A middle school math teacher's quest to teach each lesson at least a little bit better than the lesson before.

Archive for November, 2012

Play the game of Rule! to promote number sense

The #msSunFun topic of the week is promoting number sense, and I think the game of Rule! is a great way to do that.  My students love playing the game of Rule! and I love it too, for all of the practice it provides for recognizing and generalizing patterns. I believe that any time you can get kids to repeatedly manipulate numbers to look for patterns, you are building their number sense.


Rule! is an adaptation of an activity by Grayson Wheatley and George Abshire, from their book Developing Mathematical Fluency (http://www.mathematicslearning.org/index.cfm?ref=30606&ref2=12). Their book is one of those resources that all middle school math teachers should know about but I’m afraid few do.  It’s full of ready-to-use lessons and even comes with a CD of those activities. I definitely need to blog about other activities I use from this book!

Rule! can be played with a whole classroom of students and every student can stay engaged all along the way.  I typically use Rule! as a warm-up activity, although you could also use it as a time-filler if you ever have 5-10 minutes to spare.  And if you’re using it as a warm-up, you can start as soon as the first students walk into the room — no reason to wait until everyone is present to get started. One great aspect of this activity is that Rule! takes practically no planning on the part of the teacher — all you have to do is think of a function rule (younger grade teachers may think of this as an input/output table) to be the Rule! for the day.  Keep your rule in mind as you call on students around the room. If the student you call on hasn’t figured out the Rule! yet, he/she just gives you a number — this can be any number, although low numbers are usually more helpful in terms of helping others determine the Rule!  As students give you their numbers, write these on the board along with your “output” (the outcome when you apply your function rule to their number).  I write mine with an arrow in between, so for example if a student gives me the number 10, I would write 10 –> 33, and if a student gives me the number 5, I would write  5 –>18.  (Have you figured out my Rule! yet?) Since all of these numbers are written on the board, late students can jump in as they arrive.

Continue around the room calling on students in order (not by hands raised).  I usually go from group to group around the room (calling on every student) and I try to start with a different group each day because the kids hate to be in the first group I call on since they don’t have enough evidence to determine the Rule! It really kills them when they figure out the Rule! soon after you’ve called on them.  That always makes me smile because I love to see them so motivated to figure out the Rule!

If you call on a student who has determined the Rule!, instead if them giving you a number, they should say “Rule!,” which means you give them a number and they have to apply the Rule! to your number. This is what makes this activity so engaging . . . you can tell whether the student knows the Rule!, but the secret isn’t given away for students who haven’t figured it out yet.  (And I choose the number I give the student based on the mathematical ability of the student in question — you can keep it easier for a struggling student or make it harder — incorporating fractions, decimals, or negative numbers — for more advanced students.)  I usually continue calling on students until at least 4 or 5 kids have figured out the Rule! before I allow anyone to describe the rule.

At that point, I often have students who had determined a different Rule! from what is being described.  For example, with the numbers I gave earlier, one student might describe the rule as “multiply by three and then add three to the product,” while another student might say “add one and then multiply the sum by three.”  At times during the year (and as time allows) I have all students practice writing expressions based on the students’ descriptions of the rules, and sometimes we simplify the expressions to show that they are equivalent.  You can adapt what you do after the Rule! is determined to fit the needs of your class.

I hope your students love to play Rule! as much as mine do!


Problem Solving

As much as I wanted to participate, I had too much going on to start blogging during @samjshah’s New Blogger Initiation.  Ever since then I’ve been trying to find the “right” time (and just time in general) to start blogging.  That’s going to be my biggest challenge as a blogger — worry over finding the “right” topics, the “right” words, the time to do it “right.”  I’d better write the rest of this post before I decide something’s not right!

