Jackie Cooke’s K-5 Math Blog

We started out the school year on this Blog talking about how to help students develop strategies that would enable them to become better problem solvers. In the September entry you will recall that I shared a slideshow I had created to teach students a process that assisted them in communicating their thinking in written form as they worked their way through problem solving tasks.  At this time of year, we circle back to this previous theme as many students are being asked to put effort into final work sample tasks showcasing what they’ve learned how to do in math. To assist students in creating their best work samples, we reviewed the ROS²E problem solving process as mentioned above. I updated the ROSE PS Review presentation to assist students in practice scoring (using a draft revision soon to be posted to this site of the Oregon Mathematics Problem Solving Scoring Guide) for some fraction tasks, since that is an appropriate concept for grades 4 and 5 and those are the students I am working with on this project.

We have also spent time talking about developing habits of mind that will serve them well as they move on to their next grade level. I have included what I call “Motivational Training” as well. I share with students some of the statistics regarding why it is important for them to take as much math as they can. At the same time we talk about exercising habits of mind to make for strong students. Such qualities as patience, responsibility, creativity, positive thinking and perseverance are emphasized.

In previous entries I have shared with you the Elephant Lamp story contained in the “MathDifficulties” TOMT article and  the Finders Keepers story I tell students from my childhood to encourage the qualities of honesty, respecting the property of others, asking for help when math doesn’t make sense and working hard to learn math facts. For this entry, I’d like to share with you one more story. This one highlights the quality of perseverance. The story is called  Can Do.

As I mentioned above, I also share with students some of the statistics that show why it is so important to take as much math as they can. I pull these statistics from a speech I delivered variations of during the year I was named Oregon’s Teacher of the Year. As you can see from the speech, developing strong mathematical problem solving skills bodes well for students’ future performance in math classes in high school and beyond, which in turn correlates to the likelihood of a better job with more capacity to earn a decent wage. I’d love to hear about anything you do in your class to motivate students to work hard at math. The June and August issues of the Teacher to Teacher newsletter will focus around two very good books, Malcolm Gladwell’s Outliers and Daniel Coyle’s The Talent Code. Both issues will look  more deeply at the idea that effort not talent is responsible for students being successful in math.

Math Homework

I was asked to share my thoughts on homework the other day. I decided this was a conversation worth bringing up with all of you.  I’ve done a lot of thinking about this topic over the years. A great article on this topic was written by Stephen Fisher and published in TOMT magazine in September of 2008. Steve included this interesting historical look at the evolution of homework in our country.

“Homework was not always the institution it is today.  A 1901 California civil code forbade homework.   In the 1920s, five to six hours of fresh air and sunshine were considered preferable to homework.  Attitudes towards homework have changed through history; but the present role of homework as a critical tool for increasing student achievement can be traced to the 1957 Soviet launch of Sputnik.  Since then, homework has played an increasingly dominant role in school-home dynamics.”

He goes on to include the following quote regarding some research done on the effectiveness of homework.

“In their groundbreaking book, The End of Homework: How Homework Disrupts Families, Overburdens Children and Limits Learning, authors Etta Kralovec and John Buell (2000) cite a significant body of research that shatters many popular beliefs about homework’s effectiveness.  They argue that homework actually pushes students and families away from schools.  This is particularly true for students from low-income families:

‘Like tracking, homework is a practice that perpetuates the social-class inequity that seems built into schooling.  When we look at homework in the context of a poor student’s life, the practice seems almost abusive… In their world, homework simply didn’t fit in. … Homework further disadvantages these children by assuming they have a quiet, well-lit place to study, far away from the TV.’ ”

I agreed with many of the premises in the article and had done a fair amount of changing the homework assignments I gave to my students. I decided that I would work to insure that any homework I assigned would be meaningful to the students and would not cause stress to family systems that in many cases were already highly stressed. I also knew that some families continued to support their children in doing homework and it was not uncommon for me to get requests to provide more homework than what was usually assigned.

So, I decided to give students (and their families) a larger variety of choices for homework and this included math homework. My underlying belief is that when students have learning choices, they take more ownership of their learning process and are more motivated to complete it. For many years now, I have used the “Home Connections” from the Bridges in Mathematics program published by Math Learning Center. I really like these packets and think they are the right blend of application and practice of concepts taught in the classroom but I find that the large amount of reading required can be challenging to families where English is not spoken in the home. So, I also included a worksheet that practiced math fluency and test taking skills. In addition, I tried to incorporate as many opportunities as possible to integrate math in problem-solving contexts and utilized the tools of technology wherever possible, too.

