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# Posts tagged teachers

## Multiplying polynomials: Quilts from Judy Chaffee

I met Judy Chaffee at Escape from the Textbook community. She mentioned a polynomial quilt activity that was a hit with her students. This are materials for the activity, posted with Judy’s permission here since she does not have a blog yet – maybe one day!

Judy writes:

I start with showing them how regular multiplication looks (shape wise) using an array.  We discuss that when you multiply one digit by another, you get a rectangle and when you multiply one digit by itself you get a square.  We then look at how a variable looks when multiplied by itself a number.  Next they draw it on paper and then cut their scrapbook paper to match the sizes used in their drawing.

I just started quilting myself and I think this summer I will actually make a quilt using this idea.  Oh yeah, I start the lesson with talking about quilting and how some mathematicians are using quilting and math.  I’ve included my powerpoint for this part.

Here are a few of the quilts students made:

This is the summary Judy made for the students:

## Guest post: Why math really does matter to your child – and you

Edward Khoo is a teacher who enjoys teaching math at his local high school and spends most of his time apart from teaching with his cute 5 year old kid. Edward writes about technology news and updates on his blog. He asked to post this essay to the Natural Math blogs to reach its readers. I really like the multiple solutions example. What do you think?

Math is a subject that many struggle with at school. In fact, I’m sure a big proportion of us would admit to saying “I hate math!” at some point in a frustrated homework session. But there are basic skills that you can draw from math that do matter. And they’re not necessarily the same as those that teachers may give to their struggling students. Whatever they say, being less than fully familiar with Pythagorean theory is not going to condemn you to stacking shelves at the mall. And having trouble with multiplication tables isn’t going to prevent you from shopping at those very same malls.

Neither does having a shaky grasp on the intricacies of computer logic stop you from surfing the internet. When teachers wield such threats over their charges, they’re just using them as blunt tools for bludgeoning students – to get them moving on with the nitty-gritty in their math class. But put those false concerns aside. Then dig a little deeper and you’ll see that math can give your child, and you, the means to ease your way through some real-life problems – both expected and surprising.

The most obvious things that math does for your child is to make them familiar with numbers. But what good math teaching should also do is to go that step further, and make your kid confident with numbers. That doesn’t mean turning them into miniature calculators, able to do long division in their head. It means helping kids to understand that numbers are not scary.

It is the fear of numbers, often created by bad teaching experiences at an early age, which stick through to adulthood – and can make many everyday situations awkward and difficult. If you’re scared of numbers, you might not want to tip after a meal, in case you get the amount wrong. You might end up on the wrong side of a bank loan, because you never did understand how to use percentages. So losing that fear is important, to navigate through life with confidence.

An example of that ‘number confidence’ is dealing with, say, a sales tax at 20% – so that you know how to work out the dollar and cent amount, without freezing. A big problem with math as it’s often taught is that most math problems are presented as only having one solution. If you don’t get that solution, after some gentle hammering to get it into your brain, the bad teachers move on. The good teachers won’t bow out that easily – they’ll try and place a number of tools into your hand, and spend time funding the one that fits you comfortably.

So 20% of \$125 isn’t some complex equation – maybe it’s shifting the decimal place along twice, to give \$1.25, and then times-sing by 20. Or maybe it’s looking at a fifth of \$125; or how many times does 5 go into \$125; or twice a tenth of \$125. Eventually one of these will click with a pupil– and with a little practice, they’ll have cracked that fear of percentages.

That approach, of trying a method out, failing, and then changing tack until it slips into place – is a really valuable life skill. It’s something a child can take with them, and apply to lots of problems in their life, both as a child and later as an adult: don’t give up, be patient, and find what works for you. So math also teaches kids to problem-solve: learning to step back from an issue, break it down into its component pieces, and then trying out solutions.

