Estimation

Like I already told you I was going to (here), I started talking about estimation in my class last week.  We read a book called Great Estimations by Bruce Goldstone, tried out a few of the problems he posed, and then tried some of our own.

I had prepared some bags of stuff for us to estimate in advance. For each item, there were two bags: one had 10 of the thing in it, and the other had an unspecified number of the thing in it. Here you can see I used Mike ‘N Ike candy and mini marshmallows. I also had popcorn kernels, elbow macaroni, and Cheerios.

I gave a set to each table, and asked them to estimate.  They also had a piece of paper they could record their thinking on.  At the end, we shared our estimates.  We did not get an actual count of the items.  This was on purpose.  I wanted them to feel like their estimate was good enough.  Mostly their estimates were in close proximity of each other, and I complimented them on that.

The following day, I have them 2 bags and an item.  They were to put 10 in one, and count as many as they wanted for the next. I gave them Lego, glass beads, counting chips and colour tiles.  Most groups put around 30 in the bag. Just we had the day before, they traded bags with each other until they had an estimate for everything.  Then, we shared our thinking, and confirmed the count for everyone.

One group thought it would be funny to put over 100 counting chips in their bag.  They were each counting out 100, rather than working as a team, so we had a good conversation about that. When another group got that bag, they were sure it was impossible to estimate. Imagine their surprise when their estimate was within 5 of the actual number!  That group was composed of grade 3 children who are still adjusting to the idea that they are the older students in the class now, so I think this was a good confidence booster for them.

This is what I discovered:  they are mostly pretty good with recording their thinking so they can share it later.  We do need to do some work on labelling.  We also need to work on each person contributing to a group assignment or task.  In each group there was a clear leader who railroaded, or attempted to railroad, the rest of the group.

This is what they discovered: they really were training their eyes (as it says in the book) and their estimates were closer to the actual counts as we went on. In some groups, they discovered the need to label.  They had written a number, say 34, but didn’t label it as “colour tiles = 34”.  When it was time to share it was tricky to share! I like that they discovered this on their own.  I know I will need to talk about this again, but I’d say about half of them were able to identify this as an important thing to do going forward.

Using the website www.estimation180.com as inspiration, I am going to create some more provocations for my students to explore.  I want to add this as an activity for them to complete during Guided Math.  (Yes, I am still trying to figure that out!)

My hope is that these estimations activities, revisited throughout the year, will help my students develop a stronger sense of numbers.  I think they will develop a better understanding of magnitude, and that the numerical reasoning skills will improve.

Finally, here is an article from Math Solutions that has had me thinking about all of these things.

Number Talks: Week 1

I hadn’t decided yet if I wanted to start off by making this sound like I am a genius (which I totally am!) or if I was going to list all the sources of this idea and talk about how they converged into this seemingly-original idea.  But then I listened one morning this week to a podcast from Derek Rodenizer on voice.ed radio.  (You can listen to it here.  It’s only 5 minutes!)  He posted this graphic later on Twitter:

That podcast made me realize that this Number Talk idea was just that – an example of how sharpening my pedagogical sword gave me an idea that wasn’t/isn’t truly mine but is me knowing a bunch of stuff and then putting it all together and coming up with something different.

The “Introducing Multiplication” unit by Marilyn Burns starts with an activity where students are asked to work with partners to make posters about numbers.  Specifically, students are supposed to write down things that come in 2’s, or 4’s or 7’s, etc.  So this is partly that idea.  If you look on Pinterest under “math activities” you’re sure to see art projects students have made where they have written down all the ways to write a number, or what the number means to them. (If you don’t know what I am talking about, look at this.) This is partly that idea too.

On the first day of school, I told my new students I was going to write a number on the  board and they were going to tell me everything that number made them think or remember.  I then talked to them about how we put our thumb on our chest to show we have an idea, and that they could take as much time as they needed because math is not a race.  I wrote a 2 on the board.

I am not exaggerating when I say we talked about this for probably 15 minutes. Here is our result:

It was so interesting and, I thought, successful that I couldn’t wait to do it for the rest of the week.  Here are our results:

I do not think it is a coincidence that we got more information for 2 and 5 than we did for 23.  This is a grade 2/3 class and most of the ideas on 23 came from the 3’s. Also, there aren’t a lot of every day things, like bike wheels and toes, that come in 12’s and 23’s.

There were some common themes:  each day they wanted to talk about someone who was the age that matched the number.  They know people who are 2, 5 and 12 – mostly siblings.  But 23 is too young to be one of their parents, and too old to be a friend or sibling (for most.)  When we talked about 23 as an age, it was clear they don’t really know what 23 years-old looks like. They guessed that I am 23, which I totally look like I am but I am not. They then guessed that our principal is 23, which is what she actually looks like (no this is not an evaluation year for me.) We settled on “Justin Bieber and college students” after some discussion.  Everyone finally agreed that a 23 year-old is a “young grown-up”.  But I think many of them are still not 100% clear on this idea.