I developed a planner sheet that I used to help students contract with me what their homework choices for the week were going to be. I also did a search of the Internet and found some great problem solving newsletters, so I included one of those in each packet. Finally, I put together a list of math choices students could do in addition to any of the math homework included in the packet. I’ve uploaded a Homework_Sample to this web site for those who might be interested.

I did set a minimum expectation for students that required they do at least a half hour of math homework each week. Our school had a policy that students were required to spend a certain number of minutes practicing reading each night. The minutes increased as student moved up in the grades.  Also, at the elementary level there are other content areas that need to be practiced. It was like walking a tightrope trying to find a balanced approach to decide how much homework to assign.

I decided to let families and students have more of a say in how much homework was enough. I accomplished this by building in a reward system. At different times of year, I varied my reward program because I found that holding students’ interest was always a challenge. The rewards they were working for also changed periodically to shake things up and keep them interesting.  At different times of the year students could earn a certain number of points, tickets, or class cash per assignment completed. Some homework like the Bridges Home Connections projects earned more than completion of a single worksheet. The coupons, tickets, cash or points could be traded in for a variety of treats like trips to a treasure chest, earning a mid-morning snack or buying items in a class auction.

Ultimately, it was up to the family and child to decide how much homework he or she was going to complete in a week. Students and families used the planner sheet to communicate with me about what homework they were contracting to do for the week. They carried the planner sheets and their completed homework packets to and from school in a plastic folder that had a clear sleeve in the front to slip the planner sheets into.

I hope you will let me know your thoughts about homework. What are your philosophies? What great homework projects have you come up with?

Since January, I have taken on another job as a math coach for a local school district. It has been very exciting. The teachers in this district have made me feel extremely welcome. I have copied below a series of email messages that were recently sent back and forth among a couple of special education teachers and myself. You will find my response to Jim’s question at the end of this series. My hope is that others will chime in  with your thoughts, too, since middle school is not my area of greatest expertise. Here is the first message that started this conversation:

Hi Jim -
I didn’t know if this information would be of use for you, but here is one of the things the math coach at JWE has recently sent out. [Anna is referring to the weekly math notes I sent out to everyone. She forwarded it on to Jim as an attachment to this message.]  I haven’t actually met her yet, but I know some of the teachers are excited about her as a resource.
Thanks,
Anna

Here is Jim’s response to Anna and I was cc’d as well.

Thanks, Anna.

This was interesting.  My questions is more to do with math calculations and whether or not we should be putting significant time and resources into trying to get our older students (7th/8th) to learn the multistep strategies they have not been able to learn in their first 6 years of school.  Should we be prioritizing our time this way if a student is able to accomplish accurate calculations with the fluent use of a calculator.  I wonder at times if it is a wise choice of time and resources when we put math calculation goals on IEP’s for our older students who are being successful with calculations with their calculator.

Thanks for thinking of me.

Jim

Before posting the the blog, I checked in with Jim to let him know what I was thinking and asked his permission to use this question as a blog entry.

Jackie,
That sounds fine.  Again, I want to clarify that I am talking about our older students who have not been able to learn the strategies in years of instruction.  Obviously there are some brain function issues in play. If a student has missed a significant amount of instruction or is in 6th grade or lower, I think it is more appropriate to carry these goals on an IEP.  I am meeting a ton of resistance from students trying to slow them down to learn these strategies they have not learned.  Would our time be better spent by engaging students in a calculator curriculum gaining them functional skills that will allow them to independently complete regular class math curriculum?

Thanks for your help with this question.
Jim

And here is my response to Jim’s very thoughtful question.

As I mentioned in our previous correspondence, I think this is a very complex question. On the surface, I would   agree  with you that helping students utilize a powerful math tool like a calculator reaps a great harvest. I have collected many math calculator games and activities over the years because I see how vital it is for students to be able to use this tool fluently. On the other hand, I see students developing computational fluency through a multifaceted approach. As I shared in a previous blog entry, the following quote comes from Foundations for Success, the final Report of the National Math Panel.

“The mathematics curriculum must simultaneously develop conceptual understanding, computational fluency, and problem-solving skills. The development of these concepts and skills is intertwined, each supporting the other and reinforcing learning.

Teachers can help by providing students with sufficient practice distributed over time and including a conceptually rich and varied mix of problems to support their learning. In addition, teachers should encourage and support students in their efforts to master difficult mathematics content. Students who believe that effort, not just inherent talent, counts in learning mathematics can improve their performance.”

At the intermediate level and above, students need to be fluent in mental math, paper and pencil methods and using technology such as a calculator for computing answers to situations involving numbers (both whole numbers as well as fractions and decimals).  Another often overlooked or underdeveloped aspect of computational fluency is not only being able to compute in all three of the ways mentioned above, but also knowing which method is best used based on the given task. In addition, students must be able to determine if an exact answer or a close approximation (estimate) is sufficient.

