Teachers Talk

A recent blogging assignment in my grad class was this:
Blog: Your choice. What about your thinking or practice do you want to share with the world at large. This can be a record of something you’ve done, a particular activity to share, work from this course to share, or an opinion piece on an issue of the day (Khan Academy, standardized testing, Michigan education funding, ...)

They wrote interesting pieces, and I love hearing teachers' voices about that which they care most.  Four of the teachers have public blogs, so I'll point to theirs with this post. Leave them a comment, encourage them to keep blogging! Those who were blogging on Blackboard, I'll quote more extensively.  Post a comment here for them on how they should export and continue their writing!


Bill: Credo about what it takes to teach mathematics.
Also see: Bill's take on the 3 Act Story (I'm quite interested in this as a structure.  Here's a short urli.st of blogposts I've found on it.)


Eric: Textbooks. "The secret to being a lazy math teacher?  Good textbooks!..." Good hook!

Also see: HS grading

Melissa: only assignment she missed all course! Plenty to read at her blog, though.
See: Her description of her lesson planning process.

Ted: quick take on the public's perception of teachers and Summer Vacation.

Also see: his outstanding concept map for linear intercepts. (Not a blogpost, but wowser.)

Amy: Traditional Teaching
Taking this class has really got me thinking about the way that math is traditionally taught, and the way that I teach math. It has opened my eyes to the importance of teaching students how to problem solve and think critically. This has caused me to feel uncomfortable in my classroom for the past week or so. I feel that I want to make some changes, but yet it seems so overwhelming. The math department I work in is very concentrated on "everybody doing the same thing." This includes assessments and lessons. Also, the time that it takes to develop new tasks for student also is a daunting task.

So, as I reflect about changes I want to make for next year. I am thinking about making small changes. I think my first focus is going to be on formative and summative asssessments. Providing more opportunities for mastery rather than completion. I would also like to incorporate reflection on a daily basis to get students thinking about their problem solving and being able to put their thoughts into words.

In the future, I would like to work towards having more discovery and problem solving activities related to the concepts in my classes.

Erin: the switch from Michigan's High School Content to the Common Core State Standards
I have to complain for a minute about the switch from Michigan standards to the new Common Core standards. I have no problem with switching to common standards, I have no problem with the content of the standards, and I don't even have a problem with standardized tests based on the standards. My problem is this...

We are supposed to begin teaching to the new standards this year with testing based on the new standards to begin in 2 years, however, in the meantime, we are still being tested on the Michigan standards. That doesn't make any sense. The standards really are different in some ways and we are in the process of designing our curriculum based on the new standards. We have to be careful for the next two years that we also teach the HSCEs becuase that will be on the MME.

Someone tell me why we are rolling out the new and testing the old. Is testing really that important that we can't miss 2 years? Perhaps at least we can remove some of the consequences in the meantime so that we can develop a coherent curriculum without fear of the government taking over our schools.

Monica: Differentiation
When I took my Curriculum Development Class, we focused on Understanding by Design and Differentiated Instruction. Prior to that class, I had believed that differentiated instruction was synonymous with individualized instruction. And it wasn’t until the last 4 weeks or so that we started to talk more and more about what DI was, and how to adapt lessons and differentiate them.

First, some things I learned about DI:

• Quality differentiation begins with a growth mindset, moves to student-teacher connections, and evolves to community.  In a growth mindset students persist in the face of setbacks and see their effort as a pathway to mastery. Students embrace challenges, learn from criticism, and are able to find inspiration and motivation in others’ success. Rather than plateauing with skills and knowledge, students with a growth mindset reach higher levels of achievement.
• Quality DI is rooted in meaningful curriculum (not fluff!)
• DI is guided by on-going assessment which is used not for grades, but for instructional planning and providing feedback.
• and DI addresses students’ readiness, their interest, and their preferred method of learning.

