Finding Ways to Nguyen Students Over

Interview with a middle school math teacher on the Common Core

[Cross-posted from mathbabe.org, Cathy O'Neil's blog.]

Today’s post is an email interview with Fawn Nguyen, who teaches math at Mesa Union Junior High in southern California. Fawn is on the leadership team for UCSB Mathematics Project that provides professional development for teachers in the Tri-County area. She is a co-founder of the Thousand Oaks Math Teachers’ Circle. In an effort to share and learn from other math teachers, Fawn blogs at Finding Ways to Nguyen Students Over. She also started VisualPatterns.org to help students develop algebraic thinking, and more recently, she shares her students’ daily math talks to promote number sense. When Fawn is not teaching or writing, she is reading posts on mathblogging.org as one of the editors. She sleeps occasionally and dreams of becoming an architect when all this is done.

Importantly for the below interview, Fawn is not being measured via a value-added model. My questions are italicized.

——

I’ve been studying the rhetoric around the mathematics Common Core State Standard (CCSS). So far I’ve listened to Diane Ravitch stuff, I’ve interviewed Bill McCallum, the lead writer of the math CCSS, and I’ve also interviewed Kiri Soares, a New York City high school principal. They have very different views. Interestingly, McCallum distinguished three things: standards, curriculum, and testing. 

What do you think? Do teachers see those as three different things? Or is it a package deal, where all three things rolled into one in terms of how they’re presented?

I can’t speak for other teachers. I understand that the standards are not meant to be the curriculum, but the two are not mutually exclusive either. They can’t be. Standards inform the curriculum. This might be a terrible analogy, but I love food and cooking, so maybe the standards are the major ingredients, and the curriculum is the entrée that contains those ingredients. In the show Chopped on Food Network, the competing chefs must use all 4 ingredients to make a dish – and the prepared foods that end up on the plates differ widely in taste and presentation. We can’t blame the ingredients when the dish is blandly prepared any more than we can blame the standards when the curriculum is poorly written.

Similarly, the standards inform testing. Test items for a certain grade level cover the standards of that grade level. I’m not against testing. I’m against bad tests and a lot of it. By bad, I mean multiple-choice items that require more memorization than actual problem solving. But I’m confident we can create good multiple-choice tests because realistically a portion of the test needs to be of this type due to costs.

The three – standards, curriculum, and testing – are not a “package deal” in the sense that the same people are not delivering them to us. But they go together, otherwise what is school mathematics? Funny thing is we have always had the three operating in schools, but somehow the Common Core State Standards (CCSS) seem to get the all the blame for the anxieties and costs connected to testing and curriculum development.

As a teacher, what’s good and bad about the CCSS?

I see a lot of good in the CCSS. This set of standards is not perfect, but it’s much better than our state standards. We can examine the standards and see for ourselves that the integrity of the standards holds up to their claims of being embedded with mathematical focus, rigor, and coherence.

Implementation of CCSS means that students and teachers can expect consistency in what is being in taught at each grade level across state boundaries. This is a nontrivial effort in addressing equity. This consistency also helps teachers collaborate nationwide, and professional development for teachers will improve and be more relevant and effective.

I can only hope that textbooks will be much better because of the inherent focus and coherence in CCSS. A kid can move from Maine to California and not have to see different state outlines on their textbooks as if he’d taken on a new kind of mathematics in his new school. I went to a textbook publishers fair recently at our district, and I remain optimistic that better products are already on their way.

We had every state create its own assessment, now we have two consortia, PARCC and Smarter Balanced. I’ve gone through the sample assessments from the latter, and they are far better than the old multiple-choice items of the CST. Kids will have to process the question at a deeper level to show understanding. This is a good thing.

What is potentially bad about the CCSS is the improper or lack of implementation. So, this boils down to the most important element of the Common Core equation – the teacher. There is no doubt that many teachers, myself included, need sustained professional development to do the job right. And I don’t mean just PD in making math more relevant and engaging, and in how many ways we can use technology, I mean more importantly, we need PD in content knowledge.

It is a perverse notion to think that anyone with a college education can teach elementary mathematics. Teaching mathematics requires knowing mathematics. To know a concept is to understand it backward and forward, inside and outside, to recognize it in different forms and structures, to put it into context, to ask questions about it that leads to more questions, to know the mathematics beyond this concept. That reminds me just recently a 6th grader said to me as we were working on our unit of dividing by a fraction. She said, “My elementary teacher lied to me! She said we always get a smaller number when we divide two numbers.”

Just because one can make tuna casserole does not make one a chef. (Sorry, I’m hungry.)

