That's my attempt at a Glide Reflection Frieze.

This week my K-8 students were working on motions again. Using the Geogebra activities at Motion Sketches and More Motion Sketches. For K-8, there's really not a need for glide reflections because they're usually not a part of the curriculum.

This doesn't fit with Euclid's vision of motions, which was strongly tied to congruence. Any two objects are congruent if and only if there is a motion from one on to the other. (Aka rigid motion or isometry or Euclidean transformation...) This requires four motions, not just three. But a glide reflection is just a slide and a flip, you may say, we don't need it. Well all the motions can be made from just reflections, but we still teach turns and slides.

But I've been stumped as to a good way to present these glide reflections. Students can recognize them by a process of elimination and students can make a motion that is a glide reflection. The next level of knowing a motion is to be able to specify it. Students are good at finding lines of reflection, and can specify direction and distance for a translation. It is difficult for many/most to find the center of a rotation, without being told. They can do it in a dynamic environment (cf. MotionControl, a geogebra webpage) but it is difficult for them to construct. The first guess seems to be connecting corresponding points and trying where the lines cross. (Which doesn't work.) So it's really hard to get students to know how to specify a glide reflection. Mathematicians usually describe a glide reflection with a vector and a point or position of that vector. The vector indicates the direction and distance of the slide, and the line containing the vector is the line of reflection.

This sketch is my attempt at a glide reflection sketch - the goal was to create an environment where students might be able to notice things that would lead them to construct the idea. I would love it, if you try it out, to get feedback about ways to make the sketch more supportive. Thanks!

As a webpage or geogebra sketch.

## Thursday, October 28, 2010

## Friday, October 22, 2010

### Unit Rummy and Game Design

The math game is at the end of the post - feel free to skip the rambling!

This post actually started as I was thinking about my middle school bible study (online at ghbiblestudy.blogspot.com) and trying to think about a game that fits this week's study on Lazarus and the Poor Man. An hour seems to be a bit long for my students to engage in bible study, at least for now. (I haven't helped them build that capacity. Yet.) One of my favorite games occurred to me: The Great Dalmuti by Richard Garfield. (Currently in print... so buy it! Funagaingames.com is a great online game store.) It's actually an adaptation of a family of games that have been played for many years. So I was looking for the rules for playing with a regulation deck, so that my students could play at home. (It also got me thinking about making a version of the game with Monopoly money... more on that later!) Richard's version uses a pyramid deck, with one 1, two 2's, etc., that helps gameplay but is hard to simulate with a regulation deck.

In looking for the rules, I stumbled across an online archive of Richard's game design articles from

I sort of think the fourth year of HS math (for non-STEM majors, at least) should be a game playing course with Go, chess, hearts, bridge or Euchre, rummy, and Magic. (Maybe Minecraft... I'm investigating.) What else would you add?

The version of Dalmuti I wound up with for my youth group is below. No specific math content, but problem solving and strategy is strongly helpful. Maybe a good setting for probability dilemmas.

Ruler and Peasant

One of my favorite games to adapt is Rummy. (Rules here at Pagat, good resource for card game rules.) It's a game about connections, seeing the cards in your hand in relation to each other and the cards in the discard pile, those relationships and then what your opponents have. There's deduction and induction as you assemble pieces of information from what people play, discard and pick up versus the hidden information in their hand. How much more mathematical can you get?

Units of measurement is one place where many preservice teachers feel underprepared, especially with respect to metric measurement units. As we work in my K-8 geometry/measurement class on units, we use cards with a variety of units. For them I have on some weird ones (drams, furlongs, hectares) to help them think about the process of learning these units. We sort them by quantity that they measure, pair them as equivalent amounts, think of things to measure with the unit, and arrange them by relative size of the unit. We play Concentration (aka Memory) with the equivalent pairs. But the most fun, clearly, is playing Unit Rummy!

The hand is over when a player has no cards.

Instead of the weird units I have them sort and reason with, we play this game with a more elementary set of units. (Email me if you want the college version with the weird units.)

Unit Cards (Elementary)

**Adapting Games**This post actually started as I was thinking about my middle school bible study (online at ghbiblestudy.blogspot.com) and trying to think about a game that fits this week's study on Lazarus and the Poor Man. An hour seems to be a bit long for my students to engage in bible study, at least for now. (I haven't helped them build that capacity. Yet.) One of my favorite games occurred to me: The Great Dalmuti by Richard Garfield. (Currently in print... so buy it! Funagaingames.com is a great online game store.) It's actually an adaptation of a family of games that have been played for many years. So I was looking for the rules for playing with a regulation deck, so that my students could play at home. (It also got me thinking about making a version of the game with Monopoly money... more on that later!) Richard's version uses a pyramid deck, with one 1, two 2's, etc., that helps gameplay but is hard to simulate with a regulation deck.

