Sunday, October 21, 2012

The Willy Wonka Math Teacher Challenge

I love Willy Wonka and the Chocolate Factory,,,you know, the old one with Gene Wilder. The new one is too weird. The music is terrible and the Oompa Loompas are just plain disturbing. Anyway, I've used the old movie for math lessons teaching proportions (how tall is Mike TV when he gets shrunk by Wonkavision?) and for finding the volume of a sphere (how many gallons of juice are inside Violet after she chews on the blueberry gum?).

I was watching it again with my son the other day and I remembered how much I love the scene below. The students' teacher, Mr Turkentine, is introducing a lesson on percentages. Originally I was going to critique the lesson in this blog post, but I'd much rather open this up to my math teacher community. So please watch the video and come up with your own criticisms. How would you teach this differently? What would you keep the same? Leave a comment below. And don't read anyone else's comments until you've made your own. That's cheating.

Good day sir!



I said good day!

Saturday, October 13, 2012

Texting Algebraic Expressions

I try to read as many other blogs as I can. Sometimes I'll find a really cool idea and it somehow gets lodged into my brain somewhere, waiting to be used at just the right moment.

One of the posts/podcasts that I read/heard was from Dan Meyer and his lesson on scientific notation (I think I got it here). In it he explains how writing numbers in scientific notation can be related to texting. When texting, we often abbreviate or use acronyms to speed up the process (ttyl = talk to you later). Or maybe we do this because we're lazy. Or maybe we do this because it's the cool thing to do. Or maybe there is an annoying limit on how many characters you can use.

Yesterday I was teaching how to write algebraic expressions and this idea popped back into my head. We were looking at the expression "seven more than a number". This has twenty-four characters, and being the ancient 34-year-old man that I am, would take a while for me to text. (And it doesn't help that I only just got a cell phone a couple of months ago.) I then demonstrated how long it would take for me to text this expression to another teacher. I sent her the text which was a little odd for her since I didn't explain why I was sending it. She was nice enough to send back a response:


Not bad for an English teacher who curls up into a fetal position every time I talk about math. And look! She used a variable! That saved a lot of time. And only five characters were needed! Awesome! What a convenient way to write that expression! Now kids, here's 200 to do on your own. Good luck.

Saturday, September 22, 2012

Good Day or Bad Day? (Adding Integers)

I love this clip from Seinfeld where Kramer describes what happens at the dinner table when you're married. It would seem that one of the worst parts of marriage is the time when you talk about your day. Was it a good day or a bad day? Eventually this became the inspiration for how I chose to teach integer operations.



I used to teach adding integers with gang violence. There were two gangs (positives and negatives) who would fight and kill each other. The only problem with this metaphor was that it was only really good for adding integers. It didn't help me teach some of the other operations (subtracting a negative, negative times a negative).

I also had the problem of offering students too many ways to think about adding integers. I used gang violence, money, number lines, drawn positive/negative signs, and the boring rules. But it was way too much information. I was trying to offer my students choice, but many were becoming confused.

So I'd like to say that I have one way to explain integers, and it all has to do with having a good day or a bad day. I start the lesson off by showing this slide:


I ask my students how they know if they've had a good day or a bad day. What kind of things could help you move to either side of the spectrum? I then use specific examples, always in pairs (see below). I use little arrows to indicate which way we are moving on the happy face (soon to be number) line.



The first two examples are pretty obvious. If two good things happen, then it's a good day. If two bad things happen, then it's a bad day. 


This third scenario (the lost cell phone) had some debate. Maybe the loss of a cell phone was a good thing...perhaps this would convince your parents to buy a new/better phone. But the point was to show that when a bad thing happens, and then an equally good thing happens, you end up back where you started. It was neither good nor bad.


The kids laughed at this one. Not sure why hamster death is so funny. Perhaps it was the juxtaposition of the two scenarios that made the second seem so ridiculous. But this is exactly what I wanted. Sure, it was nice to find the dollar. But your hamster is dead. Finding the dollar does not make up for the fact that your hamster died, so overall, it was a bad day.


And finally, the lost/found money. This last scenario transitioned into the use of a number line (at which point my students thought, "Oh, so this is a math lesson").


Where do I go next? Basically negatives are bad things that happen and positives are good things that happen. If more bad things happen, it's a bad day. Or if there are more negatives, then the answer is negative.

I can easily see how I can relate this now to subtracting and multiplying. Taking away bad things helps make your day better (subtracting a negative). When bad things happen over and over again, it is a bad day (positive times negative). Taking away bad things over and over again makes it a good day (negative times negative).