What are the good and bad things for kids about testing?

Testing is only good for kids when it helps them learn and become more successful – that the feedback from testing should inform the teacher of next moves. Testing has become such a dirty word because we over test our kids. I’m still in the classroom after 23 years, yet I don’t have the answers. I struggle with telling my kids that I value them and their learning, yet at the end of each quarter, the narrative sum of their learning is a letter grade.

Then, in the absence of helping kids learn, testing is bad.

What are the good/bad things for the teachers with all these tests?

Ideally, a good test that measures what it’s supposed to measure should help the teacher and his students. Testing must be done in moderation. Do we really need to test kids at the start of the school year? Don’t we have the results from a few months ago, right before they left for summer vacation? Every test takes time away from learning.

I’m not sure I understand why testing is bad for teachers aside from lost instructional minutes. Again, I can’t speak for other teachers. But I do sense heightened anxiety among some teachers because CCSS is new – and newness causes us to squirm in our seats and doubt our abilities. I don’t necessarily see this as a bad thing. I see it as an opportunity to learn content at a deeper conceptual level and to implement better teaching strategies.

If we look at anything long and hard enough, we are bound to find the good and the bad. I choose to focus on the positives because I can’t make the day any longer and I can’t have fewer than 4 hours of sleep a night. I want to spend my energies working with my administrators, my colleagues, my parents to bring the best I can bring into my classroom.

Is there anything else you’d like to add?

The best things about CCSS for me are not even the standards – they are the 8 Mathematical Practices. These are life-long habits that will serve students well, in all disciplines. They’re equivalent to the essential cooking techniques, like making roux and roasting garlic and braising kale and shucking oysters. Okay, maybe not that last one, but I just got back from New Orleans, and raw oysters are awesome.

I’m excited to continue to share and collaborate with my colleagues locally and online because we now have a common language! We teachers do this very hard work – day in and day out, late into the nights and into the weekends – because we love our kids and we love teaching. But we need to be mathematically competent first and foremost to teach mathematics. I want the focus to always be about the kids and their learning. We start with them; we end with them.




TP and Taxi Rate

This jumbo roll of Charmin 2-ply toilet paper made me think about area, and I wonder if my kids know how to convert from one square unit to another. I suspect they will do this incorrectly.



Before I ask them to calculate the area of 1 roll, I ask them to:
  1. estimate the number of sheets in 1 roll
  2. estimate the dimensions, in inches, of 1 sheet
  3. calculate the total area, in square inches, of 1 roll based on your estimations above
Their estimates for the number of sheets range from 123 to 15,000. The median is 250 though. Most think the sheet is a square. So they know how to calculate the area of 1 sheet then multiply this area by the number of sheets to get the total area. No biggies.

Then I give them 328 as the actual number of sheets. I also measure the dimensions of one sheet as 4.0" by 4.2". But we'll just consider it a square of 4" by 4". I ask them to imagine if we rolled out the 328 sheets on a flat surface, what area — in SQUARE FEET — would the whole roll cover?



The results confirm my suspicion. It's okay that my 6th graders don't yet know, but I'm surprised that most of my 8th graders do not know.



**********

For today's math talk, I put up this question with a picture that I'd taken while riding in a cab in New Orleans. 

How much would it cost for a 30-mile ride in this taxi cab?



While I love how some of them were able to solve this problem mentally by knowing that it takes 8 1/8-miles to form 1 mile, so multiplying .25 by 8 to get 2 is a good strategy.

However, not one student saw the .25¢ during their calculation and discussion. Only when I asked them to look at it again more carefully that a few spotted the error.



Discount and Sales Tax



Instead of doing the above exercises in our textbook, I had my 6th graders do this:

Shopping Contest at Target

Let’s pretend Target has a contest. The contest is for shoppers to find merchandise from their online store.

Contest rules

  1. You must choose at least 10 different items.
  2. The items must come from at least 5 different departments, such as footwear, kitchen, clothing, toys, etc.
  3. You may buy more than 1 same item — you can buy 2 or more packs of athletic socks.
The winner

The winner is the shopper whose merchandise receipt totals exactly $500 or closest to it — without going over. He or she gets to keep all the merchandise!

Your task

  • Phase 1: Go to Target’s website and find some items that you’ll want to get for yourself and your family. List the items and their original prices in the table provided separately.

  • Phase 2: The store manager (really, it’s your teacher) then announces the percent discounts for different categories of merchandise. Write these down below. Apply these discounts to your items and calculate the new sale prices. 