Then there’s another side to math which goes beyond those basic questions of numeracy, and onto the issue of understanding the numbers pumped out by the media and politicians. This is a huge life skill, one that underpins our ability to take part in the debates about where our world is going. And as has often been pointed out, in relation to such numbers, there are lies, damned lies and statistics. It is the skill of knowing what politicians and the media are trying to say, through the numbers they quote, that makes you a properly engaged citizen.

So when you are told that the risks of skin cancer, from overuse of tanning salons, has gone up 50%, a strong math skill would tell you what this means. You’d be asking – “well if it’s gone up by half, what was it before, and what is it now?” Compare that to, “holy, molely, half of sun-bed users are going to die of cancer!” That is why math matters to you and your child – without it, you can make bad decisions that can seriously affect your life.

## Escape from the Textbook! March 23rd at 9:30pm

Join Henri Picciotto, involved in mathematics education since 1971, in building a safe haven for math teachers who Escape from the Textbook!

## How to attend the event

• Wednesday, March 23rd 2011 we meet in the LearnCentral online room at 6:30pm Pacific, 9:30pm Eastern time. WorldClock for your time zone.
• Click “OK” and “Accept” several times as your browser installs the software. When you see Elluminate Session Log-In, enter your name and click the “Login” button
• If this is your first time, come a few minutes earlier to check out the technology. The room opens half an hour before the event.

Math 2.0 weekly series: http://mathfuture.wikispaces.com/events

## About the “Escape from the Textbook!” community

“Escape from the Textbook” is a sharing and collaboration network for middle and high school math teachers who want to escape from the textbook for a lesson, a unit, or an entire course. Hopefully some of the 400+ members will attend this event!

While our schools are very different from each other (large and small, middle and high school, public and private), the challenges facing us are similar. The Escape from the Textbook! network can help us take up those challenges through:

• networking with like-minded teachers
• sharing of successful approaches
• multischool collaboration groups that focus on specific courses or topics
• strategies on how to complement or replace textbook material
• assessment ideas
• different lenses to analyze curricular and pedagogical ideas

The first Escape from the Textbook! conference was held on February 12th, 2011 at the Urban School of San Francisco. You can watch the conference video recordings. The speakers were:
- Jo Boaler (author of What’s Math Got to Do with It?) on pedagogy
- Paul Zeitz (author of The Art and Craft of Problem Solving) on problem-solving

## Event Host

 Henri Picciotto writes: Please visit my Math Education Page, where I share much curriculum and philosophy, particularly about tool-based learning, and my Math Education Blog, where alas I post rather irregularly. I have been involved in mathematics education since 1971, at every level from counting to calculus. I have been a teacher, a math specialist, and a department chair. I have written many books and articles, and edited the “Activities” department in The Mathematics Teacher. I have created software and video lessons, and been a presence on the Web. I am an authority on the use of manipulatives in secondary math education. I have spoken at conferences, been a consultant to schools (hundreds of workshops and presentations), and served on advisory boards for various projects. I am also involved in word puzzles, particularly cryptic crosswords. For more information about me, check out my résumé and my personal home page.

## Dig and fill: The shadow scholar

One of the  most-discussed articles this month in Chronicles of Higher Education, The Shadow Scholar introduces “Ed Dante” who writes student assignments for hire. Here’s how he describes his daily work…

“In the past year, I’ve written roughly 5,000 pages of scholarly literature, most on very tight deadlines. But you won’t find my name on a single paper.

I’ve written toward a master’s degree in cognitive psychology, a Ph.D. in sociology, and a handful of postgraduate credits in international diplomacy. I’ve worked on bachelor’s degrees in hospitality, business administration, and accounting. I’ve written for courses in history, cinema, labor relations, pharmacology, theology, sports management, maritime security, airline services, sustainability, municipal budgeting, marketing, philosophy, ethics, Eastern religion, postmodern architecture, anthropology, literature, and public administration. I’ve attended three dozen online universities. I’ve completed 12 graduate theses of 50 pages or more. All for someone else.

