# 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?!)!! 🙂

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# 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!

# Measurement systems – ho hum…

I’ve never particularily enjoyed teaching the measurement systems.  We have expectations in a few of our courses where students need to work with the imperial and metric systems and I can just never come up with good ideas to make it interesting.

In an attempt to find some resources for teachers to use in the grade 10 applied course I came across the story of the Gimli Glider, a plane that had to glide to land after running out of fuel mid-flight.   The incident happened at a time when Canada was moving to using the metric system and the lack of fuel was due to an error in conversion between imperial and metric.  The wikipedia article linked above talks about the actual error that occurred.  I also managed to find video of the CBC news story that talked about this incident.  Perhaps by sharing this video and story and having students check the calculations of the crew and pilot, students might understand the importance of understanding these measurement systems in many careers.

When I found the CBC video I also discovered a collection from their archives of a variety of news stories from the time that Canada was switching to metric.  There are stories of retailers being arrested for advertising using the wrong system, debates about which system is better, and radio shows suggesting that converting to metric was a communist plot.  I’m not sure the debates and radio show would interest students, but I bet news stories about being arrested for using the wrong system might provoke some interesting conversations.

In fact, that got me wondering – if people had to advertise in the metric system back then, have things changed?   I opened up a Home Depot flyer and was interested to find that most linear measurements and area measurements were given in imperial but volume and mass measurements were usually metric and sometimes you see both woven together such as 8mm laminate flooring for \$1.99 per sq ft.   I wondered about grocery flyers and found that often weights or mass are given in both metric and imperial but the imperial was usually written larger.  Volume again was almost always metric but you sometimes see pints or half pints of fruit.

Now I wish I had always started the unit off with this activity!! Instead of arguing with students about why they needed to learn both systems, give them a bunch of flyers and have them highlight all the instances of metric in one colour and all the instances of imperial in another.  I would think after that activity they would see why they might need to have some understanding of both systems even though Canada is metric.

# So the speed of the car is…..240000km/h

Arrrrghhh….don’t you hate it when students write down answers that are ridiculous and seem oblivious to the ridiculousness of it. I am at least happy when they write me a note that says “I know this can’t possibly be right but I can’t find my mistake…”. Then I know that they are at least reflecting on what they have done.

We have had quite a few conversations with teachers lately about students judging the reasonableness of their answers and how it is a source of frustration for many of us. So this is something I have been thinking about over the past couple of weeks.

Is it really their fault that they don’t do this enough? I have been reflecting on my own practice and specifically thought about:

• How often did I ask my students to consider and discuss what might be a reasonable answer before they began a problem?
• How often did I give them opportunities to share strategies for estimating what might be a reasonable answer?
• How often did I give them questions that ended up with unreasonable answers (because I just chose the numbers quickly without thinking) and when a student questioned the reasonableness I told them just to ignore it because I just made it up.

I also thought back to my own experiences with estimating in elementary school.  I don’t ever think I was taught strategies for estimating.   I actually recall, being asked to estimate when adding two 3-digit numbers.  I think it was suggested that we round the numbers first and then add them to find an estimate.   Unfortunately, the only strategy I really had for adding 3-digit numbers together was the traditional algorithm so it seemed ridiculous to me to round the numbers, line them up, add them together and then do it all over again with the actual numbers to get the exact answer. So I just would add the numbers together using the traditional algorithm and then round my answer and write that down for my estimate.

What started me thinking about this was the day we were in the grade 5/6 class from our last post.  The teacher started the the lesson by asking the students the following question:

” A parent brought in 100 timbits for our class of 22 students to share.  Would every student get at least 2?  Would every student get 10?”

He had students think about the first option – Would everyone get at least 2?  Students shared their strategies for checking this:  some skip counted, student by student by 2’s, others were comfortable just doubling 22 and saying that there would definitely be enough.   When asked if everyone would get 10, some students just multiplied 22 by 10 since it was a nice number to multiply by, while others counted by 10’s around the room and realized that they would hit 100 too soon.

This approach allowed students to say that the answer should be between 2 and 10.   Several students talked about how they could narrow it down more and were comfortable saying that it would be between 4 and 10.

I started thinking that this is a nicer way to think about estimating.  Why when I was estimating in elementary school did I always have to come up with a single number for my estimate.  In fact, I think that is part of what bothered me – I knew that no matter what I wrote down it would be wrong.  I think I would have been much more comforable coming up with a range of what a reasonable answer would be:  “my answer should be between 100 and 150….”.

Dan Meyer’s Three Act Math videos encourage this idea by often asking students to make predictions and write down an answer that they know will be too high and too low.

We tried the You Pour, I choose problem in a class yesterday.   However, this problem starts by asking students to guess which glass has the most pop (or soda in US english 🙂 ).   I tried to get students to write down their guess on a whiteboard.   It was really difficult to get them to commit to one.   I still think that having to write something down, that might possibly be wrong, is what was holding them back.    It is amazing how the fear of being incorrect in math, even when just estimating or guessing, stops students (and adults) in their tracks.

So, this is something I am going to be working on and thinking about more.

Intentionally talking with students about what a reasonable answer might be, before they begin the problem, and sharing strategies for coming up with their estimate.   I also want to test my theory that, when possible, asking students to come up with a range for their estimate, instead of a single number, might make them more willing to commit to putting something down on paper before they begin.