  • Phase 3: The current tax rate for our city is 8%, so you must add this to your total. In this phase, you get to add or remove items on your shopping list to reach the target goal of $500 without going over.


**********

I only did Phase 1 with the kids, my sub supposedly did Phases 2 and 3 with them. (I wanted to do the rest of the lesson with them when I return from NCTM — and after spring break — but they voted to continue the lesson with the sub as they were really into the task. I'll find out tomorrow when class resumes.)

Why this task
  1. You can change everything about it. Shop somewhere else instead of Target.
  2. Change the rules depending on time available and access to computers. Are there enough computers for each kid or do they need to work in groups of 2 or 3? My kids were in pairs. (We have enough laptops, but at least 30% of them have issues.)
  3. I like the idea of not going over a certain amount — $500 in this case — instead of "whoever is closest to the target price" because I think it keeps the kids more reserved in their shopping spree. Students understand that if there were only 2 contestants, then the one with a final receipt total of $154 would win over the other with a $501 total.
  4. Kids don't know what the exact discounts are until Phase 2, so this makes it a fun temporary secret. But they know to go over budget in Phase 1 because there will be discounts — not all departments have to have discounts either. Their totals in Phase 1 were in the $600 to $700 range because they also know that it'd be easier (faster) to remove items than add them later in Phase 3. They were also told that the discounts would be somewhat realistic, meaning Electronics will get a smaller percent discount, if any, than Clothing.
  5. Most people like to shop, and kids are people too, and it's funner when it's a contest type of shopping. But they'll become more aware of how much things cost — and how quickly they add up in the shopping cart. 
  6. They may very well decide NOT to drop out of high school because of this lesson. They'll think, Everything costs money: skateboard, AXE deodorant, sneakers, peanut butter, Oreos. I remember doing that Target lesson!
 
:)






Powdered Beignets

Due to a sequence of shitty events, I find myself at the airport three hours early for my flight.

I'm carving into my first of three overly powdered beignets.

A mother sits an arm length to my left along the wall booth. Her teenage daughter across from her. They talk about colleges. Mother attended Cornell and reflected that she was a moron for not taking a course from Carl Sagan. Daughter is here to check out Tulane; she's a junior now. They talk about her classmates and where they're considering. Nothing out west, they say, even though Stanford would be awfully nice. 

Mother: We're going to drive you there.

Daughter: I can fly. 

M: I know. But we want to drive you. Time for us to talk. I like that they want the freshman and sophomores to live on campus. 

D: We'll see.

A father-daughter pair sits next to them. The four start chatting as if they know each other from back home. One girl congratulates the other who has already spent over three hundred dollars at her future school's bookstore. This one replies, "Oh, I have to decide first! Then I'll buy all the stuff!"

They finish their entrees and the mother orders some beignets for them to share. Mother points to my plate to tell her daughter what beignets are. 

Their easy conversations. They talk about the college choices they have. It doesn't matter whether they will fly or drive. The opportunities are there. They barely touch the beignets. I'm eating my last one.

The girls and their parents are white. I think about half of my students who are not. I think about my Felix, my Andres, my Jaylene, my Israel. Kids whose parents don't always speak English. I don't imagine them having these easy conversations about colleges. Why is that. The privilege afforded to some. I get stupid teary.

Time for me to board.



Prices, Proportions, Percents

I was in Garden Grove with my son on Sunday, and he insisted that I try this smoothie place called Tastea. With Jamba Juice and Blenders and all the other juice bars around town, I was skeptical that this joint's concoctions would be anything different. He ordered a taro milk tea and I got a Thai tea, both with boba. Just one sip and I said to him, "Let's order another round! It's a long drive home!" Soooo delicious.

 

While there I saw a math lesson brewing, so I picked up their menu with prices. This is the lesson with my two classes of 6th graders.

Me: (I tell them about Tastea and how I wish it were closer.) Okay, let's start with something you might be more familiar with, Starbucks. I love that now I live within walking distance from one! Do you know the different sizes that they have there?

Class: (When I refer to "class," I don't mean the whole class, of course, but somebody in the class joins in on the conversation.) Tall, grande, venti.

M: Do you know exactly how much liquid each size holds? (They make various guesses. I bring out the 3 sizes that I got from Starbucks so they have a visual.) I normally order a tall mocha frappuccino, let's say the price is $3.50. Do you think the venti, which is twice the volume of the tall, would cost twice as much, or $7.00?

C: No.
 
M: Why not?

C: You normally get a better deal with a bigger size.

M: What do you mean by a "better deal"?

C: (All their answers show me that they understand the idea of more bang for your buck. Then finally someone says...) Lowest unit price!