Different methods of differentiating instruction include using

• Choice Boards--like a tic tac toe, where you have 9 activities listed, one in each box, and have students choose whichever 3 activities they’d like to do, as long as they make a tic-tac-toe. These activities should be rooted in the same learning objective, but address different learning types (multiple intelligences).
• Cubes or Think Dots--cubes would be using a net for a cube and having one question on each of the six sides. Think dots follow the same idea, but the students would roll a die or number cube, and then do the problem underneath the number they rolled on a worksheet (the worksheet would have 1 through 6 on the top, and the 6 problems listed under it—rather than making the cubes, now you just make a worksheet, and use a number cube).
• Sternberg’s Tri-Mind--list three different sets of directions to address the same objective. One way of addressing the objective would be analytical, one way would be practical, and another would be creative. There are tests, similar to multiple intelligence tests that students can take to see of the three they prefer.
• You can also have two similar worksheets, where one is more advanced and the other is more basic, depending on the level of the student. Students don’t know they have different leveled activities, but this is a great way to address the “just-right” problems for students on a case-by-case basis. You can make more than a basic and advanced, setting up four or five levels, but remember, differentiation is not individualizing.

In this class, I wish we would have learned earlier what differentiation was all about, and how to differentiate activities. This was most beneficial part of the class for me, and I wonder why “stuff like this” hasn’t been around longer. I feel that DI addresses issues that have been around in schools longer than solutions have, and this is something that should be included in all undergraduate teacher prep classes now (which I’m hoping it is)!

Monica's digital decimal differentiation designs are available at Scribd or by email from me.  There's quite a bit of work done on tic-tac-toe, tri-mind and cubes.

Closing Thought
Powerful stuff when teachers start sharing on Twitter or writing for sharing.

So why don't you join the conversation, you?










Photo credits: Search Engine People Blog, Cliff1066 @ Flickr

Division into Decimals - Undone

This is a pretty focused game and it is undone.  Even the fifth graders couldn't help me finish it... maybe a reader can help?  We were playing Manga High games with my preservice middle school teachers this week, and they noticed something.  The context can be completely silly.  They liked "the penguin game" in particular.  They thought it was mathematically worthwhile, teaching estimation.  They found the problems worth doing in the game and felt like it could help students improve their estimation and computation.  But we never really have to chuck penguins to safety.

There's something appropriate to the context.  We're approximating, acting quickly, there's the idea of what fraction of the way across are we... but it doesn't bear much scrutiny.  It's just silly and a bit of fun.  I think that's an element missing from some of my practice games.

The Game
The game premise is complete simplicity: race from one end of the paper to the other (25 cm) by drawing rectangles with area = 10 sq.cm.  Each turn your rectangle is determined by rolling the width with 2 dice.

That's it.

The Lesson
I shared that we were playing a game that had no name and no context.  I needed their help.  Immediately they began shouting names and ideas.  If I were a game designer, I would do this.  Just get a bunch of kids and let them throw ideas at you.  Holy cow were they enthused and specific.  Sadly, I had to tell them I wanted ideas for this game, and we'd have to play it first.  Me versus the entire class, like we usually do.

I showed the graph paper and explained that the goal was to get to the top by drawing rectangles with an area of 10 sq. cm.  I drew a 1x10 rectangle.  But that wouldn't be very interesting. So instead we're going to roll the width of the rectangle.  I rolled a 2 and asked, "how tall should it be?"  A couple students quickly jumped to 5, so I drew it and we checked - there were 10 squares.  We rolled for them.

Aside: that worked well enough that I would fake a 5 if doing this again.  (Faking a 2 is greedy.)

