In preparation for an in-service workshop I am running, I wanted to get my hands on Dr. Smalls book Big Ideas from Dr. Small. I didn’t want to purchase it, but I didn’t know anyone who owned them.
Then I remembered seeing a link for the Margaret Wilson Library in email and other correspondence with the Ontario College of Teachers, and some vague recollection that members could borrow books for free.
I decided I would give the service a try and crossed my fingers that they had the book I was looking for. They did! Here is how the service works:
First, a search for the book using the online catalogue.
Finding the book using the online catalogue was easy as pie! I found the book and “checked it out”. I had it delivered to my home, but you can have it delivered to your school if you prefer.
A couple of days later the book arrives in the mail!
It arrived quickly! It took only a couple of days. Set aside the mailing label that is tucked inside, and be sure to keep the packaging. You will need to use that to mail it back. Inside is also a bookmark with the due date marked on it.
When you are ready to send it back, prepare the pre-paid mailing label.
The mailing label needs to be cut to size to fit properly onto the label. Just cut along the dotted lines. Make sure you keep the other half though; it contains the tracking number for your package.
Put it in the post.
Voila! It was super easy.
If you are not finished with the book yet, you can check it out again online provided no one else has requested it.
Now that I know I can get books so quickly and easily from Margaret Wilson Library, my only problem is finding time to read them!
Today I posed a question to my grade 5/6 students that I came across on John Stevens Twitter feed (@jstevens009).
The question I posed was “Would you rather have a stack of quarters from the floor to the top of your head, or $200?”
I asked students to think about which they would choose, and to explain why they made the choice that they did.
This was the first time we have done a “Would you rather” question in math, and I was anticipating the mathematical discourse and thinking that this activity would prompt.
There was some interesting conversation that happened before students started on their written responses that were not captured in their written justifications. I wish there were some way I could capture that.
Here is a justification one of my students :
M, Grade 6
- This student didn’t take any chances estimating the thickness of a quarter; he asked me for one so he could measure it! I gave him two and he measured them on a ruler and determined that one quarter is about 1mm thick.
- He then used a series of ratios to determine that 1000 quarters would be one meter.
- He did a rudimentary measurement of his height using a meter stick and determined that he was 1m and 42cm tall.
- He knew that 42cm would be 420mm, or 420 quarters.
- He determined his height in quarters: 1420 quarters tall.
- Using long division he determined how much money in dollars 1420 quarters comes to by dividing it by 4. He got $355.
- He chose to take a stack of quarters to the top of his head
Unfortunately, there were many responses from my students were not justifications at all:
R, Grade 5
The question I am grappling with right now is what I am going to do to address the answers that do not include any mathematical thinking or any justification. There were more students than I wish to admit did not have any justification to their answers. Today was our first day back after Christmas break so maybe they are just a little rusty. It was also a new activity; perhaps I didn’t explain my expectations very well.
Either way, I there are some things I need to address with students. I am considering modelling a justification to this question or one similar, but I don’t want to prescribe how students are to think about the problem… the point of the activity was for them to decide on an approach not for me to choose one for them.
I am also considering creating some sort of rubric for justifications that will give feedback for improvement that somehow does not force a particular strategy. I use a checklist for questions students need to explain their thinking (Answer, Why, Examples, Generalize, Clarify, Limitations), but it doesn’t fit well for this Would You Rather activity.
I typically teach grade 5 and 6, so when I helped out in a grade one/two class and the students were asked to represent a 2-digit number four different ways, I drew a little bit of a blank. I immediately thought of representing the number with base 10 materials, and then thought of using tally marks, but that was as far as I got! What are four different ways to represent a two-digit number?
Using some of my own ideas, and watching and learning with the kids, here are some representations we came up with:
1. Drawing Shapes in a One-to-One Correspondence
Most students started with representing their number using shapes; one picture or shape until they reached their two-digit number.
It was a lot of drawing, and a lot of counting, and some kids found it challenging to count what they were drawing accurately. Most students drew their shapes in rows, which was easier than counting 30 or more shapes randomly arranged on the page, but there were still some counting errors. We talked about strategies for counting accurately, but some students still found this difficult.
There were also a number of students who thought that drawing squares and then drawing triangles counted as two different ways to represent their number. It opened up discussion that representing a number with a one-to-one shapes or drawings was one strategy, and choosing a different shape did not change the strategy.
2. Tally Charts
The second-most popular representation students used was the tally chart. I could tell that this was a representation they had been practicing. Almost all students used this representation, and used it confidently.
Many students checked their work by counting the groups and skip counting by 5s, but some counted each tick individually if they lost where they were or wanted to double check.
3. Ten Frames
Students had only very recently been introduced to the ten frames. Some students shied away from using it because they weren’t confident enough to try it even thought they identified as a representation that could be used. Others got stuck on trying to draw the ten frames, drawing a 10 frame with only 8 squares.
If they made mistakes drawing their ten frames, they typically did not catch their error because they checked their work by counting by ones instead of tens. They did not realize that a 10 frame holds ten. A rather interesting error. I thought some students might count their 10-frame as 10, but not a single student did.
4. Groups of Ten
Students made the connection to this activity and grouped shapes into sets of 10.
5. Number Sentences
Before students set off to task, the children participated in a class brainstorm and discussion to come up with some ideas about the different ways a number could be represented. One of the ideas that was shared was that number could be represented as an addition or subtraction sentence. Very few students used this representation, although many students had not finished the task before it was time for me to go. They got so caught up in the counting I think, that they forgot about number sentences.
Some students were not able to come up with 4 different representations. Some students represented them inaccurately because they mis-counted, or drew their 10-frames with only 8 squares.
How do we get students comfortable with all of these ways of representing number? And how do we get them representing number accurately so they don’t make mistakes counting or drawing 10 frames?
I created a chart to summarize the ways that grade 2 students can represent number. Feel free to use it in your classroom.