Ed Yourdon @ Flickr |

*The Duelist*, which was the print magazine dedicated to one of Richard's other games: Lost in the Shuffle: Dalmuti This article is about how games could be as big as movies. Is that what's finally happening with the gamification of the internet? (Note that Richard's current thoughts on game design, along with those of several other designers, are there to hear at gameswithgarfield.com)I sort of think the fourth year of HS math (for non-STEM majors, at least) should be a game playing course with Go, chess, hearts, bridge or Euchre, rummy, and Magic. (Maybe Minecraft... I'm investigating.) What else would you add?

The version of Dalmuti I wound up with for my youth group is below. No specific math content, but problem solving and strategy is strongly helpful. Maybe a good setting for probability dilemmas.

**The Math Game Part**gail m tang @ Flickr |

One of my favorite games to adapt is Rummy. (Rules here at Pagat, good resource for card game rules.) It's a game about connections, seeing the cards in your hand in relation to each other and the cards in the discard pile, those relationships and then what your opponents have. There's deduction and induction as you assemble pieces of information from what people play, discard and pick up versus the hidden information in their hand. How much more mathematical can you get?

Units of measurement is one place where many preservice teachers feel underprepared, especially with respect to metric measurement units. As we work in my K-8 geometry/measurement class on units, we use cards with a variety of units. For them I have on some weird ones (drams, furlongs, hectares) to help them think about the process of learning these units. We sort them by quantity that they measure, pair them as equivalent amounts, think of things to measure with the unit, and arrange them by relative size of the unit. We play Concentration (aka Memory) with the equivalent pairs. But the most fun, clearly, is playing Unit Rummy!

**Unit Rummy***Deck*: All unit cards. Players: 2 to 5*Deal*: Shuffle all cards together. Deal 5 cards to each player. Flip over one card to start the discard pile. Place the stockpile and the discard pile card in the center of the table, where all players can get to them.*Object*: To have the most played cards by the time someone goes out.*Course of play*: After the deal, all players receive time to organize their cards. The person on the dealer’s left goes first, continuing clockwise. A player may start his turn by drawing a card from the pile or picking up discards, going as far back into the discard pile as he wants. If a player goes back to a certain card, he must immediately play that card in a meld. (see “The Melds” below). You can’t discard this card. He may play any melds he wants to. It is not required to play a meld. A player must always end their turn by discarding a card (their choice). Lay the discards down in a way so each card is visible.The hand is over when a player has no cards.

*Optional Scoring*: Score one point for each card in play, and subtract one point for each card in hand. The player who goes out gets 3 bonus points.*Calling “Rummy”*: If a player discards a card that may be laid off on other cards, the first player to notice it calls “Rummy” and plays the card for his benefit. A card counts a discard once a player takes his hand off it. Once the next player draws or picks up a card it is too late to rummy.*The Melds*: a meld is a set of three or more matching cards. Cards match if they are equivalent measures or all use the same unit. For example 1 c = 8 fl oz = .25 quart for equivalent measures, or 1 ft, 1000 ft, .1 ft for all the same unit. On your turn, you can play a card that matches another player’s meld.**Cards**Instead of the weird units I have them sort and reason with, we play this game with a more elementary set of units. (Email me if you want the college version with the weird units.)

Labels:
Dalmuti,
games,
Measurement,
Rich and Poor,
Unit Rummy

## Monday, October 18, 2010

### To Understand - Book Club

The K-8 Geometry class is reading

The aspect of the book that students had trouble with is that each chapter (from 3 on) starts with the story of someone in the arts advancing their own understanding. It was difficult for some students to connect with these stories if they were unfamiliar with the artist. Last semester students did make lots of connection to what they'd read from Mem Fox;

The structure we're using is groups of up to 8, where each group has a leader - responsible for keeping things moving, questioner - for prompting discussion, recorder, and a reporter who leads the group's sharing with the whole class. After the free discussion period, the group talks about what to share, then the whole class discussion starts.