Nathan Kraft



















Tuesday, September 11, 2012

Exploiting My Son for Math

At some point I'm going to have to apologize to my four-year-old son, Emmett. Over the last year I've been using him for all sorts of math lessons - many times under the guise that I'm spending quality time with him. It all started with this video:


I posted this on facebook and lot of people thought it was cute. But many also wanted to know the answer to the question. I used the same video in class and one of my students went home and worked on the problem with her dad. She was so proud when she came in the next day with the right answer.

I quickly learned that I could use his "cuteness" to teach math. He was helping me teach probability:


He was helping me teach problem-solving:


And sometimes he'll even inspire lessons. The other night I was sitting in the kitchen doing some work. Emmett should have been in bed sleeping, but instead he was in the living room stacking cups.
Emmett and his cups.jpg

He was so proud of his design and he wanted to make an even bigger pyramid. So we went out the next day and bought a bunch of cups, and hence, a new math lesson was created.
SAM_0385.JPG

So again, I apologize to Emmett for using him this way. But I think he's having some fun in the process. Besides, how many kids have had the chance to do this?


I only hope his cuteness doesn't run out any time soon.

Nathan Kraft

Monday, September 3, 2012

PEMDAS must die!

Here is my third blog post for the new math teacher blogger/twitterer initiation. My prompt is:


2. Your students come into class rife with misconceptions. EEEP! What is a misconception that you see a lot? If you haven't done so yet, brainstorm a way to deal with that misconception so that students will leave your class with that misconception fixed. (Or if you have a misconception that you deal effectively with, how do you do that?)


True to fashion, I will now sort of answer the prompt:


My buddy Andrew Stadel just made a great song about PEMDAS.



It is tough to write a math song that is not ridiculously cheesy. In fact, most math songs drive me insane. They're usually so terrible that I can't bare to force my students to listen to them. Of course, that doesn't stop me from making my own songs and thinking they're not cheesy. (The video below was made with the help of my eighth grade students last year.)



But I'm getting a little off topic here. I love Andrew's song. It's very catchy and I think I have most of it memorized. But I still hate PEMDAS. Every year I get a few eighth graders who insist that you always multiply before you divide. And you always add before you subtract. There is no doubt in my mind that the acronym, PEMDAS, is the culprit.

Once in a while students will even go so far as to tell me that I'm wrong and their teacher from last year is right. I then hand them a piece of paper with "18/6*3" written on it, tell them to go find their teacher from last year, and have them simplify the expression. The last time I did this, the paper suspiciously got lost in transit.

It takes a ton of warm-ups and review to finally break them of this dirty habit.

So I suggest...no more PEMDAS. No more "Please Excuse My Dear Aunt Sally". It only does more harm than good. We tell them to remember left to right, but they don't. They only remember PEMDAS.

Nathan Kraft

Tuesday, August 28, 2012

Guess and Check

I used this as a warm-up activity for 8th grade today. I got this off of 101qs.com from someone named Adam Bevan. (Whoever you are, thanks!)


Here's what I loved about it:
A. Some kids noticed right away that you could use the 9 as a 6.
B. Some noticed that 1's and 2's had to be on both cubes to make 11 and 22.

And that's as far as many got. Many were paralyzed by the fact that there was no certain way to know where the rest of the numbers went. They thought that there had to be one right answer, and could not decide on how to fill the rest of the blanks.

I had to keep pushing kids. Guess. Take a guess. Do you need a 3? Of course you need a 3. You don't know where to put it? Try putting it here. Will that work? I don't know either. Keep guessing and see what happens.

So my kids learned two valuable lessons today. 1. Sometimes you solve things by guessing. 2. Sometimes there's more than one right answer.

Looking back: It would have been better for me to have a physical representation of this. Some students never saw these blocks before and couldn't really grasp the concept.

Nathan Kraft

Sunday, August 26, 2012

Working for the Man: A Cautionary Tale


This is another post for the mathtwitterblogosphere people. The topic for this one could be anything I want with the catch that I can't reveal its oddity to the reader.


As often as possible, I try to make sure that none of my posts are too vainglorious, but I was honored to have Dan Meyer pass my name on to a company as someone who could write math tasks for their new curriculum. I was hesitant at first since my writing is not very Hemmingway-esque, but was excited to do something different. I love writing lessons for my class and analyzing what works. And now I was getting paid a decent amount of money to do it! Awesomeness.

