Adventures in subitizing

The cool thing about doing Quick Images with 5th graders is that you can explain the purpose of it to them.  They understand when you explain the concept of children's mathematical development moving from counting single dots at a time, toward more sophisticated ideas like clumping dots together, or recognizing 5 dots immediately based on their knowledge of that pattern from using dice.  They also enjoy saying the word "subitize" as much as I do.

The down side of QI's and 10 year-olds is that the patterns I had to use were not at all difficult for them.  They enjoyed explaining their thinking, hearing other ideas of how to "see" the solution, and getting to play a fun game, but I found that on day one they weren't really stretching their thinking.

So on day two I modified one of the cards at the end of a string of 3.  We were working on patterns of 4 (four dot clusters).  At this point the kids have all put on their cool jackets, popped the collars, and feel like they can breeze through the quick images - having fun, but not working too hard.  I chuckled inside when I heard the collective "hey!", "what?", "was that..?", and other shrieks of doubt when I snuck this doozy in on them:

Really nothing earth-shattering, and most of them got the answer just as quick as with the others once they refocused, but I could tell they had to do some more mental gymnastics to find this one out.  I only had time to get the solution to this one from a couple of students, but it was fascinating nonetheless.  Rather than the typical counting up, adding on, or simple multiplication, this problem required multiple processes.  Here is how they did it: 

"I recognized the shapes as groups of 4, and saw there were blocks of 8, so I started to just multiply those blocks of 8 times the 4 times across.  But then I noticed some were missing, so I looked at it as 3 groups of 4 on each side (as circled above) and got 3 x 4 = 12, and there were 2 of those, so I had 24.  Then I added the 2 little groups of 3 each, so 6 more.  So I got 30."

"I noticed there were two dots missing right away, so I knew that.  I saw it as groups of 4, and it was 4 groups across and 2 groups high, so there were 8 groups of 4.  That means I had 32, and I know it was 2 less than that from the 2 that were missing, so it was really 30."

When this latter solution was re-voiced, we settled on an equation that looked like this: 2 x 4= 8 (groups), x 4 (dots/group) = 32, - 2 (the dots know to be missing) = 30.

 These kids have some pretty sophisticated strategies for solving these puzzles.  They have asked me when we are going to do some more.  I do have some plans to try some even more complicated dot puzzles like these.  I have and idea to do some with half-dots since we are working on adding fractions, and when we start our intro to geometry we are going to do some where they are given 10 little interlocking cubes on their desk and shown a quick image of a 3-D shape that they must replicate with their pieces.

These kids already have the ability to remember and manipulate the images they see in their minds - that is why some of these patterns are easier for them.  I am extremely eager to see how they might handle the more complex spatial manipulation that comes with the geometric shapes, and how they will group different parts of the shapes together - subitizing with geometry, and actually using the manipulatives in front of them to bring the shapes to life.

Feedback loop

As I teach more and more within my classroom, I am starting to get more and more feedback from my CT.  It has all been really positive and helped push me to want to take on more challenges and try new things with our students.

Here is a typical post lesson recap for us (whenever we can find some of that all-precious time): It usually starts with me talking about what I felt went well, and what I noticed that could use some polishing.  Then I ask a question or two about what she would normally do when she sees the lesson going a certain way, or how she deals with those little things that pop up.  I am usually concerned about one or two things that I noticed, and try to go through my thinking during the lesson, and talk about why I made some of the choices that I did.  I know, me over-thinking things, can you believe it?

Case in point: yesterday I was doing some math problems with my kids on their little white boards.  They are starting to learn how to add and subtract fractions.  Some students really get it already, and some are still a little fuzzy.  Two kids in particular are struggling, so I had spent the previous day working with them quite a bit.  As I was giving them the problems I would wait a bit for the kids to "show me they were done", and they were ready to present by raising their boards.  I so desperately wanted those two kids to raise their boards.  The tricky part is if I linger too long to wait for those kids, I lose the other 20 or so.  It's a balancing act.  As the tasks progressed, I still kept waiting for those two boards to raise.  I could see their brains working, but I could also see the eyes of the other kids starting to glaze over.  As we came to the last of our questions, I was determined to hold out and let those kids get their thoughts out of their heads, give them that extra time to put the answer onto their board and raise it high.

The seconds seemed to stretch on forever (it's amazing how 30 seconds feels like 25 minutes when standing in front of that board).  Finally, a miracle!  The boards came up in the air, and a smile came to their faces.  I even chose one of them to explain their thinking, and they did so flawlessly.  My decision to hold out at the risk of losing some of the kids who were farther ahead seemed justified - I now had a student who previously was in doubt, feeling and thinking of themselves as a mathematician, full of confidence.

