## Saturday, February 9, 2013

### Why Teach Problem Solving?

So, I attended a professional development session yesterday on early numeracy, and the presenter showed this video.  I almost snorted Diet Coke out my nose!  He said that when he has sessions for teachers (we were all instructional coaches), he asks them to rate on a scale of 1 to 5 how important they feel it is to let students struggle.  Usually, the ratings are pretty high.  He then asks them how that belief is reflected in their teaching practices.  Ouch.  That's usually a tougher question.  It's so hard to let kids struggle and not jump in to help.  The presenter reminded us of Vygotsky's Zone of Proximal Development and that the goal is to assign tasks just beyond what a student is comfortable with...not WAY beyond, but just beyond.

Here's an example to consider.  Say that you've taught fractions and now you're moving on to fractions on a number line.  Your first thought might be that you need to teach it.  Hmmm, well let's see.  Try this instead.  Display the number line shown below, and ask students what they think A represents.  Accept all responses. Don't forget to ask students why they responded as they did.  I will assume that no students responded correctly with 3/4.
Now, without saying anything, change the number line by drawing in what you see below.  I'll lay you odds that you'll have a chorus of ooh's, ah's, and light bulbs going off!  Have other number lines, split into different fractions, ready to go and have students come up and practice (a document camera or interactive white board would both be good for this).

Next up, fractions greater than one on the number line.  Now, this next part is assuming that the students have already explored fractions greater than one in their study of fractions and have practiced fractions less than one on the number line until they're comfortable with it.  Instead of teaching a full-blown lesson on extending to fractions greater than one on the number line, put up the number line shown below.  Huh...I wonder what A stands for here? Let the students struggle with the problem for a short while and see what happens.
Click here to grab a couple of sheets of number lines with fractions. Check out this blog post for instructions on how to use the snipping tool in Windows to insert the number lines into your own documents.  :)

1. I am just starting fractions so this is awesome!

1. Awesome, Tammy! Glad it was timely. :)

2. Perfecto! This post came at the perfect time!!! Thanks again for sharing your expertise!!!

Andrea
One Teacher's Take

1. Hey, Andrea! Love your blog post on the 1/100 day of school!! Awesome extension.

3. That's great! Love it!! :)
Jessica
Ideas By Jivey

1. Thanks, Jivey! Neat little fraction freebie over at your blog!

4. I can't wait to try this out! It makes complete sense to me this way.

1. And it does for the kiddos, too! Glad you like it. :)

5. Oh, my goodness! The snipping tool - a piece of heaven under my nose all this time! You have saved me countless hours for creating class materials. I can't believe no one has shared this before!

Thank you for all of your great tips and knowledge. You are always an inspiration. I shall be using the fraction number lines in the next week.

Yes, it is so important to let kids struggle with problem-solving. I have to keep mum and stop watching the clock and thinking how I'm not "getting anywhere" while I give them this valuable time to process and make connections.

Val

1. Yes, Val, the snipping tool is a dream come true! I literally don't think I could function without it. :)

6. The concept of 'struggling' and ZPD is interesting, and comes up at my school all the time. We received some funding this year to do PD on executive functioning. From this, many of our teachers have taken to breaking down tasks so that each child knows exactly what to do and feels successful. They color in clocks to show how long we have to work, make examples and model carefully so that all students know what to do to complete the task. In some ways, this is nice, but it is draining all of the creativity and innovation. (I think these strategies would work better if teachers practiced more gradual release of responsibility, but that's another story.) I may show them this video!

1. Yikes! That teaching strategy doesn't sound like something I'd be comfortable with. I'm a little too "outside the box" for that. It may help kiddos in the short run, but I'm not sure about the long term impact.

2. I struggle with it, too! (And I never use those colored in clocks. I could only conceive of using it as a manipulative for teaching fractions.)

Another one are these three colored placements -- Ready - Do - Done. Students need to envision what they need to get ready, what they need to do, and what it will look like when it's done. I'm all for students being prepared, and for knowing what it looks like to be hard at work, but it's stifflng! What it looks like to "work hard" may change fluidly from task to task, or from student to student with their different approaches. And, most significantly, the "done" part may look (often should look!) different from student to student or group to group.

I'm all for students being set up for success, and I would never want to deny students with executive functioning needs strategies that may help them, but how can we bring the creativity back to our classrooms? How can we prepare students with (buzzword alert) 21st century skills if we are so focused on this? If anyone has any suggestions, I'm up for hearing them.

7. I've done something similar for introducing fractions on a number line!

I have also used cuisenaire rods to teach this, particularly struggling students. I guess it would be the concrete step before the lesson you described! I generate number lines on centimeter grid paper (e.g. 12 cm long lines, from 0 to 1). We then make a quick map of what we know about fractions: are numbers, have equal parts, can be shown in multiple ways, remind us of division, etc.

