Pennsylvania's group of 22 teachers just completed the last institute of the ANI training. As one of the facilitators, I was preparing to introduce signed number operations which are always difficult for students as well as for teachers to understand. This particular training I have written and rewritten and each time we reach this part of the program, I hope I am finally reaching the outcomes that I wanted. I was fortunate this time to be able to go over the presentation with math researcher Lynda Ginsburg who was presenting with me. The peer review of what I was doing was so helpful and I would suspect, for teachers struggling to write that perfect lesson, a peer mentor might also be welcomed.
I found in my presentation that while I started with a real life context to introduce integers (temperature change), I deviated from that context as I introduced various manipulatives. While temperature change worked well with number lines and thermometers, I also wanted to introduce two sided chips and other manipulatives into the mix. It was at that point, I should have perhaps introduced debt and had the chips utilized for various money situations. I think I salvaged the lpresentation by introducing hot cubes/cold cubes to illustrate various integer operations dealing with rising and falling temperatures in a soup cauldron.
I did incorporate formative assessment opportunities by using a thumbs up, sideway thumbs, and thumbs down to assess the teacher's understanding with various manipulatives and also allowed for participants to share their thinking through demonstration. The various manipulatives were the authentic materials used to help solve integer problems
My aha’s for this rewritten presentation were that working with manipulatives that might be new for teachers or learners, needs time just for play. I am curious now what would have happened if I just passed out a particular manipulative, described how it works, and let them play and create situations that would have involved the four operations with integers rather then me giving them scenarios. Instead after demonstrating various strategies with integer operations and manipulatives, they were given isolated math integer problems to solve using various manipulatives to understand the operation with signed numbers. I think instead of isolated problems I would create some real life examples with integers and then allow the participants to choose which model would make sense to utilize (ie. number lines, thermometers, two sided disks, hot cube/cold cubes, patterns).
To our Pennsylvania folks and others on this listserv, what do you think? What have you tried in your classrooms around integer operations?
Comments
In my years working w/ students and integer operations, I've seen some students benefit from the chips as manipulatives... but many just don't make a connection.
Putting big number lines on the wall has been a big help (had a young man take a picture of it last week). It reminds me how sometimes gross motor -- moving hands in a 'big' way' up and down the line -- makes a bigger impact on students than a little number line in front of them. (I wonder if the big motor folks are the ones more likely to benefit from the manipulatives...)
As part of Power in Numbers project, I put our chapter on integers on oercommons ( https://www.oercommons.org/authoring/13198-integers-introduction-to-the-concept-with-activiti ) ...
My thinking is that the biggest problem wiht teaching integer operations is going too fast. Every. Single. Semester. I have some students who start to get confident in adding different signed numbers. Then BOOM! Subtract!" and everything explodes into smithereens. If I ran the world or designed the curriculum, I'd channel my Orton-Gillingam training that says "separate stuff that's confusing!" and teach addition... and then teach perimeter of shapes, or "applying parts and wholes principles to word problems" or ... anythign... and include frequent practice in fairly small-number addition of integers and word problems too... and *then * bring on the subtraction creature.