Speaking Mathanese

Kids butcher the Mathanese language.  I’m just sayin’.  We have all these kids who speak text just fine.  It seems to me that Mathanese should be right up their alley.  All we are doing is taking a bunch of words and converting it to symbols.  Should be easy, right?  Not so much. 

I find that kids have a tough time translating algebraic expressions to English and vice versa.  Am I alone? 

Yeah, didn’t think so. 

One of the things that I have been trying to focus on this year is to convey to students the universality of the things they are learning.  For example, cause/effect in language arts becomes input/output in math.  Conflict resolution is the same as problem solving.  Language arts has expressions and sentences, so does math.  Scientific method can compare to making a conjecture in geometry, testing it out and then using inductive logic to arrive at a conclusion (read: rule). 

So what happens when you tell them to translate: the product of 3 and the sum of x and 2?

You get: 3x+2, right? 

Not quite. 

Well I figured we needed to develop a mashup of English and Mathanese; Mathglish, if you will.  Here is what we came up with:

English to Mathanese:

This should read: The product of 2 and the sum of the product of 4 and x and 3.

Mathanese to English:

The key this time was to allow the mashup.  I live in a rural area where the Spanish speaking population is very large.  Many of my kids speak and understand Spanglish.  I have never done it this way before and the kids nailed it. 

How do you do it?

Update:  Just did a quick check for understanding 2nd period and  26/28 kids circled the bases.

How They’re the Same

I’ve kind of been off the grid lately-save following a few conversations via Twitter- due to  the birth of my son.  Thanks again to everyone for all the well wishes.  Mommy and baby are doing great.  Sleep is a precious commodity but I am blessed to be able to take a few days off to enjoy the adjustment to our little one.  I always seem to compare my approach to teaching with my approach to parenting and vice versa.  Here’s the top 10:

10.  The clientele will expose your bad habits. 

9.  You can read as many “how to” books as you want, but nothing prepares you for your first real life encounter.

8.  They like it when you act goofy.

7.  Sometimes you just have to wing it. 

6.  They get grumpy before lunch time.

5.  They get sleepy after lunch time.

4. You’re gonna lose some sleep.

3. Working at one makes you better at the other.

2.  Balance is crucial.

1.  If it stinks, change it.

It Could Be Worse

Man, I am horrible right now.  No, not the tell-people-that-I’m-bad-so-they-tell-me-I’m-good kind of horrible.  I mean really horrible. I am way behind in planning, grading and have way too many ideas and no way to implement them.  Or if I do try to implement them, they’re half-baked.  It may have a little to do with a certain visitor we are expecting.  But at the end of the day I have been feeling way scattered.  It’s not a good feeling but I know it’ll pass.  I have been doing this long enough to know better.

There, I said it!  Let the healing begin.

Jose Strikes Again

During the warm up today, I asked the question:

When does the absolute value of r equal r?

I liked the way the students handled themselves during the discussion so I took out my phone and recorded the following. After a little prodding, Jose jumped in.  I’m thinking of changing his name to Q.E.D.

Thoughts I Have While Brushing My Teeth

Being organized requires a bit of planning and takes a little more work on the front end. As a result, one is more efficient and ends up doing less work in the long run. For example:

But why do I keep doing this:

What Are You Looking At?

Today I gave my classes a survey as a way to gain some feedback on how the first quarter has gone.  One of the questions was “What would make you more comfortable asking questions in class?”

Here is the response that really pushed back:

Well, this may seem silly and childish, but you want the truth, right?
Well, when a student asks a question, you seem to direct your answer to the person who asked it, which makes me feel uncomfortabe. I mean, if other people don’t understand, then why only talk to one person, instead of the whole class? It makes me feel weird, like I’m the only one who doesn’t understand, and the teacher looking at one single student seems to cause everyone to look, making the student even MORE uncomfortable. As I read over this, I feel I want to delete it, because it seems so silly and unnecessary of mentioning. I won’t delete it, I guess, because I suppose you want to know this, no matter how silly it (mine) is.

WOW! I had never really thought of that.  Yeah, I guess if I am burning a whole through a kid with my gaze while I am answering a question, it may just make them think twice about asking another one.  I don’t think I do that, but perception is reality to these kids.  So if she says I do it, I guess I do.  Need to keep a watch out for that one. 

Where do you look when you are answering a question from a student?

I’m Telling Ya, Lesson Plans are Overrated.

This year, I have kind of introduced equation solving to my 7th graders very informally.   One way I have done this is by giving  them a few balance equations like this:

It seems like it takes the edge off when the variable isn’t there.  But today one of our warmup problems was: 5x + 1 = 2x + 7.

I have been amazed at how many of my students have been willing to attack equation solving by using a guess and check table.  I’ve never taught it that way, but some kids have just taken to it.  After today, I may start to encourage it.  One kid noticed that when you let x=1, the right side is greater than the left side.  But if you let x=10, the left side is greater.  When the balance of power shifts, you know that the answer is between your last two guesses.  Of course, typical guess and check strategy.  But the thing I like about it when dealing with these linear equations is that they are beginning to think in terms of linear systems and how the point of intersection acts as a dividing point between which equation has greater value.  They’re teaching me something. 

But Brandon took the cake.  He says, “Mr. Cox, you can tell the left side is going to be 6 because 5+1=6 and the right side is going to be 9 because 2+7=9.”

“What does x have to be for that to be true?”

“X=1.  But as we make changes to x, the other one is growing faster.”

“How fast is it growing?”

“The left side is growing by 5 and the right side is growing by 2.  So eventually, we know that the left side is going to be greater than the right side.”

“Yeah.  So when are the 1 and the 7 important?”

“Only at the beginning.”

