One Christmas long ago, I was maybe 9 or 10, my grandparents gave me a subscription to Games Magazine. While I cannot give this magazine full credit for turning me on to math (so many contributions there), it did go above and beyond in its ability to feed my brain and satiate my thirst for challenging and entertaining puzzles. It still does. What I especially love about this magazine is that there are plenty of puzzles that I cannot solve... and still plenty that I can.

When I met my first 'Paint By Number' logic puzzle, I remember the wonderful satisfaction of incremental achievements. At first, it seemed so impossible, but bit by bit, the little squares got filled in. Each success led to another inspiration and slowly a picture emerged.

And I remember the first time I introduced these puzzles to my students - 14 year old minority algebra students stamped as 'underachievers'. Sure, there were some that took one look and walked away, but I was surprised by the ones that gave it a try, and amazed at their tenacity in pursuit of success. I can still picture Nestor at a desk in my empty classroom for an hour after school!

So, my gift to you is this snowflake puzzle... for yourself and your students. If you click on the image, you will be redirected to a site where you can download it for free. I created the puzzle using Griddlers Deluxe software that I downloaded online. You can find more puzzles there and also make your own. Enjoy!

Happy Holidays!

## Monday, December 12, 2011

## Wednesday, November 30, 2011

### Making Assumptions

There's the old adage about making assumptions: If you

At this point you are hopefully feeling a bit of unease, if not downright criticism, towards my list of presuppositions... especially the last one, which I will reveal as being the original impetus for my train of thought. I have been rethinking the quadrilaterals unit (in geometry) lately. Not that it needs an overhaul or anything. In fact, it's a pretty solid unit, but my new inner voice needs to find a higher rationalization for everything.

Searching for "essential questions in geometry about quadrilaterals" brought results that seemed petty and narrow: "How do we identify various quadrilaterals?" "What types of quadrilaterals exist and what properties are unique to them?" etc. At the risk of offending, or sounding trite, I don't really care about the answers to those questions. But here's a couple questions I DO care about:

What I like about THESE questions is that they are not necessarily about geometry, or even math for that matter, but they DO get at the heart of something I am trying to convey over the entire geometry curriculum:

How much information do I need to know before I can start making assumptions?

I cannot assume my 'blue-eyed person' carries the OCA2 gene unless I know for certain that he is indeed a true blue-eyed person (and not wearing contacts, or transplants, or cosmetic pigmentation)... and for that I might need genetic proof. When it comes to quadrilaterals (or any type of defined shape), I need to know and fulfill a precise definition. I cannot assume that my 'square' has four right angles unless I know for certain that it is indeed a square... and for that, I need to meet the definition of a quadrilateral with four congruent sides and four right angles.

Are there 'good' assumptions and 'bad' assumptions, and what is the criteria for making that judgement?

I would say a 'good' assumption is a valid or proven one. 'Bad' assumptions might be invalid or simply unproven (i.e. unsure). If I am making an assumption about the diagonals of a quadrilateral bisecting one another, I'll need to be able to demonstrate sufficient evidence that my quadrilateral is a parallelogram... or run the risk of making a bad assumption.

What are the consequences of making invalid assumptions?

Ahh, the rub. Indeed, sometimes the consequences of invalid assumptions are minor. If I assume that a foul smelling item is not safe to eat, I might miss the opportunity to taste a delicacy, but in the absence of further evidence, it just may be the safest assumption to make. In math class, the consequences of making bad assumptions are usually loss of credit/points on an assessment. Like my choices about the foul smelling item, students quickly learn what is 'safe' and fail to find the motivation for further validation.

But geometry is the study of space and measurement and the world around us. It's important that our students fully understand that the true consequences of invalid assumptions can be catastrophic, expensive, deadly, or any combination of these. It is essential that they understand that even the most generic classroom exercises have implications for the world around them. I mean, I'm not really suggesting that their attention to precision on my quadrilateral homework will save the lives of hundreds of people. Or, maybe I am.

