Skemp Discussed

This semester we had the opportunity to discuss Richard Skemp's great article on Instrumental and Relational Understanding in class. (“Relational Understanding and Instrumental Understanding,” Richard Skemp, Mathematics Teaching in the Middle School, September 2006; link goes to a pdf hosted at Portland State) Students read the article, with the following home workshop. When we came to class, they discussed at their table and then made one 'slide' presentations to the class on 2 ideas.  The questions are the ones I used for an online discussion before, recorded in this blog post.

Home Workshop 16 - Learning Math
Instrumental Understanding
How do I understand?
“Relational Understanding and Instrumental Understanding,” Richard Skemp, Mathematics Teaching in the Middle School, September 2006
Discuss in class Wednesday
Objective:  TLW use specific questions to better focus on and understand relational understanding.

Schema Activation: How do you multiply fractions?  How well do you understand the multiplication of fractions?

Focus:  Directed reading.
Below in the activity are a dozen discussion questions. As you read, keep notes on your thoughts about the questions.Read through the questions before reading the article.

Even though this article was written for teachers, Dr Skemp wrote mostly for researchers, and at times the language is a wee thick. Press on!

Activity: Read the article, jotting notes on the 12 questions below.  These are just notes, and you may find you have no thoughts on a couple of them. In the reflection you will expand on your thoughts for two of them.

1) What is the point of starting off with the Faux Amis story? 
(A faux amis are two words in different languages that sound similar but mean differently.  Sopa (soup) and soap (jabón) are my favorite from Spanish.  Skemp says that the ways we use "understanding" are as different as if they were faux amis.)

2) What is your favorite example of “rule without reason”? Why?

3) Does the author’s idea of looking for your own examples and his three reasons for it make sense? Why?

4) Explain Skemp’s two kinds of mismatches (in the classroom) in your own words.

5) Of his two kinds of mismatches, which is more common? Which is more of a problem for the teacher?

6) What are Skemp’s faux amis in mathematics teaching? Is either one an issue in your math major classes here in GVSU?

7) Would you add any advantages to his list for instrumental mathematics?

8) Would you add any advantages to the list for relational mathematics?

9) Do you agree with the advantages that he lists for the two types?

10) What’s an example of relational understanding in your non-math life?

11) What’s an example of relational mathematics understanding for you? How do you know?

12) So, what about your classroom? Will you teach for one, or the other, or both? Why?

Reflection: Pick 2 questions you would like to talk about in class, and write a thoughtful response to each.

• Look over your notes/highlights/work from reading.  What did you take from it?
• In your own words describe the ideas of instrumental and relational understanding.

The other thing that we've been developing in class is the idea of questioning, both as a teacher, and the benefits of student to student questions.  This class struggles with being quiet, but by the fourth group, they've hit full class discussion mode.  I really think that conversation is the only way to work towards understanding of big ideas. I filmed the first two groups and then handed off the iPod for recording.

In their groups I asked them to share their reflection from the workshop, they discussed a bit, and then to decide on two points to present to the class. Mostly they used the questions to frame their points. We talked a bit about presentation zen, and I asked them to make a 'slide' for their two points on the board, with the idea to not have a lot of text, but to support their idea with a succinct statement or even better, a visual. They did an excellent job, and I hope you enjoy sharing in their discussion.

Group 1 focused on questions 5 and 8.

5) Of his two kinds of mismatches, which is more common? Which is more of a problem for the teacher?


8) Would you add any advantages to the list for relational mathematics?


Group 2 

2) What is your favorite example of “rule without reason”? Why?


 5) Of his two kinds of mismatches, which is more common? Which is more of a problem for the teacher?


Group 3

5) Of his two kinds of mismatches, which is more common? Which is more of a problem for the teacher?


11) What’s an example of relational mathematics understanding for you? How do you know?


Group 4

12) So, what about your classroom? Will you teach for one, or the other, or both? Why?

Group 5

1) What is the point of starting off with the Faux Amis story?

10) What’s an example of relational understanding in your non-math life?
  (Nice because it was ambiguous whether their example was relational or instrumental.)

As I listened to their discussion, I was struck by how many of the concerns of inservice teachers they already have, which is a real testament to the idea from The Teaching Gap that teaching is a cultural activity.  If we do not do something to resolve the tension between what teachers feel they are expected to do (the job) and what they want to and should do (the vocation), we're not going to make any progress.  It's almost what Skemp is talking about with the faux amis about our two ideas of learning. The same two ideas are competing and confusing us when we use the word teaching.

