Building Enduring Mathematical Understanding, one lesson at a time.

Archive for the category “Blogging Initiative”

New Blogger Initiative: Integer Context Cards

For Week 2 of the “New Math Blogger Initiative” (or is it an “initiation?” hmmmmm) I decided to learn how to embed a document using Scribd and show something that I am proud to share!

If you read my Made4Math post from last week (Math Cards) you know I like “multi-purpose” tools. I haven’t been teaching Middle School too long, in the grand scheme of things, and when I first had the opportunity to introduce Integers to a class of very low sixth graders (all Level 1 on the state test) I knew that putting them into context would make all the difference. So. . . what are some contexts for integers? Well, there’s temperature, and altitude, and money. . .? I brainstormed long and hard and came up with quite a list.

Without further ado, my very first Scribd document!

Integer Context Cards

(Hmmmm. Rats! The font changed when I uploaded the document. The original was in Herculanum, a very cool looking ALL CAPS font, so now it appears that I don’t know my capitalization rules – oh well.)

There are a total of eighteen different contexts with six cards for each one. Some are “stretching it” a bit, but still reasonable. Note: for altitude, one of the cards is for Arlington, Washington where my school is located, so you might want to “personalize” that one. 🙂

I made one copy of each page on a different color of copy paper to make them easier to sort and then had them laminated, cut, and paperclipped in sets of six. (I only made one set for the entire class, but you can certainly create duplicates, especially if you have a very large class.)

Order, Order, Order

Phase 1: After a brief exposure to a few of the cards, we started our first activity with the cards. My students were already assigned to groups of six. We actually went out into a space in the hall for this.) Each group received a different set of cards, passed them out amongst the group members, and silently “raced” to put themselves in order from least to greatest. Once a group was done, they ALL had to raise their hands. After I checked for accuracy, they turned in their set and grabbed a new one. (The first year I did this I checked the groups off on a master sheet, but the following years I just trusted their memory – “We already did that set.”) The goal was to accurately get through as many sets as they could in the time allotted. Often enough, groups were in too much of a hurry to read carefully enough to identify the “key words” that signified whether their value was positive or negative. Getting a “no” when their hands were raised definitely encouraged them to take their time a bit more.

Phase 2: The next activity took it just a bit further. Groups (of three this time) had a sheet of number lines. Each time they received a new set of cards they had to “fairly accurately” plot and label all six values (along with zero if it wasn’t in the set) on a number line. Choosing a scale was challenging for some of the sets (especially with the first group of students.)


(See the cool font? Oh, well.)

Integer Operations in Context

A few weeks later in the year, my seventh grade class was working on operations with integers. I printed the Integer Card sheets four to a page and created little packets for students to share. (See the image above.) Using a “Think Pair Share” type of model, I posed questions using the people, places, or things on the cards and students had to write a math sentence, model the problem on a number line, and find the value (first on their own paper, then with a partner on a mini-whiteboard.) A big key was writing the math sentence as opposed to just finding the value. I wanted them to make the connection so that when they saw a “naked numbers” problem they could try to connect it to the contexts we worked with in class.

We initially focused on addition and subtraction situations:
(**The white text is shown first. After the Think-Pair-Share on whiteboards I reveal the number sentence and diagram and move onto the next problem.**)

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Next we moved onto subtraction in “finding differences” contexts, as well as multiplication and division. I dropped the “number line” requirement, although we did end up sharing it on some of the more challenging multiplication/division situations. (Again, the white text for each problem is given first, TPS, then the yellow answer is revealed, discussed, and we move on to the next problem.)

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In subsequent years I used the Integer Operation activities with my sixth grade classes as we introduced this seventh grade concept in the Spring after the state test. During “Review Activities” time in class I had small groups create original “stories” along with their associated number sentences, from the cards, but I think I would like to make that a more integral part of the initial learning as well.

Ahhhhh, context! Even if it is really “pseudo context,” in this situation students have something to grab onto that helps them make sense of the integer operations as opposed to a mercurial “set of rules to follow.” On the other hand, it is THROUGH these experiences that students begin to create their OWN rules about the patterns involved in integer operations, but only when the concepts “make sense” to THEM! I think we are finding a bit of EMU right there. 🙂

Note: There are multiple slides for each operation, and if I got my act together I might copy and paste them all into one slide show and try to attach them, but that would take a higher level of understanding on Scribd than I currently possess, as the slides are all now on my iPad, and I am just sharing screenshots.


