ELL Math – Fractions
I’m still figuring lots of stuff out with my ELL mostly-refugee students and I’m loving that I get to process it all through this blog. Here’s what’s on my mind:
We started going fractions, which has been going really well. My general classroom structure is to pick a topic for the week, go over it every day, then quiz on it on Friday. Last week was just representing fractions – things like writing a fraction given shaded pieces or given a point on a divided number line. I made these Geogebra programs to help me with the practice problems and to keep the visual theme going:
Writing Fractions from Visuals: http://tube.geogebra.org/material/show/id/1754581
Writing Fractions from Number Line: http://tube.geogebra.org/material/show/id/1754587
Once we had the concept of identifying fractions, we moved onto combining fractions with common denominators. Here are some artifacts from how that went:
- If I had it to do again, I’d add a third box to the worksheets for students to draw their answers too.
- I love that I was able to circle back to positives and negatives, but now in the context of fractions – so, I get to hit the skill again, but not in a way that seems repetitive and like we’re spinning our wheels moving nowhere. It was also a really easy transition to represent fractions with +’s and -‘s instead of just shading them in, so our visual language of ‘zeroing out’ was re-used with these fractions.
- Most of problems were along the lines of: given a symbolic fraction problem, draw a picture to help answer it. However, my favorite types of problems were actually the reverse: given two pictures of fractions, write the symbolic representation and then answer it. These really helped cement the visual language I wanted them to use.
- For the most part, the students who drew pictures got correct answers, whereas the students that didn’t tended to get incorrect answers (not pictured). Even some of my more advanced students reverted to drawing pictures to check their work. Seeing that was one of the most tangible manifestation of one of my biggest overall teaching philosophies: teach a representation and rules on that representation, then let students recreate that representation to solve problems. This is basically my philosophy behind everything in this class – the symbols have a visual representation and rules on how they interact which gives you the answer. The students who took the time to create the representation tended to get correct answers – the students who moved too fast got incorrect answers.
- Some students still struggle to identify the sign of numbers without anything in front of it. For example, in the expression “3 – 5 + 4”, students are confident that the 5 is negative and the 4 is positive, but are unsure of the 3. This is curious to me and I don’t really know how to fix it other than “if there’s nothing there, it’s always positive!”, which is an arbitrary rule and is hard to communicate in the absence of a common language. For some students, I wonder if its because their native language reads right-to-left whereas they are suddenly learning a language that reads left-to-right.
The plan after this is to go into fraction multiplication, then into combining fractions with different denominators. For a while I was struggling with how to teach this visually, but this demo lesson from ST Math was invaluable in informing how I’ve been teaching fractions: http://www.mindresearch.org/play/. I use it with every student now, even my non-refugee students.
Also – I wish I could erase the part of my brain that wants to draw a circle as the default way to represent fractions. From a pedagogy standpoint, everything is much easier to teach if I default to drawing an array of rectangles of a number line (but especially an array of rectangles). A rectangle divided into fractions segues segues to fraction multiplication easier, it segues to decimals easier, and its easier to draw and manipulate if I make a mistake while drawing.
When I’m not teaching full-group lessons, my students work self-paced on Khan Academy (more info here). The self-paced aspect is working great as is the alignment between what I’m doing full-group and what they work on individually. For the earlier exercises, KA also has lots of different types of exercises for students to work on with multiple representations, so students get lots of practice on the same thing even though they’re progressing through the curriculum.
In general, I think the ‘Missions’ are pretty useless with my students – there isn’t a lot of logic to how the problems are generated and it all seems chaotic. My students get frustrated and want to give up in the face of being unsuccessful and not seeing how the previous exercises connect to the next exercises. However, if students have been working through individual exercises, then they can use the ‘Mastery Challenges’ to revisit exercises and gain ‘mastery’, which I like. I had my students do this for a week and had them skip the ones they had never seen before. It was especially interesting to see them work on the Early Math problems – many of them are explicitly language based (like these ones on ‘shorter’ and ‘longer’ and ‘bigger’ and ‘smaller’), so many of my students learned how to use Google Translate to answer these questions, which I thought was a valuable teaching moment even if it wasn’t necessarily a ‘mathematical’ teaching moment.
Khan is also really good at adding new exercises – almost as if they read my mind, they added a Multiplication Using Array’s exercise this week, which is pretty much exactly how I taught multiplication to students. But – I wish there was a place where I could see when new exercises are added or updated (I asked them on twitter, but no response). I just happened to ‘discover’ these ones – it’d be great to receive email updates or an rss feed or something when new exercises are added.
Christopher Danielson recommended this program called ST Math for this class since their philosophy is almost entirely aligned to my goals in this class: start with the visual, then add in the symbolic later. Running through the Demo lesson linked above, it seems like a pretty awesome program for this very specific demographic. I’ve even convinced my school to look into purchasing it, but we’re having a lot of trouble getting a hold of someone from the company who we can talk to about buying the program. So, if anyone from ST Math happens to read this, I’d love to get in touch to look into using this software in my classes.