All posts by Deborah Nied

A Year in Reflection

As this year comes to a close, I want to tell you, my fellow bloggers and all of those who have left responses, how much you have inspired my teaching and growth throughout this year. When I think back to the beginning of the year and our discussion of starting this blog, I think about how vital it was for all of us as teachers to foster a Growth Mindset in our students.  Now that the year is winding down, I am curious about how you feel with regards to Growth Mindset in your students.

On the first day of school, I asked my students the following questions:

  • How do you feel when you make a mistake in math?
  • What would you like me to say when you make a mistake?
  • What would you like to classmates to say when you make a mistake?
  • How do you think we can best support each other in this class, especially when someone makes a mistake?

Responses to these questions were eye-opening and varied greatly. Some students wrote comments from: mistakes turn “into an opportunity to learn better,” and “mistakes help me grow,” to mistakes make me feel “angry,” “like I am going to cry,” and “frustrated/annoyed.” As I reflect on the students who had a negative self concept when making mistakes, I am curious about how their feelings have changed since the beginning of the year, if at all.

This week, I plan to give students back their written responses from the first day of school and answer these questions again. In my observations, we have come SO far as a learning community, and I can’t wait to hear how my students respond now (I will definitely share in a future post).

I am curious about what you are all doing with regards to end of year reflections, how you feel about your progress with Growth Mindset, and what this makes you think about for next year.

Bridge to Multi-Digit Subtraction Standard Algorithm

In my 12th year of teaching, I continue to be amazed at the misconceptions surrounding multi-digit subtraction. In my experience, students who are masters at memorizing the steps of the US standard algorithm are quite successful, though have little understanding of the place value. Why do we borrow? What does that even mean?

In an effort to do a much better job this year, I am trying a new strategy. Given the problem:

634 – 368 = z

I am requiring my students to write out each number in expanded form:

634 = 600 + 30 + 4

368 = 300 + 60 + 8

The language that I am using with this strategy is intentional. Since we cannot take 8 from 4, we need to borrow a ten, thus changing the 30 to a 20 and changing the 4 to a 14. I am stressing with students that this is the same value:

600 + 30 + 4 = 600 + 20 + 14

This alone is mind-blowing!

So now, if we finish the problem, it looks like:


The power in this language is that now, rather than saying: “What is 12 – 6? ” I am now saying, “What is 120-60?” My hope is that by changing the way I speak, I am deepening students’ understanding of place value, and how we talk about it. The best part of this is that when I revert to my old ways (which I do), students immediately correct me by saying,  “Excuse me, Mrs. Nied, but that is not 12-6–it is 120-60.”  Students have completely taken over this language and I am hearing it constantly in group discussions. Woo hoo!



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.


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.