You’ve never heard of me, but there’s a good chance that you’ve read some of my work. I’m a hired gun, a doctor of everything, an academic mercenary. My customers are your students. I promise you that. Somebody in your classroom uses a service that you can’t detect, that you can’t defend against, that you may not even know exists.”

One of the most heart-wrenching parts of this story: what took Dante over the moral event horizon was his college experience. He wanted to do some real work that OTHER PEOPLE WOULD FIND USEFUL.

“I was determined to write for a living, and, moreover, to spend these extremely expensive years learning how to do so. When I completed my first novel, in the summer between sophomore and junior years, I contacted the English department about creating an independent study around editing and publishing it. I was received like a mental patient. I was told, “There’s nothing like that here.” I was told that I could go back to my classes, sit in my lectures, and fill out Scantron tests until I graduated.

I didn’t much care for my classes, though. I slept late and spent the afternoons working on my own material. Then a funny thing happened. Here I was, begging anybody in authority to take my work seriously. But my classmates did. They saw my abilities and my abundance of free time. They saw a value that the university did not.”

Some prisons and armies use a form of punishment:  first, you dig a trench. Once you are done, you fill it back in. Doing work that nobody will ever appreciate reminds me of this psychological torture.

Different people require different use/practice ratio to find their practice (learning tasks) reasonably meaningful and motivating. Everybody understands that doing work that is useful for others requires some amount of practice tasks. However, many college courses and whole program set this ratio to zero.  The prospect of current tasks being useful years into the future is way too distant to motivate most humans.

Have more tasks that have immediate use for some currently living people and communities. Students can write for collaborative open resource projects, review papers for conferences, program and distribute needful software, and otherwise pitch in where work is needed. This way, even if they hire someone to do their work (which will be less likely this way), at least the work will be useful to the society!

A few examples of immediately useful learning tasks that proved successful in my teaching

• Write a Wikipedia article
• Compose music for a clip based on an essay
• Illustrate a book
• Comment on an active popular blog or forum
• Answer questions at a topic help forum

Another big huge motivator is play. In a twisted perverted way, grades provide a game mechanics that supply a sort of motivation. But this is another story.

## Wish list: A student-invented notation wiki

Paul Libbrecht has created a very inspiring wiki collecting math notations from different countries. We are currently discussing the wiki at the Math Future email group. Here is what I dreamed up, inspired by it…

There are quite a few existing collections, in books and sites, unfortunately NOT aggregated in one place yet, of lesson plans devoted to children inventing notation. John van de Walle had written about it, for example. “Living Math” community, led by Julie Brennan, has a lot of discussions about this. I do this activity routinely, with all my students in most of the topics. When this is going on, students LOVE to look at multiple historical or modern notations, which we usually do after they’ve invented their own. This way, they see themselves as a part of the long continuum of math creators.

For the purposes of such an activity, my wish list is:
- a place like your wiki available (check!)
- a place just like that, but for student- and teacher-invented notations
- a cross-linked depository of lessons/activities using the above census items (somewhat like Joel’s http://geogebramath.org/lms/nav/index.jsp); that is, aggregation of links to activities where each notation is used on notations’ pages, and links back to the notation census from activities’ pages
- a way of commenting back and forth with people who are contributing activities (like blog comments)

The next step toward my wish list is to start a sister wiki for student notation.

## 3 Large Math Applet Communities

This is a guest post for Technology Integration in Education. I review three large active online communities centered on sharing applets, widgets, interactive models, and other pieces of “executable mathematics.” What other applet-making communities are currently alive, active and promising? Please add to the list!

1. Scratch from MIT, the most popular descendant of Logo, currently has about 130 million applets. There are two very good reasons for this popularity. First, the programming environment itself is visual, intuitive, and so simple I used it with kids under three and they got it. It looks and feels like building with Legos.