Turns out, not so much.

I started by working on something I already felt comfortable with: rates and speed limits. I used this video as the introduction for the activity and quickly started working on conforming a lesson to the company's format. But it wasn't as easy as I thought it would be. The format itself felt restricting and I continuously asked myself, "Is this really what they want?" I slowly started to think that this would be a huge time-sucker and it would definitely take away from what I was trying to plan for my regular teaching job. 

Once I was getting paid to write math lessons and had restrictions placed on how that should be done, what used to be fun became work. My school doesn't pay me extra for investing all of my time into my own lessons. And I love the freedom of creating my own work in the format that I choose. Eventually I had to resign the task-writing job.

Since then, I've been reading a book called "Drive" by Dan Pink which was suggested by a former colleague (and about fifty other people). The whole point of the book seems to be that extrinsic rewards stifle creativity. That people do their best when the task itself is rewarding and actually underperform once a reward is offered. There seems to be a huge parallel between the ideas in the book and my experiences this summer. And I've begun to think about how this can also be applied to my classroom. How can I offer similar kinds of freedom when my students are exploring math and problem-solving? How can I limit the use of extrinsic rewards (grades) as a motivator? Will students see my use of standards-based grading as a way for them to control their own learning, or will they see it as another carrot-and-stick routine?

The avuncular, Nathan O Kraft

(Secret message to you know who (not Voldemort): You might be thinking, hey, he didn't use okra. Look at my name again. Snap.)

Sunday, August 19, 2012

I Still Suck at Teaching (and how I'm going to fix that)

Disclaimer: I'm writing this post in response to Sam Shah's and the mathtwitterblogosphere's initiative to get more teachers blogging. If you're a new math blogger like myself, you should check them out.

I once heard someone say that when you enter your sixth year of teaching, you'd feel confident with your ability to teach. That you would "know what you're doing". Well I'd like to thank Dan Meyer, Steve Leinwand, Fawn Nguyen, and Andrew Stadel who, through their expertise and great ideas, have proven to me that I still suck at teaching.

There are two big ways that I will be improving my teaching this year. The first is the use of standards-based grading. I was first introduced to SBG through Shawn Cornally at ThinkThankThunk. I've also been influenced by Robert Marzano's Classroom Assessment and Grading That Works, a presentation by Grant Wiggins (Understanding by Design), some blog posts by Dan Meyer, and a bunch of emails back and forth with Fawn Nguyen and Andrew Stadel (two great teachers who are also trying to unravel this beast).

Everything about SBG makes perfect sense. It helps students retain what they've learned by allowing them to self-monitor, re-learn, and re-assess. It also helps me, the teacher, focus on what is essential. The biggest issue will be acceptance (from administration, students, and parents) of the grading process (how do I assess, how do I assign a letter grade). I think everyone is so entrenched in the tradition of point systems and letter grades, that this will meet some resistance at first. The trick will be properly explaining it (which I'm attempting to do here).

The other big change for this year will be the use of more problem-solving sessions, especially in the three act format as explained by Dan Meyer. I found out about Dan through his TED talk and I was immediately blown away. I then found his blog and 101qs website and started using his format with great success. It's amazing to me how student motivation can be intensified through the use of proper media and real real-world problems. (I say real real-world because I've found that many textbook examples of "real-world" are made up. How do they get away with that?) I've made my own three act lessons here, but my favorite is Andrew Stadel's File Cabinet. To me, this is the standard to which all other three acts should be judged.

As the year progresses, I will be sure to use my blog to report on these changes. It seems like these are very new concepts for many people and we're all trying to figure them out at the same time. Sharing this learning is the best way for all of us to become better teachers (and maybe not suck so much).

Nathan Kraft

Two things:

1. I briefly met Dan Meyer in Philadelphia at the NCTM conference in April. He suggested I start blogging. I thought he was crazy.
2. I've also been very influenced by Steve Leinwand. Even if you're not interested in SBG or Three Act Lessons, you should watch this very short video and think about your own practices. He also has a great book called Accessible Mathematics which expands on what he says in this video.

Wednesday, August 8, 2012

More Integer Operations


I was writing this in another person's blog entry about rules for integers. Figured I might as well put it on my blog.

For subtracting, I like to talk about owing money. If you’re at the restaurant, and you have a bill for $45 (-45), and I come along and say, “Hey, let me pay that for you.”, I would take away that negative: -45 – (-45) = 0. You now owe nothing.

