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

Archive for the ‘#msmath’ Category

Want a good discussion? Ask a good question.

I’ve seen a lot of MTBoS buzz about the book 5 Practices for Orchestrating Productive Mathematics Discussions  by Mary Kay Stein and Margaret Schwan Smith. It’s a great book that offers practical advice on getting the most out of discussions in the classroom.  But in order to have a great discussion, you need to start with a great question.  Where do you find good questions to ask?

One of my favorite resources for good, open-ended questions is the book Good Questions for Math Teaching: Why Ask Them and What to Ask, Grades 5-8.  (And there is a K-6 version as well.)  Here’s the description from the back of the book:

“Good questions”—or open-ended questions—promote students’ mathematical thinking, understanding, and proficiency. By asking careful, purposeful questions, teachers create dynamic learning environments, help students make sense of math, and unravel misconceptions.” 

My own copy currently has sticky notes marking a bunch of questions I want to use or that I’ve used in the past.  (Am I the only crazy person who will forget to use a resource if I don’t have a visual reminder of it?!)


The book is 195 pages long, and the questions start on page 17 — this is a book FULL of questions you can use right now in your classroom.  But if you’re looking for a resource for questions with an answer key, this isn’t it.   I love that, because teachers should be anticipating their students’ strategies and misconceptions (the first of the 5 Practices).  It’s so much easier to anticipate what students will do if you’ve gone through the process of answering the question yourself!  But the book does offer commentary after each question, outlining potential problem areas, giving justification for using the problem, or offering suggestions for implementing the question effectively.

The questions are organized by seven strands that should easily align with CCSS standards:  Number Relationships; Multiplication and Proportional Reasoning; Fractions, Decimals and Percents; Geometry; Algebraic Thinking; Data Analysis and Probability; and Measurement.  Within those strands, questions are subdivided into suggested grade levels (5-6 or 7-8), but don’t limit yourself to “your” grade level when identifying good questions to use!

This question led to a very prouctive discussion in my 6th grade classroom last year. It would have also made a great journal or INB prompt:

Which of the following problems has the largest product? Try to figure it out by solving as few problems as possible.  How did you choose which problems to do or not to do?

42 x 17

24 x 12

52 x 11

40 x 20

50 x 24

43 x 16

36 x 36

12 x 14

42 x 42

This week’s #MSSunFun prompt was to share a “go to” blog or website, but this book is such a good resource that I wanted people to know about it.  Where do you find good questions for discussions or journal prompts?



Is math class always going to be this fun? – First Day Quote

I just finished my first week of the school year.  This was the 12th first week of my career, and I actually slept well the night before school started for the first time ever!  I’m not sure why, especially since I’m teaching a new grade level (7th), at a new school, in a new school district this year — so you’d think I would have been more anxious than usual.  Maybe I was just extra exhausted from the week leading up to the first day.

My first day actually extended over two days.  At my new school we teach on an A-day, B-day schedule, which means that I only see my students every other day, for 85 minutes per class period.  Eighty-Five Minutes!  For the first time I can remember, I’ve actually had a couple of instances this week when I’ve finished class with a few minutes to spare.  I don’t think that will happen too often. The real challenge will be adjusting my pacing to cover all of the material I need to cover in only 1/2 of the instructional days!

Having 85 minute class periods allowed me to pack in a smorgasbord of first day activities.  I started and ended the class with a name tag idea I got from @rachelrosales’s blog, Purple Pronto Pups.  I’m so glad that her school year started earlier in August and that she blogged about her awesome First Day name tags!  For several years I’ve used name tags, but these name tags have an interesting addition.  The front of the folded 8.5 x 11 card stock has the student’s name, but the back is cut into three sections.  Each day for the first three days of class, students write an “I notice” and an “I wonder” statement about me or my class.  Then I respond to every student’s noticing or wondering.  All 138 of them.  (This is one time I’m relieved to be on an A-day/B-day schedule.  I’m not sure I could handle responding to 138 students all in one day! Especially not for three days in a row.)  But I really feel like I’m already getting some great information about my students and what they value.

After the students created their name tags, I introduced myself with my Me, by the Numbers keynote.  Then I explained their homework assignment, which was their own Me, by the Numbers list.  I really enjoy reading what students have to say about themselves.  It’s a great way for me to make connections with them and to start getting to know each of them — not the easiest thing to do with so many students whom I only see once every two school days!