M: Right! That's why so many families go to Costco. Buying in bulk normally saves us money because the item has the best unit price. Well, we're talking about Starbucks now, so buying more is a better deal, but drinking more is not so good for our body. Let's fill in this sheet. (I pass out this 2-page worksheet.) How do we calculate unit price? What place value should we round it to? How do you write thirty-one-cents-per-ounce?




M: Tastea has three sizes: mini, gigantic, and even more. Their teas can also be purchased by the "partea jug," which holds a gallon. I've given you the prices of the 10-ounce minis for the three different types of drinks, your job is to figure out the prices of the other sizes. You'll work in small groups to figure out these out. So, do you think the gigantic will cost twice as much as the mini because it holds twice as much?

C: No. It'll cost less.

M: How much less? Well, that's your group's job to come up with the best estimate. We have Starbucks' prices for their three sizes, you could look at how they price their drinks. But here's the sweet deal for you. You and your group mates do the math that you need to, then write down your first estimation right here in this column. Bring your paper up to me (only the "captain's" paper), give me a few seconds to figure out the percentage that your estimation is off by, and I'll write it in this column and give you back your paper. What percent do you want to see, large or small? What if your estimation were the actual price — what percent would I write there?



I tell them that they could figure out the actual price of the drink if they knew how I calculated the percentage of error. So, work work work. Think think think. What makes sense? Oh, I remind them that the percentage does not indicate if their estimation is too high or too low. So, again, what makes sense?

We also note that prices generally end in a 0, 5, or 9. So, even if the calculation tells them the price should be $4.23, they might want to change that to $4.25 or $4.20.

When they bring up their paper again with the second estimation, all I do is write their estimation again in pen and circle it — this is so they can't change their answer and I know that I've seen it. I do NOT fill in the "Actual Price" column at this time because the groups are working at different rates, and in a crowded room, it's easy for kids to see each other's papers, even inadvertently, and the game of estimation is over if they saw the actual price beforehand.



They simply move on to the next size to make a first estimation again. We repeat the process. 

When all groups are done with estimations for the first type of juice — smoothies — I tell them what the actual prices are. 

Now, it's their turn to figure out the percentage of error. I give the groups about 10 minutes to do so without help from me. At the end of the 10 minutes, either there's at least one group that knows how to do so and can show it to the class, or no group knows how, then I'll walk them through the calculation by asking them questions to figure this out.



They continue in the same manner for the Slushy Freeze and Specialteas on page 2. This time hopefully they'll be able to work backward from the error percentage that I give them after their first estimation.



Reasons I'm proud of this lesson:
  1. It's about proportions, but many priced items in real life are not directly proportional. The kids knew this coming in because they've been consumers.
  2. We get to talk about business strategies that entice people to buy the larger sizes while still make a profit. (Starbucks calls it "tall" because it rhymes with "small," but clearly the word tall naturally elongates the imagination.)
  3. Students get to make estimations throughout, but they know these aren't "wild-ass guesses." They start with the calculation of proportions and adjust the prices accordingly. They get to critique and argue with their group mates to come up with the best estimations.
  4. I get kids to think about percentage in a context that they can wrap their heads around. And they want to know how because their second estimation could be dead on if they knew.
  5. It's fun that the error percentage does not indicate if their estimation is too high or too low. A few groups do go farther in the wrong direction. Oh, well — good to learn that now.
  6. It'd be fun for me to get Starbucks or Jamba Juice for the group with the lowest total in percentage errors.







April 05, 2014: I got some thoughtful reflections on this lesson, I'll just share two:

I learned how to work backwards with percentages and try to get the number spot on. I also learned how business would price things by dropping the price by the perfect amount. My number sense got a lot better from all the multiplying, dividing, and reasoning. It was very difficult, which I'm very happy about. 

The teamwork was probably the hardest part of the project. M and I are very competitive, and we got different answers a lot. I learned how to work together with others a lot better, and it doesn't move your team along to place blame and argue. I'm really grateful we did this project because it was very hard and worthwhile. It was a great use of three days!

***

I learned how to use different data to get answers. Also, we have to see a pattern. This Tastea assignment was really fun. I enjoyed it and look forward to another. Teamwork is really important even though people can't agree, you got to support it. If your group gets it wrong, but your answer was right, you can't blame someone or put them down because probably they will get some right for you. So always stay positive to your teammates and encourage them.




What the Kids Thought

Recently Dan Meyer asks Mathalicious which of these three questions is "real world"?



Karim Anifounder of Mathalicious, and others have opined without consensus on this particular question and on the general notion of real-world vs. fake-world problems.

