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.

]]>One up-to-date and reliable data set that I thought would matter and be accessible to middle school students was per pupil expenditures by state. I found it at the Kids Count Data Center. It made sense to use these numbers because a data set of fifty is manageable and the sixth grade learning sequence demands understanding of distributions, box plots and quartiles. The class was in its third day of a statistics unit.

I proposed to the class that they would be using statistics to decide for themselves whether or not they agreed that their teachers should walk out to demand more money for schools. I passed out forty-nine numbers on index cards: one number for each state. I held our state of Washington face down in my hand. I told them that they were holding one, two or three states in their hands and that the numbers were in thousands. I didn’t write the names of the states on the cards so that the curious could ask about the states they held. One lay way outside the bulk of the numbers, drawing envy and attention to the state of Vermont which spends close to $19,000 per year per student. The class quickly determined that Utah had the lowest spending at a little under $7,000 per year per student.

I showed the five number summary I had calculated in advance:

- Minimum: 7
- Maximum: 19
- Median (or middle:) 10.5
- 1st Quartile (middle of the bottom half:) 9
- 3rd Quartile (middle of the top half:) 13.5

I stood in the middle of a line on the floor and proclaimed myself the median (the middle number when numbers are arranged in order,) and asked everyone who had a card below the median (in the bottom half) to place it on the floor. Then those between the median (middle) and third quartile (the top fourth) of the states came forward to place their cards. Finally, I stood at the third quartile and asked the big spenders (the top fourth of the states) to come forward.

Then I stepped out of the line and asked if anyone had any questions. Hands shot up as several students blurted, “Where is Washington?” I said, “I am still holding that card. Where do you think we belong in this distribution?” A few pointed to the minimum, assuming that we keep company with Utah, but most pointed to near the middle of the line.

I stepped forward with my card with the nine on it and stood at the first quartile which marks the top of the bottom fourth of state spending. I asked students what that meant. They replied with dismay, “Three-fourths of the states spend more on kids than we do.” Then I stepped out of the line again and took my place just eight states from the bottom. I said, “Yes you are right, and with a total of about six states spending around $9,000 per student, we are about eight states from the bottom. Do we have a problem? Talk to your partner about whether we have a problem.”

I didn’t ask students to write down their replies because I want them to be free to decide for themselves if Washington State has a school funding problem. I decided that asking them to share would put them too much on the spot. The pressure of writing something down didn’t feel right: It is my job to provide information, not to persuade.

When they shared out, most agreed that we have a problem. One pair dissented and wisely said, “We think we need more information. It depends what they spend the money on, like tests or teachers or field trips or sports.” I told them that another data set I could have brought was teacher salaries. “I am fine,” I said, “but a beginning teacher makes the same hourly wage as my eighteen year old son does building trails in the state forest.” Another student made the argument that his grandma was a teacher and things must be fine because she has good retirement, but then he caught himself, “Wait, she lives in Vermont.”

I agree with the pair of student skeptics. We do need more information. How districts spend the money does matter. But without the money to spend, those closest to the students — the principals and the teachers — won’t even have the option to choose to spend wisely. I don’t plan on moving to Vermont, but I do know my talented student teacher is leaving the state for Oregon, where spending is near the median.

I don’t know how many families will join us on Tuesday, but the parent volunteer in the room was swayed by the data to reach out via email to the families of my class. For those willing, let’s meet at Percival Landing at noon, Tuesday, May 26, all dressed in red and walk to the Capital Building for our kids!

]]>I have taught math in public schools for thirteen years. Each new batch of students comes with a handful of parents who grapple with how best to help their children wrestle with math homework. You, yourself, may be one of those parents. When faced with math tasks asking for explanation of reasoning, or when the new-fangled curriculum wields a new strategy in place of familiar algorithms, the kettle of parents minds starts to bubble, roil, and whistle.

Dear Parents,

I have good news, a simple mandate which will both alleviate this internal pressure to perform math miracles AND will be better for your child’s math. The advice? BE LESS HELPFUL.

Don’t answer their questions. Don’t show them shortcuts. Above all, DO NOT teach them the old tried and true shortcuts of the standard algorithm.

I am not giving you a free pass, here. Notice, I did not say, “Make them do their math all by themselves.” Rather, I want you to enter their world of confusion with them. Pull up a chair. Grab some scratch paper. Do the problem with them. Not for them. With them. Answer their question with more questions. Verbalize when you get stuck. Normalize the struggle.

One common deficit in our students is their inability or unwillingness to ask questions. But if a student can verbalize what they don’t understand, if they can formulate a question, then they are in an infinitely better place than if they quit before really diving head-long into the pool of problems. Your best way of helping them is to role model these skills. Make them teach you.