Make and Take

I'm falling behind on blog-writing, but have to share this game. Definite keeper, with great potential. Easy rules, great mathematical situations and pretty fun.

The game grew out of a meeting with Nick Smith, one of our novice teachers with a good game eye. He was looking for a way to make a game with number operations and maybe order of operations that had us using cards and trying to make a target. Wasn't quite working out.

Finally it occurred to me that if you were setting the target for your opponents... started trying it with cards and BAM! A game. It's simple enough, probably someone else has come across it before.  Basically, you deal 5 cards to each player/team, each team picks one card for the other team to make by combining their remaining cards with operations.





(Direct link to document.)

To launch it with the 5th graders today,  the teacher and I started to play. I put up the values for Ace, Jack, Queen and King on the board - which was a good idea as students consulted it frequently. Today was 11-11-11, and Jake, one of the students, had a birthday... his 11th! (He was in the local paper last night.) So we renamed the Jack the Jake in his honor, since the Jack is worth 11. Sometimes stuff just works out.

After about three turns of demonstration, the students were clamoring to play. Who are we to stand in the way of a math game?  Students were engaged, making interesting combinations, and making more complicated combinations as play went on.  It adapted well to students at different levels, as they were choosing combinations, and I was able to see automaticity with subtraction improving in students who have some math struggles at the same time as self-identified math whizzes were challenged to find fascinating 4 card combos, like Jack - 4/4, divided by 2 to get 5.

I tried the game with younger learners previously, and they got good addition and subtraction practice, and think it would extend well to middle school as support for order of operations.  (Write down your combination and check it on a scientific calculator.)  It was nice that sometimes the game called for easier combinations and had moments of challenge. Students were actively searching for new people to play and telling stories about games and combos. Very fun.

To finish our time, we discussed the combos (I had recorded some of the better ones on the board, such as Q, 5, 4, 4 -> 10), the strategy and the name.  I like how the game becomes a context for some pretty good problems.  The students were split on what made a difficult target. The majority felt like middle cards were harder to make, but a few thought the smallest cards. I actually don't know! One aspect of the game that I like a lot is that you gain information about the opponent's hand as you play. A strategy that many came across was reusing target numbers that your opponent couldn't make.

There weren't a lot of suggestions for names... math war and math attack had some support.  More suggestions about the game would be welcome, also.

Having written those game design commentaries lately (one & two), I can't resist thinking about the game using it.

1. Goal(s). See numbers as related by operations. This game is great for that.
2. Structure.The game reflects the goal by using a shifting set of cards. The slow turn over allows students to build relationships and more and more complicated sets of computations.
3. Strategy. The selection of targets and which cards to keep to make combinations is the first level. Taking into account your opponents' cards is a whole 'nother level.
4. Interaction. Choosing the target for your opponents and having to make their target offers lots of interaction.
5. Surprise. The cards you draw and the target you're trying to make.
6. Catch-Up. This could be a weak area. Once kids are good enough, it's rare to miss the target, which means it's hard to catch up. That's when you switch to the four card variation, which can be very challenging.
   
7. Inertia. Kids were divided on the 10 card winning condition. Some thought it should be lower. One student who loved the game suggested 13 cards!
8. Rules. Big win for this game. Very simple.
9. Context. No context, but the game did seem to have a pretty fun level of gameplay for students.

Image credits: qthomasbaker, ames sf @ Flickr

Game Design: 6-10

Mark Rosewater, head designer for Magic: the Gathering, has up the his intro to game design article, so it must be time for my 2nd part of my commentary thinking about educational games.

The first five principles were:

1. Goal(s). Easiest part for educational games.
2. Rules.  
3. Interaction. 
4. Catch-Up. Most subtle, maybe because so many games lack this.
5. Inertia. Hard for teachers.

The second five:

6. Surprise. The game should have some unpredictability for players.

To me, this connects strongly to Interaction and Catch-Up. One way to get surprise is hidden information - which often can contribute to interaction amongst players.  Information can be hidden from both or the players can hide it from each other. The new game Flip Out has a good element of this with two sided cards of which each player sees different sides. A benefit for math and literacy is that this makes inference a part of the game.