Some notes from our discussion:

*To Understand*this semester. This book is Ellin Oliver Keene's follow up to her mega-book with Susan Zimmerman,*Mosaic of Thought*. Yes they're reading a reading book. But this book is really about comprehension, much like Mosaic of Thought, and most of these preservice teachers will be in elementary school, and teaching reading. This is the second semester using this book, which had mixed results last semester, but overall positive reviews from students.The aspect of the book that students had trouble with is that each chapter (from 3 on) starts with the story of someone in the arts advancing their own understanding. It was difficult for some students to connect with these stories if they were unfamiliar with the artist. Last semester students did make lots of connection to what they'd read from Mem Fox;

*Reading Magic*is on my list, but I haven't gotten to it yet.The structure we're using is groups of up to 8, where each group has a leader - responsible for keeping things moving, questioner - for prompting discussion, recorder, and a reporter who leads the group's sharing with the whole class. After the free discussion period, the group talks about what to share, then the whole class discussion starts.

Some notes from our discussion:

**Chapter 1**: Rethinking Understanding &**Chapter 2**: Seeking Understanding in our minds, our lives- Teach concepts for greater depth. Current understanding is the key to understanding in the future. Don't do too many topics.
- Even in a calc class, you can move on before students have a conceptual understanding.
- Quality vs quantity is a dangerous choice. Eg. pizza parties for number of books vs what was learned.
- Teach in different styles with kids having choice of structure. In elementary and HS everything was more structured vs in college when the info is presented and you do with it what you will. Elementary school needs more choice and freedom.
- Everything has trial and error. Repetition is everywhere, for example math workbooks that ask many of exactly the same form of question.
- Testing is crappy, and we focus more on scores than understanding. The test generate so much pressure from so many sources. Home, school, etc. You never get them back, so you can't learn from it. Test too much, at least standardized testing. What if the kid doesn't have supportive conditions (sleep, food, etc.)
- All children can achieve greatness. A principal who says "someone's got to flip the burgers..."; this kid just isn't good at that. Beliefs will be communicated to students. Vs. believing in all students.
- "We can extinguish the notion that some kids are going to make it and some kids won't." Psych study that showed teacher beliefs have a huge impact on achievement.
- After Ellin described a great day, her husband David asked: "why isn't everyday like that?" Easy to make excuses. Ultimately it's because we don't expect them to. Belief motivates.
- What will you do about barriers to eliminate the real reasons for student struggle? Eg. a teacher who brought in bagels and cream cheese before a long standardized test.
- Also be careful not to expect too much, so students feel defeated.
- In elementary kids are excited to go to school. They will/do lose that if it's not supported.
- Do teachers care more about noise or intellectual activity?
- One student hated the book so far because it felt like the author posed questions without answering them. Another student said that that was creating a desire to read on and a curiosity. That maybe the whole book was about what it means to understand.

Cartoon from the always amusing Speed Bump.

**For my teaching**- First workshop (focused on determining importance) was more beneficial than the 2nd (stopping at each page to monitor your comprehension). Better recall of Ch 1. Stopping at each page broke the flow of reading. (This was ironic because last semester it was reverse, so I had changed Chapter 3 workshop to be like Chapter 2. But I let them know they can always choose a different way to meet the objective for the workshop.)
- This workshop (Ch 2) can be an example of what not to do. If it becomes a task to do the motions, then it defeats its own purpose of looking for comprehension.

## Thursday, October 14, 2010

### Resource Round Up - October

Photo por pepetomo @ Flickr

**Carnivals**

Math and Multimedia Carnival 4 has been up for a bit. Lots of good things, interesting entries and great 4 images. I feel challenged, as it will be here next month. (Quick, think of cool 5 stuff!) I'm already getting some submissions, and I encourage you to submit your own items, of course, but also to share posts on blogs you follow that are good examples of the organizing idea. What is the organizing idea? Work that shows one or more of the following:

1. Connection between and among different mathematical conceptsSubmit an article.

2. Connections between math and real life; use of real-life contexts to explain mathematical concepts

3. Clear and intuitive explanation of topics not discussed intextbooks, hard to understand, or difficult to teach

4. Proofs of mathematical theorems in which the difficulty of the explanation is accessible to high school students

5. Intuitive explanation of higher math topics, in which the difficulty is accessible to high school students

6. Software introduction, review or tutorials

7. Integration of technology (Web 2.0, Teaching 2.0, Classroom 2.0), in teaching mathematics

Math Teachers at Play is appearing tomorrow at Homeschool Bytes. She did a really nice Adventure Edition. With one of Denise's Alexandria Jones adventures, Mimi Yang's deduction without p's and q's, and Guillermo Bautista gets back to the root of probability in lotteries... check it out!

The puntifical Frank and Ernest, of course.

**Inservice Links**

I get the opportunity to work on occasion with the secondary math teachers in Allendale, which is a 1:1 school (laptop:student). Mostly it's trying to support them as they integrate technology for their efficiency and student support. I don't know as much about their tech as they do, mostly, and obviously they know their students better, too. They do a great job at using the time together to mutually develop tech skills, share what they're doing and how, and fit in some rare 6-12 content discussions while they're at it.