But still, I am wracked with guilt that the kids who understood the math had to wait an extra ten seconds.  I feel like I have let them down.  This is when my CT chuckles and says "Huh, I thought that went really well - you gave ______ a chance to show her classmates what she knows.  She seemed excited, that was great.  But you know what you could do?  You can write some difficult 'challenge problems' over on the side of the board for those kids who want the extra work - that'll keep them busy while we let the kids who need the extra time work through the math".  Genius, pure genius.  She always seems to have the most simple, perfect solution for my concerns ready and waiting for me.

I guess this is my long way of saying that the best feedback I get from my CT is the acknowledgement that I am noticing the right things and making sound choices.  I am seeing the tricky areas that I'm not so sure about and am not afraid to admit I don't think something was working.  I appreciate that she is eager to share her ideas on how I could get those sticky parts unstuck, and how she lets me know that every teacher runs into those same "difficult spots".

The feedback I get allows me to enter each experience with a fearlessness and willingness to try things out; to not be so worried about doing something wrong, but excited to do those things that I think are right.

That sinking feeling: My science inquiry project about quicksand

My students are currently working in the Land & Water science kit.  They have been learning about groundwater, absorption, runoff, stream patterns, and rain.  They have big tubs of different types of soil components and they add water.

Really we have just learning about the world around us by making, and playing with, mud each day (how is this not the best profession ever?).

This adding of water to clay and sand got me to thinking about quicksand.  It seemed like a fun thing to play with and it tied in nicely with the whole land and water theme.  I have always wondered if the weight and size of a person made a difference in how fast they sunk in quicksand (there's a sleight chance I may have watched too many jungle explorer movies, or played a few too many games of Pitfall as a kid).  Does the small, wiry scientist sink just as fast as the hulking, neck-less poacher?
 
In order to set up an experiment to find this out I would need some quicksand.  Apparently, in the scientific world, we do our experiments about quicksand using Gooblex.  It's a corn-starch and water mix, also known as a "non-Newtonian fluid", that behaves just like quicksand; quick movements meet fierce resistance, while slow movements are allowed.  I decided to make some in order to see if it would both be doable by my students, and serve our quicksand needs.

  I mixed 16oz of cornstarch with 2 cups of water (green food coloring optional).  The whole thing only took about ten minutes, so it met my time criteria, and it seemingly will work for my quicksand tests.  Even if you don't want to do the science, it's pretty weird stuff and fun to make.

Now that I knew I had a viable quicksand, I could form a question for the kids to explore.  I'm still working of this part, so bear with me, but here's my idea so far:

Question:
Does the size, weight, and shape of an object affect how fast it will sink in quicksand?

My plan is to have the kids make their "quicksand" in a 500ml container (the ones we are using are approx. 5" across and 8" tall).  They can then use masking tape to mark a point 3" below the top line of their "quicksand".  I am thinking that if we use the same size and shape objects (our fixed variable) of different weights (changed variable), we can chart the data on a table like this:

From there we can alter the fixed and changed variables to see the effects of size and shape. 

I may need to shorten this to simply being about weight.  Making the "quicksand" and conducting this first phase of the experiment would be a full days lesson in itself.  The "non-Newtonian" fluid doesn't last overnight, I checked.  The water and corn starch separate from one another and the water evaporates, leaving a hard crusty substance behind.

Well, there it is so far.  I will keep you updated on changes and progress.  Let me know what you think, or if you have any ideas.

Learning to lead

Now that I am back in my main placement, I have been looking back at my time in my Dyad and have noticed some differences about how the kids are learning.

The kids at my main school are fantastic at group work.  They seem to really enjoy doing collaborative tasks.  I have seen these kids on many occasions lead their own own group work.  I believe this is a function of having been given this responsibility throughout the years in their schooling.  With only one teacher in the room, this has been an imperative that they can function on their own.

However, I have also seen these same kids very reluctant to drop this independence and raise their hands to ask for help when they are unsure of something.  It is great that they have been given the sense of empowerment that they can do things on their own without the teachers help at all times, but it almost seems to me that they will attempt to struggle on through until I see them stuck and come to help, when they could ask sooner.

Back at my Dyad placement school, things were quite the reverse.  It was a parent cooperative, and there were always adults around to help the kids.  The kids were used to having help at their disposal and were quick to ask for it.  They often did so before they thoroughly attempted to solve a problem on their own.