I wonder what my 8th grade geometry kids think of this question.

I give them Version A on a strip of paper and ask them to work on it alone for 5 minutes. I tell them that I'm interested in learning if they understand the question as is, therefore I'm not answering any clarifying questions about it. After the 5 minutes, I put them into random groups, and they work on the problem for another 10 minutes.

Then I show them Versions B and C and ask for their preference and reason for each version.

Version A: 18 likes, 14 dislikes

Highlighted reasons for LIKING:
    • It gives every detail you need to know. It tells you directly all of the information. It also seems easiest to solve.
    • It isn't as confusing as looking at fast-motion pictures of a circle and a square. Doing math is more exact than visual guessing. [Did he translate the animation to mean guessing?]
    • I am able to make my own diagram and I can try to solve an equation to find out the answer.
    • It is simple and not confusing. It allows me to think the way I want to and not be misled by a moving picture.
    • With the given information, you could construct the two shapes by working backwards.
    • There's enough given information to make the problem interesting and hard.
    • I like this one the most because you can actually read the problem and refer back to it.
    • I think it can be solved using an equation, and be solved more easily than B and C.
    • It's a more accurate way to find their areas and make them equal.
    • I like this version because I understand the problem.
    • Instead of a picture of it on paper, you have to visualize it in your head first.
    • You can get an exact answer. It is challenging.
Version B: 18 likes, 14 dislikes

Highlighted reasons for LIKING:
    • It would be cool to build the animation in GSP and solve it that way.
    • It's a lot more simple. It provides an image and idea of what it looks like.
    • I can visually see when they are equal. It will be easier to see when they are equal instead of having to do a load of math.
    • It is visual.
    • I feel I have a higher chance of answering the question with a right answer.
    • You can easily see when the shapes have the same area.
Version C: 15 likes, 17 dislikes

Highlighted reasons for LIKING:
    • It seems easier because you can just count the candies and see if they're equal.
    • Anyone can count how many candies there are, then subtract the extra space to get the correct area.
Some students like and dislike more than one version. My takeaway on their responses:

Version A
LIKES: (see above)
DISLIKES: Not understanding the question, or "I'm a visual learner, so I like Version B better."

Version B
LIKES: It's visual. It's easy.
DISLIKES: Too fast and hard to follow. One student, "The movement is distracting and confusing. I feel like it's too abrasive and violent. Math should be more elegant than this." 

Version C
LIKES: You just count the number of candies. It's visual.
DISLIKES: Too fast to follow. It seems too easy. There's space between the candies. One student, "You can't get the exact answer... And the leftover space in one shape may be more than the leftover space in the other."

I collect all their papers before telling them which version I like. I like Version A for its simplicity. I'm curious if the stated question is enough information for them to understand. This student's reason nails it for me: "It allows me to think the way I want to and not be misled by a moving picture." 

We all have students who struggle with word problems. I don't think this means we should give them fewer word problems. I think it means we should give them better word problems — ones that are written with just enough information and not embedded in contrived contexts that either confuse or insult the students. And for students who need help with the question, they get to hear an explanation from a classmate. 

Version B is okay, but I don't want to start with it because I feel I'd be wasting a perfectly good question in Version A! I'd reach for a piece of string to explain this question, if needed. Version C gives me a headache.

At least one person in each group understands the question, and they do their best to make sense of it in just the short 10 minutes that we have. They're trying. And making mistakes.

  


Our whole-class discussion at this 8th grade level:
  1. That "arbitrary" point P is pretty darn close to the middle of A and B. You can roughly tell from using a piece of string. Or you can tell from arriving at y2/(4pi) = x2/16 (where x is distance AP forming the square's perimeter, and y is distance PB forming the circle's circumference) — the denominators are almost the same.
  2. Likewise, P cannot be at the center because pi doesn't reconcile nicely in the equation. 
  3. We can solve for x and y using some arbitrary distance AB, and we find y to be slightly shorter. 
  4. We can ask a related question: A circle's circumference and a square's perimeter are equal, what is the area enclosed by each? Kids can certainly think about optimization and do a little bit of calculation outside of a formal calculus class.
In addition to asking the students which versions they like, I also pose Dan's exact question to them: Which of these is a "real world" math problem? Or is none of them a real-world math problem?

Their answers vary as widely as those of math educators'. However, I find this correlation that doesn't surprise me: kids who like math more do not care if the problem is real-world or not. 

This [Version C] is the most "real-world" solely because of the fact that it involves a material object which in this case is the candy. However, the thing you're solving for in this question is not very "real-world" at all. Personally, I don't care at all if a problem is "real-world" or not; I just like to solve problems.