Their requests ahead of time were about using Posterous and Examview. On a trip to another local 1:1 school, someone had advocated Posterous, raising their curiosity. Posterous' advantage over blogger and wordpress is the ability to update a blog through email. An advantage for student projects or live blogging, but not really for a teacher's blog. I found a few resources:

- One teacher's use of Posterous
- Posterous in education wiki
- The official Posterous blog

Examview is a program for generating assessments that allows you to create variant exams easily, and supports both review and test modes, as well as automatically gathering student data. The resources I found:

- A neat Examview collection of samples aligned by NY content standards (close to ours)
- Examview support forums
- Examview sample question banks. There's one for Michigan. (Not math specific, but I'm assuming at least part of it is math.)

Other possibilities I shared for investigating:

- Teacher LED IWB resources (http://www.teacherled.com/all-interactive-whiteboard-resources/)
- Two wikis with IWB resources for K-8: SMART measurement and SMART geometry

- Blogs: Teach Science and Math with some good technology information;

Dan Meyer, HS teacher with great curriculum resources, problems and ideas. Shares everything. Becoming famous quickly - partly because of his TEDx talk;

Kate Nowak - amazingly honest teacher. Great math teaching ideas.

Maria Anderson - Muskegon CC teacher and amazing tech for math leader. For example, check out her interactive mindmap on playing in math. (LOTS of resources.)

- Consider using Twitter. I know it might sound dumb, but it has connected me with almost too many great resources and ideas.

Decent list of math teacher twitterers to check out.

Angela Maier's guide to Twitter use.

- Math teacher wiki. Rapidly growing resource of lessons, games, and ideas. Check out the virtual filing cabinets if nothing else.
- My week at math/tech camp this summer

- Are you looking for computer math games? Try Waker, Entanglement or a whole suite of them at Manga High. (Last actually has some real curriculum potential. Offers tracking data, easy class enrollment and lots of offered help.)

- If we don't get a chance to watch it in session: Sugata Misra's TED talk on students self-learning with computers.

## Wednesday, October 13, 2010

### Practice Problems

I'm trying to shift towards standards based grading (SBG) this semester for the content grade in my geometry class, and so I'm paying a lot of attention to the kinds of problems I write. My tendency is to write big, open, sprawling problems that cross many standards, but I haven't yet worked out how to do that and SBG. I know that I want problems where students can show understanding without necessarily getting to a correct answer. Mathematicians are wrong a lot. What distinguishes us, I think, is that we often know whether we are wrong or right.

My students asked to have the time before the test to practice, instead of the book group I wanted to do. (110 min class and a 1 hour exam.) Very reasonable. But that meant that I had to come up with practice problems! I'll post those now, then the test when all students have taken it.

Photo by scrappy annie @ Flickr

Do you give students practice? What's the relationship between practice and the assessment problems? This was a big topic in my student teacher observation this morning also.

2-D geometric objects: number of edges; side length; angle size; parallel; perpendicular; convexity, etc.

Try your choice of the following problems. Look for problems that allow you to problem solve and share your thinking.

Which standards could you demonstrate on which problems?

1) Connect each side property to an angle property and draw a different polygon to match each pair.

Draw 3 different connections and try again!

2) Sometimes quadrilaterals are defined by their diagonals rather than by sides and angles. Determine which quadrilateral goes with which definition below, and make your argument. If no quadrilateral goes with a definition, state why.

a. Diagonals both bisect each other.

b. At least one diagonal bisects the other.

c. Diagonals are equal length.

d. Diagonals are perpendicular.

e. Diagonals do not intersect.

f. Diagonals are perpendicular bisectors.

3) On our Area on a Grid class workshop, find the areas of the shapes using formulas.

Area on a Grid

4) On graph paper, divide a square up into exactly 7 triangles with as many different triangle types as possible. Can you get all 7 types?

5) Area

a) Make an area formula for a trapezoid, or prove the one you know, using the formulas for rectangles and triangles.

b) Make an area formula for a kite.

c) Make an area formula for a chevron, using the diagonal lengths.

6) On graph paper, find squares with areas listed or argue why you can’t: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

a. Note: 5 is possible.

b. Find the edge length for each square you found using the Pythagorean Theorem. What do you notice?

7) Draw a 2 circle Venn diagram with the labels only in your mind (eg. a line of symmetry and at least one pair of parallel sides). Write in the quadrilateral types where they go, and give to a tablemate to figure out the labels.

