## Candy Math

If one buys a bag of Sour Patch Kids, will there be an equal distribution of the good colours and the gross colours?  Because if I could buy a bag of just red and avoid the gastly green, and blue, I’d be happy about that!  The children in my class disagreed and hoped that there would be an abundance of blue.  But we all agreed that it should be equal.  Time to test it out!

I put a bag of Sour Patch kids on every table, and told the students they could pick their own groups.  This doesn’t happen often for us, but I wanted to see if they would distribute themselves evenly. One group of 3 got a whole bag to themselves because nobody else wanted to work with them. That left one big group of 6 to share a bag.  I think the kids in that group will think twice before they settle on a group next time (cause you know there will be a next time!)

I was very interested in the strategies students would use.  I had predicted that there would be some organizing into groups by colour, and I was right for all but one table.  It took them a few minutes of debate before they all agreed to do this.  At first, each was starting his/her own groups, stealing from the others to try and create one pile for each colour, except they were all trying to create the pile right in front of themselves.  They had 4 red piles, 4 blue, etc. Finally they realized, with a tiny bit of prompting, that one pile for each colour would suffice.

Since September, we have been talking about how organizing into groups of 5 makes counting a lot easier.  But…still…lots of kids were counting by 1’s.  *sigh*

Over time, however, they switched to grouping, usually by 2’s, but at least it was grouping.  One group, the group I would least expect to struggle with this, organized each colour by a different number.  Then they couldn’t figure out how to make that into a graph.  We had a very interesting conversation about this, so I’m counting it as a win.

The graphing was fun to watch too.  We’d talked about how we don’t always have to count by ones on a graph, but we clearly have some growth to do in this  understanding.  Though I’d given them a task that required more than the number of rows I had given them on the graph template, they still thought they could just fudge it.

Let me interpret this for you:  this group had 21 Red, 17 Green, 15 Orange, 13 yellow, 22 Blue, and 1 split (I have no idea why, but one child was obsessed with the possibility that 2 colours might have melted together and therefore 1 SP kid fell in to 2 categories AT THE SAME TIME! This did not actually happen, but this student really wanted me to believe it was not only a possibility, but an inevitability!)    I have just one picture of this to share, but it happened all over the place.  I eventually pulled them together and reminded them about choosing a scale.  The grade 3’s got it after that, but the grade 2’s not so much.  It’s not one of their expectations anyway, so I’m not worrying about it. They could read the graphs the grade 3’s created, so we’re good!

This week, assuming the one day I have to do some actual teaching actually doesn’t get interrupted by the unexpected, we are going to find out if we get more caramel popcorn or more cheese popcorn in a bag of Chicago Mix.  We are going to compare the store brand to the Orville Redenbacher brand and see if one is more even than the other.  We are also going to compare the Humpty Dumpty brand “party mix” and the Doritos brand “party mix” to see if I am truly being ripped of, as I suspect, and getting more than half a bag of pretzels.  I firmly believe there should be a “No Pretzels!” option here, just like there is a “No peanuts!” option for a can of mixed nuts.  Am I alone in this?

I used to do food math all the time.  Froot Loops, Smarties, M & Ms – they all make great math manipulative.  But for several years I have had students with food allergies and bring any sort of food into class was so stressful for me that I avoided it at all costs.  This year I have very little of that to contend with, so I’ve been going for it.

Oh…almost forgot!  One group got two half Sour Patch Kids.  They weren’t sure what to do about it.  We had a great conversation about the 2 halves making a whole.  They were reluctant to believe me, which just shows that I didn’t do quite enough with fractions this year.  I’ll have to rectify that next year.

## Finding Connections

A few weeks ago, I sort of made my husband famous when I wrote about how he and I had each solved a problem about a good deal.  I had used an elegant solution, to quote Cathy Fosnot, and he had used the long division algorithm, which was just fine, also to quote Cathy Fosnot.  (You can hear the whole thing here.)

I had occasion to ask my husband to solve some math problems again this week, and I thought I should make sure that everyone knows he is my go-to double-checker. His methods may be old-fashioned, but he gets the job done.

I have applied for a Teacher Learning Leadership Project grant (TLLP) and am pretty close to getting it approved (I HOPE!!)  At the beginning of the month, I received a request for some clarifications about the project, which is apparently what they do.  Makes sense.  I am asking for just shy of the maximum allowed amount of money, so I’m actually glad to know they are making sure people are being fiscally responsible with this money.