Second, Scratch has excellent tools satisfying all principles of community building. Applets are shareable with one click of the “share” button, as you make them. Each applet gets its own linkable and taggable page with comments, information about the author, ratings, tags and other community feature.  Applets are remixable, with automatic tracking of previous authors. I use Scratch to show kids what is open source software, as you can open any applet’s code in your editor with a click of a button on its web page. Also, applets are easily embeddable. Here is one of my students’ favorites:

Because Scratch community does not have any top-down taxonomies, categories, or quality controls, it may be hard to find applets that satisfy a particular set of topic, level and quality requirements – what teachers need for lesson planning. On the other hand, kids can usually find what they need, such as games with particular game mechanics.
There is also an educator community ScratchEd, and International Scratch Day celebrated in May by hundreds of local communities, and online.

2. GeoGebra is a powerful platform for making sketches and animations, solving problems, and supporting rapid development of math interactives. Its particular strength is the dynamic connection between a computer algebra system and geometric constructions. Drawing with Euclidian tools is automatically or easily linked with graphs, functions, formulas and tables of values. A lovable feature is the ability to remember any multi-step geometric construction as a custom tool. For example, once you build a star using Euclid’s construction axioms, your next star can be created with a click of a button.

GeoGebra community has a strong support for local groups through GeoGebra institutes and regional conferences, and has applet-sharing wikis in 27 languages. Applets are shareable, embeddable, and remixable, but community tools, such as an active forum, are not applet-specific. This promotes creation of many different communities for aggregation and discussion of GeoGebra content in groups, such as the wiki where I first met the above animation.

3. Wolfram Demonstrations requires software with a somewhat steep learning curve. This means every Demonstration is made by a math geek, with the obvious implications for quality. Mathematica software is not free (the viewer is), but volunteers for Wolfram used to get a free copy. Volunteer tasks are accessible to novices.Demonstrations are linkable, and their code is available for remixing. There are no other social features, such as discussions of applets, tags or ratings. The site uses top-down categories for browsing applets. The sense of community is sustained largely through academic connections of authors, and isn’t apparent from the site.

## Lectures rock

I love lectures. Of certain kinds, of course. For people who don’t like lectures of any sort, like myself a couple of years ago, I have two words:
Khan
Storytelling

Storytelling should be named “fabletelling” though. What Khan does are stories, on the other hand.

Definition 1. A lecture is a non-interactive (broadcast) delivery of content that includes the voice of one person.

Definition 2. A good lecture is a lecture with the following characteristics:

• The lecture is under 7 minutes. Longer good entities have non-lecture breaks or invite the pause button, making them a mini-series of lectures.
• The lecture delivers one unit of content (story) or of meaning (fable). What is a “unit” is determined by the field and the level.
• The pace is matched to cognitive patterns specific to its content. This requires the lecturer to have strong knowledge of content and of other learners and practitioners, that is, pedagogical content knowledge.
• The voice and the other media, if any, are in rhythm and content harmonies. Rules such as, “No more than three words per slide” attempt to enforce this principle rather feebly.
• The lecture has depth and breadth, because the lecturer loves the content and is intimate with it in several ways.

There is a need to develop more good lecture types though, for different learners. Hence “not complex enough” tag here.

Tanya’s case studies in “Math Careers and Choices” are more fables than stories.

## Simple, illegal in 50 states?

There is now a blog carnival about assessment. What an interesting idea!

Which reminds me to write up how I assess, if needed. It is connected with how I design tasks. I know it is unlike the way it’s usually done, and contradicts regulations of most institutions, and is hard to implement with reluctant learners/young kids/space aliens, and several more “yes-buts.”

Everything is public. If a student can’t share the work with the world when it’s done, it’s not worth the time. Also, students who aren’t sure how to start get to see examples from the early birds.

I want high production/consumption ratio for each assignment. That means the outcomes of tasks can’t be too boring, repetitious, or otherwise unneeded by the world, because who would want to produce something like that?! This rule excludes exercises from being assignments in their own right, though exercises may be required for an assignment’s success. “This is not a drill.”

The scope of assignments is infinitely scalable up and down, but with a cost in effort, both ways. Scaling tasks down (aiming for efficiency) is not easy, and produces valuable learning. Students can add their own contexts to tasks, and are invited to increase personal meaning and significance of tasks. In short, tasks are open.