Their roll - a 7!  That's how wide it should be, but how tall? "3!" kids are quick to say. Hmm - that makes for an area of 21.  "1!"  That's only 7.  "1 and a half!" Let's check.  They talk me through 1.5x7. (Not a few couldn't recall.) Hmm, 10.5, too much.  "1.4!" They walk me through that... so close! 9.8. "1.45?" We try it and get 10.15.  Wait, I say, aren't we really doing division?  What times 7 equals 10? We started an saw that 7 goes into ten once.  "It doesn't work," someone said.  Another student said, "add a zero!" I used the notation my son has been using, which I like.  We got to 1.428, and then realized that was close enough for drawing the rectangle.  "Is it point-4-2-8 repeating?" "Will it ever repeat?"

We finished our practice game, and recorded our calculations on the white board.  When they played for themselves (mostly 2 on 2, with a few 1 on 1's), they referred to the ones we had calculated, but had also several others that needed to figured out.

In the summary, they thought the game was fun enough to play, and had good math.  25 cm made for about the right length of game.  (Hah!) They really got into trying to come up with a context. REALLY.  "Monkeys climbing!" "Monkeys escaping from sharks!" "Tiger sharks!" "People escaping from sharks!"  "You're escaping evil aliens."  "It's an alien trying to climb to the moon!" "It's ..."

OK... it should connect somehow to what we're doing.  I see why the climbing.  But why sharks?  Where is the shark chasing?  Could it connect to what we're stacking? What would you stack that has different sizes? "Students escaping an evil teacher by stacking homework!" "Books!" 

OK... maybe the game remains undone.  Maybe it should be unfinished?

Reflection
The game made a good context for modeling the division on which they needed practice.  The representation of the rectangles was supportive.  We talked about the connection between 10/2 and 10/4, for example.  They got the indirect variation aspect of it, and were rooting for small numbers for themselves and large numbers for me.  The familiarity with common division computations was good.  For such a vanilla game, it might be a nice aspect to keep it vanilla, and keep the game design aspect to the lesson.

If I was developing this into a video game, the subsequent levels would introduce some variability of the dividend also.  I also think you could have kids estimate the division.  If they overestimate, no block.  Increased decimal places would get you farther faster.  One variation that I ruled out was to have the game be a race to 2.5, so that the kids were dividing 1 instead of 10.  But then the area is .1 sq. units - not nearly as intuitive.  My next lesson would be how you can use these results to get at other computations, and to strengthen that multiplication connection.  I'm convinced that 99.9% of kids do not see the partial product connection with the division algorithm.

My 1cm graphpaper (pdf) is at Scribd if you'd like it.

Product Game... again!

It is no secret to my students how much I love the Product Game.  It is fun, not just fun for, you know, a math class.  The strategy required is at least as good as Connect Four, which is a surprisingly deep game.  The practice value is huge, as students have to compute many, many more products than would ever be done on worksheets.  The mathematics has connections, as the products lead to factorization, which enhances the strategies available.  But even the pedagogical structure is nice, as you consider moving one factor leads to considering families of multiples that are good for learning multiplication facts in a way that promotes fluency and efficiency.  The first I saw of it was from the Middle Grades Mathematics Project, the precursor of the excellent Connected Mathematics Project middle school curriculum.  (Which still has the game.)

So I love to adapt it.  It's never quite as good as the original, but often the great structure of the original allows new features to come to light.  Here's a previous handout that has the original Product Game and and Integer variation.

The fifth grade class I'm working with is beginning multiplication of decimals, by considering whole number times decimals that include tenths.  They're starting the whole counting up the decimal places routine, without much though of unitization.  If you have 5 bags with 4 apples each, you've got 20 apples.  If you have 5 groups with 4 tenths each, you've got 20 tenths... it's just that we don't often look at 2-point-OH as 20 tenths.  With my class I'd be looking for a context to start at this - probably money.

This class expects games from me, though.  I thought we had played the product game already (that was last year, Mr. Golden!) - oops.  In this version of the game, there's markers to make for your team.  I didn't have my usual two color counters available, but I've also learned that making markers or game pieces is a point of engagement and pride for some of the students.  Others are content with a quickly scrawled initial, and that's okay, too.  Mr. Schiller set a time limit of 3 minutes for making your markers, which was a good idea and the right amount of time.

