OAME 2013 Presentation: Talk is Cheap

Thanks to those who were able to join us at OAME2013.  We really enjoyed the discussion with both groups that we worked with.  Here are our slides and links to some of the resources, videos, books, articles, etc. that we shared or referenced.

Talk is Cheap OAME 2013

“How Quiet Helps at School” from Prelinger Archives

matching patterns cards

Resources/Books referenced:

Malcolm Swan’s Standards Unit
Jo Boaler’s What’s Math Got to Do with It
From Patterns to Algebra Resource

Blogs we referenced:

Dan Meyer
Matt Vaudrey
Mind Shift: Annie Murphy Paul

Introducing (or re-introducing) solving equations!

I was recently co-planning a lesson with a group of teachers for a grade 9 academic class.  The class had worked a bit on writing expressions but the teacher wanted to introduce equations and solving them!  We looked at a few different problems and then settled on this one:

Capture

from http://www.edugains.ca/resources/LearningMaterials/ContinuumConnection/SolvingEquations.pdf

We weren’t sure how kids would approach it… would kids just do trial and error?  Would they start writing equations?  How long would it take them?

We made the decision to take the part about finding 2 ways to solve the problem out.  In hindsight, we should have left it in.  We ended up asking most groups to show it another way anyway, and often that was the part that challenged the students and lead them to writing equations.

As students worked we monitored to see what strategies were happening.  There was a great variety of strategies!  Not many students did trial and error.  Most did a type of “undoing” of the mathematics (there are 2 frogs and 2 lions, I know the frogs are each 7 so that is 14, 30-14=16, 16/2=8, therefore the lion is 8).  Some were more formal about their work.  We made a decision to put the problem on the board but not give students their own copy.  This really encouraged the use of variables.  Some pairs of students redrew the table but used f’s, l’s, b’s and h’s to represent the different animals.  Some went further and wrote out expressions.  One or two groups wrote an equation and solved it.

As we were monitoring the groups, we tried to organize a sequence that would make sense for students to present their strategies.  Our learning goal (we had this in mind… we didn’t tell the students this at the beginning) was for students to see how the situation could be described by an equation, understand what an equation is and what it means to solve an equation, and start to understand the balance model for solving equations.  With this in mind we found three groups to present to (hopefully) make these concepts clear to the class.

The first group that presented had used the undoing strategy that I mentioned earlier.  This seemed to be a good one to start with because almost everyone in the class could understand what they were doing.

Next we had a group that originally had used the undoing strategy but then extended their work by writing the expressions that represented each row (2L+2F, etc.).  Originally when I saw their work there was no equal sign, but by the time they got up to present they had added the equal sign and the total to each of their expressions.  They hadn’t solved the equations, but they had them written there so that allowed a nice class discussion about why those equations described the situation, what the difference is between an equation and expression, and what an equal sign represents (one student yelled out “it means what is the answer” when I asked this question, but after some turn and talk time he changed his mind).  We were also able to show how solving those equations related to the undoing method the first group had used, and introduced the balance model!

Finally we showed a totally different approach to the problem.  This group had decided that the sum of the rows had to be equal to the sum of the columns, so found the total of column 1 by doing (28+30+20+16)-(19+20+30).  This didn’t relate directly to the algebra we were discussing, but it was so cool and such a neat way of thinking about it we had to share!

It was great to try this problem as an introduction to solving equations.  I think it’s a nice way to initiate some of those important conversations that need to happen for students to understand this concept.

A conversation about 50/50

I was recently working with a grade 6 teacher and his class. They had been working on probability and we gave them a fairly standard question to consider:

  • What is the probability of rolling a 5 on a typical 6 sided die?
  • If you rolled the die 30 times, how many times would you expect a 5 to come up?

Every pair in the class answered the first part of the question correctly but many of them struggled with the second question. What I found fascinating was that many of them believed the answer to the second question was 15/30. It wasn’t until I sat and had a conversation with one of the groups that I realized why. They said “well, there is a 50/50 chance of getting a 5 so that’s why its 15, it can either be a 5 or not be a 5”.