My students asked to have the time before the test to practice, instead of the book group I wanted to do. (110 min class and a 1 hour exam.) Very reasonable. But that meant that I had to come up with practice problems! I'll post those now, then the test when all students have taken it.

Photo by scrappy annie @ Flickr

Do you give students practice? What's the relationship between practice and the assessment problems? This was a big topic in my student teacher observation this morning also.

**322 Midterm Practice***The Standards*

**A**. Analysis of characteristics and properties of2-D geometric objects: number of edges; side length; angle size; parallel; perpendicular; convexity, etc.

1. concept: definition and recognition

2. application: use to sort and characterize, build or draw

3. combination: consider multiple properties in a single object

4. familiarity with examples: triangles, quadrilaterals, polygons

**D**. Distance, area, angle1. concept: definition and key properties

2. formulas: use and derivation

3. connections among formulas

Try your choice of the following problems. Look for problems that allow you to problem solve and share your thinking.

Which standards could you demonstrate on which problems?

1) Connect each side property to an angle property and draw a different polygon to match each pair.

(at least) 3 congruent sides | no adjacent congruent angles | |

pair of perpendicular sides | (at least) 1 angle > 180 degrees | |

3 parallel sides | pair of adjacent congruent angles |

2) Sometimes quadrilaterals are defined by their diagonals rather than by sides and angles. Determine which quadrilateral goes with which definition below, and make your argument. If no quadrilateral goes with a definition, state why.

a. Diagonals both bisect each other.

b. At least one diagonal bisects the other.

c. Diagonals are equal length.

d. Diagonals are perpendicular.

e. Diagonals do not intersect.

f. Diagonals are perpendicular bisectors.

3) On our Area on a Grid class workshop, find the areas of the shapes using formulas.

4) On graph paper, divide a square up into exactly 7 triangles with as many different triangle types as possible. Can you get all 7 types?

5) Area

a) Make an area formula for a trapezoid, or prove the one you know, using the formulas for rectangles and triangles.

b) Make an area formula for a kite.

c) Make an area formula for a chevron, using the diagonal lengths.

6) On graph paper, find squares with areas listed or argue why you can’t: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

a. Note: 5 is possible.

b. Find the edge length for each square you found using the Pythagorean Theorem. What do you notice?

7) Draw a 2 circle Venn diagram with the labels only in your mind (eg. a line of symmetry and at least one pair of parallel sides). Write in the quadrilateral types where they go, and give to a tablemate to figure out the labels.

Labels:
area,
assessment,
problem,
quadrilaterals,
Triangles

## Wednesday, October 6, 2010

### Pythagorean Puzzler

Today my students are investigating the Pythagorean triangle relationships. The first question is can we tell the angle type of a triangle just by the side lengths? (A previous post shares that investigation.) Then they'll look at problems using the Pythagorean Theorem (same post - it was a long one!). Finally, they'll look at developing some reasons that the theorem is true. One visual geogebra proof is here - that's the easiest to extend to an algebraic proof. But my favorite visual is the puzzle proof, where the medium square is cut up to make the square on the hypotenuse with the smallest square. Here's the sketch to explore that:

I was going for a feltboard look - what do you think? It's also available as a standalone webpage or the original Geogebra file.

My notes on embedding geogebra in a webpage are here, based on Kate Nowak's instructions.

I was going for a feltboard look - what do you think? It's also available as a standalone webpage or the original Geogebra file.

My notes on embedding geogebra in a webpage are here, based on Kate Nowak's instructions.

## Tuesday, October 5, 2010

### Differentiation

Differentiated Instruction can get a bad rap because it has a name. So people give their spin on it, and then people can trow rocks at specific ideas and say, "see, I knew it was bad." But at the heart of it, we know our students are diverse, and have different ways they learn best as well as different ways to express understanding. One size fits all does not fit me, and if it did, it would not fit you.

Coach Ginsburg has two recent posts about this, and they're both quick worthwhile reads. I whole-heartedly agree that differentiation needs to be practical and sustainable. Making three different activities for multiple days a week is neither. Choice and optional levels of support might be feasible.

Our student teachers have been thinking about this and I thought I'd just share what they wrote. This is in response to reading an article of their choice. The bibliography is at the end. As you will be able to tell, there are some pretty sharp prospects in this bunch!

Cartoon from Cornered. Photo from herzogbr@flickr.

Less of a specific differentiated lesson, but another way to differentiate to ELL’s is to speak more slowly/clearly than one would otherwise. From there, assignments, tasks, or activities can be designed specifically to practice speaking and explaining new concepts in approprate mathematical language. This would benefit all students, but would also help the ELL’s by exposing them to more mathematically charged English. An idea that I got from the text would be to cover and explain important concepts before defining new terms/words.