Maybe I’ve been waiting for just the “right” motivation.  That came in the form of this week’s #msmath #msSunFun topic, problem solving in the mathematics classroom.  Problem solving has been a focus of mine since I started teaching — and even before that, really.  As a student at the University of Florida, I was very fortunate to work as an office assistant for a math education professor, Dr. Mary Grace Kantowski. She had done her dissertation on the topic of problem solving in mathematics, and her focus on teaching math through problem solving profoundly influenced who I am as a teacher today.
So, on to the topic of problem solving in my own classroom (and thanks for sticking around if you’ve gotten this far).  If you’re in a hurry and just want to know what I’m doing this year, skip to the last paragraph!  If you’d like to know how I got to where I am today, read on.

In 1993, my second year of teaching, I began giving a weekly problem solving assignment to my academically gifted fourth, fifth, and sixth graders.  Each week they had 5 non-routine problems to solve.  In addition, they had to correct any mistakes they had from the previous week.  At the beginning of the year the problem sets were organized by suggested solution strategies (e.g., make an organized list).  To help familiarize students with various solution methods, we would work on one or two of the problems together each week.  Later in the year, there were no suggested strategies, just random problems, and students did all of the work on their own.  The students kept their work in spiral notebooks (and in more recent years, composition books) so that they could refer back to previous work whenever they wanted. Back then it was tough coming up with 5 new problems per week for each grade level.  There are so many more resources for those types of problems these days!

Quick story: a couple of weeks ago I was at the local high school homecoming game and saw some of my former students who were in town for their 10th high school reunion.  These “kids” were some of my very favorite students from my favorite class ever, so it was great to see them all grown up!  Several of them came over to reminisce about being in my class, and each one of them brought up those problem solving notebooks.  A couple of them said their they still have their notebooks from my class! I choose to take that as a sign that after so many years, they are still proud of the work they did!

Over the years, I tweaked parts of the assignment every year — replacing problems that were too easy or too challenging, requiring students to write a written explanation for one of the problems each week, having students identify the strategy they used to solve each problem. Below is a copy of the guidelines students had last year.  (I tried to figure out how to get these to show and not just give you a link, but couldn’t figure it out and I’m trying to get this posted TODAY! Anyone interested in tutoring me in WordPress?) These were taped into the front of the problem solving notebook, along with a problem solving strategy list.

Problem Solving Guidelines 2011
This was a great assignment for my students, but every year I have more students than I did the year before.  (This year, 106 of them.) So over the years, the time it takes me to grade the notebooks, making thoughtful comments and questions to every student, has gone from time-consuming yet manageable to overwhelming to completely unrealistic.  Last year, as I was struggling with how I could continue to provide students with regular problem solving experience but still have time to sleep at night, I was also looking for a way to use the problem solving assignment to also encourage written communication of thinking and sharing of various solution methods into the experience.

I decided to choose one or two of my favorite problems per week (depending on time constraints for the week) and have students show their solutions (including the requirement of some sort of model that helps explain their thinking) on separate sheets of copy paper.  They turn these in at least a day before our “problem solving day” (Friday), which gives me time to quickly sort through the papers to look for correct and incorrect answers and to identify various solution methods.  I assign students to a discussion group (4-5 students) which typically consists of some students with correct answers and some with incorrect answers, as well as at least one student who used a solution method that’s different from the others.  Groups are required to come to a consensus on the answer and also to share their solution strategies.  They are also asked to look for ways that thinking was conveyed in a clear, concise manner as well as ways that thinking could have been communicated more clearly (either by themselves or others).  The following week they are graded on the revision of their original solution, so for some students this means starting over completely, while other students are just refining their work and/or their explanations.  I use a very basic rubric for this, and I’d include it here but this post is already way too long. I am still refining the process, but so far I like how it’s going.  Here are samples of student work from last year (their original versions – I wish I’d thought to scan their revised work). And once again, I can’t figure out how to make the PDFs show below.  Basically, when I give students their problem, the question is typed at the top, the majority of the page is blank, and there’s a place to fill in the final answer at the bottom.  Getting the right answer isn’t really the main focus of this assignment, but it does help to see the students’ answers, all written in the same location on their papers, so that I can sort them into groups quickly.
I’m looking forward to reading other #msmath #msSunFun posts on this topic, and I would love to hear what you think about what I’m doing this year!