I never anticipated this misunderstanding (perhaps because I’ve never taught grade 6) and wondered how many of my senior students have been confused about this but couldn’t express it as well as these young girls. Is there confusion about 50/50 because it is a term that is used in everyday language and isn’t always used with the same precision that we use in math?

After having a conversation with the other teachers that were working in the room that day with us, it became evident that there were quite a few students dealing with this same misconception and we decided we needed to stop and have a class discussion about 50/50. I am going to do my best to recount the dialogue because I found it fascinating, but it was over a week ago now so there will be some paraphrasing and I can’t quite remember all the names (they aren’t my students).

We first asked the students to talk to the person next to them about “What does 50/50 mean?” Once they had a chance to chat with a partner we had them share their ideas with the whole class. I wrote down what they said on the board.

Student A: it means half and half
Me: Can you explain that a bit more?
Student A: well, 50 represents 50 out of 100 which is 50% which is the same as half
Aiden: Its like if I was walking down the street with my friend and we go past a store and we might go in, or we might not – so there is a 50/50 chance that we will go in
(I wondered if I should question this a bit more – is it really 50/50, does it depend on what kind of store it is…..but I left it for now)
RJ: When I get home from school, there is a 50/50 chance that I will get to watch TV.
Me: What do you do if you don’t watch TV?
RJ: Chores
Me: So, over the course of 4 days, how many days would you get to watch TV?
RJ: (thinks for a bit) about 1 or 2
Owen: If you draw a target and cut it in half, you would put chores on one side and tv on the other.
Me: (I draw the target as the student describes)
Student B: It has to be equal percentages
Me: (I draw another target with chores as 3/4 of it and TV as 1/4) So is this 50/50?
Student B: no – that is 75/25 because chores is 75% of the circle and TV is only 25%
Me (asking RJ): Is this more what your situation is like?
RJ: smiles and says yes!

We move on to the next question: “Is the probability of rolling a 5 on a regular die 50/50?” and give them time to talk with a partner about it.

Student C: “No, because there are 6 different numbers you can get and the 5 is only one of them”
Tiffany: “It would only be 50/50 if there were three 5’s on the die”
Me: “What do the rest of you think about what Tiffany just said” (several nods) So, are you saying that if the die had numbers 1, 2, 3 and then 5, 5, 5, that it would be 50/50?
Tiffany nods yes but hands are frantically going up as I say it.
Syndan: I disagree with Tiffany. I think it would have to be three of one number and three of another number.
Me: So the die might have the numbers 3, 3, 3, and 5, 5, 5?
Tiffany: Yeah – I disagree with myself too. I think its what Syndan said.
Me: Can someone explain why?
Tiffany: Because for it to be 50/50, half needs to be one number and half needs to be another number.
Student D: (Comes up and draws a target next to the other ones) If you drew a target for a regular die there would be six pieces and the 5 is only one of them so its not 50/50.

At that point our time was up for the day and I walked away amazed at the level of their discussion, willingness to put out their theories and challenge one another. The entire conversation fascinated me and exhausted me because the things they were saying I had not anticipated. I didn’t expect them to be so clever, especially about changing the die.

I’m so glad we decided to stop our plan and take time for the conversation!

Symbaloo – a nice way to organize resources!

One thing about 2013 that I love is the mass of information and resources that are available to us as teachers on the web.  The problem that I’ve had though is that it is almost overwhelming!  I have had a HUGE list of bookmarks that I try to go through every so often, but that is time-consuming and I often can’t remember which are my favourites of my “favourites” on my browser.  I just discovered Symbaloo that lets you make a homepage with all of your bookmarks on it.  The thing that makes me happy about this, is that I will be able to find these sites on any computer, not only my work computer.  I took a few minutes this afternoon and sorted through my favourites.  Here’s the link for what I’ve come up with so far:     Math Sites

Equations, expressions and inequalities, oh my!