Reflections: Why “findingEMU?”

There is a blurb on my “about” page that describes the title of my blog:

Enduring Mathematical Understanding is found within, built upon your own foundation, framed by your current perceptions, and constructed from your experiences. By sharing my thoughts, ideas, and ramblings with others, I hope to encourage the growth of positive dialogue towards this lifelong goal for myself and others.

Wow! I really came up with that on my “first day” blogging, what seems like ages but was just over three weeks ago! (I did say “ramblings,” and that’s sure the truth.)

I have thought about writing a blog for quite awhile now. (Still to come – my “From Lurking to Learning” post.) Over time I toyed with different choices:

“MathMom” because I was a part time teacher / part time stay at home mom. My kids were (ARE) both math fanatics, and I would like to share some stories about watching them grow as mathematicians. (While I don’t exactly have their permission, I have put a few memories down on “paper” over the past few weeks.) However, “Math Mama Writes,” a blog that I have read, appreciated, and enjoyed for many years now, already had a claim on that type of title, so I decided to move on. (In addition, my kids have grown and I don’t find nearly as any opportunities to “be” a Math Mom anymore 😦 .)

My next thought was “Making Math Meaningful” (shortened to M^3….ooh!) While it has a nice ring to it, and it IS a part of what I try to do in my classroom, what does “meaningful” mean? Definition: full of meaning (duh!) significance, purpose, or value; purposeful (you mean, like “full of purpose”?); significant. I DO want to help students find meaning in math, see its value, know that it has purpose. However, my lessons are not “chock full of 3-Acts,” so it didn’t seem to quite capture the “spirit” of my classroom.

Recently, the phrase “Enduring Understandings” seems to have taken its place alongside “Standards,” or “Essential Learnings,” but I really like the use of the term “Enduring.” Teachers complain all the time about lack of retention – students can learn something for a week or a month, but it is no where to be found after that. What is it that makes the learning endure?

Now, “Understanding,” that’s a whole ‘nother can of worms. It seems like last spring, but it was only in June that I read Richard Skemp’s paper on Relational Understanding and Instrumental Understanding. I ran across it by linking from a Math Mama Writes post to The Republic of Math blog. (I can’t imagine why I didn’t read it when it first came out – oh yes, I was still in Elementary School!) It is a powerful piece, written 36 years ago, that, for me, emphasizes the difference between when a student says, “I understand!” and when they truly do. “Conceptual Understanding” vs “Procedural or Algorithmic Understanding” are more commonly used today, but I feel that, in reality only “Conceptual” actually reflects true “Understanding.”

So, that’s what it all boils down to for me. How DO I help my students FIND Enduring Mathematical Understanding? I can’t find it FOR them – sometimes I have a hard enough time finding it for ME. I don’t want the “I can do it!” math, I want the deep comprehension down in their core that builds and branches off from what they DO really know and understand. What I do in the classroom: the questions I ask, the answers I “don’t give,” but the way in which I respond to questions, the student dialogue I promote, the activities I provide that lead without “dragging them along,” and the culture of the classroom that I help establish, are all ways that I can help them on their journey.

I love the crafty stuff, notebook organization, games and activities, and “monster” whiteboarding plans (tooooooo Monday links to post!) that I have been bombarded with over the past few weeks, (I think I have at LEAST tripled my Google Reader) and the discussions about SBG, (or sbar?) how to deal with the homework hassles, and uses for technology on Twitter. However, what I really cherish from the past three weeks are the conversations about teaching fraction division with/for understanding (“I hate “magic math”,”) introducing integer operations with meaning (“I abhor – is that too strong a word? – the “follow the rules” approach”, and re-defining slope (NOT just “rise over run” – KA), running across David Coffey’s ideas about flipping homework that gives “practice problems” an entirely different level of meaning from back in February, and finding Michael Pershan’s fabulous new (in addition to his insightful “regular” one) blog, Math Mistakes where readers are faced with the tasks of identifying not only “what don’t they know,” but what DO they understand and thoughts on remediation – how can I meet them where they’re at?

Deep breath. One sentence?! I’m not editing. (Except to add MORE words, I suppose.)

findingEMU – oh, like “Finding Nemo!” Well, yeah, I went for the “catchy” goofy title that resembles a wonderful little Pixar movie, but with an odd looking flightless bird instead of an adorable clownfish.

Only I guess I really didn’t, because I really AM trying to find ways to reach EMU.

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