If a problem didn't have to do with "real-world" I will still do it if I like it. It doesn't really matter.

I don't think any of these problems are "real world" math problems. I like how they make me think. But I don't think I need them in the "real world."

I wouldn't care if it is a real-world problem because I was there to learn. I think all versions can be a real-world problem because it can be needed in some situations.

I feel like all of these problems are real world... But honestly it doesn't matter at all to me. It doesn't matter if it's real world or not, it doesn't affect me wanting to solve the problem.

So, one from the kids.



Teaching Absolute Value

I know Common Core does not have absolute value in grade 8, but I'm teaching it anyway because we're still doing "algebra 1" this year. (A year ago Raymond Johnson looked into the inclusion of this topic in the different grade levels.)

My 8th graders know that the absolute value of 5 is 5, and the absolute value of -5 is also 5. Some recall that it's the distance from 0 on the number line. 

We begin by solving a few of these: abs(x) = 4, abs(a) = 0, abs(w) = -3. A few trip up on the last one but recover quickly and move along.

Me: What is the distance between these two points?


Class: Eight.

M: How did you get eight? 

C: Subtract. 

M: What about this one? The points are at -3 and 9.


C: The distance is twelve.

M: This one?


C: Thirty-two.

M: Good. Distance is always positive... How did you find the distance between the points again? What operation did you use?

C: Subtraction!

M: Then I'm going to add an equation below each number line showing subtraction. Is that okay?


M: So, the distance between 5 and 13 is 8. Then, what is the distance between 13 and 5? 

C: Eight

M: Woah! It's the same? Meaning I can write the equation either way?


Kids agree that subtraction can be commutative when it's inside the absolute value bars because we’re just measuring distance. The distance from Johnny to Julie is the same distance as from Julie to Johnny. I'm not going to argue.

Me: Given two points, you can tell me the distance between them. So now I’m going to give you just one of the two points but tell you the distance between them, and you find the missing point x.

C: x is ten.

M: Yea, ten works. Let's try to read this open sentence. How would you say it?

C: x minus six... The absolute value of x minus six is four.

M: Hmmm. Oh, you say the words 'absolute value' because they're there. Let's try again without saying those words. Use the word 'distance' instead.

C: The distance of x minus six equals four.

M: Let me show you again the first one that I'd asked you. I remember just asking you, 'What is the distance between 5 and 13?' What did I not say even though it's there?


C: Minus.

M: Right. Let's not add stuff we don't need. You know naturally that finding distance implies subtraction. So, say the equation again.

C: The distance between x and six is four.

M: Or you could say...? Can we switch the points around?

C: The distance between six and some point x is four.

M: Alright. Is 10 the only answer for x? We are trying to find a point on the line that makes the equation true. So, let's use the number line to solve this. Because we know 6 is one of the points, let's locate it. We need to find the other point that would be a distance of 4 away from 6. So, it could be to the right of 6, or to the left of 6. Where does this put us at?


M: Oh, why isn't the point -6? I see a 'minus six' in the equation.

C: Remember, that minus is for subtracting. We need it there to find distance.

M: I remember. We need it.

We do a few more of these. Enough to bore us, need something new. 

M: Let's try this.


C: No subtraction sign. 

M: And you said we needed it. Then create it. Make it happen without changing the problem of course.

C: Change it to minus minus...


M: What does this problem say now?

C: The distance between x and negative eight equals five.


We do a few more of these. Enough to bore us, need something new again. 

M: What about this?


M: Nothing terribly exciting. The other point(s) that we find is now worth 2x, so we just need to solve for x.


Then we do a batch of these: 


Hey, what about these, where there's more stuff stuck around the absolute value quantity. Oh, we just need to first isolate the absolute value, then it's business as usual.


We spend the next whole class using Desmos to explore the shifts/changes to the parent absolute value function. Students need to write down their predictions first before graphing. One student was very excited when she got the V-shape to turn upside down.


We discuss some real-life scenarios that may involve absolute values: margins of error, ranges of measurements: distances, scores, speed, temperatures, pH levels, elevations, etc. 

Not proud to admit that I spent a lot of hours in college playing pool instead of studying, but never once did I associate the path of the ball as an absolute value function. Consider me odd if I always thought of angles instead.



Solving absolute value inequalities start similarly enough. 



Kids know from graphing inequalities that there's "shading" involved. They also know the difference between open and closed points. So I just have the kids use their thumb and forefinger to indicate the distance between the two points, then if the inequality says less than [or equal to], then it's natural to pinch their fingers closer together, indicating that the region inside the points need to be shaded.