I was in a grade 9 class today trying out a lesson we co-planned about equations.

The discussion with the teachers when planning the lesson was around whether or not students understand what an equation is, and if they understand what it means to solve an equation.

This is what we did:

  • Students, working in pairs, a set of cards revised algebra sorting cards. Students were given the instruction to organize the cards into groupings that made sense to them. I reinforced that their groupings had to make sense, in that, every card in the group should be there for an obvious reason- they should all have something in common.
  • We then asked the students to visit 3 other teams and try to figure out how they grouped their cards. This was an interesting task and involved a lot of arguing.
  • We then did a stay and stray. Each pair of students choose one person to stay with their groupings and then the other partner visited the other teams to check their guesses for how their grouped their cards and to ask for any clarification.

The groupings that students came up with included: grouping by type of numbers in the expressions (these have fractions, these don’t), by whether or not there were variables, by the number of terms on either side of the equation, by formulas, just numbers, just one variable, etc. What was impressive was the level of engagement on this task and the conversations that they were having. They were talking about equations, exponents, fractions, binomials, distributive property, etc. They were really thinking about how they wanted to group them and were often struggling with where to put the last few cards they had.

Students all returned to their seats and then I shared with them how I had grouped the cards and asked them to try to figure out my thinking.

This is what I showed them: algebra card groupings. (note: there are a few equations in the first column that are actually identities and I thought of putting them in a separate group but decided not to have this conversation yet. I think a nice follow up to this conversation would be to give them cards with just equations and talk about which ones are sometimes true, always true and never true) They had a tough time with this. I gave them some time to talk with their partner about it and then we did some sharing. A few of them took some guesses but we often would find one of the cards out of place based on their guesses. For example, a student said that “group 3 has more than one thing on both sides of the equation” but then realized that there was one like this in group 1 too. Eventually we got to group 1 being “things we can solve”, group 2 being “things that don’t have equal signs” and group 3 being “things that aren’t true”. I helped them with the terminology equations, expressions and inequalities.

We spent a bit of time talking about the card that said “x + 1 = x + 2” and why that wasn’t an equation. One student explained that “since the x’s both have to be the same number it is impossible – you can’t add 1 to a number and then 2 to a number and get the same answer”.

We moved the conversation to the first column of “things that can be solved” and talked about what does it mean to solve. Students actually seemed to be very comfortable with the idea that a solution to an equation is an x value that makes the statement true.

The last task was to choose one of the following: a) make up an equation that has a solution of x =2 b) make up an equation that has a solution of x = -3 or c) make up an equation that has a solution of x = 1/2.

Most students chose to do a), a few did b) and no one tackled c) – but it is early in the unit and the teacher will likely have them return to this and try to do c) later.

I was impressed with the complexity of some of the equations they came up with. Here are some pics:

equations with solution x = 2

soltuion x = 2 more solution x = 2 more solution x = 2

equations with solution x = -3

equations wtih solution x = 3

Considering that the students haven’t actually worked with solving equations yet this year they seemed to have a pretty good understanding of how to create equations with a specific solution.

Calculators or Not?

This question has been nagging in the back of my mind for the last couple of months.  Calculator technology is readily available to us… whether it be a simple computation calculator or a complex graphing calculator, we literally have the technology at our fingertips.  “But what about when you are in the real world?”  you say.  “You won’t have your graphing calculator with you in the middle of the grocery store, will you?”  But we do!  We have our phones, our ipods, our ipads… we are almost always within reach of a digital device that has some type of calculating power!  So is there really any good reason we should ask kids to not use their calculators? 

This has been my struggle… but my thought right now is yes!  Yes, there are times that telling kids not to use a calculator can help develop there number sense.  After all, a calculator is only as intelligent as the person using it, right?  If students have no sense of the operations that need to be done, or if they have no sense of the reasonableness of an answer, the calculator may be of no use at all. 

