# Perseverance through the Field Trip scenario

When I think about what excites my students and me most about teaching and learning math, I think of real-world challenging scenarios like the Field Trip problem, which requires students to work collaboratively to solve a problem, evaluate their strategy, and produce a poster that stands alone (shows clearly labeled strategy and answer) that would ultimately be shard in a Math Congress (intentional group discussion) at the culmination of the lesson.

The scenario: A 4th grade class traveled on a field trip in four separate cars. The school provided a lunch of submarine sandwiches for each group. When they stopped for lunch, the subs were cut and shared as followed:

• The first had 4 students and shared 3 subs equally.
• The second group had 5 students and shared 4 subs equally.
• The third group had 8 students and shared 7 subs equally.
• The last group had 5 students and shared 3 subs equally.

When they returned from the field trip, the students began to argue that the distribution had not been fair, that some students got more to eat than the others. Your job today is to determine whether or not this distribution was fair. Did each student get the same amount of sandwich, no matter what group they were in?

I intentionally chose this problem for several reasons. First, it aligns perfectly with Common Core Standards including, but not limited to, 4.NF.A2 (comparing fractions with unlike denominators), 4.NF.B.3B (decomposing a fraction using visual fraction model), and the Standard for Mathematical Process of making sense of a word and persevering in solving. Next, I knew that I needed to step beyond the adopted Bridges program to provide an opportunity for a group work task-one that truly requires numerous strengths, as opposed to one or two. Finally, it’s fun. Doing a well-written problem with a group is a unique experience that includes encouraging, disagreeing, and thinking about strategies. When I am not instructing in the front of the room as the “all-knowing sage” and when there is not one “right” way to find the solution, the beauty of math emerges for everyone.

As an introduction to the problem, I shared with students 2 goals that I had for the day: #1- That I would not be helpful (I sometimes have a problem being way too helpful thereby inadvertently “doing the thinking for them” ) and #2- That I hoped students would really struggle with this problem.

Now that I had the class completely wide-eyed and alert, I shared a list of 10 requirements for being successful in this problem, including reading carefully and having the patience to re-read, thinking about fractions as division, dividing wholes into equal pieces, comparing fractions (>, <, =), and justifying (proving your thinking) with clearly labeled work. Not everyone has strength in each area, thus the requirement and beauty of valuing unique individual strengths though collaboration in a small group.

As groups began to work, my class took absolute ownership of problem solving strategies. Looking across the room, I noticed a group struggling. When I stepped in to help, one student quickly said, “Mrs. Nied—You said you wouldn’t help us. Give us a chance to figure it out on our own!” Could it get any better than that? Additionally, I observed groups with heads together, using language like, “I like how you said that,” “Could you help me understand your strategy,” “Don’t forget that we need everything labeled,” and “Yes! I figured it out!”

The most exciting outcome was that one of my students who typically has low status and would unlikely be asked for his ideas was literally the only one in the class who knew exactly what to do right away. His strategy:

It was incredibly empowering to my student to be able to say to other groups who were absolutely stuck to go ask him to describe his strategy, and he was elated to share!

Through this problem, I observed two common misconceptions that occur when working with fractions. Using play-doh to build the sandwiches, many students began with first dividing each sandwich in half and giving each student in the group a half. Then, students took the remaining halves, broke them in half, until each student had the same about of play-doh pieces. Beyond the ½ of the sandwich, students lost track of what a ½ of a ½ is, and what a ½ of that is, and what a ½ of that, is etc. With the question requiring students to know how much each student was given, students did not know what the model that they correctly built represented. Another misconception occurred when students had to order fractions. Many students indicated on their posters that 7/8 is less than 5/6. When asked their reasoning, they said that the whole is broken into smaller pieces, so the fraction must be smaller, without taking into account the information provided by the numerator. Both of these misconceptions were discussed at the heart of our Math Congress.

In teaching and learning, there is nothing more powerful than when students take complete ownership of their work. As I think about how we started the year sharing how we feel about math, I am amazed at how far we have come! When I asked students today how they feel about math, they responded, “excited,” “like people listened to me,” and “happy,” whereas at the beginning of the year they said, “confused,” “sad,” “frustrated.” What a dramatic difference!!! It makes a significant impact when I say, “Remember the Field Trip problem” to students who are struggling in other areas as a reminder of what can be accomplished through perseverance.

# Why School Districts’ Adoption of “Common Core Curricula” is not Common Core

This year, our school district adopted a new math curriculum.  According to several of the teachers who served on the adoption committee, the chosen materials were head and shoulders above and beyond what other curricula were providing to meet the mathematical demands of the new Common Core.

When I received the materials, I looked forward to a program that could satisfy the Common Core’s demands that kids problem solve, persevere, seek short cuts, and critique mathematical ideas.  But during the trainings, and in several communications from our principal and data team leaders, we received the expectation that we teach the program “with fidelity.”  This included an expectation that we teach the same material on the same day as the other grade level teachers in our school.  Ironically, I was even asked to skip an entire module so that I could begin the next unit in tandem with my fellow fourth grade teachers.