 



The textbook will tell kids to set up the "two cases" to solve these inequalities (same thing with equations). Then, kids are asked to graph the solution.



But if kids learned to solve using the number line itself, then there would be nothing to memorize because they learned distance way back when they started learning to crawl. And since the graph shows the solution, then writing what that solution is is easy because it matches the graph. Like below, x lives on the green line between -8 and -2, being ≥ -8 and ≤ -2.



For greater than inequalities, the student would naturally spread his fingers apart to indicate shading outside of the points. 


 



I don't know why there seems to be a lot of rules when learning to solve absolute value equations — which inequality sign for when it's and or when it's or. Oy.






Moon Badges

When you watch too much Khan Academy you earn moon badges
When you earn moon badges you can tweet that you earn moon badges
When you tweet that you earn moon badges your followers say what the hell
When your followers say what the hell you feel you need to explain
When you feel you need to explain you make no sense in 140 characters
And when you make no sense in 140 characters you lose all your followers
So don't lose all your followers, get rid of videos and upgrade to direct instruction

All kidding aside, I'm having kids spend no more than 5 minutes of class time to write one of these for a math trick or math something. They may illustrate it outside of class if they want. Here's the first one from a student who was in the room when I was thinking of this idea aloud.

When you cross multiply to solve a proportion you waste a lot of time
When you waste a lot of time you don't finish your test
When you don't finish your test you get a bad grade
When you get a bad grade you can't get into college
When you can't get into college you can't get a job
And when you can't get a job you become homeless
So don't become homeless, get rid of cross multiplication and upgrade to algebra





[Added 03/13/14 and later...]


When you don't struggle in math you get bored
When you get bored you fall asleep in class
When you fall asleep in class you get sent to the principal's office
When you get sent to the principal's office your parents start yelling at you
When your parents start yelling at you you get sent to military school
And when you get sent to military school you get stuck in a bunk with someone who never showers
So don't get stuck in a bunk with someone who never showers, get rid of not struggling and upgrade to Mrs. Nguyen's math class! :)


When you don't label your units people don't understand you
When people don't understand you they think you're illiterate
When people think you're illiterate they don't talk to you
When people don't talk to you you go insane
When you go insane you start eating things off the street
And when you start eating things off the street you get rabies
So don't get rabies, stop not labeling units and upgrade to labeling units


When you guess you don't get an exact answer
When you don't get an exact answer you have to do twice the math
When you do twice the math you get tired
When you get tired you start sleeping in on school days
When you start sleeping in on school days your mom is late for work
And when your mom is late for work she loses her job
So don't make your mom lose her job, get rid of guessing and upgrade to exact answers


When you FOIL you can only multiply two numbers in each expression
When you can only multiply two numbers in each expression you can't complete algebra
When you can't complete algebra you can't move onto geometry
When you can't move onto geometry you get held back in freshman year of high school
When you get held back in freshman year of high school, you don't have friends
And when you don't have friends you stab yourself with a fork
So don't stab yourself with a fork, get rid of FOILING and upgrade to double distribute

When you don't memorize theorems you don't know what to write in your proofs
When you don't know what to write in your proofs you fail your geometry class
When you fail your geometry class you lose all your confidence
When you lose all your confidence you decide to work at a dead-end job
When you decide to work at a dead-end job you get hired to become a low-budget amusement park mascot
And when you get hired to become a low-budget amusement park mascot you get tackled by an immature middle aged man
So don't get tackled by an immature middle aged man, get your theorems memorized and upgrade to a math tutor

When you get straight A's people call you a genius
When you get called a genius you work to make yourself a genius
When you work to make yourself a genius you become the next Albert Einstein
When you become the next Albert Einstein you make theories
When you make theories you want to test Einstein's theories
And when you want to test Einstein's theories you get sucked into a black hole
So don't get sucked into a black hole, ignore people who call you a genius and upgrade to a critic


When you don't do your math homework you don't understand the subject
When you don't understand the subject you fail the class
When you fail the class you flunk the grade
When you flunk the grade you feel depressed
When you feel depressed you continue to flunk
And when you continue to flunk you become a twenty-five year old man in a fourth-grade class
So don't become a twenty-five year old man in a fourth-grade class, do your homework and upgrade to the next grade level


When you say math isn't  your forte Mrs Nguyen gets mad
When Mrs Nguyen gets mad she yells at you
When she yells at you you feel like a loser
When you feel like a loser you don't succeed in life
When you don't succeed in life you can't get a job
And when you can't get a job you work on a smelly old fishing boat
So don't work on a smelly old fishing boat, get rid of saying math isn't your forte and upgrade to loving math