In thinking about the situation you have described with your middle school students, there are a couple things that concern me. I worry about what is underneath the reason why these students are resistant to learning strategies to help them learn to compute accurately. Are you confident that they are truly fluent using a calculator?  Would they be able to explain how they are planning to use the calculator to solve a multi-step, complex problem solving task? Are your students able to explain their thinking orally well enough that you are  confident they understand what the task is asking and can apply the correct math concepts to reach a solution?  In that case  I would say utilizing the calculator may indeed be the appropriate choice because they may be bogged down with some kind of motor coordination or organizational issue that is separate from their ability to compute fluently.

Are they resistant because of a negative attitude towards math? A friend and I just did a workshop for middle and high school teachers on motivating unmotivated students. Perhaps what you are seeing is not a learning issue at all but a lack of seeing the relevance or importance of learning math or a student’s lack of belief in his or her own capability to do math?

Another concern is around whether your students may have given up a long time ago because they have to work much harder at math than other students because they are actually doing a harder kind of math. If the only strategy a student has is some variation on counting, they have to work extremely hard to solve any math problem since counting is such an inefficient strategy. There is a very good discussion on this topic on this blog dated April 2009 around the issue of students not having good number sense. Truly, I believe this is at the root of why so many struggle at the middle and high school level.

A few years ago, I wrote an article that was published in TOMT magazine on the topic of helping students with math dyscalculia. If these were my students, I would want to spend time finding out how they are thinking through problems and about math in general before I made a decision to drop one of the IEP goals from their plan. How about the rest of you? Anyone out there who might have some other thoughts on this topic that you’d care to contribute? Are there any of you experienced middle or high school folks who could share what you do to help your students learn to compute fluently?

The March Teacher to Teacher newsletter also discusses more insight on the importance of computational fluency and how to get students to reach that goal.  Click on the Teacher to Teacher link in the Blog Roll at the right to go to the Teacher to Teacher website.

Happy New Year Everyone,

Over the winter break, I was able to spend some time putting together  a web page that is intended to provide links to supplemental resources that are aligned to Oregon’s CORE Standards. As you may know, every grade level starts with this statement:

It is essential that these standards be addressed in contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations.

I’ve found some great internet resources aligned to the process standards (especially mathematical problem solving) and the grade level CORE standards, too. I wanted to share them with all of you. This site is only a beginning, hence there is a note at the top of the page informing readers that it is currently under construction. I’d  love to hear from you if you’ve found some internet resources that you believe align well with one of Oregon’s Process Standards or a particular grade level CORE standard. Together we can create a resource that is truly useful to teachers everywhere since Oregon’s CORE standards are founded on the NCTM Curriculum Focal Points.

For more great resources, click on the Teacher to Teacher link in the Blog Roll at the right to go to the Teacher to Teacher website.

This last week I had the privilege of being a facilitator for several different groups of teachers who  were interested in increasing their mathematics content knowledge and exploring effective mathematics instructional practices. One of these opportunities was as a facilitator for the Building Mathematics Leadership (BML) Across Oregon training workshops.  These workshops were held in Ashland, Salem and a third one is scheduled for the Pendleton area next week. This series of workshops is co-sponsored by Oregon Department of Education and Oregon Council of Teachers of Mathematics’ Professional Development Cadre.  There will be two more of the  BML  series of workshops in February and March. The BML link above includes registration information. Saturday found me in the Education Building at Portland State University teaching the final face-to-face session for a PrISM Oregon course called Deepening Algebraic Reasoning in the Elementary Classroom.

In both the BML and PSU PrISM sessions a question came up about how to better promote mathematical problem solving in schools.  During the BML session in Salem a teacher shared how she was using some of the tasks we’d shared at previous sessions with all the teachers at  her grade level. These grade level teachers jointly created a bulletin board they were using to display student responses. They were pleased with the amount of interest and discussion this board was generating with their students.  I also shared an idea. In the late 90′s a couple of  my sisters and I were all teaching at the same elementary school in Gresham. We decided to create a Math Problem Solving Contest schoolwide. Each month we would post a new problem. The principal would include the tasks   in the weekly newsletter that was sent home to parents. Students submitted their solutions to any one of us. Students who had solutions that worked were given small prizes (free popcorn at lunch, a coupon to the student store for a pencil, a special bookmark, etc.) At the end of the year, all students who participated in the contest got to participate in a Math Celebration. (It was similar to a field day  except all events had  a math component.) Washington State Mathematics Council’s Everything in Its Proper Place “Apples:  Part 1 and Part 2“,  Noyce Foundation’s “Problems of the Month, and PDC’s Penny’s Ancestors Math Lesson Design are examples of  tasks students solved in these contests. Any of the  Teacher to Teacher Problem Solving Tasks would also work well. What do you do to promote problem solving in your school? We’d love to hear from you.