We were to start by playing me vs. the class, so I could model some of the multiplying strategies I wanted to share.  But my son made me some excellent University of Michigan and Michigan State markers, and, being a proud alum, I had to be State.  There were students who couldn't bring themselves to being on U of M's team, and how could I argue?  So we played Spartans vs. Wolverines.



Product Game Decimal

The group play got us through all the rules and allowed us to model a lot of the whole x tenths.  But we never got up to the hundredths.  So we discussed how to get those.  I think they knew on some level it was tenths times tenths, but had a bit of the 'we haven't been taught this yet' syndrome.  I used the analogy of a dime being a tenth of a dollar, so what's a tenth of a dime? "A penny!" And what part of a dollar is a penny?  How many does it take to make one dollar?  It is terrific that they are used to seeing one cent written as .01

It was interesting seeing them play.  They started out almost entirely in whole x whole, and then were forced to the whole x tenths by the game play.  And if the game went on long enough, to the hundredths.  Several people got calculators to explore this, and a couple got to the calculators and then beyond them in the space of the hour.  Some students were done with the game after one session, but others were definitely up for more.  Hope you get a chance to try it and get a little bit addicted.

Photo credits: From Flickr, jeff_golden, 24oranges.nl and fireflythegreat.

Battle of the Century

Dorky little skit I wrote a while ago. Should we film it?


Fractions vs. Decimals

The Battle of the Century

Ringside Announcer (RA): Welcome ladies and gentlemen to the Battle of the Century: Fractions vs. Decimals!

Old Man Fractions has been king of the hill for so long he can remember the pharaohs. But relative new-comer decimals has been rocketing through the ranks past previous contenders like Mixed Numbers and Percents, buoyed by the rise of science and handheld technology. Tonight they settle the issue once and for all, mano a mano.

Color Commentator (CC): That’s right, Jim. And they have both clearly prepared. Fractions has developed his upper body so much he looks positively improper. Decimals has emphasized speed work, and is awfully quick to the point. Hey, looks like they’re ready to start.

RA: They come out swinging! Fractions looks like his strategy is to corner decimals and work his weaker visual representations. Oh there’s a pie model and a fraction strip combo! Decimals finally lands a 100 grid haymaker and gets back out to the center of the ring.

CC: Looks like that speed work is paying off, Jim. Decimals is coldly calculating without having to hit any special menu buttons on the calc, if you know what I mean.

RA: Not really, Howard, but I’m used to it. Oh! Decimal made a rounding error and Fractions lands an uppercut.

CC: That’s exactly the answer, Kid Decimals!

RA: The traditionalists are out of their seats, cheering on Fractions. Even the French are into it!

CC: He’s certainly got that je ne sais quoi, eh, Jim?

RA: Huh? Back to the action, Fractions is pressing his advantage. But decimals sees an opportunity and – oh! The referee calls time!

CC: I don’t think it was intentional, but that was definitely below the vinculum.

RA: The referee gives Decimals a warning and they’re back in. Fractions still looks a little wobbly, and Decimals presses the advantage, really working over Fraction’s arcane and misunderstood algorithms.

CC: Invert and multiply that! Whew!

RA: Fractions gives a nice example of unit fraction multiplication and is back in the fight. Oh, and lands a nice left hand on a complicated long-division problem.

CC: Decimals looks like he doesn’t know if his point is going left or right, Jim.

RA: It’s back and forth at this point folks. Fractions simplifies nicely, and catches Decimals a good one. Decimals lands a nice easy comparison, but Fractions hits a unit confusion counter-punch.

CC: That’s half of something, alright.

RA: Then Decimals comes right back with a repeating combination! Oh, and a non-terminating, non-repeating wallop! Fractions has no answer for that.

CC: Right in the Pi hole! Practically transcendental ring work, Jim.

