Category Archives: Thoughts on Learning

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!



(Un)Fighting Standardized Tests

I teach sixth grade and I find the MAP (Measures of Academic Progress) standardized test frustrating. Schools use it in ways I wish we didn’t and we can’t use it in ways we wish we could.  I didn’t realize how conflicted I really felt until my students bombed the test when I had them take it ‘just for practice.’ 

Learning Testing

I had asked that my students be excused from the second of three yearly rounds of testing. When I made my pitch to my principal, I explained that I can’t really use the MAP to inform my teaching because the information I receive from it is so generalized:  it doesn’t help me pinpoint what students know and don’t know.  Also, many of the sixth graders in my class read too much into their score and lose confidence in themselves as mathematicians if it doesn’t rise dramatically.

We decided on a compromise:  Take the test, but tell them it was just for practice.  Students wouldn’t freak out. And they wouldn’t lose out on a opportunity — and this is the part of how we use the test that I don’t like —  to experience how the school would determine their math placement for the coming grade.  They’d just calmly give it their best. Was I in for a surprise.

When I saw their scores, my heart sank.  They relaxed, too much. Much of the class didn’t do as well as they did when they took the test in fifth grade, in spite of six intervening months of learning. These results were completely out of line with every formal and informal observation I’ve made of their learning so far this year.  They also don’t match students’ performance in prior years and I have some of the most powerful student growth I’ve ever had this year.  Also the experience was still toxic, at least for some of them. One student came into class the next day begging to be taught algebra so he could do better next time.  Another was frustrated that she couldn’t explain her answers.  She loves that part of how we do math.   A third said it was a waste of time because he didn’t care about the MAP because tests like that aren’t going to determine who he is.

To learn more about what happened, I gave my students an anonymous survey.  Their responses to my questions and our discussion shed light on the results.  Here is what l learned:   When they know that a test is tied to an opportunity, like getting into a class, they do their very best. In fact that is what 80 % of them had done in fifth grade when they knew the test would determine their middle school placement. This time, for the practice test, about seventy percent of them reported not working very hard, because, they reasoned, “Why bother?  It was just for practice.” This was about the same percentage as those whose scores dropped.  They also said they wouldn’t ever work that hard if it was just practice. The test taking experience is unpleasant, it takes too long, and no one reads their work anyway.

My sixth graders taught me that what students understand about how test results are used matters to them.  They knew that in the 5th grade, their placement in math class would be determined by their score on the MAP and they responded accordingly.  I, along with many educators, have mixed feelings about this, and wish that students’ opportunities weren’t determined by test scores alone. In fact, as professionals with the best interest of children in mind, we can and do work with families to find the best fit for students. Sometimes, we can advocate successfully for a child who hasn’t scored well.   In the past I have shared this with students, emphasizing caring adults’ role in advocating for them.  But in reality those conversations are the exception, not the rule.  Usually it’s a score that determines a child’s math placement. In their world, the numbers do count. Test results matter to students’ future selves. Many of my students will find that doors open for them when they perform well on standardized tests.  Others will lose opportunities when they don’t do well.

I have gone into test days too wishful for years, and my wishes have skewed the way I present the experience to my students.  I look at the scores and wish I could learn more about my students from them, and I wish the scores didn’t impact my school and students so much.  My first wish is silly.  I already know what I need to know from spending day after day with these young people.  As for the latter, I suspect my nuanced understanding of human development as far more than a test score has been getting in the way of my students’ performance.  On test day, I have reassured them that I know how smart they have become through their hard work.  They believe in my belief in them and probably trust that I will be there if they slip up like I have been before.

Next time, instead of reminding them of how much I care about them, I will help them see the doors that open and close based on their scores. I will tell them that doing well means having choices and having more power and agency to determine the next steps in their own lives. I will tell them that I care about their results because I care about them. I will help them see that the tests are the only way that people who don’t have the time to come into our classroom and listen to how amazing they are can find out what they know.  To the extent that I can, I will help my living-in-the-present eleven year olds see past the people they already know to imagine those they don’t know yet and to have those people, including their future selves, matter.

Solving Problems that Matter

Math class goes like this: Teachers give problems; students do problems; teachers give the answers.  Teachers give more problems; students do more problems; teachers give more answers. Repeat.  Eventually, students get a grade. And then they know if they are good at math.  My students know this is how it is by the time they start middle school.  But in my class it doesn’t work that way.  Sure, I give problems, and yes, students do problems. And I have answers, but I keep them to myself. Knowing their answers make sense has to come from students themselves. My job is not to let them know whether they are right or wrong, but instead my job is to convince them that they have power and control over their own problem solving.

This may seem strange.  You may wonder, “How will the students know they are right if the teacher doesn’t give the answers?”  Here’s what I wonder “The last time you had to figure something out, how did you know you were right?”  I’m guessing you probably didn’t find someone in a position of authority and ask them to look it up in an answer key to confirm that you are a good problem-solver.  Genuine problems come our way without solutions.  If we had solutions, they wouldn’t be problems.  I’d be willing to bet you did one of the following as you solved the last problem mattered to you:

  • You didn’t what do do and you talked it over with your friends.
  • You slept on it, hoping it would go away and woke up ready to work on solutions.
  • You had an idea, tried it out mentally, and satisfied it might work, you took action.
  • You had an idea, shared it with a trusted friend or co-worker or family member to make sure it made sense to them too, and if it didn’t, you adjusted accordingly.
  • You tried your solution, saw it didn’t work, learned from your mistake and tried something else instead.

My math class offers students the opportunity to do what we all do when we have problems.  And if they knew I would provide answers as soon as they get stuck, the game would be over.  They wouldn’t that they are in charge of solving the  problems that matter to them.


“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.


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.