We then try to find 1/2, 2/3, 3/4, 5/6, 11/12, etc., on a number line that is 12cm long. Students quickly release that the dark green/6cm cuisenaire rod fits along the 12cm line twice, dividing it into halves. The tricky part will then we what do we label: the interval or the line. (We remind ourselves what we know about number lines, and how they look.) Students can also generalize about which fractions will work best on this 12cm number line. Some students also notice that I always chose a fraction that is '1 jump away from 1,' and that the numbers are getting progressively larger.

We then move onto a 10cm number line. It could be done without the grid paper, but it really helps the students who are struggling, and also allows some students to make the jump to the purely representational (e.g. this number line is 15 cm long, so jumps of 5cm will divide it into thirds because 15 ÷ 3 = 5).

1. Another great activity! Very visual, and that's what it's all about. Thanks for sharing, Jenna!

8. LOVE the questioning strategies!!!! (And regularly use this video in talking with teachers about problem solving...LOVE IT, TOO!) :)

You always have the most amazing posts. Thank you!

1. Thanks, Cindy. It's all about questioning, really. A well-placed "why?" or "I wonder?" are magical. :)

9. Would like to think that I let kids struggle 'just' enough and not too much. But if the students are really 'stuck' we do need to step in...but that doesn't mean we need to step in with an algorithm or process. Questioning techniques can certainly guide the students to discovery and thereby, modeling how to attack a problem.

Thank you for sharing your examples. Really like the visuals and think kids will have a better understanding.

Thank you.

1. Yes!!! Step in with a question and not step-by-step instructions. That is so true, Nancy!! Thanks so much for the comment. :)

10. Thanks for stopping by my blog today! I LOVE your blog--follow by email to make sure I don't miss anything! :) As you can see, our class is right smack in the middle of our fraction studies--and I am a FIRM believer in coaching students through the struggle! In our class we have a saying...I start by saying, "We can figure this out eventually because..." and the students yell out "MATH MAKES SENSE!" I truly believe we can do that for our kids if we slow down, dig deeper, ask the right questions, and LET THEM STRUGGLE and make meaning.

1. Love your comment and your philosophy. You've got a great series of fraction posts over at your blog. Lots of excellent stuff!

11. You are a star for sharing this today. I have my observation this week with the principal and fractions on a number line is my topic!! Thank you for sharing...this helped a bunch!

Stephanie Ann

1. Wow! Talk about great timing, Stephanie Ann! I'm sure you'll rock it. :)

Julie

1. Welcome, Julie! I'm following your blog now, too. :)

13. Wait! You mean I can't fill the number line with 1, 2, 3, 4 and then not figure out what the one on the right side means. Oh thank you for the bar graph/number line connection! I needed that light bulb...so hopefully my students get a light bulb! The video was also spot on, and hilarious.

Laurie

1. What? You mean your kiddos do that with number lines, too? I thought that was just a Texas thing. Ha ha. :)

2. If light bulbs were audible you would have heard my class, all the way from Oregon, during math yesterday. Thanks for helping me....a 32 year old learn something new...which then helps 30 kiddos have an aha moment!

Laurie

3. Woo hoo!! LOVE to read comments like that! Thanks for taking the time to comment. :)

4. Here's a little educational trivia: did you know that in Singapore they NEVER teach the area model for fractions? They only use the number line! I also love that video clip! I've used it at several pd sessions I have facilitated! It's hilarious and eye opening all at the same time!

14. This is great! Thank you so much! Sometimes it's the little things that help them visualize it (and me too!). Next up, I'll have to modify this for mixed numbers on a number line and then decimals! Thanks for the great starting point!

Amanda
Collaboration Cuties

1. I'm with you about the visualization, Amanda!

15. Thank you for your GREAT blog! I love all your ideas. I thought I had done a decent job teaching fractions on a number line, until they took the test yesterday....I'm going to try it once more and see if your ideas don't get them all in gear! Thanks again!

1. Glad I gave you another strategy to try! Good luck! :)

16. I regularly follow your blog and especially appreciate examples like this on how to help teachers make the paradigm shift to allowing students to take risks rather than feeding them algorithms. My role is changing to include coaching so my question is what training has proven most valuable to you in your role as coach?

1. Interestingly, Wendy, our district has provided more content training to the coaches that "coach" training. We did have Jim Knight come in and give a presentation to the coaches last year, and that was helpful. I'm now participating in a book study using Marzano's book on coaching. I'm really liking that and I like the book study format over "stand and deliver" professional development.