One thing  I was asked to do was to make sure my budget aligns with the project goals.  I went over the entire thing with a fine-toothed comb, making sure I had the right number of days, and had figured it all out properly.  I have to account for the number of days each member of the project will be out of the classroom, and how much it will cost to provide coverage for the class.  My principal is joining, so I have also had to account for the extra money paid to a “teacher in charge”.  One member of our group doesn’t actually teach at my school, so I have factored in some mileage for her.  Even though I’ve been over it a few times, I needed someone to double-check it all for me.

I suppose you could say we worked as a team here.  I gathered the information we needed, and organized it into problems.  My husband, new to using an iPhone took one look at the work I needed him to do, pointed at his phone and asked, “Does this thing have a calculator on it?”  I showed him, and he proceeded to answer each question  on it’s own.

Again, the interesting thing here is that he didn’t see that he could solve one and use it to solve the others.  The 4 day option is double the 2 day option, but he didn’t use this. I think I am paying extra close attention to this right now because in my class we have been talking about splitting.  If we know that 40 + 50 = 90, then we don’t have to start over to solve 44+50.  We know it is 4 more than 90!  For some people, this might not be a huge revelation.  But for me, when I first learned to do math without using algorithms, these important connections between problems were completely missing. The only time I used anything like this was when I figured out 3 x 7 = 21 (for example) and then found all the 3 x 7 or 7 x 3 on a Mad Minute and wrote 21.  But I wouldn’t, for example, notice that if I knew 3 x 7 = 21 I could use that to help me with 3 x 8.

Connecting is one of the 7 mathematical practices in the Ontario Mathematics Curriculum: Grades 1-8 (2005).   On page 16, it says:

Experiences that allow students to make connections – to see, for example, how concepts and skills from one strand of mathematics are related to those from another – will help them to grasp general mathematical principles. As they continue to make such connections, students begin to see that mathematics is more than a series of isolated skills and concepts and that they can use their learning in one area of mathematics to understand another. Seeing the relationships among procedures and concepts also helps develop mathematical understanding. The more connections students make, the deeper their understanding. In addition, making connections between the mathematics they learn at school and its applications in their everyday lives not only helps students understand mathematics but also allows them to see how useful and relevant it is in the world beyond the classroom.

Since I have started to focus on teaching students to see the connections in math, I have noticed an increase in their over all number sense.  For many, as soon as they see a connection, it’s like a switch was flipped and they “get it”.

## They heard me. They really did!

Last week, I was ending the week feeling like I may have spent a few days talking to the walls. (You can read about it here.)   This weekend, I feel much better.

We spent the week working on building an understanding of number lines. After making a measuring strips, in groups of 5’s and 10’s, and measuring some things, we needed to start thinking about how a person could skip around on that number line and use it for adding.  When I taped a 100 strip to the board and started asking kids to tell me the number of a certain cube on that number line, it was like a miracle had occurred.  Because nobody could reach the number line to touch each square, and because we’d talked a lot in our math congresses about how we could use the 5 and 10 structure of the paper number line to skip count, they started actually using the number line tool and the skip counting strategy to find the answers I was seeking.  THEY ACTUALLY DID!

Oh, and no big deal, but they were finally counting on from a known number instead of starting back at zero every time.  Seriously.  I’m not even exaggerating to make myself look/feel better.

Here’s the lesson for me:

1. Trust Cathy Fosnot.
2. Sometimes moving forward helps some kids who appeared to not be ready to move on.  I thought I would do a quick number string, sort out who needed some more help with skip counting and counting on, and then make up some Math Workshop groups.  But, low and behold, some of the kids who haven’t been counting on started counting on!  And many who had been fully committed to counting by ones were using the 5s and 10s.

So there you have it:  Valentine’s Day, Winter Electives, and a field trip, all in the same week, and we still moved around on the Landscape of Learning!

math

## What to Look For: Chapter 1

This Summer some colleagues and I are reading a great book together! This is the first time in a while I have done a face-to-face book club, so I am kind of excited about it.  We’re getting together in a few weeks to talk about it! Writing helps me to process my thinking, so I am going to write about each chapter as I read.

There aren’t many pages in this chapter.  I read it in under ten minutes.  But I think there are some important things here.

Alex Lawson defines strategies  and key ideas in this chapter.  These are important to understanding the rest of the book, and really they help with any math teaching conversation, not to mention actual math teaching. They are helpful for that too.