Earlier tasks come up again in later tasks. The idea is to help someone finally understand earlier concepts through later ones, not to punish people for missing something. This takes some redesign of curricula, especially in math, to provide grounding rather than “toppling towers.” Heads up: it’s a lot of work.

Here lies a toppled god
His fall was not a small one
We did but built his pedestal
A narrow and tall one
“Dune”

Tasks have built-in,  intrinsic quality requirements. Something both obvious and important depends on Arete.

Task feedback is provided by the world (since tasks are public and live) and by class peers, and is a part of assignments. Students submit produced content into appropriate public channels where constructive feedback is likely. Teacher feedback is on-demand, as needed for assignment success and requested by students.

If assignments are set up as described above, which is complex, assessment matching it can be extremely simple. Just tally finished tasks. All the sophistication usually reserved for assessment is instead built into the task design.

## Problems and their camp followers

What is a problem for one person can be a puzzle or an exercise for another. A smart teacher can turn an exercise into a problem by being “less helpful.” Many curricula and educators also work hard on turning all problems and puzzles into exercises by attaching a step-by-step guide to every one, and by formalizing math before solving as such begins. Lockhart comments on the results:  “I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.”

Video: Dan Meyer on being less helpful.

• Problems are mathematical questions for which the solver has no readily available methods of solutions, but has ways and the intrinsic intellectual need to figure them out. In particular, solvers do not know all mathematical concepts or formulas they will use.
• Open-ended problems have multiple correct solutions, often infinitely many of them. The correctness of open-ended problem answers is determined by the mathematical qualities of the solution, such as rigor, logic, definitions, and aesthetics.
• Exercises are mathematical questions for which the solver knows what methods, concepts and formulas will lead to the solution.
• Closed-ended problems and exercises have answers known exactly ahead of time. For example, multiple-choice questions are always closed-ended.
• Puzzles in mathematics are similar to problems in that the solver does not know which methods, concepts and formulas to use. The difference is that problem solving involves eventually developing a mathematical method for solution, whereas puzzles require intuition, finding a trick, or guessing, and may not involve any methods at all. There is no clear line between puzzles and problems, and most collections have a mix of both.

Relationships between problems and puzzles are complicated. Consider the nine-dot puzzle, a classic. Join all nine dots, using four straight lines, without lifting your pencil from the paper.

The difficulty of this puzzle is psychological: most people assume that lines have to start and end inside the assumed square. Once solvers start to experiment with lines that go beyond the square, the solution usually presents itself soon. There is no general math method involved. Unlike another classic, “the wolf, the goat and the cabbage,” or the nine-coin puzzle that open the door to the whole class of interesting math problems and investigations, the nine-dot puzzle stands by itself.

The nine-dot puzzle helps to make a powerful problem-solving point about assumptions. Puzzles may have a powerful role in mathematical problem-solving, similar to “the Mozart effect.” Puzzles support mathematical values and develop the mathematical sophistication, on a meta-level. I do not consider puzzle-solving the same activity as problem-solving. Evidence: quite a few people like one and hate the other, or are good at one and not the other.

What to make of pastimes like Sudoku and KenKen? Their combinatorial complexity quickly overwhelms human processing capacities, such as memory and attention. Human solvers are forced to develop and use intuition-based strategies, which is a puzzle trait. Yet there are also consistent, formulated strategies and rules solvers develop and use, which is a problem-solving trait. Finally, individual steps are routine exercises. The balance between puzzle, problem-solving and exercise traits in Sudoku, in particular, hits the sweet spot for millions of people. Chess and several other abstract games are similar in their balance.

I have a gut feeling, not yet supported by readings or experiments, that there is some sort of 3d flow channel among problems, puzzles and exercises. Each person has particular needs for the balance between the three to develop mathematically. The balance shifts with time. I have no idea how to visualize 3d flow channels.