The other easy way to add surprise is random events - which can contribute to making catch up possible.  The only thing that makes Monopoly playable is the dice rolling.  In Euchre, no matter how good you are, you need cards to play. The math benefit is the addition of probability, even if informal, to game play. It's no surprise that the two most common game pieces are dice and cards.

7. Strategy.

Interesting to me that this is so low on the list, which makes me wonder what he was ordering them to achieve.

by mhuang @ Flickr

This is the biggest add-on for educational games over other activities. The problem solving inherent in any game with strategy is such fantastic grist for mathematics.  Mathematicians often see math as a game because of this strong connection. How do we achieve a result with allowable moves? Using games with K-12 students,asking for their strategies always makes for an amazing summary and unearths most of the math content of the games. It also helps build Inertia as then students are more interested in playing again, trying our others' strategies or designing ways to beat them.

There's a natural tension between Surprise and Strategy.  If things are too random, strategy loses all impact. If things aren't random at all, it is chess or go.  Both great games, obviously, but also both games that struggle with Catch-Up and Inertia for many players. Plug for Magic: the balance of these two elements is a large part of what makes the game so bloody amazing. Also applies to Bridge, to a lesser extent. (Yes, I'm claiming Magic > Bridge.)

8. Fun.

I struggle with this. Because I find interaction, surprise and strategy so engaging, I love games in general. I'll play anything. But what makes a game fun to kids is often a surprise to me. It's not uncommon for me to take a game to kids, and let them add the context. I wrote about this a bit with my Division into Decimals game.  Games like Decimal Point Pickle and Power Up had this in spades. Probably this is the difference between a game being good, and the game being a smash hit.

To some extent I think the last two principles are really subcategories of this one. Did they get pulled out to make ten or - more likely - is there something I'm missing that makes them truly distinct?

9. Flavor.


10. Hook.

Flavor is about the context and setting for your game, which heavily influences the fun aspect for players, in my experience. At least on entry, and Mark connects this to the barrier or entry cost to your game. The other principles determine long term fun. In our house, this gets us to play a game fr the first time, but won't sustain interest. One neat point he makes about flavor, though, is how it can influence design. My youth Bible study is making a return of the Lord card game based on the 10 bridesmaids parable. (Yes, really.) But the context for the game is inspiring a four horseman of the apocalypse feature that will definitely add interest to the game. Probably shouldn't have shared this story.

The idea of constructive flavor reminds me of my colleague Jacqui Melinn talking about integrated units.  A marine biology integrated unit is not when you put your math practice problems on a whale-themed sheet, it's when your questions about whales require math to think about and solve. Good flavor isn't an add on, but supports the game mechanics. For a math game, this gets at the structure of the game supporting the mathematical objective. Cheap flavor is the hallmark of flashcard/drill math games. "Look you're doing lots of multiplication, but it's on a baseball diamond!"

Hook is what gets people to try your game. This is less important in educational games to me as we have a built-in market (students), but I'm also not trying to sell my games to a publisher. (So maybe my hook is that my games are free?)  However it does remind me of Dan Meyer talking about a hook for a lesson, and could well be linked to engagement. I just don't know how to tease it out from flavor and fun. Maybe hook is a measure of whether the game has things that make you wonder?

My Nine
Looking over the list, I think I'd order them more like the following to get at my process.

1. Goal(s). Design starts with objectives.
2. Structure. (Not in his list!) What is the essential nature of your set of learning objectives and how can that show up in the game?
3. Strategy. These three have to go into the primary design phase as well, or the game will just not have them.
4. Interaction.
5. Surprise.
6. Catch-Up. As you start to playtest, these two are important to attend to for good design.
   
7. Inertia.
8. Rules. For me this comes late; kind of a synthesis step as you think about how to communicate the game. It will often result in design revision, though.
9. Context: Fun-Flavor-Hook. To me this can't really be evaluated fully till you're out with the intended audience. You need a first take on this before that, but should be open to major changes in this area.

Boy, I enjoyed these two articles. Thanks, Mark, for  writing them, and giving us fledgling designers something to think about. Note that Mark has a Tumblr where he answers many questions and gives the behind the scenes story at Wizards of the Coast. I asked, for example, about the order of his list, and he wrote: "I left it up to my subconscious. That was the order that felt organically correct. I’ll be honest, that I can’t exactly explain why."

Images: All Magic: the Gathering stuff is very heavily (c)'d by Hasbro.