You can do similar things for multiplying.
3(5) means I give you three 5 dollar bills: net result, add 15 (+15)
3(-5) means I give you three IOU slips (each owing $5): net result, lose 15 (-15)
-3(5) means I take away three of your 5 dollar bills: net result, lose 15 (-15)
-3(-5) means I take away three IOU slips: net result, gain 15 (+15)
OR
getting something good is good
getting something bad is bad
losing something good is bad
losing something bad is good
I learned that last one from a student, and who knows where he picked it up.
Nathan Kraft

Sunday, July 29, 2012

Draw a Picture, You Idiot!

I'd like to make my lesson on writing algebraic expressions more concrete for students. My hope is that students' understanding of simple expressions will lead to better interpretation of equation word problems. I was inspired by Steve Leinwand's book, Accessible Mathematics, where he argues that diagrams should be drawn as much as possible to help students conceptualize material. Here is an excerpt:

"Without question, one of the most common responses I have when sitting in the back of a mathematics class is screaming under my breath, 'Draw a picture!' or 'Use a number line!' or 'Ask them what it looks like!'" (page something or other...I don't know, it's on my Kindle app, how am I supposed to cite these things?)

I think he really meant to say, "Draw a picture, you idiot!" (hence, my post title). So I figured, why not apply it to this lesson:
What does everyone think about teaching this way? Will this help students gain a better understanding of variables and operations? Should I preface this with diagramming of numerical expressions first? What about expressions that don't lend themselves well to a diagram (such as 5 divided by n)? Any suggestions?

Nathan Kraft

Saturday, July 28, 2012

Gang Violence and Adding Integers

This is my borderline inappropriate way to teach integer addition to students. It was inspired by a presentation I saw by Dr Kadhir Rajagopal on solving equations (NCTM 2009, DC). Check out his website here.

Algebra tiles are a great way to teach integer addition. But I like to represent positives and negatives as members of two rival gangs.
The yellow gang member is basically the same as a yellow algebra tile (+1) and the red gang member is the same as a red algebra tile (-1). Yellow gang members get along with other yellow gang members. Same goes for the reds. But when a yellow and red meet up, bad things happen.
To a 13 year old child, this explanation is much more satisfying than some nonsense about "zero-pairs". And when one of my students is confused about an addition problem, all I have to say is "gang violence", and they know exactly what to do.

Say a student is presented with a problem like this: -4 + 7. I have to ask two questions: Which gang will win? (the positives) How many will be left? (three)

To me, this is much better than some silly rule that students have to memorize. They are visualizing the numbers. And they are seeing positives and negatives as opposites that cancel one another out. And best of all, they remember it.

Credits: Graphics are from the Smart Notebook software. I'm sure the people at Smart Technologies appreciate that I've found this use for them.

Nathan Kraft

Sunday, July 22, 2012

Measuring S'more Gooeyness

One day I was making S'mores in the microwave.

I noticed that the directions on the package of chocolate bars and package of marshmallows disagreed with each other. Chocolate: 10 to 15 seconds on medium. Marshmallows: 15 to 20 seconds on high. This was especially odd since both packages suggested using the other product to create the S'mores. (Honey Maid Graham Crackers remained neutral with no suggestions for how to make your S'more.)


I saw some potential for this in a math lesson. So I made S'mores with varying times in the microwave (10 seconds on high, 20 seconds on high, and 1 minute on high).

I took height measurements for each scenario. First 0 seconds...

Then 10 seconds...


Then 20 seconds...


Then 1 minute...


But none of these measurements seemed like a satisfying way to measure gooeyness, especially since 10 seconds and 20 seconds appeared to have the same height. This would have to be a qualitative experiment. So I ate them.


I suppose it comes down to personal preference. For me, 10 seconds (top) was perfect. 20 seconds (bottom left) was too messy, and 1 minute destroyed my tongue. Unfortunately, I don't see this as something I can use in math class. Any suggestions?

Nathan Kraft

Monday, June 4, 2012

People who "like math"


Sometimes an adult will tell me, “I like math.” And I look at them incredulously, and I say, “Really?” “Yes, it always came so easy to me.”

This is a horrible reason to like math.

Students sometimes ask me if I like math. They wonder whether or not I go home and solve equations all night. And I say, “If you think that’s what math is, then no, I don’t like math. I don’t go home and solve tons of equations every night. What I do like is problem solving. And if I have a reason for using math, then I enjoy it very much.”

When you design your instruction, remember that your students probably feel the same way, with the possible exception of those who “like math”.