*assume,*it makes an "ASS of U and ME." I might argue though, that I make assumptions all the time, and that assumptions are vital to the efficiency of my day:- I see an 'open' sign in a window, and I assume I am welcome to enter.
- I hear a siren, so I assume that there is an emergency situation.
- I smell something foul, so I assume that it is not safe to eat.
- I see a person with blue eyes, so I assume they carry the OCA2 gene.
- I see a square, so I assume that it has four right angles.

At this point you are hopefully feeling a bit of unease, if not downright criticism, towards my list of presuppositions... especially the last one, which I will reveal as being the original impetus for my train of thought. I have been rethinking the quadrilaterals unit (in geometry) lately. Not that it needs an overhaul or anything. In fact, it's a pretty solid unit, but my new inner voice needs to find a higher rationalization for everything.

Searching for "essential questions in geometry about quadrilaterals" brought results that seemed petty and narrow: "How do we identify various quadrilaterals?" "What types of quadrilaterals exist and what properties are unique to them?" etc. At the risk of offending, or sounding trite, I don't really care about the answers to those questions. But here's a couple questions I DO care about:

- How much information do I need to know before I can start making assumptions?
- Are there 'good' assumptions and 'bad' assumptions, and what is the criteria for making that judgement?
- What are the consequences of making invalid assumptions?

What I like about THESE questions is that they are not necessarily about geometry, or even math for that matter, but they DO get at the heart of something I am trying to convey over the entire geometry curriculum:

**truth is the result of a string of proven facts (or unanimously approved axioms)**. In the context of the quadrilaterals unit, this idea gets a chance to settle in a little more deeply.How much information do I need to know before I can start making assumptions?

I cannot assume my 'blue-eyed person' carries the OCA2 gene unless I know for certain that he is indeed a true blue-eyed person (and not wearing contacts, or transplants, or cosmetic pigmentation)... and for that I might need genetic proof. When it comes to quadrilaterals (or any type of defined shape), I need to know and fulfill a precise definition. I cannot assume that my 'square' has four right angles unless I know for certain that it is indeed a square... and for that, I need to meet the definition of a quadrilateral with four congruent sides and four right angles.

Are there 'good' assumptions and 'bad' assumptions, and what is the criteria for making that judgement?

I would say a 'good' assumption is a valid or proven one. 'Bad' assumptions might be invalid or simply unproven (i.e. unsure). If I am making an assumption about the diagonals of a quadrilateral bisecting one another, I'll need to be able to demonstrate sufficient evidence that my quadrilateral is a parallelogram... or run the risk of making a bad assumption.

What are the consequences of making invalid assumptions?

Ahh, the rub. Indeed, sometimes the consequences of invalid assumptions are minor. If I assume that a foul smelling item is not safe to eat, I might miss the opportunity to taste a delicacy, but in the absence of further evidence, it just may be the safest assumption to make. In math class, the consequences of making bad assumptions are usually loss of credit/points on an assessment. Like my choices about the foul smelling item, students quickly learn what is 'safe' and fail to find the motivation for further validation.

But geometry is the study of space and measurement and the world around us. It's important that our students fully understand that the true consequences of invalid assumptions can be catastrophic, expensive, deadly, or any combination of these. It is essential that they understand that even the most generic classroom exercises have implications for the world around them. I mean, I'm not really suggesting that their attention to precision on my quadrilateral homework will save the lives of hundreds of people. Or, maybe I am.

Labels:
assumptions,
essential questions,
proof,
quadrilaterals

## Sunday, November 27, 2011

### Cyber Monday Sale

Teachers Pay Teachers is offering a sitewide discount of 10% off all prices (that's all reagular priced AND sale priced items) on Monday, November 28.

In my store, all items are already 20% off through Tuesday, so with the combination of the deals in the store and the the additional discount (don't forget to use promo code CMS28), that's a combined discount of 28%: the largest discount you'll ever see at TpT!

Swing on by the store on Monday! I have a couple of new items, including a fun activity for geometry students in the quadrilaterals unit.

-Emily

## Saturday, November 12, 2011

### Show your work!

One of my colleagues recently said to me "You know, nobody does proofs anymore." Seriously? I think that would be a catastrophic move in the wrong direction, and it reminds me of a nagging problem that I have avoided.

