Learning Mathematics: Web vs. Ladder?

I have had several conversations recently about the teaching of math that focus on what students can do and what they need to do next.  Many times in education, we worry that students cannot do something until they have a prerequisite skill.  I think of this as a "ladder" approach.  Students need to achieve the first rung of the ladder before reaching for the next rung above it.

While I certainly believe in the value of learning progressions, it seems that we can get too caught up in individual skills and overlook the big picture.  I think of mathematical knowledge in terms of a "web" rather than a "ladder".  Providing students with a rich mathematical task (defined by Jo Boaler and others as having a low floor and a high ceiling), students have the opportunity to use what they already know about mathematics to construct further knowledge.  While the teacher has a learning target for the lesson, if students are allowed space and opportunity, they might connect mathematical knowledge that the teacher had not considered.  Knowing learning progressions will help the teacher support the students in making connections and guiding them towards the learning target - but the students should be the ones making sense of the mathematics.  This idea is interwoven throughout the Standards for Mathematical Practice when students are expected to construct viable arguments, look for and make use of structure, and make sense of problems.

A perfect example of this occurred last week during a course that I teach at Otterbein University.  The students are working towards a 4-6 endorsement for mathematics.  We were exploring Sums of Consecutive Counting Numbers.  After defining the problem, here is a sample of what students discovered:

We looked at at numbers through 50 but you can get a sense of the problem in this chart.  Students observed that for some numbers, such as 1, 2, 4, 8, and 16, we were unable to find consecutive counting numbers that would create those sums.  Students also noted that some of the sums could be created in more than one way with consecutive counting numbers.

Then, a student made an observation that I had never heard.  I have facilitated this problem with students in grade 3, 4, and 5, with undergraduate students, with graduate students, and with practicing teachers.  Each time I think that I have heard all of the possible observations, someone surprises me by making an observation I hadn't seen before!  A student thought of the consecutive numbers as blocks and visualized adding another column.  She was using geometry to make sense of the problem - this seems like such a natural way to view the problem yet no one had ever mentioned it to me before!

After describing how she visualized the problem, students made a conjecture that they could find every other number beginning with 3 if they used 2 addends.  They could find every 3rd number beginning with 3 if they used 3 addends and every 4th number if they used 4 addends.  They thought this pattern might continue.  I then asked them to consider why this pattern might exist.  What in the structure of the numbers created this pattern?  This led to another interesting discussion.

My learning target for this lesson was to have the students engage in a rich mathematical task.  I asked them to share the mathematical content they felt could be explored with this problem.  They listed:

1. finding sums
2. divisibility 
3. looking for patterns/algebraic thinking
4. exploring multiples
5. problem solving

I would have a difficult time putting our discussion on a "ladder" - however, I can visualize an interconnected web.  The mathematical topics are known in advance but the way in which they will be connected is the work of the students.  The students were all able to participate in the problem and share their thinking.  When I have used this problem in classrooms in an elementary school, all students are able to participate and fully contribute to the conversation without needing to group them based on ability. How might thinking of mathematics as a "web" change our instruction?  What shifts need to occur in classrooms for this type of thinking about mathematics to occur? 

What a math classroom should look like, sound like, and feel like

During my first class of the semester last Thursday, I asked students in my Advanced Pedagogical Content Knowledge for Intermediate Mathematics course to describe their experiences as a student of mathematics.  Here are some of their comments:

I find it frustrating that these students - who are young enough to have been in my own 4th grade class - had these types of experiences in math class.  Students thought that perhaps their memories were stronger for high school math classes rather than elementary school classes.  Are these experiences listed above what we want for our high school students?

It was refreshing to hear what my students thought a math classroom should look like, sound like, and feel like.  Here are some of their thoughts:

I am looking forward to expanding on these ideas with my class this semester!

Principles to Action: Using ideas for professional development

I have been reading NCTM's Principles to Actions: Ensuring Mathematical Success for All.

 This book is a wonderful resource for educators wanting to learn more about mathematics education based on current educational research.  One of the areas emphasized is "Facilitate Meaningful Mathematical Discourse".  While the focus is on students, I have also used this idea to facilitate meaningful conversation among colleagues about our mathematics instruction.  We had 20 classroom teachers attend an early morning "Math Summit" at our building in November.  Out of the conversation, we identified retention of basic facts as a common source of frustration and a topic we wanted to explore further.

In December, we met again and sorted the standards for grades K-5 related to number sense, computation, and fluency. After discussion around the progression of learning across grade levels and observing that students should know from memory the basic facts for addition, subtraction, multiplication, and division by the end of grade 3, we read Gabriel T. Matney's article "Early Mathematics Fluency with CCSS-M".  Currently, we are discussing in an on-line forum how we might define fluency and how we use number of the day, number talks, and problem solving to build fluency.  Teachers are expanding their definition for fluency far beyond speed with recall of basic facts.  We've had great discussions!

In Principles to Actions, there is a chart that describes the actions students are taking when participating in meaningful mathematical discourse.  These include:

• Presenting and explaining ideas, reasoning, and representations to one another in pair, small-group, and whole-class discourse.
• Listening carefully to and critiquing the reasoning of peers, using examples to support or counterexamples to refute arguments.
• Seeking to understand the approaches used by peers by asking clarifying questions, trying out others' strategies, and describing the approaches used by others.
• Identifying how different approaches to solving a task are the same and how they are different.

I am excited by the energy and enthusiasm teachers bring to our conversation about fluency.  They are engaging in all of the behaviors of meaningful mathematical discourse!  Our next question for discussion is to describe an application of a number of the day, number talk, and/or problem solving that is used to connect to fluency.  I'm looking forward to the sharing of ideas!