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!

Looking to Strengthen Your Lessons?

There are so many lessons available to teachers - it can be both a blessing and a curse.  Finding different resources allows educators to personalize the learning - both for their students and themselves.  How do you know whether the lessons are high-quality?  What might work well in the lesson?  What might need to be revised to better fit your needs?  Rubrics might be one way you can use to help you preview lessons you find and determine how they might best fit your needs.

I have been so fortunate to work with Achieve's EQuIP (Educators Evaluating Quality Instructional Products) over the past year.  Our group is comprised of 55 educators from across the country.  We look at lessons for mathematics and English language arts.  Using the EQuIP rubric, we review lessons and units, giving feedback to the developers about what works well within the lesson/unit and what might be improved.  During our meeting in Washington D.C. this summer, the Teaching Channel came to record our work.  You can see more about Achieve's work here: https://www.teachingchannel.org/videos/better-common-core-lessons-equip

If you are interested in using the rubric, here is a link to the EQuIP rubric:
http://www.achieve.org/files/EQuIPmathrubric-06-17-13_1.pdf (for mathematics)
http://www.achieve.org/files/EQuIP-ELArubric-06-24-13-FINAL.pdf (for ELA/literacy)
http://www.achieve.org/files/K-2ELALiteracyEQuIPRubric-07-18-13_1.pdf (for K-2 ELA/literacy)

Lessons and units that are judged to be "Exemplary" or "Exemplary if Improved" are shared on the EQuIP website and are free for anyone to use.  When developers submit lessons to Achieve, they agree that their work will be shared with others if it receives the "E" or "E/I" rating.  You can find those lessons here: http://www.achieve.org/EQuIP

Ohio has created a similar rubric, based on the EQuIP rubric, to help educators evaluate lessons they might use in their classrooms.  You can find the Ohio Quality Review Rubric for mathematics here: https://education.ohio.gov/getattachment/Topics/Academic-Content-Standards/Mathematics/Resources-Ohio-s-New-Learning-Standards-K-12-Mathe/Mathematics-Quality-Review-Rubric.pdf.aspx

I hope that you will use some of these tools to talk with your colleagues about the lessons you are teaching.  Many teachers no longer use textbooks and these rubrics are one of several available tools to help teachers know that they are using high-quality resources.

Seeing Through the Eyes of your Students

During my coursework at Ohio State, I read a powerful article that made me think more deeply about what questions we might ask students.  The article was written by Shelly Harkness who is currently a professor of mathematics education at University of Cincinnati.  Entitled "Social constructivism and the Believing Game: a mathematics teacher's practice and its implications', the article examined how teachers responded to students when the teacher took the position that what the student said was correct - regardless of the accuracy of the student's statement.  You can access the full article here: http://bit.ly/1rgRAEY

Harkness writes of a student who responds "false" to the statement that All triangles have three sides.  The student is asked to come to the board and draw a figure to represent what she sees.  She draws the following:

The teacher responded, "No.  That's not a triangle.  It's not flat.  The answer must be true."  The conversation ended and the class moved on.

Harkness writes that the drawing above made her reconsider her own thinking about the mathematics.  She wondered if Kayla looked at each side individually, thinking each of the four sides was flat.  What questions might you ask Kayla to explore three-dimensional space? What discussion might you have about sides and faces?  Or about geometric language?

What does Kayla know about triangles?  How might this knowledge be influencing her interpretation of the question?

As we begin this new school year, I encourage you to assume all answers your students give are correct.  At least from their point of view.  Children do not intentionally give incorrect answers - the answer they give has validity from their point of view.  Working to understand why they believe they are correct is the first step to uncovering their misconceptions and working to build their understanding.  This school year, work to build meaning together WITH your students.

How I spent my summer vacation.....

We are getting ready to begin another school year - and I am excited about the opportunities I have had over the past few weeks that will influence my work for the upcoming year!  It has been a busy summer and too long since I've written on this blog.  Here are a few highlights:

• 4 days in Columbus, OH with the Network of Regional Leaders (NRLs) and the Ohio Department of Education.  One day in June was spent debriefing the previous year with the Mathematics Cohort.  Then we spent 3 days in July with NRLs from all disciplines at the Summer Leadership Academy.
• 1 1/2 days in Washington D.C. with EQuIP peer reviewers reviewing lessons for Achieve and sharing our work the the Teaching Channel.
• Many hours working on a parent website as a resource for parents in our district.
• Many more hours working on math resources for 5th grade teachers in our district.
• Meetings to plan professional development for using ST Math - our district is so fortunate to be part of a consortium that won a Straight A Fund grant from the State of Ohio to provide this opportunity for our students!
• Tutoring 2 wonderful girls who are going into 5th grade!!!
• Presenting at the Innovative Learning Environments 2014 conference in Hilliard, OH.
• Participating in Jo Boaler's "How to Learn Math" course through Standford University.
• Reviewing manuscripts for the Ohio Journal of School Mathematics.
• Reading great articles about teaching mathematics!!!

It wasn't ALL work!  We did manage to get away for a family vacation and I enjoyed time with family and friends this summer.  I feel so fortunate to have had these opportunities to spend time with so many talented educators!  I hope to share what I have learned with you in the upcoming weeks.

I hope you have had the opportunity to recharge this summer - and that you have a WONDERFUL start to this school year!

Using Number Pieces for Work with Decimals

I've written before about Number Pieces - one of my favorite apps!  (It is free - I hope you give it a try!)  While there are obvious connections to the primary grades for this app, how might you use it in the intermediate grades?

One possibility is for addition and subtraction of decimals.  Using base-10 blocks for decimals is a shift for students.  Since early school experiences, many students have thought of the unit (or small cube) as "one".  Before using base-10 blocks with decimals, it is important to give students opportunities to explore changing the meaning of "one".  If "one" is the long/rod, what does the unit cube represent?  If "one" is the flat, what does the long/rod represent?

After students have a chance to explore these relationships, the base-10 blocks can be a useful tool.  For example, in the problem "Mary bought 9 pens.  Each pen cost $0.35.  How much did Mary spend on the pens?" students might use repeated addition to solve the problem.  They might set up the problem to show that the flat will represent $1.00 and 3 tenths and 5 hundredths represent the cost of one pen.

Then students might draw 9 representations of 3 tenths and 5 hundredths to show the 9 pens.  They can show the repeated addition as 0.35 x 9.

If you use the "lasso" tool, you can group all of the blocks together and then click the "join" button.  This will group the tenths into ones, as well as the hundredths into tenths.

Using the "lasso" again, the app will join the tenths into ones.  The student can then see that 0.35 x 9 is equal to 3.15.

This app provides students with a tool to visualize joining situations (and separating situations if you use the different colors) with decimals.  This can be an engaging tool for students and help students create mental representations of the problems they are solving!