Welcome! We are so glad to have you here for our summer book study of Teaching Student-Centered Mathematics by Van de Walle, Lovin, Karp, and Bay-Williams. If you are just arriving on our blog today, we recommend you get yourself a copy of the book if you are teaching primary students. Even if you are a teacher of older elementary students, an understanding of developmentally appropriate mathematics instruction at younger ages is essential. If you do not have a copy of the book, we will be doing our best to summarize while sharing our thoughts as well. We consider the contents of much of this book to be fundamental—A LOT lies within its pages. Realistically, we cannot address everything, but rest assured you will come away from this book study feeling affirmed, having your eyes-opened, and feeling inspired to help the kids in your charge.
So let’s begin!
In the preface, the authors begin by sharing their belief that children can learn mathematics with understanding. When teachers provide opportunities for their students to “interact with and struggle with the mathematics using their ideas and their strategies—a student-centered approach–the mathematics they learn will be connected with other mathematics and to their world.” The first part of the book, chapters 1-7, help teachers in creating this student-centered environment.
Chapter 1: Teaching Mathematics for Understanding
When I read the first few paragraphs of chapter one, I was taken back to a time when teaching fifth grade that I had an in-depth conversation with my students about the difference between simply doing mathematics and using mathematics. Each year I would have an abundance of students enter fifth grade knowing procedures/steps to multiply and divide, yet they lacked the conceptual understanding of what it meant to multiply and divide–and for that matter, when they should multiply or divide. The following expression became a central focus in our classroom, as it was posted for all to see and remember, “Become a user of math, not just a doer of math!” Taking those fifth graders “back” in a sense (to the conceptual understanding of operations) had to be done, but it also had to be done in a way that provided students with affirmation of what they already knew and would help them develop relationships between their current knowledge and new concepts. Understanding had to become the first focus!
In chapter one, the authors stress the importance of students not only having procedural fluency but conceptual understanding of mathematics as well.
All of the following were cited as sources (containing process standards/strands) for helping educators think about how to engage students in procedural and conceptual understanding of content, keeping in mind that the practices/strands shared should not be addressed in isolation. They are ways for children to learn any mathematics.
- The National Council of Teachers of Mathematics Process Standards (2000) — Here is a PDF that overviews the Process Standards as well as Six Principles for School Mathematics.
- The National Research Council’s research review, Adding It Up – Here is a PDF outlining the Strands of Mathematical Proficiency (p 115-156)
- The Common Cores Standards for Mathematical Practice – www.corestandards.org/Math/Practice/
The authors go on to give readers a look into a second grade classroom in order to illustrate teaching for understanding–relational understanding. Students were given a story problem at the beginning of the lesson and were then given the opportunity to solve the problem in their own ways. Students relied on previous learning about numbers and place value and were able to make connection between their learning and the problem at hand. Various strategies were used, and a natural expectation was for students to explain what they did, why they did what they did, and how they knew their solutions made sense. The teacher’s instruction of a specific strategy was not at the center. In such an environment, students benefit greatly from listening to the thinking of others and, at times, discovering ways of thinking they themselves have not thought of—strategies that may be more efficient than their own. In this way, students are able to expand their repertoire. There will also be students who have not developed some previous learning essential for understanding the problem and/or the thinking shared by their fellow classmates. They may not be able to make the connections others are able to make. By making all of these observations, a teacher is able to assess students’ learning and plan for future explorations.
To end chapter one, the authors briefly discuss three common types of instruction; direct instruction, fascilitative methods, and coaching. There is no clear-cut answer for which type of instruction is most appropriate. I think this quote is helpful in thinking about what the primary focus of instruction should be, “The essence of developing relational understanding is to keep the children’s ideas at the forefront of classroom activities by emphasizing the process standards, mathematical proficiencies, and the Standards for Mathematical Practice.”
Furthermore, in a mathematics classroom that promotes understanding….
- Children’s ideas are key.
- Opportunities for children to talk about mathematics are common.
- Multiple approaches are encouraged.
- Mistakes are good opportunities for learning.
- Math makes sense.
