Bottoms Up to Conceptually Understanding Numbers

This is just awesome! I am working my way down to the younger grades, and learning about this foundational pieces has been an eye opening experiences. I do not know if it is the length of time since I was this age, or the lack of experience in teaching these grades, but these are pieces that I just assumed we came equipped with. Learning that it is both purposeful and intentional to give students this experience plays out in so many fundamental ways, it also helps me think of the students I had in high school and college that did not experience this earlier success.

As I grow in my understanding of how students (and all peoples for that matter) learn and acquire mathematical knowledge, the more important these foundational pieces become. Funny how something as seemingly simple as an inversion to the number line creates understanding, like vertical number lines, these are difficulties we need to attend to, and we need to be careful how we accomplish these tasks.

All that to say, this is a great read and I am reblogging, more for personal reference than anything else, it acts as a reminder for that which I discussed above.

Questioning My Metacognition

The concept of 10 more/less is a beast in the primary grades.  Last week I realized that I’ve been feeding the monster that I’m continually trying to defeat. Almost every day, in every K-2 classroom across the United States students encounter this guy:

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I made a conscious effort to pop my head in every K-2 classroom in the schools I visited this week.  It was great to see that every classroom had a traditional 0-99 or hundreds chart posted like one above. 

While visiting, one of the teachers asked if I could come back today and model a lesson using a 0-99 chart because her students “just weren’t getting it.”  I gladly accepted her invitation and showed up with this guy:

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I didn’t use manipulatives for this lesson because I was specifically focusing on the rote counting process which precedes one-to-one cardinality when counting by ones OR tens.

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Continents, Math Explorers’ Club, and “I use math for…”

One of my favorite mathematicians and a really awesome guy too!

Math Munch

Welcome to this week’s Math Munch!

stevestrogatz Steven Strogatz.

All of our munches this week come from the recent tweets of mathematician, author, and friend of the blog Steven Strogatz. Steve works at Cornell University as an applied mathematician, tackling questions like “If people shared taxis with strangers, how much money could be saved?” and “What caused London’s Millennium Bridge to wobble on its opening day?”

On top of his research, Steve is great at sharing math with others. (This week I learned one great piece of math from him, and then another, and suddenly there was a very clear theme to my post!) Steve has written for the New York Times and was recently awarded the Lewis Thomas Prize as someone “whose voice and vision can tell us about science’s aesthetic and philosophical dimensions, providing not merely new information but cause for reflection, even revelation.”

NMFLogo_Horiz_RGB_300DPI2This Saturday, Steve will be presenting at the first-ever

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Fractions, Sam Loyd, and a MArTH Journal

I was looking for some interesting and fun MaRTh for some fourth to sixth graders I have the pleasure of working with. As I was digging around the interwebs, I found one of my favorite haunts,, had just what I needed. I will be using this article to have some fun and I’ll see what happens. Look back here soon for results from this activity….oh, and I think I know an extension of this for some very bright seventh graders I work with, I bet they will love this.

Math Munch

Welcome to this week’s Math Munch!

Check out this awesome graph:

What is it?  It’s a graph of the Farey Fractions, with the denominator of the (simplified) fraction on the vertical axis and the value of the fraction on the horizontal axis, made by mathematician and professor at Wheelock College Debra K. Borkovitz (previously).  Now, I’d never heard of Farey Fractions before I saw this image – but the graph was so cool that I wanted to learn all about them!

So, what are Farey Fractions, you ask?  Debra writes all about them and the cool visual patterns they make in this post.  To make a list of Farey Fractions you first pick a number – say, 5.  Then, you list all of the fractions between 0 and 1 whose denominators are less than or equal to the number you picked.  So, as Debra writes in…

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Technology Enhanced in Math Lesson

What an exciting time to be a teacher, a student, and a learner!

I have a tentative understanding, at best, of human nature, but I do believe that most people find change uncomfortable and difficult to deal with both emotionally and intellectually. I am not immune to these feelings, but I like change. I like to embrace the unknown, I thrive off of dreaming about the possibilities that could be.

One of the most exciting things about the change of standards to the Common Core State Standards in Mathematics is the depth to which students are expecting to dive in their understanding. With this in mind, the fun of being a teacher is to create learning opportunities that facilitate students’ thinking and facilitate opportunities to dive deeper in their thinking.

The original idea for this lesson is hard to nail down, but I do know a large part of it came from the wonderful book Fostering Algebraic Thinking. The results are shown below:

I tried the lesson with sixth and seventh graders, and the biggest thing I realized is that there is a bunch of background knowledge with respect to the technology that students need to be familiar with to take advantage of the lesson. On the other hand, this seems like a natural way to introduce the use of technology in a lesson which has its own advantages.

What would you do differently? What would make this lesson better?