Last year when I was teaching second grade, I believe I taught what would be considered a “typical” elementary classroom. I had 23 second grade students. Three of my students were identified with 504 plans for ADHD. One had an IEP for special education and three for speech and language services. Eight of my students were English Language Learners, one had an IQ in the low 80’s and 6 were reading beyond 3rd Grade level. Recently I came across the following quote in the book What Successful Math Teachers Do, Grades Pre-K- 5: Research Based Strategies for Standards-Based Classrooms by Edward S. Wall and Alfred Posamennier.

“Although a teacher may be tempted to have all children begin with the most efficient computational strategy, all children do not come into the classroom with the same skills and prior understanding. Children need opportunities to build on their own understandings and to publicly compare and contrast their strategies and those of their peers.”

The author was talking about this in regards to the teacher simply showing students strategies to solve problems rather then creating opportunities for students to construct their own strategies as they work towards solutions to problem solving tasks and then present their thinking to the class. I’ve used the idea expressed in this quote as the foundation for deciding which students I’d ask to share their thinking and in what order. When studenst are engaged in working through a task, I circulate around the room looking at student work and asking students to justify and explain what they are thinking. As I walk around, I hand out stickies to students with a number written on them letting them know the order I am going to ask them to present. Sequentially, I look for students who have elements that are on the right track and represent the thinking of those students who are still on a concrete level of understanding. Then I progress through the continuum. The final student presenting will represent those who are working at a more complex or abstract level of thought.

But their are other organizational strategies that can be considered. Recently, I attended a session by Jo Boaler at the NCTM Regional Conference in Boston, MA. During this sesson I watched a video clip of students engaged in solving a task. After working privately for a very short time, the teacher stopped the group and asked a student to present his thinking so far and then raise any questions he still had. There was not time for any students to work through to the solution. What ensued was a very lively discussion by the group with other students coming up to respond to the first student and demonstrate their own thoughts and strategies as the whole group worked together to come up with a communal solutiion.

Part of the craft of teaching is to make these kinds of instructional decisions such as how to structure the sharing out afterwards. The following is a list of a few possible ways to organize how the discourse around solutions will occur:

• Pair student work that contains common misconceptions with work containing correct solutions to bring out in discussion the similarities, differences, and contradictions.
• Present work with communication gaps to get student questions and feedback that will help clarify and fill in the gaps.
• If a problem lends itself to using a variety of math manipulatives, drawings, math tools, etc., the presentations could be focused on showing the variety of representations.
• Select students who represent the most common way the class chose to solve a task and then proceed through the continuum to the least common.
• Havie a student present a divergent way of thinking about a problem to broaden the group’s thinking.
• Have the whole class display their work on a table, in the hallway, or hung on the classroom walls and have the class do a gallery walk in random order to view all the work. Students record their questions and comments on sticky notes.

The key here is to use the student presentations to deepen the collective understanding of the core mathematics presented in the lesson. Gail Gerdeman of OSU’s Scientists and Teachers in Education Partnerships (STEPs) Program has designed a handout called a “Student Response Planning Tool” that talks about this subject in more depth. She
has also created a Sharing Template sheet that helps teachers record their plan for student presentations. I’d love to hear your thoughts on this topic or any other suggestions you may have regarding how to organize student presentations.

Click on the Teacher to Teacher link in the Blog Roll at the right to go to the Teacher to Teacher website. Currently there’s a great newsletter posted that shares more about Jo Boaler’s research.

It’s the start of a new school year. I hope you all have had as successful and fun a start to your year as I have. This school year finds me at a new school and in a new teaching assignment. I am now working as the Title I Technology Teacher at Highland Elementary School in the Gresham Barlow School District. My job description this year will be to support the math and literacy goals of the school using the tools of technology. My schedule includes weekly rotations of all 1st through 5th grade classes through the lab in 30 minute blocks. In addition, I have two half hour blocks devoted to a 5th Grade Tag Literacy Group and a 4th Grade Math Intervention Group. If you are interested in finding out more about what types of projects I am doing in the lab, visit my web page and follow the new link I’ve added to the Computer Lab page. My plan for this blog is to continue to share any issues around mathematical problem solving that may surface during the year. Hopefully I’d also like to share with you any useful technology enhanced math activities that I come across.Here’s to the start of a great year.