What does chapter one mean for me as a teacher and my students as a learner? I agree with everything that was shared in this chapter—yet a classroom that promotes relational understanding does not happen magically. It takes work in establishing procedures and routines so students know exactly what is expected, feel comfortable, and develop perseverance. One of the most valuable things we do at the beginning of the school year is talk about what mathematicians do. Students are asked to think about times when they use math in and outside of school. General questions are also asked such as, How do you feel? Why do we use math? What helps you be successful? It is amazing how many wonderful ideas students come up with that are important for understanding. Click her to view a list of the ideas generated by last year’s students. When students generate ideas themselves, ownership is established. Much time is also spent at the beginning of the year helping students understand what to do when someone’s idea/thinking does not mirror their own. Questions are asked, How can we respect ideas that are not the same as ours? How can we praise the thinking of others? How can we encourage others to help them grow without helping too much? We then create an anchor chart that lists positive behaviors for sharing. I model the behaviors whenever appropriate and draw special attention to students who are good examples of the behaviors as well. As a teacher, I also need to constantly remind myself to let my students maneuver through problems/tasks and not guide them too much. I think we all want our students to be able to think and act independently, yet we can actually hinder this goal by imposing one way or leading students to a strategy that they may not understand. The anchor charts are for me just as much as my students. They serve as reminders for all of us! I appreciated how the authors ended the chapter by reminding me that I am a learner right along with my students—students are able to deepen their understanding when I am able to deepen my own.
Chapter Two: Teaching Mathematics Through Problem Solving
All of chapter two focuses on understanding how to teach through problem solving, not simply for problem solving. I will highlight aspects of the chapter that mean the most to me as I reflect on my students and my own practices.
First, what is the difference between teaching through problem solving and for problem solving? Teaching for problem solving involves the teacher directly instructing a specific skill, giving the students the opportunity to practice the skill, and then solving word problems using the skill. Teaching through problem solving begins with the problem itself. A problem, specifically designed/chosen by the teacher helps students learn new mathematics while creating relationships/making connections to previously learned concepts.
I find guided math groups to be an ideal setting for differentiating problems/tasks for students (ways to differentiate problems will be discussed in chapter four). In an intimate, small group setting I am able to work closely with students as they problem solve, asking thought provoking questions and listening the discussions among members of the group. I feel differentiated guided math groups to be equally as important for more advanced students as those who are struggling to understand a concept or lack some foundational knowledge. All students benefit! While I am working with a small group, students are then engaged in meaningful tasks.
Knowing what makes an effective problem/task is also helpful:
- The problem should engage children where they are in their current understanding. The knowledge students have before solving the problem/engaging in the task can be used, yet the problem should lead to new learning.
- The problematic or engaging aspect of the problem must be a result of the mathematics that the children are to learn. Students are focused on making sense the concepts inherit in the problem to help them gain new understanding.
- The problem must require justification and explanations for answers and methods. The solution or the process are not straightforward, therefore students must determine if their solution/process is reasonable, and they are responsible for justifying their thinking.
After reading this chapter, I took some time to reflect upon many of the problems I designed for students to explore in the whole group setting, in small guided math groups, and collaboratively with a partner in the past school year. I considered the three components of effective problems as shared above. How did they measure up? Some were fabulous, others could be revised to be more effective, and some did not “pass the test”. Where students are in their current understanding and requiring justification are always on my mind when designing problems for my students, yet there are those that fall short in helping students gain new understanding.
In second grade, we always encourage students to represent their thinking in different ways. There are an abundance of ways from using words and numbers to drawing the tools used to solve a problem. Therefore, I found the “rules of thumb” for using representations to be especially important:
- Introduce new representations and tools carefully. Make sure you show how the representation or tool can be used to illustrate a specific idea.
- In most cases, let students select the tools they want to solve problems. Make a variety of tools available.
- Encourage students to create their own representations. Do not direct student to a particular tool (unless as noted in the next rule).
- Encourage the use of a specific representation or tool if you feel it will help a student who is struggling.
- Ask students to use a representation or tool when explaining their thinking to you an/or their classmates.
Chapter two also provides guidance in using manipulatives (download some free here) and presents a three-phase lesson format for teaching through problem solving. If you do not have a copy of Teaching Student-Centered Mathematics PreK-2, we recommend you get yourself a copy so you can reap its benefits fully and take the time to absorb each chapter’s contents. In the mean time, you might be interested in learning more about three-phase lesson planning to teach math through problem solving by visiting this site.
We would love for you to leave your thoughts and comments about chapters one and two. If the comment field is not visible, just click on the title of this post and you will see the comment field at the bottom. Please feel free to share what the chapters mean for you as a teacher and your students as learners. Also, we encourage you to visit our fellow bloggers who are participating in the book study along with us. You can link up to their blogs at the end of this post!
Thank so much or your time, and we will see you on Wednesday for a discussion of chapters two and three!
All the best–