Fix It

I appreciate Marilyn Burns’ presence on Twitter and her blog. She offers great contributions to the math education community.

Marilyn's recent blog post, entitled Fix It: An Activity for Ordering Fractions, is a well-written reflection of a lesson she taught to fifth graders. Marilyn describes how she used an engaging format and includes details of some masterful scaffolding for a student who needed help. It's worth reading the whole thing.

Fix It Fractions reminds me of the Clothesline Fractions activity Rebekah Thomas did with her summer school class. Before class, she’d hang a set of fractions (using clothespins) on a length of string in her classroom. A different student was in charge of “fixing” them each day (putting the fractions in order). The student presented his or her work to the class along with an explanation.

In Marilyn’s activity, students create their own Fix It Fractions sets for others to try.

There is something inherently compelling about fixing. It is different than doing math work that involves simply performing a calculation or solving a problem. Students look at the work of someone else that may be intentionally or unintentionally incorrect and perform an error analysis. There are lots of formats for doing this, including My Favorite No, or presenting two or more solutions and asking students which is correct and why.

I always appreciate when I see error analysis type tasks in Eureka Math lessons, Smarter Balanced Assessments (both of which we use in our district), and other materials. When I think back to different jobs I have held in my life, I think the majority of my time was spent interpreting, analyzing, adapting, correcting, and reimagining work that had been begun by others.

Let’s think about how to include more of this type of approach for students, and see if it improves their school experience.

NOLA Greats

There is a treasure trove of greatness at NCSM New Orleans. See below for conference tweets that will link you to wonderful minds and resources.

Choose a problem and anticipate student responses. T/F: 80/4=(80/2)+(80/2). Kazemi & Hintz

Rethink homework! Purpose is not to give a child a grade, ever. What's your homework protocol? @tkanold tkanold.blogspot.com

Go beyond checking for understanding. Students must get feedback, take action. @tkanold

Working on our practice. Teach the same lesson twice with collaborative reflection in between. Nice! @maxmathforum http://mathforum.org/pps/

Number sense is acquired; you don't teach it, you nurture its development. @SkipFennell mathspecialists.org

Set it up so kids are asking the next question of the teacher, using the lang. of our discipline. @jgough http://jplgough.wordpress.com/

Stop making excuses. Implement what we know works. @steve_leinwand http://steveleinwand.com/

We say all students can learn, now we need to act that way. Cathy Seeley (via @Maryvfitz)

Estimation Jars

Today I’m creating Estimation Jars for Math Night at John J. Flynn Elementary School. This is always a highlight at Math Night and it’s fun to watch kids at work on their estimate. I have one jar full of Rolos (there is something appealing about the gold wrapping), a smaller jar containing Skittles, and a very small jar with unpopped popcorn.

This year, I plan to accept all estimates within a reasonable range for each jar and then randomly draw a winner. That’s different than past Math Nights when the person who estimated the number closest to the actual number won the jar. I am hoping this helps promote a more correct understanding of estimation, with the goal being to produce a reasonable estimate, not to land on or closest to the exact number.

It is a good idea to have a few jars containing different sized objects so students can think about how the relative size of the objects affects the number that fit in a space. Ongoing math explorations can be done using estimation jars outside of an event like Math Night. Here is an excellent write-up and video about how classroom teachers can use estimation jars with their students, with a focus on the all-important concepts of doubling and halving.

Where Assessment is Headed

As we move toward the new Common Core State Standards, the issue of how to incorporate the all-important Standards of Practice tends to elicit lots of discussion and differing opinions, especially when it comes to assessment.

I found this interesting new post by Dan Meyer about the role of computers in mathematics assessment: What Silicon Valley Gets Wrong about Math Education Again and Again. Here’s a quote:

‘Explain whether 4/3 or 3/4 is closer to 1, and how you know.’
It's simple enough to write down an explanation. It's also simple to speak that explanation out loud so that somebody can assess its meaning. In 2012, it is impossible for a computer to assess that argument at anywhere near the same level of meaning. Those meaningful problems are then defined out of ‘mathematics.’

This is a good read to follow Conrad Wolfram’s talk I mentioned earlier in this blog.

I like the Roman mosaic question Dan includes in his post. This would be fun to give to students. I, personally, would need plenty of time to come up with a coherent answer.

Fractions and My Brother

Man changing stock quotes on a chalk board. NYPL Digital Gallery

My brother, Zack, was here visiting the other night. My 12 year old son had a question about his fraction homework, and this sparked a conversation between Zack and me about how fractions are taught in school. 

Zack said, “Fractions are a useless implementation of decimals.” 

Hmm. Zack worked in finance for many years as a derivatives trader. I never really understood what he did besides the fact that it entailed making split second decisions involving lots of money and using insane spreadsheets. Zack is still my go-to person when I need help writing a really complicated spreadsheet formula or macro. Not only does he understand what I’m saying, but he has the answer in .9 of a second or less.

Here is a video of Zack sharing his thoughts on fractions. I hope it sparks some interesting conversation for readers of this blog. I brought this up with my colleague, Penny Stearns, who expressed a completely different perspective from Zack. Perhaps she will star in a video rebuttal at some point. Professor Tim Whiteford weighed in with still more to think about. What an excellent topic for students to question and debate in order to deepen their understanding of the application of fractions and decimals in different contexts.

Math and the iPad

 

I have a new iPad and I’m looking for good math apps.

Lee Orlando has a new iPad, too. She loves trying new things and is already way ahead of me on using the iPad in math class. This is from a recent email:

This weekend I bought an adapter for my iPad so I can hook it up to the LCD projector, and I also got a wireless keyboard.  Today, I went into school to try it out.  It was so cool to see the iPad screen projected and to sit at the back of the room (or any place in the room for that matter) and see the text appear.

This was soon followed by another email:

I just found some awesome free apps for the iPad.  All are from "Mathtappers" and the three that I downloaded involve placing numbers (including rational numbers) on a number line and finding equivalent fractions.  All games are designed for three levels of play.  I've been trying them out at each of the levels, and they get pretty challenging at the highest level.  However, the easiest level is well within the ability of fifth graders.

...these apps look like a great way to engage students in a whole-class warm-up activity/discussion.   I had made up my own number line activity using a sketching app that I have... the kids' attention level goes WAY high when they get to come up and draw on the iPad!  Getting that adapter for the LCD projector may have been the best investment I have made in a long time.

Later, I ran into Lee at school and she showed me how she’d photographed a piece of graph paper to use as the background of her sketching app, so that students could draw arrays and fractions with the aid of the grid. What a great idea!

I hunted around a bit and found some other useful apps. My favorite so far is Sketchpad Explorer. If it’s been awhile since you contemplated the Pythagorean theorem, you’ll enjoy the Getting Started screen, which allows you to drag right and non-right triangles around to see the theorem in action.

The real fun, though, comes when you touch the little book icon in the lower left corner of the screen. Choose “Elementary Mathematics” and Sketchpad Explorer presents you with a suite of eight activities involving symmetry, triangles, fractions, decimals, multiples, and volume. There is even a logic game which gives less than and greater than clues to find an unknown number. Sketchpad Explorer’s creativity and nice, clean graphics are appealing. At first glance, there seems to be a wealth of resources and lesson ideas for teachers on the website. I can’t wait to try these with students.