I Can't Afford Not To

When I share with teachers what my students do outside of the textbook/curriculum, I get the familiar and reasonable concern from them that there's not enough time to cover the content as is, how is it possible to do "other stuff," such as:
  1. Math Munch
  2. Problem Solving (weekly, in-class, group)
  3. Math Talks (including Visual Patterns)
My reasons for doing the above, respectively, are: 
  1. I want my kids to take pleasure in seeing how beautiful math is, to appreciate the elegant proofs, to imagine the possibilities.
  2. I want my kids to think deeply, to struggle, to persevere, to honor the process of problem solving instead of just answer getting. These skills directly help students with content material.
  3. I want my kids to share their thinking out loud because a quiet math classroom is a scary place. 
It's very simple in my mind why I do what I do. If my administrators told me tomorrow that I could no longer do any of these, then I know it would be time for me to leave the classroom. I don't follow fads and reforms and jargon. I don't enter into those conversations in real life or online because I don't know how or have anything to say.

I want to follow these reflections that my kids (6th graders) write:

First of all, the equations are coming so much easier to me! I think what helped me is opening up my mind to other methods, and trying out methods that I've seen other people use. I feel that the reason I have trouble with some problems is that in my mind, I make the problem seem so much harder than it actually is. After doing them week after week, everything is coming to me a lot easier than they used to.

This week I have learned a lot. I've learned new methods of solving problems and reviewed old ones too. Math Talks have helped me a lot in homework and class work too. God I'm brilliant.



So I have a lot to reflect on because there were a lot of Math Talks. I really liked the shopping problem because, you know, I'm a girl, and I like shopping. Sometimes I don't get anything about a problem or pattern at all, then someone explains it really well and I get it. And I think, "Why didn't I see that?" That happens to me a lot.

The next time I think I could think a bit more because this time, now I thought I should have thought about it a little more, seeing all the other people's answers got me to think that I should have done better. Yesterday I wish I raised my hand before some other people because I had it up and other people were saying the same thing as me.

To improve I could get more time and after talking to Mrs Nguyen I understand more when she explains it better. The thing I found most difficult is problem solving & solve the patterns because it is hard to finish up mentally.

After talking to others it always makes more sense and they help. To improve I could get here earlier.

I think I'm getting stronger each day doing mental math and patterns. My favorite math talk this week was the pattern on 2-26-14, I thought it was clever because towards the end it wasn't the rule, it was your rule.

I really loved the math talk on last week's Friday. It was the buy one get one free or get 45% off. Because after you told us the answer and it could go either way, I thought to myself like how didn't I figure that out. So there's no right or wrong to the question.

I thought the guess-and-check that Cristian did was very helpful and it really helped me learned the strategy.

This week of math talks was very fun. I like the problem solving puzzles and equations. After talking to Skylar, she helped me understand the problems more.

This week I had better understanding with math talks. The one that really helped me was Janae's; it was different and extravagant.

My favorite day this week was "would you rather get 45% off everything or buy one get one free." Although it was simple, it was fun to share our different opinions.

One of this week's math talks equation that really helped me understand was Seth's equation from Thursday. It was when you divide money but you ignore the decimal and add it in later. This one helped me a lot.

On 2/21/14 math talk I did not get it at all, then when Diego explained how you were supposed to do it, I got it and that was good.

I am going to use different kinds of math strategies that everyone was using because some of the strategies help me solve a problem faster.

...

What I'm reading is that they value their classmates' different ways of doing mathematics; they benefit from them.

A month ago Sam Shah started Explore Mathematics! with his Advanced Precalculus class, then last week Sam shared with me what one student had written:




I choose to dedicate some of our math time to explore out-of-content mathematics because I can't afford not to.



Blessed

Some people make six-figure salaries, but I bet they don't get to feel as rich and as blessed as I do tonight when I read these two Friday reflections.



I feel like geometry is more of a puzzle rather than a chore. Having to find ways of proving something seems more like fun than a class. Learning about shapes and things I thought to be impossible are really cool. I can't wait for what amazing discoveries I'll learn next. [Grade 8]



You've really inspired me and now (more than ever) I want to be a teacher. [Grade 6]


I'm crying and thinking I should quit teaching because it can't get any better than this — like quit while I'm ahead. 

But you know what though. You, you, and you, and you over there, yes, you too — have helped me do this really hard job better than I ever could on my own. This is the best time to be a math teacher, and I thank you.



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