We have English Language Learners in our classroom this semester, so any other ideas would be greatly appreciated, party people!

I also was reminded from reading the article, Dynamic Concrete Instruction, that even when teaching all of our students the same thing at the same time we can differeniate our lesson by remembering to include different learning styles. I think that idea is so basic, yet not so easy, and an idea that I have learned for so long it has been put on the back burner or maybe kind of second nature by now and does not get that much attention. However it is still a very important part of differentiation in the classroom.

Coach Ginsburg has two recent posts about this, and they're both quick worthwhile reads. I whole-heartedly agree that differentiation needs to be practical and sustainable. Making three different activities for multiple days a week is neither. Choice and optional levels of support might be feasible.

Our student teachers have been thinking about this and I thought I'd just share what they wrote. This is in response to reading an article of their choice. The bibliography is at the end. As you will be able to tell, there are some pretty sharp prospects in this bunch!

Cartoon from Cornered. Photo from herzogbr@flickr.

**Differentiation Ideas - Fall 2010**

**Class Discussion**:*Question*: The idea is good - individual attention and opportunities. But what about the gifted? How do you grade students that are working on different levels of work? If one makes a video and one a test? How do you grade fairly?*Discussion*: Mastery learning. Compare with the objectives. With extensions ready, the students can work on ungraded material. They want to be busy. Extra credit even, if it's necessary.*Question*: I like choice: but how do I explain the objective for each project? How do I set it up?*Discussion*: that’s what I see rubrics doing. They convey how to evaluate your work; clearly, concisely... this is what it looks like.*Question*: How could we implement this in our classroom? Everything is as a whole, everything is at once and it’s just too fast for some students. Makes me not to want to do the clicker, but they want that, too. They’ve mostly been doing clickers and notes.*Discussion*: Are there ways to provide different levels of support for during the clicker questions? Hints? Though that can annoy people who’ve already clicked in. Without individual work, it’s hard to differentiate. Work on the questions first? Don’t enable clickers until after work time. Our cooperating teacher (CT) wants this quick.

**Individual thoughts***Bre Betz*- I have already seen this differentiation a lot in my classroom. We have a complete class of lower level students who this greatly effects. We also have a few “gifted students” in our room. For these students, I think that giving them some kind of extra credit or an extension into higher level math on certain problems is a really good idea. If they are not challenged very much they will eventually loose interest. On the other hand, one of our “gifted students” actually takes this idea into his own hands. Once in a while he will try and go further with a question in the book, usually he doesn’t quite make the correct assumptions, but at least he is trying and thinking outside the box.*Autumn Langlois*- I love the idea of differentiating in the classroom. I also love the idea of giving the students choices as much as possible. My goal is to have two homework assignments per class. This would include the normal one that i would have handed out to every student and the other for the gifted. Those that just want a challenge can also chose the harder one. But the catch is once you pick up the harder one you can not go back and get the easier one. I love the idea of pulling students out to look at the material more deeply but in the school that I am in now, I do not see that as a possibility. The gifted article also hit the nail on the head when they started to talk about how schools are putting a lot of emphasis on the lower level students when the upper level students deserve just as much of a challenge as they are getting. Scary but true. The article really opened my eyes and I will be more aware of what I am saying to those that always have their hand up in the air.*Stephanie Bares*- I was always unsure on how to help the gifted students with out giving them extra problems of the same thing. These students need a challenge and I was afraid if you gave them too much that they would ask why don’t other kids have to do this but it seems that they actually look for a challenge. the students also do not want to be pointed out as the real smart student they want the challenge to be there for anyone one so they are not that one special person. I like the differentiated task that Mrs.Baker in the article “The Gifted student” did. I liked it because students that were not gifted still could work through it and the gifted students still felt there was a challenge and also were able to share with others there ideas which in turn may of help others. I would hope I could make activities that would be able to include everyone like that one.*Jacob Dunklee*- One way that I like differentiation in the classroom is to assign the same homework assignment to all students, but have extra, more challenging problems for the gifted students to work on after they are done; and for these problems give them extra credit and maybe a little treat if they complete so many of them. I also would offer more support to the struggling students by getting together in small groups to work on the homework together. I would try to at least pair them up with one of the students that in the gifted category or are able to understand the topics and explain them to others. This would allow for everyone to get something out of the instruction without killing myself by making up three different assignments for every lesson. As a teacher we can only take on so much extra work on top of everything else. But if we put in the time in the first few years of teaching it would be easier because we would have most of the assignments saved because of technology in this day and age so the extra work load would decrease eventually. But there will always be different levels that we would have to prepare for.*Kady Dingman*- Planning lessons that excite and engage all students in the classroom at all levels can be challenging for a teacher. One way that my CT tries to engage gifted students in her classroom is that she sometimes has an extension activity for the students who fly through the lesson that she puts questions from the extension on the tests as extra credit problems, so that those students have some incentive try the extension activities. I think this is a great incentive for these students to try the extension. Another way I might use differentiation in the classroom is to give the students some choice in the assignments they complete, so that it’s not necessarily more work, just more challenging tasks that the students would’ve chosen to complete themselves.*Matt Compher*- I specifically read about differentiation for English Language Learners. One thing that can be done in the classroom to support differentiation is to allow students to work with one another on a given task. Pairing ELL’s and non-ELL’s would benefit the ELL’s by giving them a second source of instruction, and the non-ELL’s would be given an opportunity to practice speaking and explaining mathematical concepts.Less of a specific differentiated lesson, but another way to differentiate to ELL’s is to speak more slowly/clearly than one would otherwise. From there, assignments, tasks, or activities can be designed specifically to practice speaking and explaining new concepts in approprate mathematical language. This would benefit all students, but would also help the ELL’s by exposing them to more mathematically charged English. An idea that I got from the text would be to cover and explain important concepts before defining new terms/words.