Advice for Solving Equations

Quick post to share the teaching idea. The preservice high school teachers read 

“Advice for Solving Equations,” Steuben and Torbert, Mathematics Teacher, April 2006

The reflection was to give your advice. While I think the advice they give is solid, what I like is the giving of advice.

John G:
1) remember the meaning of the =. These things are the same thing, though they look different.
2) remember your purpose and for what you’re solving.
3) think about the best representation for solving this equation.

Ellen B:
1. Don’t forget about substitution! I wouldn’t have thought about this in the problem in the reading, but it made the problem so much easier.
2. Could you change the form of the equation to make it easier to solve?
3. Know what you’re solving for; what should the solution be in terms of?

Alyssa B:
1. Look at the problem before you start; what do you already know or what is important to note?
2. Make sure your answers make mathematical sense!
3. If you’re stuck, try rearranging/changing the form of the equation.

Greg O:
1. Never give up on a problem, if your stuck try rearranging it to make more sense to you.
2. Always look for things that you know in a problem so that you can maybe substitute something for it.
3. Make sure you can reverse the process so that you can get the original answer, a way of checking to make sure your answer is correct.

Jordan D:
1. It’s alright if you can’t get it right at first. You learn by spending time on the equation and from mistakes.
2. Experience helps you solve challenging equations, there are many steps that do not seem natural until we use them.
3.Always check your solution, make sure that it makes sense and works.

Emily W:
1. “You will get much more out of a problem if you work on it for 15 to 20 minutes and fail than if you turn to the solution after only 3 minutes.”
2. When you first start a problem, notice everything you can about the problem (Can x = 0? How many solutions will this equation give? Etc.).
3. If the method you are comfortable with does not come up with the correct solution, don’t be afraid to use one you are less comfortable with.

Mike Simon
1.) When solving a problem that looks difficult, begin by stating what you know.
2.) Work with what you know and if you get to a point where you are not sure if there is a next step, look over your work.
3.) After checking over your work, a make a guess and see where it leads you. There is not shame in being wrong.

Matt York
1. The whole purpose of the problem is defeated when you give the solution after just 3 minutes.
2.  As you are solving, always keep in mind the goal... never forget the purpose of the problem.
3. If you have no idea where to start, try some numbers which make sense in the context of the problem, it might lead you somewhere.

Joe Freiman
1. It’s better to work for 15 minutes and fail, than to look at the answer after a minute.
2. Using your past experience is the best way to solve difficult equations, so by practicing constantly and gaining more experience is the best way to get better at solving complex equations.
3. Check your work, make sure your answer makes sense.

EDIT: some end-of-the-semester catch-up-additions...

Brandi Stewart
1) It is ok to be wrong and have to try many different ways.
2) Check your work
3) It is not bad to ask for help, especially if you have tried the problem on your own and do not know how to figure it out.

Courtney Johnson
Mine are kind of specific but...
1) Try putting all variables on one side and setting equal to zero to see if using the quadratic formula is a possibilty
2) See if any substitutions can be made (e.g. trig identities)
3) Try removing a common factor to simplify

Amanda Hoezee
1) You will get much more out of a problem if you work on  it for15 to 20 minutes and fail than if you turn to the solution after only 3 minutes.
2) One does not need to be talented to solve challenging equations; one only needs experience.
 
Mitchell Brady
1) If you are stuck on a problem keep it simple and work with what you already know about the problem a solution that may help.
2) If you are stuck go back through your work to see if something sparks your memory to continue.
3) Finally, do not be afraid to continue and be wrong, just learn from your mistakes and try another method, but first figure out why it was wrong so you do not make the same mistake twice.

Ryan Warner
1) Failing isn’t always the worst thing, you can learn a lot sometimes by failing
2) There is almost always more than one way to solve a math problem, when solving equations this is definitely true so try different ways if you can’t figure it out right away.
3) Learn everything you can about the equation first before you try to solve it.

Shannon Penix
1) It is important to look at the problem all the way through before trying to solve it, noticing things you already know.
2) It is alright to take a while to get through a problem and even not succeed. At least there is learning involved.
3) Take the time to go back over your work if you get stuck. There may be earlier steps that could trigger your thinking to continue.

Jeremy Sheaffer
1) Be fair to both sides of the equation.
2) Check your work
3) The harder something is to learn, the more chance there is that you will remember it.

Image credit: dullhunk @ Flickr