Magic

“The universe is the teacher of all things.”

–Maria Montessori

Dear Parents,

I believe that my job as a teacher is to teach kids the magic of math; to carry on a long and noble tradition of inquiry.  You see, math emerged as a religious quest to uncover the mysteries of the universe.  “Sacred Geometry” emerged as mathematicians realized a mystical and crystalline structures in everything around them.  I do not believe that they found this beauty through closely observing mathematical worksheets that their teacher gave them.

I recently read an amazing Popular Science article, “Behind New York City’s Macroscopic Snowflakes“.  Even if you don’t read the article, you should absolutely check out the photographs of the snowflakes.  There are the most amazing photographs of snowflakes, but they look alien and bizarre compared to the crystals you have pictured in your mind.   There is also a great deal of learning to be done through the close observation of these fractal polygons.  How many of us [math teachers] stop to smell these roses?

It instantly reminded me of an podcast I’d listened to on Radiolab.  In their piece, “Crystal Bliss,” they introduce  Wilson Bentley, the first human to photograph a snowflake.  And even though you have likely seen his iconic photos, he received a lot of criticism from other photographers and scientists of the time.  Apparently, Bentley doctored his photos.  Apparently, his main goal was to capture the “perfect” snowflake.  In this way, he was more of an artist than a scientist.  We have much to learn from appreciating the world through artistic lenses (both real and metaphorical).  It is a shame that some see science and art as exclusive.  I hope to inspire my students to blur this distinction.

And so, dear parents, I hope that you take the quest that I am on with my own students: take a look around you.  There are miracles, both evolving and crystallized, all around us.  And, they are just waiting to teach us.

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.

Who Does Homework Benefit?

As I reflect about many conversations that I have had with students and their families this year about homework, I have found myself deeply questioning who is benefiting from math homework. With current homework assignments, the students who are on track to meet end of the year standards will continue to be on track, while the students who are struggling have a negative self-concept about their mathematical abilities and intelligence reinforced night after night.

This week alone, I have had enlightening conversations with two families dedicated to helping their child in any way possible. One family stated that they are endlessly researching the Internet to understand strategies taught in class, and the other flat out stated that she did not know where to begin with helping her child. How I have appreciated the honesty!!

As a teacher, I attend trainings and have the daily support of my team to help me become proficient in strategies and thinking that the Common Core is requiring. I have had to relearn the way in which I approach math completely. I absolutely celebrate this fact! As a child, I was gifted at memorizing facts and steps, but that only got me so far. When I see the mathematical thinking that my students are doing, I am elated that they must know the “why,” and there is no right way to do math.

Having said all of this, I was curious about how research supports homework. While I know that the benefit of homework is that students gain additional practice in what they are learning in class, I still could not help wondering whether or not there are significant gains in learning. In the research that I have read, I found that there is evidence that homework is beneficial at the elementary age level, but can actually be detrimental if homework assignments are not purposely chosen (according to Robert Marzano and Debra Pickering, whose viewpoints and insights I greatly admire).

So, what to do? In my view, the Common Core State Standards are asking our students to do amazingly deep thinking—a great thing! I cannot, however, expect my families to absorb and celebrate learning, for example, the array model for multiplication, when the algorithm that we were all taught is viewed as so much more efficient and easy to understand. For now, with the standards and our newly adopted math program (Bridges) being so new, I have decided that I will no longer send math homework that requires new strategies that become frustrating to families and students. Instead, I will send home assignments that reinforce concepts that my students have already learned.

We are missing an important opportunity to excite families and students about math. Even though I will not send home the same kind of homework, I will continue to organize “Math Nights” for families so that they can understand the strategies that their students are learning in math. My biggest hope is that by keeping new strategies in class, at least for now, perceptions about our new program and the Common Core State Standards will begin to shift for the better.

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.

Division Story Problems

Students wrote some interesting division story problems today. We talked about making them real – though that can be tough – and we talked about making them meaningful – though that is certainly a challenge. I proposed to them that math should either be rather interesting or rather useful. Though story problems can fall into this category, they often don’t. For some senseless math check out what happens when we do stuff to numbers… because that’s what we do in math (not really). In the end they came up with a number of interesting and engaging problems that produced more to think about in their creation than in their solving. Here are a few:

There are 160 people at a party and there are 40 pies. Each pie is cut into 7 pieces. How many pieces would each person get?

Sasha made 117 cupcakes. She had platters that held 12 cupcakes each. How many platters does she need?

There are 157 students going on a field trip to Nisqually. There weren’t any buses, so they had to order “10 person” vans. One person brought their mom, dad, brother, and sister. Three teachers came as well. How many vans do they need to order?

The clock factory produces 20 clocks per hour. 143 clocks in the clock factory are finished. They are packed into creates that hold 44 clocks each. How many crates do they need? How many clocks are in the last crate? How long did it take to make the clocks in the first crate filled?

There were 260 3rd graders at a summer camp. Half of them were going to Great Wolf Lodge. A quarter of them were going swimming. The last quarter was staying at camp. The larger group took mini-buses that held 13 campers. The smaller groups took cars that held 5. How many buses and cars did they need?

I think some of these students could write for textbook companies! It’s interesting to see their worlds reflected in their math problems, and I was particularly happy to see how engaged they were in the process.