Ours is to Wonder Why

It began almost accidentally. After fooling around with some interesting magic squares recently I decided to put a couple in front of my students. I wanted to step away from our work with division of fractions momentarily and ignite their curiosities. Below is the fist magic square I showed them.

ms1I won’t bore you by telling you about all of the amazing patterns inside this square – they are yours to discover. In class we shared things we noticed. One of my students noted something like “It goes 8 – 5 – 2 diagonally: each number three less than the one before?” (Notice the question mark?) I said “Are you asking me? You’ve got to make a statement! You’ve made a discovery! You’ve got to stand up and shout EUREKA! So from here on out the class made really exciting discoveries punctuated by shouts of “Eureka!”

Then we looked at the following magic square.

ms2What do you notice? We made one discovery after another – the class was reaching a fevered pitch: math was rocking the house, and the students’ excitement and engagement was running into the red. Time to change gears and use some of this fuel on division of fractions.

Division of fractions – something like 2/3 ÷ 1/4 – is an odd thing to get your head around. We’ve been working with visual models for a little while but it still takes some mental work to see this stuff. In the old days there used to be this saying in elementary math about dividing fractions: “Ours is not to wonder why, just invert and multiply.” Catchy, but this takes all of the understanding right out of the equation (math joke!). Maybe the problem looks like this:

grid1Here we see a rectangle that is cut into thirds vertically and fourths horizontally. Each column represents one-third and each row is one-fourth. So the question is: How many of the shaded areas will fit into the green area? If we rotate the one-fourth, we can fit two full fourths into the green area (below). In the remaining green area we can only fit 2/3 of a quarter strip.

grid-div2So the number of fourths that can fit is 2 and 2/3. You can think about it this way: How many groups of 1/4 can I make with 2/3 of a whole? Scoop, scoop, little scoop.

If we look at the equations we can make some sense of these as well. Notice that in the equation below, we just combine the fractions into one fraction and the result is another way of looking at the picture above! How many 3/4 can fit into 2 wholes? – except this time the whole is actually just one-quarter! Do you see it?

eq1Maybe you don’t like that one… OK, I can live with that. But what if we did something interesting like found a common denominator for the fractions. 2/3 = 8/12 and 1/4 = 3/12.

eq2In this equation, the quotient of the first expression gives us 1 in the denominator! So it effectively becomes 8 ÷ 3 or 8/3, which is 2 and 2/3. This technique just melts the fraction division away.

All of this happened in one class. And there were more problems than just this one and more voices than just mine. Students were explaining what they saw and why it made sense, and they were shouting “Eureka!” every time they understood something new. Many students were standing, and hands were waving all around the room. I can tell you that at times it got a little crazy. But this is how, at 12:20pm, after a long morning of classes culminating in a math lesson on the division of fractions, a group of 25 fifth-graders spontaneously erupted into applause. It certainly doesn’t happen every day.

Ours is definitely to wonder why.

My Fear of Math Phobias


There is one idea that I would love to disappear completely from math discussions: “math phobia.” Vamoose! Vanish! Evanesco! Of all the ideas that are harmful or destructive to students’ acquisition of math ideas, “math phobia” is one of the worst.

Yet, I hear it all the time. I hear it from adults all the time. I hear it from parents. What is worst of all is that I hear it from other teachers. “I am not good at math”; “I never did understand that”; “go ask your dad”; and the darndest of them all: “I hate math.” To beat all, they say it right in front of kids–their own kids and students.  Sometimes, I’m not sure if they are calling themselves or the math itself stupid.

Now, I am not saying that there is no such thing as math phobia. I am sure that some people actually run in fear when they see numbers on street signs. I am certain that somewhere, men and women are cowering, shivering in their closets because they read “¼ cup of sugar” in a recipe, or a telephone number flashed up on the screen when they were watching the television.

What I am saying, though, is that these phrases and attitudes give kids a free pass. They hear the adults walking around dismissing their own mental power all the time. So they do it, too. Kids become lethargic. Kids start saying, jokingly at first, that they aren’t good at math. But by the fifth grade, I see many kids completely checked out. It’s not that they couldn’t understand the math problems. They simply have become so used to not putting in the required brain power (which is not that much voltage, incidentally).

So parents, I implore you: zip those math-phobic lips! Pretend that you love math DESPITE all the damaging math classes you had as a youth. Refuse to pass the buck to your partner who “has the math smarts in the family.”  Instead, each time you notice the beauty in a flower or the symmetry in a piece of artwork, declare, “Ah! Look at the geometry on that puppy!”

This may drastically change the way you interact with your students around math and around homework. Instead of the instructional coach, you are now fellow math adventurer. Talk less about how to do the math and more about what it makes you think of. Where do you see the patterns in the real world? Where does geometry show up in art and architecture? Where do fractals sprout up in the coral reefs and forests?

One more thing, don’t be afraid of pointing out short cuts. If your kid is spending much extra time and effort on a problem which has a more direct and elegant solution, go ahead and say, “Son, have you ever thought of [blah blah blah]?” Or, “Daughter of mine, I see that you are stumped here, but maybe you could solve an easier problem.”  What this will roll model is that mathematicians aren’t into punishing themselves. We are all about making work easier; making the world around us easier to understand. What is so scary about that?

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.

Places and Bases

Recently in math class we have been working with decimals. This is one of the areas in math where a solid foundation in place value will take you a long way. If for instance you want to know the product of 15 and 3.2 it’s nice to know that 10 x 3.2 is simply a decimal move and 5 x 3.2 is just half again. I am always surprised to find that place value is not well understood by 5th grade. I guess I need to stop being surprised. But perhaps place value is just a little more complicated than we give it credit for, and maybe there is just not enough emphasis in the earlier grades.

Let’s look at the base-ten place value table:

Thousands Hundreds Tens Ones Tenths Hundredths
1000 100 10 1 1/10 1/100
103 102 101 100 10-1 10-2

The reason for the ease of computations with 10 and powers of 10 is that ten is the base. This is why when you multiply 10 times anything “you just add a zero.” I try to teach my students to use the more mathematically accurate phrasing of “you just move the decimal,” but it’s no more challenging. The ones, or units, are the center of the place value system – all other places are exponentially larger or smaller than the base to the zero power.

Exponent review:

  • All numbers to the zero power are equal to 1
  • Negative exponents are just powers of base (ten in this case) in the denominator.

To understand place value, I find that it is helpful to examine what life is like in other bases. Let’s look at a base-five place value table like the one above.

One hundred twenty-fives Twenty-fives Fives Ones Fifths Twenty-fifths
1000 100 10 1 1/10 1/100
53 52 51 50 5-1 5-2

In base 5 there is no numeral 6, or 5 for that matter. There are only 0, 1, 2, 3, and 4. Once you get to 5 the notation is 10 or more clearly written, 10five. The number 12ten is 22five, and the number 78ten is 303five.

When we want to multiply any number in base five times five all we have to do is move the decimal! 115 (which is 6 to us) times 105 (which is 5)  is 110five (which is 30 to us). The product is still true regardless of the base. Likewise 22.2five (which is 12 and two-fifths) times 5 is 222five (12.4 * 5 = 62).

The place value system is dense, but when we understand the places and bases of our number system everything makes a whole lot more sense (or perhaps cents or pence if you live on the other side of the pond fence).