and the number zero makes them giggle. The fact that you can start any counting sequence with zero groups of anything and get zero is a hoot for them. Last week, I decided to try out some of my photos I have posted under the Math in the World tab. I logged in on the classroom computer and pulled up the blog to show the photos through the projector.

I scrolled through all of them stopping for a moment to consider the math that was there in each one. The were intrigued by the children's classroom in Nicaragua, and overwhelmed by the crowd of people at the UT game.

We stopped on the photo on the cut brownies. I love this one because many fraction problems and array problems talk about a "tray of brownies". But let's face it nobody except a chef can cut a pan of brownies straight. My photo comes from a time my daughter made brownies and of course there is a chunk out of the middle too. She probably decided to take a taste!

When I asked how many brownies were there (they are just starting to talk about multiplication), they were excited. One student said he could count by ones and would have to double check that a few times. Then a couples of kids said they could count by twos (the 3 x 5 array was not immediately obvious to most of them--), and showed me how that would work. Then another child showed how he could group them by 5's, and make a ten, then 5 more. Next someone said, count by 5's! I kept probing for more and asking the children to show how each strategy would work.  By the end of the discussion, the children saw that 3 x 5 was the same as 5 x 3, and the whole array (pun intended) of strategies that could be used to solve the problem.

I think using an authentic photo like this kept the problem real and fun. The discussions were light, and kids felt free to offer all kinds of ideas.  I am not sure how this would work for fractions though, since the pieces are not cut evenly.