Open Inquiry in Math?

In reflecting about this year of inquiry, I have been all over the map!  I initially felt that focusing on math inquiry would be an interesting way to steer my thinking.  There are so many topics that naturally lend themselves to literacy, I am always adapting them to math.  Then, I thought an easier route would be social studies and science inquiry, which really lend themselves to a more open inquiry approach... not easy at all!  Now I am to the conclusion that I perhaps have been a little too reflective!  Sometimes I dig myself into a hole of thought and have trouble executing a plan for fear of failing. So, I am going to continue with plans for integrating inquiry into math through the end of the year and focus on successes and challenges there.

Our last inquiry work with Michelle was around open inquiry, which is by far the most demanding in some ways because it requires me to be so flexible with plans.  It all depends on where kids are at in the process of learning.  Jamie and I are taking this approach to our current unit on Colorado History.  I don't think this is an approach that can be used in math.  An open inquiry approach in math would do little to lay the foundation that kids need in order to execute sound math thinking.  Am I wrong?  My current approach in math is curricular inquiry, where students participate in inquiry around a specific foundational topic.  Right now, that topic is perimeter and area.

Chalk Talk

As I shared in my last post, students have been thinking about math thinking and created chalk-talk posters for each of the Standards for Mathematical Thinking indicating what each standard meant to them and/or an example of how they have demonstrated that standard.  We then used sticky notes to indicate which of the cognitive thinking strategies we need in order to successfully demonstrate each standard.  The posters are now hanging in the classroom under the thinking strategies.  My next step is to physically connect each standard to its corresponding thinking strategy using yarn.  I am going to use the topic of perimeter and area to challenge students to think about how they think to solve problems.

Hallway Polygons

I am planning this week using the most logical approach.... searching 'perimeter area' on Pinterest.  I found some great inquiry activities, but do these activities lend themselves to true curricular inquiry?  Will students be discovering the meaning of each by engaging in activities that really push their thinking.  Here is what I have so far:

• Students will make 'hallway polygons' using the 1 x 1ft. tiles on the floor and painter's tape.  This is Tony's favorite!  In the past I have made these polygons and had students use red, 1 ft. strips of paper to measure the perimeter and green 1ft. square sheets to measure area.  I am thinking that I will make one and have student make 6 or 7 more for the rest of the class to measure.  I found that this is a great way for kids to 'walk' the perimeter and get very kinesthetic with the concept.
• Using Google Maps and the ruler tool to find perimeter and area.  Students find the perimeter of our school, the US, the Pentagon, the state of Colorado, Lake Superior.  This will be a great way to really problem solve the perimeter and area of irregular shapes and integrate tech. 
• Find the area of your footprint using graph paper.  Straight out of Everyday Math, this is another great one for looking a irregular shapes and solving problems.
• Measuring Penny, a read aloud about measurement that also has an activity where students have to design a dog house and a dog run to maximize area for the dog to have space to run around.
• Students will make a visual representation of the meaning of perimeter and area to hang on the wall.  The visual can be anything (tool, example, non-example, picture) to cement the meaning of each in the reader's mind.

My inquiry questions for my students:

• How does what we measure influence how we measure?
• How do we find areas of rectangles?
• How do we find perimeters of rectangles?
• How can we find rectangles’ lengths if we know their areas and widths?
• How is area connected to multiplication?
• Why does area matter?  Why does perimeter matter?
• Who uses perimeter and area in their lives besides fourth grade math students and their teacher?

How does this relate the Standards for Mathematical Practice?  I am going to ask students to identify which of the standards they are using when the compute area and perimeter.  According to my scope and sequence, they should focus on:

1. Make sense of problems and persevere in solving them.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.

Standards for Mathematical Practice and Inquiry

I am on a continued quest to embed inquiry into my math lessons with fourth graders, and I am finding that I am continuing to ask, "What does inquiry look like?"  As I read our book and learn about the different types of inquiry, I don't see any that fit for math. Is it enough for me to be a better questioner, or do I need to go full on Open Inquiry?  How can I do an Open or Curricular Inquiry when there are specific skills that students need to learn in math as they progress through fourth grade? 

Obviously, there is a continuum when implementing inquiry into the classroom.  Some of what I do is much more inquiry based than others.  I used to think that all inquiry was open inquiry, with students choosing their own path for learning.  With the demands of standards (CC or otherwise) now I think that most of what I do in math is going to be curricular inquiry.  Although, that doesn't really seem to fit either.  Now I am thinking that the inquiry piece may be around how to build math THINKERS in my classroom.

My current focus from now to the end of the year in math is how to effectively integrate the Standards for Mathematical Practice into our thinking strategies for math because I think that if I make these my focus, it will change my entire approach.  It will also honor the different learning styles in my classroom and allow me to take a thinking approach to whatever type of math I am teaching.