My original goal was to start using Number Talk routines – mainly the one with the thumbs up instead of hands up.  I also wanted to start talking about them building on each other’s ideas, which they did.  I was busy building routines, not taking notes.  But I still feel like I got some important information from some of my new students.  This will help me going forward.  I had originally planned on doing dot-number talks for the week.  I have already prepared slides to use for this using Smart Notebook software, even though I do not have a SmartBoard in my class. I kind of chickened out because the tech-gremlins have been busy at my school over the summer and things were randomly not working.  I didn’t want to start off my school year muttering curses at them while repeatedly jabbing buttons.  I’ve saved that for the coming week just to give myself something to look forward to!

To read more about how a dot number talk can be used to start the year, check out this great blog post!  https://heidiallum.wordpress.com/2017/09/09/the-power-of-dot-number-talks/

math

New School Year Eve

Tomorrow is the first day of school for teachers in my board, and really most of the province.  The students don’t start for 6 more sleeps, but tomorrow morning school libraries across this vast city will be full of educators ready to discuss how we are going to make this year the best one yet.

Or, maybe, full of people hoping we’ll have a few minutes at the end of the day to make sure the classroom is ready.

From home, on rainy days, I have been making plans.  I think I know what I am doing on the first 2 days.  I’ve spent a lot of time thinking about how I am going to start math this year.  Every August I feel like I have never taught before.  I can’t remember anything I did on the first day in previous years. I have vague memories of starting off with art projects and activities related to identifying 2D and 3D shapes, or of asking students to play math games.

The awesome math facilitators in my board have created a “First X Days” of school document for students in junior and intermediate grades, and are working on a primary version.  I attended workshop in June and so have decided to arrange my first 2 weeks recommended in this document.

I am really excited about 2 of the activities! I mean, I’m excited about the whole thing, but REALLY excited about 2 specific things.

First, Amy Krause Rosenthal and Tom Lichtenheld wrote a really great book called “Duck! Rabbit”.  Throughout the book, there is one drawing and unseen people are arguing about whether it is a duck or a rabbit.  (I’m assuming they are people.  It doesn’t really say.)

Each gives his/her/its well supported opinion. I think it really shows an important idea we need children to consider in mathematics:  There isn’t always one right answer. Two people can have different opinions and still be right.  The important thing is being able to justify one’s opinion.  (PS:  I know that’s not just in math.  I’m trying to stay focused here though.)

After, we are going to read this awesome book:

I’m waiting for some permission to photocopy a page for students to write on.  I am going to ask the students which one doesn’t belong, and then let pairs of students talk for 10 minutes about their opinion. Then they will present to the larger group.  My goal with this day’s activity is to have the students really think about justifying their thinking.  I want to introduce that word to them early on.  I’ve read Cathy Fosnot’s book “Conferring with Young Mathematicians” and she talks a lot about asking good questions that get the students to really reason their way through and around an answer so they understand their thinking deeply, rather than just accidentally stumbling upon the right answers and moving on. That should get us through one day of the first week.

The second thing I am excited about is also related to a read-aloud (because that’s how I do things!)  This one is called “Great Estimations” by Bruce Goldstein.  Inside, there are pictures  of every day items.  A small amount is sectioned off and labeled, a larger amount is also sectioned off and labeled, and then there is a large amount of the same item and readers are asked to estimate how many there are.  Here is an example:

After we talk about the pictures in the book, which will be too far away from the students to count by ones using their fingers to touch each one therefore “forcing” them to estimate, I am going to send them off to their tables.  On each table I will have 2 jars of everyday items.  One will be full, the other will have 10 of that item.  I am going to ask them to work in pairs and estimate how many are in each jar, record their answers, and then move on to at least 3 more sets of jars.  I am thinking this will take 2 days.  I’d like them to have to work out a strategy, and then use it again after a good night’s sleep to make sure they are really getting it.  I’m also hoping this will reinforce the idea that we must record our thinking clearly!  The estimation jars are not my own original idea.  I know I have read about them in an Effective Guide to Mathematics Instruction, but I can’t remember which one and I am not going down the long and windy road of www.edugains.ca to try and find it for you.  You’re on your own! Or you could just take my word for it. I’ve used them many different times, sometimes even having a new set of jars every month and revisiting this idea of estimation often.  I like it, the kids like it, and they develop a pretty good sense of magnitude, comparing, and counting if you ask me. Oh, and estimating.  They get pretty good at estimating.

In conclusion, I think this really will be my best start yet.  I am loving this idea of focusing on setting up the class so that our first 9 conversations are all about problem solving as a good thing, mistakes as an important part of learning, and about being able to participate in a productive argument about your math.  I’m glad I had someone else guide me in that direction.