It took all the self control I could muster to keep from talking about initial condition or rate of change at this point.  I’m glad I didn’t because I think I would have ruined an authentic learning moment for this kid.  The thing I wanted to encourage the most in him was the fact that he looked for patterns and then asked questions to help make sense of those patterns. 

One warmup which I expected to spend 5 minutes on turns into 20 minutes of slope, y-intercept, linear systems and problem solving strategies all because a few students took an approach I’ve never taught. 

Another example of the kids re-writing the lesson plan.

Monologue to Dialogue

“When you add a positive integer with a negative integer, how do you know if your answer is positive or negative?”

 ”Well if the negative number is bigger, then the answer is going to negative.  If the positive number is bigger, then the answer is positive.”

“Aren’t all positive numbers bigger than negative numbers?”

“Well, yeah.  But if you take the sign off the negative and it’s bigger than the positive, then the answer will be negative.”

“Why are  you taking the sign off the negative number?  What rule allows you to do that?”

“Uhh…”

“I know that I can give you 20 addition problems and you will probably get all 20 right, but I want you to explain to me why this works the way it does.  Come talk to me when you think you have an answer.”

*10 minutes goes by*

“Alright, I think I’ve got it.  If the negative number is farther down the number line than the positive number, then the answer is going to be negative.”

“Farther down the number line?”

“Yeah, it’s more negative than the positive number is positive.”

“How do you know that?”

“It’s farther from zero?”

“Oh, what do we call that when a number is farther from zero than another number?”

“Uhh…”

*5 minutes later*

“ABSOLUTE VALUE!.  If  the negative number has a greater absolute value, then the answer is negative.  If the positive number has the greater absolute value, then the answer is positive.”

“That is correct young grasshopper.  You have done well.  You may now enter into the realm of proficiency.”

I have had this conversation about 10 times over the last few days.  Our current system has students take a common formative assessment (CFA)which is very closely aligned to our state’s standards.  It’s a multiple choice test that has questions that look an awful lot like the same questions they’ll be seeing in April when we take the CST test.  Based on their score, they have a set of activities to do before they can re-assess.  Re-assessment may look like the conversation above.  I think I am really going to like this system because it allows for dialogue between teacher and student.  I have the opportunity to ask them about the why and actually tie it to their grade.  The benefit to this is that students have choice in how they demonstrate their proficiency the second time.  The first time, it’s a multiple choice test.  However, the second time may be written, oral or heck, they may even draw a picture. One of the best things about this is that the students are taking more ownership of their learning because they have to direct some of the activities.  They actually have choice.  And that’s empowering.  They aren’t waiting for me to give them another hoop to jump through. 

The parents are coming along slowly.  Many of them didn’t understand how their student could score 100% on the CFA and yet the score in the grade book shows up as 80%.  Last night was Back to School Night and I got the chance to explain that each standards’ assessment is two parts.  The first part is multiple choice and the second part depends on the student.  Once they realized that their child’s grade quits improving when they quit trying, I think they got it. 

It’d be nice if we could focus less on the grade and more on learning, but…

…baby steps.

What’s the point?

One of my favorite activities is to have students draw a point on a paper and see how many distinct lines they can draw through the point.  I usually set it up as a competition to see who can get the most lines inside of 15 seconds or so.  On your mark, get set, GO! Pencils start flying.

Then to bring the lesson home, I say, “Alright, flip the paper over and put two points on the page.  Now we’re gonna see who can get the most lines through both points.” 

Ready, set, GO!

They get the first line fast.  Then they panic as they move the ruler and pencil searching for that elusive second line.  Most of ‘em end up looking something like this:

I tease them a bit and we all get a good chuck out of it. I know, I know.  It’s not nice to take advantage of these trusting impressionable children.  But I don’t care who you are, that thar’s funny!

 

And, they never forget it.

How Close is Close Enough?

For my 8th graders, homework for day 2 consisted of a worksheet where students determined which set(s) included given numbers. Pretty easy stuff. But I threw one of my favorite problems at ‘em to see what they’d do with it.

What’s the sum of 1/2 + 1/4 + 1/8…?

At first they’re thinking, “not possible ’cause it goes on forever.” I told them to try it anyway.

The two most popular answers were “.99…” and “1.” But those who answered “1″ were quick to admit that they just rounded off. We open the discussion and I was very pleased with how thoughtful and respectful everyone was. These kids were really interested in getting to the bottom of this. It was a great opportunity to demonstrate that often times drawing a picture will allow you to see things in a problem that you may not otherwise catch.

So we draw a square on the board and shade 1/2. Then we shade 1/4, then 1/8 and so on. They soon see that the square will eventually be full.

Me: “So is it 1 or is it just really close?”

“Really close. Because the square is never completely full. You always have half of the remaining area that is unshaded.”

Good. So let’s see how they handle this.

“Alright, what’s 1/3 as a decimal?”

“.333…”

“Ok, and how about 2/3?”

“.666…”

“what’s 1/3 + 2/3?”

“1.”

“And what’s .333…+ .666…”

“.999…”

“So does .999… = 1 or is it just really close?”

At this point they admit that it looks like it’s equal but it just doesn’t make sense. Time to to talk about what it means to be infinitely close to something. This is always a fascinating discussion.  We discussed the idea of a neighborhood and how if .999… does not equal 1, then there must be a number between them. 

“Give me the number and I’ll shut up”, I tell them. 

One kid says,” How about .0 with a repetend, then a 1?”

But another student catches this, “If the zero goes forever, when do we add the 1?”

 It amazes me how these kids can grapple with the real “stuff” that is mathematics.  These same questions that got me hooked as I was taking my analysis classes in college are finding their way into the minds of 8th graders.  And you know what?  They get it…at least as much as they possibly can. 

Man, I love this job!