The Standard for Mathematical Practice:

describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).   --http://www.corestandards.org/Math/Practice/

The idea of working thinking routines into math class is not a new one to me. Nor is an emphasis on discourse and writing.  But I am frustrated by the way new things seem to come down the pike, without any connection to what is already in place.  With a continued concern that kids cannot transfer their knowledge from content to content, I am interested in streamlining what I am doing in terms of setting expectations for thinking and understanding in fourth grade.

My plan is to lead students in an inquiry lesson where they pull from their schema about the thinking strategies and connect the SMP to each.  We are going to do one (Questioning) together as a class and then students will do the Chalk Talk routine to connect them.

How does this all connect to inquiry?  I am thinking that if I can do a better job of setting the expectation that, "These are the ways you will be expected to think in math," then that will lead us to much richer, inquiry-based math learning.

Letting Go

My group, The Weavers, has been nothing but inspirational.  Our monthly meetings provide me with a much needed place to work through ideas and thoughts that seem somewhat muddled in my own brain.  One thing I realized is that I have been scaffolding my student's ability to be capable of inquiry since day one.  Our belief in the explicit teaching of the thinking strategies has provided our student's with the cognitive abilities to blossom when given an inquiry situation.  So I began my dive into math inquiry by reminding them of the knowledge they already possess!

 I used to think that math inquiry was going to lead me on this great, amazing, inspiring teaching journey!  Then I realized how HARD it was going to be.  With all the pressure of testing etc., it was terrifying to think of letting my students come to understanding through their own inquiry process.  Through the support of my amazing friends and colleagues, now I think that this inquiry journey is exactly what I need as a teacher and specifically what my students need as learners.  Each time I provide them with a "mini" inquiry into a math topic, I am in awe of what they come away with learning.  And not because I told them it is true, or the right way to do it, because they came to that conclusion on their own.  This is the type of learning they will remember!

Math Inquiry

In my group, the Weavers, I have committed to trying inquiry in Math.  In the last month, I have been very close to giving up that goal.  I was really not sure how I could possibly integrate inquiry into math.  There are too many foundational skills that have to be mastered before students can move forward.  This work is far too important for me to give up as much control as inquiry requires me to.  Then I had a conversation with a math teacher in the district who is finding success with presenting her students with one cognitively challenging task and allowing them time to explore the problem with an understanding that students will not necessarily complete the task with success.  Her focus is on the process that they go through as they collaborate in an attempt to find a solution to the task.
Talking to her made me realize that I don't have to approach inquiry in math as I might in science.  Math inquiry can be truly 'mini,' focusing on a very small objective that I want the students to master.  I decided to give it a shot.


My students have been learning about fractions.  They have been composing and decomposing fractions, learning about equivalence, and learning about the reasons why we add and subtract them the way we do.  I decided to try allowing my students the opportunity to "discover" how to multiply a fraction and a whole number.

We started by activating our schema around how we multiply.  Collectively, we found four strategies for multiplying: by writing out our understanding that multiplication is x, counted y times (skip counting), creating equal groups, using arrays and by adding repeatedly.  Once we made that list, I shared the objective with students: They will learn how to multiply a whole number and a fraction and apply a rule to these kinds of problems.  We brainstormed three of these types of problems.  Then I set them off to work together using any tools in the room to help them (pattern blocks, base 10 blocks, rulers, number lines, paper pencil.)  They worked for about 15 minutes individually and in small groups of their choosing.  At the end of the 15 minutes, some were still struggling with the concept of applying what they already knew to fraction multiplication while others had grasped that and had successfully created a rule.  We debriefed the process and recorded our findings. 

I was very encouraged by this process. I think it gave me hope that inquiry can be an essential element in students better understanding math.  Overall, my students were more engaged in this process than if I had done a mini lesson on the rule then given them time to practice.  I had a student, who generally struggles in math, ask me that if we use repeated addition to multiply, wouldn't we be able to use repeated subtraction to divide?  I had another student think that she had found the rule, but could only apply it to two of our three problems.  After much discussion, where I had to force myself to sit back and NOT say anything, she realized that, in fact, the rule did work, it just resulted in an improper fraction rather than the equivalent mixed number she came up with.  She came upon this by asking questions of herself and her partner, not by having it delivered to her by her teacher (not that I didn't want to.)

I used to think that inquiry had to consist of complex, backward designed lessons.  Now I know that I can approach inquiry as an innovative way to tweak my math workshop where I put the ownness of learning squarely on the shoulders of my students... AND they are totally capable of rising to that challenge.  I also used to think that inquiry didn't have a place in math.  Now I know that it fits into my beliefs about the way students best learn math.

My next step is to have students discover the process of multiplying two fractions in the same way.  I want to see if I can replicate the success I had with this lesson. Another next step, or a missed step, is to assess this process to see how well my students understand how to multiply a fraction and a whole number.