Since the beginning of my teaching career, there are a handful in every class: those students who, despite numerous pleas to "show work," refuse to write anything but the answer on the page. I give what I think is a thoughtful (and lengthy) practice set as homework, only to find that student who has successfully reduced it to a smattering of digits that fills up only a quarter of a notebook page!

So then comes the blow: the thing I've been forcing myself into these past months. I ask myself "WHY?" And I don't mean "Why do these kinds of students frustrate me?" I

That particular question has evaded me many times, probably because of a lingering fear that the answer might be "It isn't." Thankfully, I don't think I'll need to go there.

At the heart of the issue is the realization that there is a quintessential difference between an

Student A has successfully shown that 4 is AN answer to this problem (I took the liberty of writing what I believe is going on in Student A's head, because usually this version of the solution is simply a scrawled 4 on the page) Student B, on the other hand has successfully shown (and verified) that 4 is THE answer to this problem. The difference may seem subtle and unnecessary for a typical beginning algebra student, but it quickly becomes

The result of treating 'work' as 'proof' could very well be an improvement in reasoning and logic in students across the board. Certainly it would make a difference to my geometry students when they are first confronted with conditional statements, laws of reasoning, and proofs. It seems that more and more, students (and people in general) are forgiven for their inability to write a logical and convincing argument.

So, is my colleague right? ARE we moving away from proof? Are YOU? Seriously, I want to know.

Since the beginning of my teaching career, there are a handful in every class: those students who, despite numerous pleas to "show work," refuse to write anything but the answer on the page. I give what I think is a thoughtful (and lengthy) practice set as homework, only to find that student who has successfully reduced it to a smattering of digits that fills up only a quarter of a notebook page!

"You need to show your work," I'll say.At this point, blank stares or shrugs are about all I ever get, but depending upon my mood, I might launch into one of the following attacks:

"I did it in my head," he replies.

"You didwhatin your head?" I'm not letting him off, I don't care if he's a genius or not. "You need to write down what you did."

- "You need to know what you've done, so you can correct yourself if mistakes are made." (
)**You're not as clever as you think.**

- "A good mathematician must show diligence in precision, and exercise good habits of mind." (
)**You'll never amount to anything.**

- "It may seem easy now, but it won't always be and showing steps now will make the steps more natural in the future." (
)**Pleeeease... I need to validate my existence.**

- "You only get partial (or no) credit. You need to show your work to get full credit." (
)**You will do it because I SAID so!**

So then comes the blow: the thing I've been forcing myself into these past months. I ask myself "WHY?" And I don't mean "Why do these kinds of students frustrate me?" I

**know**the answer to that question. The real question is**"Why is it important to show your work?"**That particular question has evaded me many times, probably because of a lingering fear that the answer might be "It isn't." Thankfully, I don't think I'll need to go there.

At the heart of the issue is the realization that there is a quintessential difference between an

**answer**and a**solution**to a problem. A solution illustrates, generalizes, communicates, and verifies the results. An answer is just a number. In mathematics it is important to find the correct answer, sure, but more importantly I want my students to gain skills in writing accurate and convincing solutions. It's not enough to just know the answer, you need to be able to prove it beyond a doubt to yourself and others. Take this example:Student A has successfully shown that 4 is AN answer to this problem (I took the liberty of writing what I believe is going on in Student A's head, because usually this version of the solution is simply a scrawled 4 on the page) Student B, on the other hand has successfully shown (and verified) that 4 is THE answer to this problem. The difference may seem subtle and unnecessary for a typical beginning algebra student, but it quickly becomes

*essential*, like in the case where more than one solution exists or when extraneous solutions occur.The result of treating 'work' as 'proof' could very well be an improvement in reasoning and logic in students across the board. Certainly it would make a difference to my geometry students when they are first confronted with conditional statements, laws of reasoning, and proofs. It seems that more and more, students (and people in general) are forgiven for their inability to write a logical and convincing argument.

So, is my colleague right? ARE we moving away from proof? Are YOU? Seriously, I want to know.

## Thursday, November 3, 2011

### Designing Algebra

I am conflicted.

