Category Archives: Math Curricula

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:

studentwork1It 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.

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.

Shape Classification

image001This month we are working on classifying quadrilaterals. Creating these overlapping hierarchies is  always a challenge. Relationships that work one way, don’t work the other way, and there is a lot of specific vocabulary that needs to be learned and applied in novel situations.

It’s a little like this: everyone who lives in Olympia lives in Washington, but not everyone who lives in Washington lives in Olympia. Bringing it back to the shapes – all squares are rectangles, but not all rectangles are squares.

To do this work, we need to know the defining characteristics of the quadrilaterals that we are classifying – and here we run into yet another difficulty… mathematicians don’t always agree on these! But the work of classification goes on. We just learn the specific characteristics and create the relationships accordingly. Here is a table that tries to communicate these overlapping relationships.

quads-3Under the inclusive definition of trapezoids, all six shapes a the top of this post are trapezoids, under the exclusive definition only three of them are… can you spot them?



What’s the first question that comes to your mind?

Dan Meyer, prolific blogger, high school math teacher, and current Ph.D. student at Stanford wants to know. Dan has decided that there are way too many artificial scenarios in his high school math texts that pretend to model the world. He wants to see the real world in his math class because he sees the math in the real world. is the brainchild of Dan Meyer. On this site, anyone can post an image or a short video and let the world ask the math questions that naturally arise from the scene.

From this, he and many of his followers have created lessons where the students can generate the questions (probably the ones that the teacher has in mind), ask for any relevant information they need, and solve real problems the way they would in the real world.

It’s fantastic – check it out, but don’t be surprised if you find yourself seeing math problems everywhere.

How about this one?66-the-ticket-roll

How many do you think there are? What number is just too many? What number is too few? Be brave.

What do you need to know?