We have English Language Learners in our classroom this semester, so any other ideas would be greatly appreciated, party people!

*Emily Holth*- I really like the idea of differentiation in the classroom. I have learned a lot from my CT since he is teaching by the GLCE’s. [Michigan's middle school content expectations] So, if some students are having trouble with a particular GLCE, then during group work, those students can have instruction focused on that GLCE. Also, for those students who are not having any trouble with the content, my CT provides them with challenging activities. Sometimes it will be an extra credit worksheet that asks them to apply what they have learned and extend it to answer a question. Other activities that are available to these students include the question of the week. Each week my CT posts a question that is optional for all students, and those who get the correct answer receive extra credit. These questions are also challenging and require further thought than what was necessary for the lesson. I think by separating students by what they understand and what they are struggling with, we can really focus with them on that particular area. Some students may not need the reteaching for a particular topic, so they should be able to work on something else that will give them more of a challenge, rather than simply focusing on the students who are at a lower level. I think this is a great way to accommodate for all students.*Tess Wells*- I really like the idea of regrouping students according to their struggles, at least for part of a class period. At my placement, a formative assessment (sort of a mini quiz) is given at the end of every investigation. I think it would be a good idea to look them over and group them into different categories of certain struggles if they seem apparent and then come up with an activity to help that certain struggle that students can work on the next day in class in those groups. I know time is always a huge issue so maybe that could take the place of a warm up or something some days at the beginning of class.I also was reminded from reading the article, Dynamic Concrete Instruction, that even when teaching all of our students the same thing at the same time we can differeniate our lesson by remembering to include different learning styles. I think that idea is so basic, yet not so easy, and an idea that I have learned for so long it has been put on the back burner or maybe kind of second nature by now and does not get that much attention. However it is still a very important part of differentiation in the classroom.

*Gerald Smolka*- There are so many challenges that we as young teachers must face. While I have already been contemplating how to keep extra attention focused on the student’s who are struggling, I hadn’t given much thought on those students who are excelling with the material. Fortunately, I have been seeing differentiation modelled for the ‘gifted’ students in my FST class. Mrs. Robbert always makes sure to have some kind of supplemental activity reading for students who finish an assignment or project before the rest of the class. The big question with the extra activity is how to grade it. But I don’t feel its necessary to always grade materials that are given to students. I believe that students want to be challenged and enjoy activities. As a matter of fact, removing the grade from an activity might serve to motivate them to explore it without the possibility of failure. My ED 310 prof, Mr. Stockton, is using this approach with a couple books that he would like us to read. They are on the required textbook list, but he isn’t testing us on any of the material and isn’t giving us a schedule to have things read. He has left it up to us as to whether we will read it or not. Well, now since I don’t have to read it I want to read it at a pace that is comfortable to me.*Katelyn Marckini*- I read the article about differentiating Instruction for the English Language Learners. This could be particularly useful in the school that I am working in now We have a few students who I can tell speak Spanish at home. Therefore, they have broken English in the classroom at times. I think that I could help these students by giving directions slowly, repeating directions, allowing ELL students to work with non-ELL students (have all students pair up), etc. Usually when I am working one-on-one with students I will have them read the question that they are asking about out loud. This way they get a chance to verbalize math and I can make sure that they are capable of doing so. I may try to start having students pair up to work on things in the classroom and see how it goes.*Ashley Eastman*- we see a lot of differentiation in our classroom because we have many students at different levels. we have one class full of students with learning disabilities so we change our lessons for them even though they are in the same course as the other three blocks our teacher teaches. then in our normal class we have a couple of gifted students who go above and beyond what we expect. these students will usually look at the problems in a different way then most kids to challenge themselves, especially one of our students. they usually start their homework in class so they aren’t ever just sitting there they are always doing something.**Bibliography**:- “The Gifted Student”. Kathryn Chval and Jane Davis,
*Mathematics Teaching in the Middle School*, December 2008/January 2009, pp 267-274. - “Differentiating Instruction in Mathematics for the English Language Learner”. Deandrea Murrey,
*Mathematics Teaching in the Middle School*, October 2008, pp 146-153. - “Dynamic Concrete Instruction in an Inclusive Classroom”. Bradley Witzel and David Allsopp,
*Mathematics Teaching in the Middle School*, November 2007, pp 244-248. - “Building Responsibility for Learning in Students with Special Needs”, Karen Karp and Philip Howell,
*Teaching Children Mathematics*, October 2004, pp 118-126.