RA: They’re really taking a beating out there. Howard, I think the crowd’s getting confused about what’s important here.

CC: I think you’re right, Jim, there’s kind of a baffled silence. Not that unusual at a rational battle like this one, though!

RA: That’s time. The fighters move to their corners. The judges communicate their decision to the ref. It’s pretty close on my scorecard, Howard. What do you think?

CC: Did you double check your answer, Jim? Nothing would surprise me –

RA: The ref is ready and brings both fighters to the center of the ring.… he pulls up both fighter’s hands! It’s a draw!

CC: The judges have called them equivalent! Oh, man! Looks like we’re in for a rematch.

Photo credit: tanita1 @ Flickr

Decimal Race

by easylocum @ Flickr

This is a game that I got to play with the fifth graders, and it was pretty fun.  The game is nothing revolutionary, just moving along in 5 hundredths increments.  What made it more interesting is the kids designed their own spinners.  It actually raises some really nice probability questions.  While the expected value is made manageable for 5th graders by being done as a sum, with higher grades you could do a first day on expected value, which raises good decimal multiplication questions.  I also let them have the option of designing their own board, but all but one just used my sample.  If you want to force board design, don't let them have it!  Their boards didn't have to be with a money context, but did have to increment by .05 to 2.00.

There was quite a bit of content value to students adding their move to their position, and a few that struggle with decimal operations really took to the idea of nickels, dimes and quarters.  When demonstrating or playing with students, I encourage you to show different ways to figure out how far your move takes you.




DecimalRace-MakeSpinners

I modeled for them a couple ways to design.  Put in 9 values, then figure out the last to make 2.00; adjusting from a previous spinner design by moving around tenths or 5 hundredths; or thinking of having $2, and spreading it out among the slots.  Many students were enthralled by the 9 zeroes spinner, but no one chose the 10 times .2 spinner.  One team asked if it was okay to design multiple spinners and switch during the game.  I said if it was okay with their opponents, since it was mathematically okay, if they were all fair spinners.



DecimalRace-BlankSpinners

If you don't have clear overlay spinners, you can get by just as easily with a bobby pin.  Hold the pin at the center with a pen or pencil and spin away.  (The fifth graders found bobby pins to be an interesting subject... go figure.) 

The squares along the bottom are for making game pieces to move around the board.  Some students really got into it because they could decorate their spinner and game piece.  That makes me think of the whole player psychographic thing that game designers think of, and wonder why I have never applied it to teaching before.  (First guess: many unengaged students are Vorthos, and many teachers design lessons for Melvins.)



Decimal Race Board

It does occur to me that different spinners are better or worse depending on the game's special squares or conditions.  Of course, if students are thinking about this, more power to them.  They deserve to get to Candyland.

Feedback always appreciated! (Seldom received.  Sigh.)

Triangle Puzzle

Have you ever had a nice problem that you just thought about at odd moments?  Boring meeting, stuck waiting somewhere, few surprise extra minutes in a day?

For a while now, my favorite problem like that has been finding a nice way to divide up a square into the seven triangle types.  I love tangrams, and I like Pierre Van Hiele's mosaic puzzle even better.  If you do too, stop reading right now and try this problem.  It's fun and worth a surprising amount of thought.  (For me, anyway.)  Then suddenly this week, one of my little thumbnail sketches worked out.  I don't know whether to be happy or sad.  Being a geogebra nerd, I wanted to make a sketch of it, and that led to making a puzzle out of it.

You can print this picture of the pieces to try in real life, or try it with the Geogebra file or as a webpage.    (A solution is an option on the file or webpage.)

But... now I'm left wondering what to think about in those rare extra moments.  Then on Twitter, Justin Lanier (@j_lanier) tweets:

Had an insight in the shower this morning. Example: .717171... = .717171.../1 = .717171.../.999999... = 71/99 (!)

