When I read visual multiplication post from Hard Enough Problems, I thought that the blogger (he/she) was describing box or lattice multiplication.
I was delighted to see another take on visual multiplication. I remember to when I saw Dr. Deborah Ball for the first time about 10 years ago at a NCTM conference. I hung on every perfectly crafted word she said. I specifically remember that it was the FIRST time I had seen alternative multiplication methods in my mathematical existence. I remember Dr. Ball showing methods from other countries. As I was watching, I was furiously trying out these noveau methods on my lined paper and then double checking them with the “normal” 2 digit multiplication algorithm that I grew up with. Dr. Ball’s talk was actually about her research into the depth of mathematical understanding of teachers and how it affects the learners.
I remember being introduced to “lattice” multiplication and again being befuddled trying to learn it. I was working with a 6th grader last year who was having issues with decimal multiplication, and I showed her the lattice method. It seemed, though, that it was a little late because she (like me) had been heavily trained in the standard algorithm. Even though the algorithm wasn’t working for her because of errors of lining up numbers or carrying, lattice multiplication just didn’t feel like multiplying to her.
I guess it’s because math should be hard. always.
A while ago I watched an interesting youtube video that dealt with student misconceptions specifically in physics. It resonated with me so much I just spent a few minutes trying to find it to share on this blog. The point of Derek Muller’s research is that students often think that they already know what they are learning. I have done many lessons where I start with a class discussion of what students know about a particular topic. Especially teaching 9th through 11th graders, many student have had Algebra I multiple times which unfortunately can reinforce certain misconceptions. Last year, I had an 11th grader who regularly announced that this was her fourth time and she was going to “pass it.”
So I just erased my 2nd paragraph, and I am starting again, so it will be shorter!
One misconception that I noticed every year with my students was subtracting integers and whether adding negative integers were positive. I have vacillated about how to deal with the misconception over the years. At this point, I usually have two responses depending on the student. If the student seems ready for the concept of subtraction, then I will use money or a vertical timeline with the student to solidify their understanding. If the student is still struggling with the concept of subtraction, then I coach the student to change subtraction to adding the inverse.
For example A) -5 -8 is the same as B) -5 + -8
I avoided the above method for a while because it doesn’t really address the concept of subtraction. However, most students can solve B) whereas many students have issues with A). I think that “converting subtracting to addition” is a shortcut that is useful for some students.
I dealt with the misconception during whole class discussions by highlighting what operation was being used when combining integers. When students were able to understand a negative and a negative is a positive was dependent on the mathematical operation, I could usually address the faulty understanding by asking them what operation they were using.
I’ve had many class discussions where students insisted that a “negative and a negative is a positive.” I try to have other students verbalize why this is sometimes true. This is a case when memorization of a key concept that is missing the condition can lead students astray.
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I am attaching one of my favorite creations for factoring quadratics. Originally, it only worked when a = 1 and then I realized that if I added a box then it would be an universal graphic organizer (GO). (Sorry, I just can’t call it a worksheet :)) Working with high school students, many of them resisted working with tiles (to factor) because they associated manipulatives with elementary school. I found that a paper/pencil GO was more attractive to teenagers than the tactile experience. However, I do agree that tactile is important, but that’s another post.
The first time I introduced this GO was in 2010 with an Algebra I class that needed extra support. Some students used the GO exclusively and were quite adept at it, other students saw the patterns and factored without the GO. When I first introduced it, each student had the GO in a page protector with a dry erase pen. I projected the GO and wrote directly on the image. It was successful, but I was surprised when students received the assessment that they weren’t sure what “factor” meant. I coached quite a few students by saying, “Remember the box?” Most students were able to factor when given the cue. So, fast forward to 2011, I really emphasized that using the “box” was meant we were Factoring! Definitely worked better.
And this is how the graphic organizer looks filled out, this is from 2010 before I figured out how to put it in the Mobi gallery, so it’s a snapshot of my computer screen. It is also before I figured out how to delete pages before I made them into PDF’s, so scroll down to the 2nd page to see the actual work. I am using the Mobi (mobile mouse), which influenced my handwriting.
So…it took me two days to embed the above document (the second one was SO fast). I wanted to resist Scribd because I have over 100 accounts with passwords and usernames, argh! However, I couldn’t seem to get it to work from Google docs which apparently now is Google Drive (will the changes ever end?). I only saw a link, but not a way to embed. I tried a few times and now am a Scribd member. It is always interesting to find out who is friends with whom in the tech arean. WP and Scribd link so easily together.
I started a blog in 2007, but even then, I wasn’t quite sure what to write about. It just seemed cool to have a blog. Then it became clear that no one was reading it. Somehow that fact changed the experience for me. Similar to the tree falling in the forest, “Was I writing a blog if no one, but me, was reading it?”
Fast forward to Fall 2011, I decided that I would write a professional blog since I had begun to follow all of these great math teachers on twitter and read their blogs. I decided to get permission from my district which took a while, and then I took a while….and then I moved! So as the summer meanders to a weird finish since I am not revving up for teaching, I saw a twitter post about the blog challenge. I thought it would be perfect to give it the go!
I have predominately been applying for positions locally which seems to take hours. I am also trying to go a different direction, which means I have been applying at colleges. On Friday, I applied for a position as a Saturday Academy Instructor with Upward Bound at Sonoma State. Wow! They wanted a course outline (written in engaging language), CA standard, assessment plan and cost of field trips!
I decided to go with the Common Core Standard F-IF.4 which is about interpreting functions. It was strange to create a written lesson plan for an unseen audience. I found the “stacking cup” lesson on the internet (thankfully!). Students have 5 styrofoam cups (upside down) and need to graph the function and estimate how many cups it takes to reach to the ceiling. It is a great project to visualize slope and y-intercept. I worked through this lesson at a workshop and loved how many componets it had. The students can explore “rate of change” with something physical.
Hmm…how do you end a blog post? TTFN