Nathan Kraft

Saturday, May 26, 2012

Students' Views on 101qs

I've been very much into 101qs.com and Dan Meyer's work. 101qs is a website where users upload photos and videos and others determine whether or not these items are perplexing. If they ask a question about the photo or video, the perplexity score of the item goes up. These questions are typically asked by math teachers such as myself.

However, math teachers are not the intended audience for these items. Math students are. While 101qs does a very good job of sorting out the good items from the bad, it seemed to me that the scoring system was more indicative of what teachers saw as perplexing - especially if they can immediately see the math behind the problem. Students don't look at problems this way. They seem to take more interest in the problems that they think are "cool".

Before you look at the data, let me explain a couple of things. I used two different presentations on two different days. Each presentation included twenty items from 101qs. I tried to pick things that were either highlighted on Dan's blog or belonged to users who regularly show up in the top 10 acts. I told the students to write either a "yes" or "no" for each item. A "yes" would indicate that they found the problem interesting and would like to solve it. The presentations were shown to thirty upper-level sixth grade students and forty-six lower-level eighth grade students. Because the presentations were shown on two different days, I did not mix the results of the first presentation with the second. This made sense as the students did not appear to be as enthusiastic the second day (not sure why). Finally, I didn't do this in a very scientific way, and there certainly may be flaws with my experiment. For instance, there were a few items where some students loudly showed their appreciation/distaste and I thought this might have skewed the results.

I must insist that this data does not condemn the work on 101qs. It simply adds to the discussion already happening on Dan's website. For me, it has given a little more insight into what might interest my students.

Click this link for the data.

Some things I noticed...

1. Timon's dominoes was still number 1. The kids loved it.


2. Although these do well on 101qs, the kids did not like these problems. I find all of them to be interesting, and I think with the right presentation, they would do well in a classroom. But at first glance, the kids gave the thumbs down.
Too good to be trueThe Ticket Roll
Ball of Mackerel

3. Griffy did much better with my kids (especially the sixth graders) than with 101qs. I bookmarked this video immediately after watching it. I love the concept and I can't wait to use it in the classroom. I was surprised to see the perplexity score so low for this one.


4. Futurama sells. I suspect showing any cartoon that the kids like and has math in it will interest them (no pun intended). This might also have done well because some of the students thought a curse was bleeped out.


5. Upper-level sixth graders were typically much more interested in the problems than lower-level eighth graders. This was to be expected.

Nathan Kraft


Sunday, May 6, 2012

Why I hate assigning homework.

Here are a list of reasons why I hate homework:

It wastes class time. Students are not motivated to review homework, especially if it's mindless practice. Most don't pay attention, especially if they didn't actually do it. This time could be spent actually learning something.

I don't have the time to really check it.

While a few actually do it, most either copy from someone else or don't do it at all. Students who copy get a bonus to their grade (for not actually learning anything) and students who don't do it see their grades plummet and ask, "why even bother trying?" I'm also forced to give students disapproving looks when they don't complete it - yech!

I've read that there is very little research to show that homework improves achievement in the middle grades. I can understand this, especially if it is given to students who don't fully understand what it is that they're supposed to be practicing.

What if the student tries, but doesn't really understand what he or she is doing? Isn't that student just practicing something wrong?

I think homework could be useful if there is some intrinsic motivation to get it done. This means that the problems need to be interesting. The only time homework works without intrinsic motivation is when the student understands the concepts well and the parents are diligent about making sure that it gets done. But how many students fit in that category?

Nathan Kraft

Thursday, May 3, 2012

More ways to remember complementary and supplementary

You probably already have 50 ways to remember the difference between complementary and supplementary. Here's one more that I made up and I think it's pretty cool. I recognize that it's very possible (although unlikely) that someone else also make up this trick somewhere in the universe. Let's just hope that I came up with it first.

I always tell my students to take the co in complementary, and simply draw a line on the c so that it becomes 90. Therefore, complementary angles add up to 90 degrees. Supplementary doesn't work as well, but I like it anyway. Take the su, draw a 1 next to the s, put a slash through the s to make it look like an 8, and top off the u to make it look like a zero. Voila! 180

Nathan Kraft

Monday, April 30, 2012

Out Rockin' Constantly

One of the ways I teach students linear equations is to have them remember out rockin' constantly. I break the equation down into the output (out) = rate of change (roc) times the input (in) plus the constant. This has always been an easy way for them to remember where everything goes when setting up many word problems.

Nathan Kraft