 Hmmm.  Really?  Maybe it's a coincidence, because 100 times .717171... minus the original leaves you 99... hmm.  Would it work for .717171.../.6666... ?  It does.  Tweet back:

@j_lanier cool. So is .a_1 a_2...a_n repeating / .xxx... =a_1...a_n/xx...x (n times) for any x? Or divided by .b_1 b_2... b_m repeating ...

Which connects to another problem (from Dave Coffey) I like thinking about: how many digits does it take 1/17 to repeat and how can you tell?  In general?

OK.  Deep breath.  There's always more problems.

Decimal Point Pickle

I came up with a variation on what was already a decent game and got to pilot it with Mr. Schiller's 5th grade this week.  Esther Billings introduced me to the game she had found in the book, Nimble with Numbers by Leigh Childs and Laura Choate, Dale Seymour Pub., 1998

The 5th grade came up with two names, Destination Elimination (which I like because it rhymes), and Decimal Pickle.  This suggested by a student who's answer for everything is pickle.  (I'm sure you know a student like that.)  But here, it reminded me of a childhood baseball game that none of the kids knew but kind of fits.  (The baseball game Pickle.)

 My Favorite Pickle

Decimal Point Pickle

Set Up:
1.    2 or more teams or players.
2.    Get a deck of cards and remove the Kings, Queens, 10s and Jokers.  Jacks stay in.
3.    Each player or team makes a path with 10 spaces.  It can be straight and rectangles, or it can be curvy and circles, but it needs to have 10 spaces and a clear beginning and end.
4.    Shuffle the cards.

Playing:  Idea is that you’re going to fill in your path from small to big, flipping over cards to get possibilities.
1.    On your turn, flip over a card.  If it’s red, flip over another card.  If it’s red, flip over another card.  But you never flip more than three.  If you run out of cards, shuffle up the used cards.
2.    Arrange those cards to make a decimal number.  Jacks are the zeros. The smallest number you can make is .000, and the largest is .999.  Say your number.
3.    Fill in your decimal number somewhere on the path.  But it can’t go before a smaller number or after a bigger number.  Your path has to start small and end big.  If there’s no place to fill in your number, you don’t. 
4.    Winner is the first person to completely fill in their path, with all the numbers in order.

Examples:
1.    J ♥, 3 ♣.  You can make .03 or .30. 
2.    5 ♥ hearts, so you flip 2 ♦, so you flip 7 ♥ hearts.  (You stop because you can’t have more than three.)  You can make one of .275, .275, .527, .572, .725 or .752.  Which you want depends on your path. 
3.    Sample filled in path below.

Variations: 
1.    Simpler:  Play where you always flip over 2 or 3 cards.
2.    Play cooperatively.  Two players work together to fill in one path.
3.    More complex:  Play with 10s, which fill in 2 places.  So 10 ♦, 5 ♠ can be .105 or .510.
4.    More complex:  Play without the three card limit.  You could hit a 10 digit long decimal or longer!  (Pretty unlikely, but still…)
5.    Make 12 space paths.
6.    Play with Jokers as a wild digit.

Teaching Notes:  As often with a new game I played me vs the class first.  It was clear that the blackjack-esque possibility of extra cards was exciting, and they quickly got the idea that it was a big advantage.  I didn't castigate anyone for saying "point two three" but often asked "so how do you say that number?"  I shared how I thought about getting numbers close together and they really ran with it.  In general hitting on lots of ideas about where to put numbers, how to divide up the path, etc.  In their 2 on 2 games, there was a lot of good discussion about strategy, how to leave space, and what they wanted to turn over.  There was a lot of excellent comparing of decimals of different length.  (One amazing discussion comparing .1 to .065)  Students got very creative with their paths and I was quite glad I hadn't brought any preprinted ones.  We actually wound up playing with everyday math cards, which thankfully came in black and blue.  Whew!

If you give it a try, please let me know what you think.


<--- 2nd="2nd" favorite="favorite" p="p" pickles="pickles">
---> PDF of the game.