This word, strategies, is one I hear a lot in teaching.  I like this definition.  It think it applies to reading and writing instruction too, expect there I think we learn more about which strategies students are using when we have conversations with them.  They don’t necessarily show their thinking on paper during reading like they do during math.  )(Bonus:  Here is what Jennifer Serravallo says about strategies and skills…all related to literacy work of course, but still an interesting perspective if you ask me.)  In literacy, the goal is that the strategies eventually shift to the background and the child can now be said to have a skill.  For example, when an emergent reader encounters an unknown word, s/he will use some strategies such as “skip ahead” or “consider context” to figure out the word.  S/he will actively consider which strategy to use in a given situation, try a new strategy if the first didn’t work, etc. But by the time that same child is a reading proficiently s/he doesn’t have to consciously consider which strategy to use. Figuring out new words just happens while the child focuses on the meaning of the text. I’m not sure if this happens in math.  I guess it does.  Normal people, not teachers who are analyzing their own work to think about the strategy they used, must just add things up and subtract things without thinking about what they do.  I should ask someone who isn’t a teacher about this.  I wrote last week about asking my niece and nephew math questions.   (here) Maybe this is something for me to pursue with them! (I know!  Don’t you wish you could come to a family dinner at my house?)

I’m rambling again.

My love for Cathy Fosnot runs deep, so I went to look for her definition of “strategies” and found it is much the same. “Strategies can be observed. They are the organizational schemes children use to solve a problem; for example, they might count by ones, skip-count, or use doubles.”  (Found here.)   Her Landscapes of Learning are detailed lists of strategies one would expect to see students use while problem solving in a variety of areas.  In this book, Alex Lawson’s book, the one I am supposed to be writing about, there are great summaries of all the accompanying videos.  Each summary lists the strategies the child in the video used to solve the problem.  On Page 15, for example, the child in that video used counting three times, counting on/counting back, counting on from the larger number, etc.  I’m looking forward to understanding more strategies that students might use in my class.  I think I’m getting pretty good at this, but I still tend to think of one strategy first and forget to think about other ways to solve problems.  This is especially true if I happen to be faced with some math that is actually matches my personal ability level as a mathematician.

In chapter 1, Lawson also gives us her definition of “key ideas.”  (Fosnot calls these “big ideas”) .

The key ideas shown in each video are also listed on the charts in the book, and defined in various places.  I went in search of some other information about key ideas, or big ideas.  The Ontario Math Curriculum defines some of them in the glossary, but they are mixed in with all the other terms rather than highlighted.  It’s still a good resource, if you ask me.  It starts on page 120.  Cathy Fosnot says this here.

“Underlying these strategies are big ideas. Big ideas are “the central, organizing ideas of mathematics—principles that define mathematical order”(Schifter and Fosnot 1993, 35). Big ideas are deeply connected to the structures of mathematics. They are also characteristic of shifts in learners’ reasoning—shifts in perspective, in logic, in the mathematical relationships they set up. As such, they are connected to part-whole relations—to the structure of thought in general (Piaget 1977). In fact, that is why they are connected to the structures of mathematics. Through the centuries and across cultures as mathematical big ideas developed, the advances were often characterized by paradigmatic shifts in reasoning. That is because these structural shifts in thought characterize the learning process in general. Thus, these ideas are “big” because they are critical ideas in mathematics itself and because they are big leaps in the development of the structure of children’s reasoning. Some of the big ideas you will see children constructing as they work with the materials in this package are unitizing, compensation, and equivalence.”

I know Cathy Fosnot is not the author of this book, but I am trying to connect to what I already know, love, and understand (more or less.) So here is a video of her talking about  “Big Ideas, Strategies and Modelling”.  I love how she says that developing Big Ideas represents a cognitive shift for the student.  They show that a student is now understanding and using something they didn’t know before. It causes them to undo previous ideas and reorganize their thinking.   If you want to here Cathy Fosnot talk more about this, try “week three” in the Not a Book Study webcasts. (here)

So there you have it.  My summary and thinking which is probably longer than the chapter itself.  I’m anxious to get going on the videos and reading in the remaining chapters of the book, and very excited about the Teacher’s Math Kit in the second half of the book (and I use “half” here to mean the second part. I haven’t actually mathematized this book to see if it is truly the second HALF.)

A word about the book club: occasionally people I know in my real life say they are reading this blog.  If you are, and you’re thinking, “Hey!  I want to be in the book club!!”  then send me a note.  Comment here, or find me on Twitter (@LisaCorbett0261), or use my board e-mail.  It’s going to be happening on August 9, 2017 in my back yard.  Unless it rains like it has most of this Summer.  Then we’ll find an alternate spot.  Either way, kids are invited and it’s a pot luck lunch. And if I don’t know you in real life and you want to come anyway, well, we can probably work something out.