Now set aside your concerns that skipping modules is the antithesis of “fidelity”.  Set aside your doubts about the district, who in recent years has provided training on formative assessment and how that guides differentiated instruction.

The point that I am trying to make is that the way our district adopted this “Common Core” curriculum was decidedly not aligned with the values and skills the common core is designed to teach.  First of all, the Common Core asks students to “use tools and make strategies.”  The district is implicitly asking teachers not to create any new tools for student measurement and not to use other teacher strategies they may have picked up in their years or decades of teaching service.

Secondly, students are asked to “look for shortcuts”.  Certainly, there have been several lessons where I think to myself, “Why are you teaching this if these students, who are clearly bored, already get this?”  The answer is because the district wants standardized adoption and the principal has insisted that we teach in synch with our colleagues, regardless of the needs and abilities of our students.

Finally, the Common Core asks kids to “make sense”, to “argue”, and to “critique reasons.”  In this vein, when my colleagues and I have tried to make sense of these demands, we scratch our heads.  When we critique these policies, when we argue our points, we are rebuffed.  In other words, math is a dialogue, not a dictate.

As parents, I hope that you understand that the Common Core asks our students to be able to do some pretty sophisticated and amazing things.  All I am asking from our district is that teachers be allowed the latitude, flexibility, creativity, and autonomy to teach in ways that students need.

# YouCubed (and more on fluency)

Jo Boaler, a math education researcher and professor at Stanford University, launched a new website this past year (YouCubed) to help teachers, students, and parents navigate math education. She recently published a short paper on math fluency. In it she discusses the problems with associating math fluency with speed or memorization.

Interestingly, the Common Core intends this de-emphasis on speed but the word “fluency” is often misunderstood by textbook publishers. The newly adopted Bridges Curriculum seems not fall into this category – their strategy for building fluency is based strongly on number sense. (Read an excerpt from Jo’s paper below)

This past September the Conservative education minister for England, a man with no education experience, insisted that all students in England memorize all their times tables up to 12 x 12 by the age of 9. This requirement has now been placed into the UK’s mathematics curriculum and will result, I predict, in rising levels of math anxiety and students turning away from mathematics in record numbers. The US is moving in the opposite direction, as the new Common Core State Standards (CCSS) de-emphasize the rote memorization of math facts. Unfortunately misinterpretations of the meaning of the word ‘fluency’ in the CCSS are commonplace and publishers continue to emphasize rote memorization, encouraging the persistence of damaging classroom practices across the United States.

# Math Smarts

“Teaching is mostly listening, learning is mostly telling.”
-Deborah Meier

As we dive in to the new school year, those of us in the Olympia School district will be meeting our new math curriculum, Bridges, for the first time.  Many parents will be trying to decode the words “Common Core” in the context of this new way of teaching.  The most difficult part for many will be allowing their children to struggle with (and through) the math.  This will require us to redefine the job of teachers.  This will also force us to redefine what “math smarts” are.  Even though this may be challenging, the upside is that more kids will have a stronger understanding of math concepts and more will identify themselves as mathematicians.

The job of teachers is shifting.  No longer are we the “Holders of Knowledge” whose purpose is bestowing computation fluency upon our pupils through direct instruction.  Actually, good teaching requires just the opposite: teachers are no longer the center of the classroom.  Good teachers coach their classes to work collaboratively, provide group worthy tasks that help them access key mathematical concepts, and then retreat to observe and assess, intervening only when necessary.

The idea of being “math smart” is now outdated.  First of all, as Carol Dweck points out, it leads to a fixed mindset.  If people say that Johnny has math smarts, it leads him to believe that it is an innate power that he has always had.  You either have it, or you don’t.  Most people think they don’t.  Also, many people mistakenly place far too much emphasis on getting correct answers in math, especially in computation.  This definition must now change to show that math is a verb.  It is something that we do.  The list  of math skills must now include listening, communicating clearly, helping others, asking good questions, improving reading skills, flexible thinking, struggling, persisting, and persevering.

In our new era of “Common Core” and Bridges, the most important thing that parents can do is to avoid telling kids how to do their math.  Rather, model the curiosity and inquiry that we are asking of our kids.  Please, resist the urge to show your kids the shortcuts.  No more “this is the way I learned to do it when I was a kid.”  Instead, take the time to just listen to your kid explain their thinking.  Ask them “why?”  Who knows, maybe you’ll learn something, too.

Sources:
Carol Dweck, Mindset: The New Psychology of Success.
Jo Boaler, How to Teach Maths.
Featherstone et al., Smarter Together.

# Mathematical Practices

Mathematical Practices

1. Make sense of problems and persevere in solving them
1. Reason abstractly and quantitatively
• Decontextualize and contextualize situations
1. Construct viable arguments and critique reasoning of others
1. Model with mathematics
• Mathematize the world
1. Use appropriate tools strategically
1. Attend to precision
1. Look for and make use of structure
1. Look for and express regularity in repeated reasoning