I am a faithful follower of the school of thought that places a high value on design: aesthetic appeal, artistic creativity, and user friendliness. I come from a family of artists, I am well-educated in art and design, and I will admit that I spend as much energy on the design of my classroom materials as I do on their content. This balance of focus comes from a deep rooted belief that a good design enhances the content of the material. AND I feel validated in my belief system. Many wonderful and successful math teachers will back me up: Dan Meyer and Edward Burger are two of my biggest idols.

Unfortunately, the conflict in me comes from my own experience. If I pause to reflect and remember the math teachers in my own life that were inspiring, engaging, and effective... these are definitely NOT the ones with the flashy materials. I recall handwritten worksheets and problems scrawled across blackboards and overhead projectors. I remember sloppy handwriting and chaotic classroom designs. I remember poorly photocopied graph paper and random threads of impromptu discussions. But I also remember the intrigue, focus, and fascination with all things mathematical. I remember struggle, laughter, embarrassment, and pride.

I'm willing to accept that perhaps my innate abilities and predisposed comfort with mathematics put me in a class of students that differed from the majority, but I cannot shake the idea that perhaps there is something bigger at the core of good math instruction. And dare I suggest that perhaps design and structure can actually detract from the spontaneity and chaos that is seated in the heart of truly excellent math instruction?

I, for one, am not ready to abandon my presentation values, but I'm feeling a tug, and I'm starting to more fully appreciate the wisdom of the vast base of experience out there. As I examine my own curricula, I find myself paying closer attention to content. I also am now beginning to look for open spaces to exploit... making sure there is room for spontaneous exploration, encouraging tangents, and hoping that for every carefully designed solution, new unanswered questions will emerge. This is a challenging, but thrilling task for me.

I'll leave with these thoughts, knowing full well that my ideas are evolving. I hope you will add your wisdom:

I am a faithful follower of the school of thought that places a high value on design: aesthetic appeal, artistic creativity, and user friendliness. I come from a family of artists, I am well-educated in art and design, and I will admit that I spend as much energy on the design of my classroom materials as I do on their content. This balance of focus comes from a deep rooted belief that a good design enhances the content of the material. AND I feel validated in my belief system. Many wonderful and successful math teachers will back me up: Dan Meyer and Edward Burger are two of my biggest idols.

Unfortunately, the conflict in me comes from my own experience. If I pause to reflect and remember the math teachers in my own life that were inspiring, engaging, and effective... these are definitely NOT the ones with the flashy materials. I recall handwritten worksheets and problems scrawled across blackboards and overhead projectors. I remember sloppy handwriting and chaotic classroom designs. I remember poorly photocopied graph paper and random threads of impromptu discussions. But I also remember the intrigue, focus, and fascination with all things mathematical. I remember struggle, laughter, embarrassment, and pride.

I'm willing to accept that perhaps my innate abilities and predisposed comfort with mathematics put me in a class of students that differed from the majority, but I cannot shake the idea that perhaps there is something bigger at the core of good math instruction. And dare I suggest that perhaps design and structure can actually detract from the spontaneity and chaos that is seated in the heart of truly excellent math instruction?

I, for one, am not ready to abandon my presentation values, but I'm feeling a tug, and I'm starting to more fully appreciate the wisdom of the vast base of experience out there. As I examine my own curricula, I find myself paying closer attention to content. I also am now beginning to look for open spaces to exploit... making sure there is room for spontaneous exploration, encouraging tangents, and hoping that for every carefully designed solution, new unanswered questions will emerge. This is a challenging, but thrilling task for me.

I'll leave with these thoughts, knowing full well that my ideas are evolving. I hope you will add your wisdom:

- Good design can enhance good content, but does not add value to poor content.
- Quality design is especially useful to enhance approach-ability, which for many students is the biggest blockade in their math education.
- Presentation skills are important life skills - necessary for good teachers and students.
- Mathematics is chaotic. Removing chaos and spontaneity from the mathematics classroom is detrimental to the educational atmosphere. Adding the right kind of chaos is an under-appreciated art.

## Wednesday, November 2, 2011

### Linking Up

Check out the Lesson Cloud for links to lots of fantastic math and science resources.