When you get stuck on a math problem, don’t give up. Persist. If you have no idea how to do a problem, guess. Make an assumption and run with it. If that assumption helps you solve the problem, hallelujah! If not, even better. Now you can go back to the assumption and say, “Now I know that this is wrong! Therefore, it must mean *blah blah blah* instead!” Then off you go, down another rabbit hole of problem solving.

Another deficit in our math students is learned helplessness. Many are unwilling to even try. Show them how to make mistakes. *Gracefully! With aplomb! With pride! * What’s more, celebrate those moments, for this is where true learning lies. Do a happy dance! Pump your fists! Cheer!

Now, if on a particular problem, you have reached the dead-end of this logical alleyway, you and your student are in the prime position to express what you don’t understand. Have your kid write down this question in place of an answer. Have them staple together all their failed attempts to turn in. If your math teacher is worth their mustard, they will honor this work and highly appreciate it. This makes their job far easier: they no longer have to guess why their kids aren’t getting this.

In conclusion, I would like to say thank you. Thank you for being involved in your children’s education. If you are reading this blog, then you are truly exceptional in your level of commitment to their education. You already know that helping kids with their academics is difficult. It is a *problem*. It is *your* homework problem. Math is the process of identifying problems, then methodically going about solving them. This is the perfect opportunity to practice what I’m preaching.

So don’t look for short cuts. Don’t look for easy fixes. Look for a solution. And this part is probably self obvious (but for some reason, not always in math): don’t give up; persist; the solution is out there somewhere, sitting right next to your kid.

]]>Below is the “pool” sequence. The gray tiles are the border and the white tiles are the surface area of the water in the pool.

This was one of the first sequences we looked at. Not the first mind you, but it surprised me how soon we got into functions that were non-linear. Actually, this non-linear function was easier for the kids to decode than the linear one that accompanied it. First students shared their observations:

- The water gets bigger by 3 (the first time)
- Next it goes up by 5
- Pool area goes up by increasing odd numbers (+1, +3, +5…)
- Pool area is 1×1, 2×2, 3×3…
- Odd arrangements have a center tile
- Sides of the pool get bigger by 1
- Arrangement number is 2 less than side length
- Pool and border expand at different constant [sic] rates
- Arrangement 1 fits into the “white” of arrangement 3
- It looks like arrangement 3 fits into 5’s pool
- All odd arrangements fit into next odd ‘s white

Next we looked at the patterns that happen in the numbers. Below is a table of arrangement numbers (x), the number of border tiles (y), and some other observations.

Arrangement (x) | Pool Border (y) | Observations | Rule? |
---|---|---|---|

1 | 8 | 5x + 3 | |

2 | 12 | +4 | 5x + 2 |

3 | 16 | +4 | 5x + 1 |

4 | 20 | +4 | 5x + 0 |

5 | 24 | +4 | 5x - 1 |

One student got really excited to see her pattern continue indefinitely. We looked back at the tile models to see how we could “see” the growth by 4 that shows up in the table. This yielded some really neat connections (and I think we came to a more formal rule). Next came the data for the water tiles. This one was snatched up easily, but the significance of our departure from linear relationships was not clear until we examined the graph.

Arrangement (x) | Pool Area (y) | Observations | Rule? |
---|---|---|---|

1 | 1 | x * x | |

2 | 4 | +3 | x * x |

3 | 9 | +5 | x * x |

4 | 16 | +7 | x * x |

5 | 25 | +9 | x * x |

Up until now, students had been seeing even steps – constant rates of change. Our outputs had always gone up by the same amount in each iteration of the sequence. With the pool area, each subsequent arrangement was significantly larger than the previous. The steps (up the y-axis) were getting increasingly tall.

The green line is the graph of the border and the red line is the graph of the pool area.

After putting all of this together, students were able to make predictions about how many tiles would be in some distant arrangement, and even find out which arrangement would have a certain number of tiles. Try it out for yourself!

1. How many border tiles will be in *any *arrangement? How many pool tiles will be in *any* arrangement?

2. Which arrangement will have 196 tiles total for both border and pool?*

*I can’t say if my 5th graders solved this one yet… we’ll have to come back to this after spring break.

]]>I 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.

What 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:

Here 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.

So 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?

Maybe 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.

In 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.

]]>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?

]]>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!

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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.

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

Thousands | Hundreds | Tens | Ones |
Tenths | Hundredths |

1000 | 100 | 10 | 1 |
1/10 | 1/100 |

10^{3} |
10^{2} |
10^{1} |
10^{0} |
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 |

5^{3} |
5^{2} |
5^{1} |
5^{0} |
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, 10_{five}. The number 12_{ten} is 22_{five}, and the number 78_{ten} is 303_{five}.

When we want to multiply any number in base five times five all we have to do is move the decimal! 11_{5} (which is 6 to us) times 10_{5} (which is 5) is 110_{five} (which is 30 to us). The product is still true regardless of the base. Likewise 22.2_{five} (which is 12 and two-fifths) times 5 is 222_{five} (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).

]]>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.

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