The main thing that stuck out for me between the two classrooms, as far as the kids go, was the lack of ability of my Dyad kids to function in a group on their own.  My cooperating teacher and I talked about it.  I wonder if the fact that the parent volunteers run all of the small group activities has taken away some of their ability to lead themselves.  It is harder for them to do because they have never had to learn how.

I'm not sure if I feel that one way is better than another, but it's interesting to see how the different school models may be affecting how kids learn.

This made me happy

I invested a lot of time in creating my Thematic Mini Unit.  Man did I ever.  But pouring so much of my imagination and creativity into the project produced something that was fun to read, but boring as all get out to look at.  Writing can be like that sometimes.  Black and white text just isn't visually stimulating.

So I loaded up the Thematic Unit section into Tagxedo.  It's like Wordle, but with lot's of fun shapes.  Here's what came up.  It's a satisfying visual for all my hard work.  I may have even smiled a bit.

It's like heartbreak & puppies.

I moved on from my Dyad placement kids this past week.  It sucks.  It's like losing 25 tiny little adorable friends, and it hurts.

Is this what it will be like every year?  Unless you are in a looped classroom, in which case you get a one year reprieve followed by twice the pain and heartache the next.  Now I know how those who foster puppies must feel: you welcome a bunch of sweet, innocent, little creatures into your life.  You nurture them, and teach them, and watch them grow and mature, then you give them an awkward hug or pat on the shoulder as they walk away from you forever.

To be fair, it was me doing the walking away this time, but the feeling is still the same.  I was wondering how I would feel when this day came.  I have talked with other educators about it, and they have told me it is a difficult part of the profession.  It's a tough feeling.  The only way I have to describe it is that it reminds me of that gut-wrenching feeling of stark panic I get when my mind flash-forwards 16 years and I hear my about-to-graduate-high-school son springing on me his epiphany to "skip college and live in South America", his time-table for return: unknown.

Not that the image of my son (quite strapping at that age, btw) tromping around the wilds of the Amazon frightens me, it is the sense of loss that tears at my heart.  We will be pouring so much of ourselves into these students of ours, it's painful to think about them moving on from us.

I was okay when I parted from my main placement kids the first time around.  I think deep down I knew I would be coming back.  It was easy to be that stolid, rock of a teacher.  After this experience with my 3rd graders, knowing them for only a short time, I begin to question if I will be able to keep up that facade when forced to say goodbye to my fifth graders.  They will be my first true classroom.  I will be spending so many wonderful upcoming days with them, getting to know them so well, helping them to reach for greater heights.

But that's the point isn't it;  in order for them to go after that brilliant future we have been painstakingly preparing them for, we have to let them go.

Manipulation

I've been on this math manipulative kick lately.

Funny thing is, I used to hate them.  I am a product of numbers-only curriculum.  Most of us probably are.  If we didn't get a math concept as kids, we did 100 more problems until we did.  If we still didn't get it, the teacher explained it louder in hopes that we would.  Naturally I was skeptical of any little toy, widget, or cog that was supposed to explain math.  I knew math as a series of numbers and rules; rigid, strict and to be followed without fail.

Flash forward, like, 50 years.  I get how to do that old-skool math, but I am falling in love with those silly little plastic strips and poker chips.  As you all know, my adult self is All About The Process.  I love the idea of getting my hands "dirty" while learning, and want to share this with my students.  I will do this with those in my classroom.  I want them to actively act like mathematicians while doing math.  Manipulatives give me a way to get them elbow deep in mathematics.

That's right; Mr. K's kids will literally be up to their elbows in math.  Getting physical with our numbers.  Kinesthetic cognition at it's finest.

Numbers aren't what math is all about for me anymore.  Sure, they get the short term job done, but in the long run I feel like I have run into problems because just knowing the numbers and rules have let me down.  At some point they fall short.  I was never taught to seek out those patterns that I could transfer over to suit my needs.  Inevitably you end up stuck and your "rules" falter.  There is nowhere for you to go but discourageville.

Discourageville is a sad and dirty little place that I don't want my students visiting.

If by shifting around poker chips for an hour kids can unlock the mystery of a function of mathematics, then it is time well spent.  Yet, from my observations, those poor little cubes and slats and cards only seem to see that light of day when a child is struggling.  I don't get this.  It seems a natural progression to use them to introduce elements of math in order to show concepts, then move onto the numbers.

Why only bring them in to try to "save a child" when their early use could have prevented the crisis in the first place?