## Thursday, October 13, 2011

### Why Wonder Why?

It is my prime objective this year to take a critical look at my teaching practices. It is this self-reflection that has always taken a back seat to what one colleague has referred to as my 'production values.' It is so easy to get wrapped up in producing something for the next class, the next day, the next unit... looking backwards always gets the shaft. No more!

So here's my idea. I pull out something that I have: something good, established, thoughtful. Maybe it's not even my own material, but I like it for its value in my classroom. And then I ask myself, "WHY?"

"Why is this piece of content important? Why does this lesson design seem good to me? Why should my students care about this?" etc.

So far the thought experiments have been eye-opening.

I looked at a unit on writing linear equations in slope intercept form. "Of course writing linear equations is important," I thought to myself. "If they can't write linear equations, then they won't be able to write quadratic, or exponential, or rational, or radical equations. It's the foundation of equation writing!"

Whoa. Who cares? Honestly, I hate to admit it, but I don't even care. Outside of math class, how often do even I find occasion to write and solve an equation? Sure: I'm pretty sure some engineers, or physicists or scientists of one sort or another write equations every day... but what about everyone else? After all, algebra is required for everyone! And so, it turns out, this is an excellent question for me: "Why ARE linear equations important?"

My thoughts eventually turned to recognizing and describing patterns. I notice something seems to be happening in the same way over and over again, so I predict it will happen that way again next time. But when I am able to generalize the pattern, I can not only predict what's next, but what will happen at ANY point in the future and that's powerful. And then my brain takes me a step further to the importance of communication. If I can take my generalized pattern and explain it to you, then we can both predict the future. And if the computations are tough, I can even communicate my pattern to a machine with no brain... as long as I speak in a language that it is programmed to recognize.

And suddenly my unit on writing linear equations in slope intercept form becomes a unit on predicting the future and building robots. And THAT'S something that interests me.

I've had other game-changing moments lately too:

So here's my idea. I pull out something that I have: something good, established, thoughtful. Maybe it's not even my own material, but I like it for its value in my classroom. And then I ask myself, "WHY?"

"Why is this piece of content important? Why does this lesson design seem good to me? Why should my students care about this?" etc.

So far the thought experiments have been eye-opening.

I looked at a unit on writing linear equations in slope intercept form. "Of course writing linear equations is important," I thought to myself. "If they can't write linear equations, then they won't be able to write quadratic, or exponential, or rational, or radical equations. It's the foundation of equation writing!"

Whoa. Who cares? Honestly, I hate to admit it, but I don't even care. Outside of math class, how often do even I find occasion to write and solve an equation? Sure: I'm pretty sure some engineers, or physicists or scientists of one sort or another write equations every day... but what about everyone else? After all, algebra is required for everyone! And so, it turns out, this is an excellent question for me: "Why ARE linear equations important?"

My thoughts eventually turned to recognizing and describing patterns. I notice something seems to be happening in the same way over and over again, so I predict it will happen that way again next time. But when I am able to generalize the pattern, I can not only predict what's next, but what will happen at ANY point in the future and that's powerful. And then my brain takes me a step further to the importance of communication. If I can take my generalized pattern and explain it to you, then we can both predict the future. And if the computations are tough, I can even communicate my pattern to a machine with no brain... as long as I speak in a language that it is programmed to recognize.

And suddenly my unit on writing linear equations in slope intercept form becomes a unit on predicting the future and building robots. And THAT'S something that interests me.

I've had other game-changing moments lately too:

- An assessment about coordinate geometry and triangle properties (how DO you rationalize incenter, orthocenter, circumcenter, and centroid?) became an end-cap to a unit about apprenticeship and tool mastery.
- An already rich unit on multiple solution methods got even richer when I refocused on presentation skills: a solution to a problem illustrates, generalizes, communicates, and verifies the results. An answer is just a number.
- And a simple lesson on using LCD to simplify rational equations became an integral part of a unit about elegance.

Labels:
backwards design,
essential questions,
linear equations,
reflection,
UbD

## Monday, September 26, 2011

### Elegance

Every year I promise myself that I'll be better at organization and post-lesson analysis, but yet, I always seem to be too wrapped up in production to have time to reflect. So this year, I find myself with a new focus: take a look at what I've already produced and do some serious self-reflection.