## Friday, October 1, 2010

### Mathematical Learning Inventory

I was first exposed to the Mathematical Learning Styles Inventory last year at Math in Action, our local math education conference. (It's the end of registration season for that - if you're close to Grand Rapids, MI give it a thought if you could come present! Fillable pdf speaker form.) The teaching center folks from Central Michigan University presented it, and had us take it and discuss. An interesting bit for me was having a group of my students there. When they had us move to tables based on our strongest style, all my students were seated at my lowest style! Hmmm - was I providing them with appropriate work and activity.

Photo by michaelcardus@flickr

I know learning styles are a controversial topic in some arenas. But I think of them as preferences and predilections. I do not see them as exclusive or predetermined or limiting. As with most assessments, they provide data about what your students prefer or think they prefer. They might indicate areas where I need to provide more or more explicit support for my students. Hopefully they indicate areas where students can become stronger with more experience and application.Strong, Silver and Perrini do a good job at laying out their inventory in the 2001 ED Leadership article, Understanding Student Differences. One of the things they describe are five great suggestions from their research:

- Have simple, deep standards
- Differentiate
- Increase the role of assessment and feedback
- Start writing curriculum that appeals to students
- Collaborate with colleague

- Mastery learners want to learn practical information and procedures (what)
- Interpersonal learners want to dialogue and collaborate (how)
- Self-expressive learners want to use their imagination to explore (how)
- Understanding learners want to learn why things work (what)

I reworked the inventory for an inservice with middle school teachers because I thought that most secondary students would see it as repetitive and onerous. Especially if they have to score their own inventory. Also, people often complained about hard choices and having to choose between equally weighted (to them) options. I thought about just giving nine points to distribute among the choices, but that would take even longer for students. (Although I like the elegance.) So ultimately I decided to break up the questions and have students indicate strong, partial or dis- agreement by giving items 2,1 or 0 points. That turns this into an informal assessment vs. a research-based inventory. Sorry! But I can share:

I gave it to my preservice K-8 teachers, and they had interesting results. The different structure seemed to lead to more balanced styles numbers, though still indicating a preference. It sounds like some of the middle school teachers are going to give it to their students, and I'll share their feedback on the assessment if they do.

Of course, if you try it with your students, I would love to hear how it goes!

I also made an easy data recording sheet to get a glimpse of your students' results quickly. (Plus bonus line plot lesson!) Out of my cold-depleted class, I had strongest traits as follows: 7 mastery, 9 Inter-personal, 0 Understanding and 10 Self-expressive. (Ties I counted in both categories.) I expected more mastery learners, as that's what most math teachers tend to be. I personally have high scores in everything but mastery, so you can see why I worry that my preferences keep me from seeing my students. But even more interesting (and relevant for planning) to me were the distributions.

The relatively high interpersonal scores fit with the engagement level during discussions, and the relatively low understanding scores fit with the lack of engagement when more formal reasoning is the topic. The split on self-expressive seems to fit my crazier assignments, where some dive in and some ... do not. I tend to struggle with what to do for mastery learners, as that doesn't fit my view of mathematics well.

I do want to reiterate what I put on the assessment: This shows the ways you are comfortable to learn math now. People can learn new ways to learn math and try ways that other people like. Everyone can learn to do math better.

In other words, remember the importance of a growth mindset.

PS> I've added a new post describing my use of this inventory and the results from a group of preservice teachers.

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