Then we moved on to Fawn Nguyen’s Noah’s Ark Problem.  I introduced the idea of noticing and wondering about this problem, and asked students to put the challenge into their own words.  (What *I* noticed in going through that process is that a lot of kids missed the fact that they were supposed to figure out how many seals should replace the question mark.  If I hadn’t gone through that process, I think I would have been busy explaining to nearly every group that 1 polar bear doesn’t answer the question!) The kids worked with their seating groups of 3-4 students.  I gave each group a large whiteboard, dry erase markers, and a copy of the problem put into a plastic sleeve so they could use a dry erase marker on the outside.  They were SO engaged.  They were SO challenged.  They were SO disappointed to not have an answer by the time class was over.  It is SO awesome when students don’t want to stop working on a problem!

I heard a great quote from one of my students as her group was working.  She was having trouble following another student’s logic, so she said, “I’m still a little bit confused.  Can you explain how you got that again?”  This actually gave me the idea to start a Communication Quote of the Week.  I’ll have to blog about it once I get all of the details worked out, but the gist is that I’ll highlight things I overhear students say (or something a student writes) that will serve as exemplars for others. If my little plan works, I’ll have students trying really hard (by using effective communication techniques) to be one of the students quoted!

We finished up the first class with students’ noticing and wondering on the back of the name tags.

Some of the things they noticed:

  • I notice you are very organized.
  • I notice it’s important to you that we explain our thinking.
  • I notice you expect us to treat each other respectfully.
  • I notice you really want to get to know your students.
  • I notice our class is noisy.
  • I notice our class is quiet.   Those last two were from students in the same class!  I agree with the previous student. 

Some of the things they wondered:

  • I wonder if you give much homework.
  • I wonder why you have a Texas A&M cup on your shelf, since you went to the University of Florida.
  • I wonder if we will do a lot of group work like we did today.
  • I wonder if we will work on many problems like today’s problem.
  • I wonder if math class will always be this fun.   

I actually got that last ‘wonder’ from quite a few students.  First days don’t get much better than that!

I can’t wait to read about other people’s First Days (or First Weeks) through this week’s #msSunFun theme!


First Day Icebreaker — Me, by the Numbers

Today’s Sunday Funday topic is Icebreakers. I guess Icebreakers are typically activities that allow participants to get to know each other — so I’m not sure if this really counts as an Icebreaker, but it’s what I did the first day of school last year to introduce myself to my students and to learn more about them.  I think I heard about Me, by the Numbers in a Global Math meeting last summer, but I’m not sure who presented it.  Maybe it was from someone’s blog.  I need to keep better records of who I’m stealing from!

I started with a Keynote of numbers that would let students know a little about me.  I included my favorite number (42 – the answer to everything!) and then a countdown from 10 to 1.  I hesitated to use my favorite number as one of “my” numbers because that when students use their favorite numbers as one of their numbers it typically doesn’t tell much about them unless they have a reason it’s their favorite number and/or elaborate on their choice.  But I had to include it because 42 is a number they’ll hear a lot this year, because when students ask me the answer to a question, my answer is almost always 42 or a variation of 42 (e.g. 4.2 miles, $42,000, 420 unicorns).  It’s my way of saying I’m not going to give them the answer!

Mrs. Royster – By the Numbers

After I going through my Keynote, I asked students to come up with 5-10 numbers (any numbers — they didn’t have to be in a countdown format) that would help me get to know them.  In the school where I taught last year, students had MacBook laptops that they could use for this assignment, so it was easy for them to include a picture of themselves on their Me, by the Numbers page.  They printed their pages and I kept them to read and to quiz myself on names and faces.

I had to keep reminding students to choose numbers that would really tell me something about themselves.  This was a lot harder for students to understand than I anticipated.  I got a lot of numbers like 1 – the number of pets I have and 3 – how many meals I eat every day.  But I also got some really great responses, such as 1 – is the number of parents I live with.  My parents are divorced and I live with my mom and my sister. And 26 – the number of times I’ve been to Disney World. (I should have anticipated the fact that she’d miss a week of school to make that 27 times!) Another one I loved, in hindsight, is 10,000,000,000,000 – is a lot of doughnuts.  At first that one seemed like a really random, unhelpful number to share.  But as I got to know that student I realized it was a great example of his off-the-wall personality.

Sample Student Work

This year I’ve changed school districts and I’ll be teaching students who don’t have laptops. After teaching with the laptops for 5 years, it’s going to be a challenge to adapt my lessons to the lack of technology access. Of course students can list their numbers on paper, but having that picture was really helpful for me to learn students’ names.  Most of my students will have access to computers at home, but I guess I’m about to find out how well they do with assignments that require technology!

Do you have suggestions for how to increase student-to-student interaction for this activity?

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!