And so, in this spirit, I pulled out an old lesson I created for my beginning algebra students on using the lowest common denominator to solve rational equations. It's a nice lesson. The student's always chuckle at my creature graphics and at the end of the day, they can do the math.

But with fresh eyes, I wonder: this particular skill is

It's about elegance.

How many times do we ask our students to make choices about using the best method or presenting the best argument? But this concept of 'best' is elusive. Ask any student, and they will surely tell you that the

And so, in this spirit, I pulled out an old lesson I created for my beginning algebra students on using the lowest common denominator to solve rational equations. It's a nice lesson. The student's always chuckle at my creature graphics and at the end of the day, they can do the math.

But with fresh eyes, I wonder: this particular skill is

*useful*, because the 'trick' can take a muddy equation and clean it up a bit. But it isn't*essential*, because the students already have methods that can solve these types of problems. And yes, I know they'll really need this method when they get to more difficult rational equations with variable expressions in the denominators. But they don't care about THAT! So what's the**big idea**? And then it comes to me...It's about elegance.

**And I love this question, because I don't have an easy answer, but I care. I WANT to be able to write things that others will look at and say, "Wow, that's really elegant." Plus, it opens up doors in all directions: multiple methods for solving systems of equations, solving by graphing, calculating area of unusual shapes, proofs, trig identities... my brain is already jumping all over the place.**__What makes one solution more elegant than another__?How many times do we ask our students to make choices about using the best method or presenting the best argument? But this concept of 'best' is elusive. Ask any student, and they will surely tell you that the

**best**solution is the**correct**solution. But I know that not all correct solutions are created equal. And this is where mathematics resembles art. While there are many paths that are*correct*, we can evaluate the merits of one solution over another by considering craft, execution... and elegance.**Disclaimer (and shameless plug):***I am proud of my original materials, especially in their revised forms, and so as I review and analyze them, I am posting them to share on the online marketplace: Teachers Pay Teachers. Some are free and some have a nominal fee. You can see them all by clicking the link to my store on the right. If you wish to become a seller too, click*here*and you will award me referral points. Happy sharing!*
Labels:
common denominator,
elegance,
LCD,
rational equations

### Good Math Teachers Are Alive and Well!

The state of mathematics education in this country is not as hopeless as we are led to believe. I’m here to say that good math teachers are alive and well in the United States. Critical thinking, intrigue, enthusiasm, real applications, and in-depth problems abound in high school math classrooms across the country. I’ve seen it, and I will testify to its existence. So, let’s make some noise. Contribute, share, grow. That’s what I have in mind.

My first thought is to start with big ideas and see where that goes. The Understanding by Design (UbD) framework calls these big ideas “Essential Questions.” A google search of ‘essential questions in math’ led me to a list of 150+essential questions in math. Seems like a good place to start, but alas, as I read through these, my suspicions are confirmed. Many of these 'essential' questions seem to fall into one (or both) of these two categories:

- I can rattle the answer off in a couple of sentences.
- I don’t care.

How about a list of questions that is intriguing to me AND to a typical high school kid? How about some questions that rise above a particular piece of content and apply to a whole variety of mathematical topics? I think the list will be shorter than 150, but who knows?

So, I start this blog with a healthy dose of humility and a shout out to all those who have mentored and inspired me in this particular endeavor:

To Ron, whose excitement and enthusiasm for math and teaching led thousands of his students and colleagues to a love of mathematics. To Rich and John, who showed me that the supply of deep and intriguing problems is endless and not as elusive as I thought. To Isaac, Mary, Steve, Andrew, Margie, Rich, and so many others who never let me forget that collaboration is so much more fun (and effective) than independent work. To Mark, who shows me that there is no shortcut through hard work. To Alice, who teaches me that real genius is not an act; it’s an action. To Colleen, who constantly reminds me that I can be better. To Jen, who showed me how to make anything

*look*fun. To Diana, Bob, Abdellah, Brian, Ivan, Yuri, Ann, Nick, and Joan who taught me how to make math*sound*reasonable. And to my parents, who taught me that if higher knowledge is the goal, then critical and creative thinking is the path.And thanks to you too. I look forward to working with you.

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