I have two examples to support my position (although I am up for debate about this!):

1)  Fractions:  What if I told you a race course, 26 miles long, had water stations at every 1/8th of the course. How far did you run when you had come to the 5th water station?  (Problem courtesy of Cathy Fosnot).  If you had a calculator you would solve this problem very differently than without a calculator.  Both are correct, both are valid, but as a teacher I need to make a decision based on the number sense I want to bring out.  If it’s decimals, or fractions to decimals, I would say using a calculator is the way to go with this problem (or at least leaving it as an option).  However, if I want kids to explore and play with fractions then telling them not to use a calculator for this problem will encourage fractional reasoning. 

2)  Factoring:  I’m jumping up to grade 10 now.  I’ve always enjoyed teaching factoring, but there is technology available now that will factor for you.  All I have to do is type the function in, and voila!  It’s factored!  So maybe I should just skip factoring and use the calculators to help us explore the deeper concepts of roots, curve sketching, etc.  Although I do feel that factoring with calculators can be extremely useful in some cases (so that the factoring doesn’t get in the way of those deeper concepts), I think that it is important that students understand the relationship of factoring to multiplication and division.  Algebra tiles are a great way at getting at that understanding.  I love how we can relate the array model of multiplication to expanding and factoring polynomials.  If we don’t let kids play with these concepts then they won’t have the opportunity to see the connections in the mathematics. 

Alright, that’s my argument for now.  Feel free to debate me!  My opinion is fairly fragile so I need some conversation to convince me (just like how kids reinforce math concepts, get it?!)!! 🙂

   

Presenting the Problem… some new thoughts (for me!)

I watched a video the other day that made me think.  It was a video from one of Cathy Fosnot’s units (anything from her tends to make me think in a way I never have before) and showed a grade 6 teacher presenting a problem to his class.  He started off telling the students about his kitten and the special kitten cat food he needs to eat.  He told them about two different stores in his neighborhood that sold this cat food, but had two different pricing “deals”.  The teacher said he needed the students’ help to find out which was better.  Right off the bat, I like this approach.  Relating a problem to the teacher’s or students’ world is definitely a point towards engagement (if it is authentic, or at least seems authentic).  But the part of the video that made me think was what happened next:  The teacher paused for a moment after presenting the problem and then students, without being prompted or asked, started raising their hands.  When he called on them they proceeded to talk about the strategy that was coming to their mind about how to solve the problem.  That was kind of cool!  It not only gave the teacher a look into some of the initial ideas the students had, but also let other students gain an entry point.  In fact, after a few students had said their brief and rough strategy thoughts, the teacher asked “Does everyone feel they have a place they can enter this problem?”

I’ve never done that before! I think I’ve always thought that it would “give away” too much, or lead kids in a direction they might not have gone otherwise.  But I love that it gives everyone a starting point!  I’m pretty sure it wouldn’t be appropriate with every problem, but I am excited to try it with a few problems and get a feel for when it works best.

On the same note, I have been thinking about the importance of making sure kids understand what a question is asking before sending them off to work.  I’ve realized recently that the contexts that seem very familiar to us as adults, aren’t necessary as obvious to students.  I’ve been looking for strategies to help students with this and here are some things I’ve seen.

-presenting the problem and then asking students to turn and talk to their partner about what the problem is asking.

-asking students to take a minute and write down any questions they have about the problem (the teacher than has them take those questions to their partner and they answer them together)

-taking specific words and asking a student in the class to explain what it is (e.g. what is orange concentrate anyway?)

I think literacy teachers call this “decoding”.  I feel like mathematics teachers are doing that twice: we need to make sure that students understand the language and context of the problem and we also want them to have an entry point somewhere to get into the rich mathematics of the problem.  And then there’s also Dan Meyer’s philosophy about engaging through perplexity…. we need to get that in there too! So much to think about!