#WhyIStay #MfAProud

So this afternoon, I had an experience that reminded me why I love to teach math to middle school students. I’m a pretty patient guy, and I am a big believer in Jo Boaler’s ideas about depth and not speed and growth mindset. I have no issues with students who need more time to master ideas and concepts, and I’m often at school late and available to help.

In my 7th grade classes, today we took a quiz on operations with rational numbers: fractions, decimals, and mixed numbers were being added, subtracted, multiplied, and divided (including in yucky complicated fractions with expressions in the numerators and denominators). One of my students is a little bit anxious about math – his mom confessed to me at parent/teacher conferences that he often can’t sleep the night before a test or a quiz. Today, when the quiz ended, he seemed upset and asked, “Can I see you after school today?” and I clarified, “Do you need more time on the quiz?” and he said no, I just want to talk to you about it because I don’t think I did very well. I said okay, and he went off to 8th period.

Meanwhile, I had a prep 8th period, so I decided to start grading the quizzes from his class, and I was in the middle of grading his quiz when he returned after dismissal. I finished grading it, and we reviewed the mistakes. Some were silly mistakes – like 21 – 13 = 7 (which he admitted rushing through) or losing a negative in a few places. We checked out a minor calculation error in one of the problems, and then we addressed a major conceptual misunderstanding of dealing with subtracting involving going from positive to negative when there were fractions involved. For example, if we do 5 3/4 – 9 1/2, he was getting – 4 1/4, which is not correct. It should be -3 3/4. I drew a number line and we did it in stages:

start at 5 3/4, go back 5 to 3/4. Then go back 1 to -1/4. Then go back another 3 to – 3 1/4. That’s -9, and now go back 1/2 to -3 3/4.

Alternatively, we talked about how you could subtract the whole numbers and the fractions separately, but keeping their signs:

(5 – 9) + (3/4 – 1/2)

-4 + 1/4 = -3 3/4

From this, I even shared with him that I did a problem like this on the board in the other 7th grade class and I actually messed it up because I always get a little turned around right by the 0 (when we switch from having + 3/4 to having – 1/4), so that’s why on my number line, I jump down by the whole number part to +3/4, then jump by 1 to cross the 0, and then jump the rest of the whole number part.

He seemed thankful that I had explained things clearly to him – and he was clearly understanding the math better – but he was still nervous/upset about the grade on the quiz (which by the way, was an 18/25, a 72%). He said “What can I do about the quiz grade?” and seemed near tears. I decided I would let him redo the quiz right then and there. So I handed him the other version of the quiz (I always do two versions to avoid students copying) and he confidently redid it. His new score is a 23/25 because he only had one minor error – everything else was completely correct (He left without seeing his new score, but just knowing that he felt more confident about it).

I came home to find this email in my inbox:

Dear Mr. G,

I would just like you to know how much it meant to me today when you let me retake our rational number quiz after school. This year has been really different from last year because of the many times you have sat down with me after school, and helped me with what I have been having trouble with. Last year I had a math teacher who wasn’t the nicest person. So with that, and math not being my strongest subject, math was kind of a struggle. I would just like you to know how much I appreciate all of this, and how much this is affecting me in a positive way.

Thanks, <student name>

The positive impact I had on him from something so simple as spending 15 minutes explaining a concept to him one-on-one, the fact that I’m sending him the message that we’re not done learning math when the first quiz happens, the fact that I’m starting to change his mindset from fixed to growth – all of these are why I stay. I feel appreciated by my students this year, and I love them all so much.

Advertisements

Depth of Understanding vs Coverage of Content #MTBOS

I mentioned in my twitter that I’ve got about 5 different blog posts in my head that I haven’t managed to put to “paper” yet – and the one I’m taking the time to jot down tonight isn’t even one of them! (I’ve also got like two or three different drafts saved on here of ones I began but haven’t finished!).

Anyway, the one that I’m thinking about right now is about the constant battle that I find myself in: do I ensure that my students have a deep rich understanding of the topics at hand or do I make sure I cover every topic before the state exam? Without even the slightest hesitation, I tell you that I always opt for depth of understanding and mastery of the skills over coverage of content.

One thing I’m enjoying about my new school is that the other 6th grade teacher (there are only two of us) has been teaching the same content for many years (I actually observed her teaching in the same classroom during my graduate school placement back in 2009!), and she’s admitted she feels somewhat burned out and is excited to be working with me for all of the new ideas she gets from me. I receive some of her resources (and her advice/notes about the topics/order of the units/skills), and in turn, share with her my ideas for revisions and for next year. Some things she’s taken to more quickly than others, but I’m enjoying collaborating with her a lot, especially because she’s a mostly enthusiastic recipient of my ideas.

Our current topic is order of operations. She shared with me a bunch of her resources of things she’s done – some I really like/already knew (like Four 4s and Bowling), some I think are useful so I don’t have to make ten thousand expressions of my own (i.e. practice handouts), but some I think are repetitive and dull and not really helping students to grow their brains. I’ve been researching the problem with PEMDAS. I told my students it was banned, and if they HAD to use something, they should use GERMDAS, which I wrote as follows:

  1. Grouping Symbols (and gave examples, like the parentheses, brackets, braces, and absolute value bars, as well as the vinculum or the fraction bar)
  2. Exponents and Radicals
  3. Multiplication and Division
  4. Addition and Subtraction

>>>>>>>>>>>>>>>>>>>>>>>>>>> (from left to right)

We discussed that the paired operations were inverse relationships, and should be done from left to right. I thought I had done a pretty decent job of training my students to go beyond the basics that trip up high school and college students (stuff like 10 – 2 + 5, where they do 10 – 7 = 3 instead of 8 + 5 = 13). And then I kept reading and digging, and realized there are a couple more things I want to get my students to understand.

I want them to understand that the order of operations isn’t truly arbitrary! And that we can rewrite expressions to be more clear and less ambiguous by eliminating subtraction and division and rewriting them as addition of the additive inverse and multiplication by the reciprocal (also called the multiplicative inverse – in fact, I’m not honestly certain what the difference is between a reciprocal and the MI). So I decided tomorrow’s lesson is going to give the students some expressions that are intentionally ambiguous to uncover some of the potential problems they might still face AND also some problems where we can discover/use the “Boss Triangle” to talk about the hierarchy of operations and why it goes in that order.

I’m taking ideas from a bunch of different places:

http://mathforgrownups.com/the-problems-with-pemdas-and-a-solution/

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=16&cad=rja&uact=8&ved=0ahUKEwjBo5jolfDQAhWqBcAKHaT6CxoQFghrMA8&url=https%3A%2F%2Fwww.montana.edu%2Fem%2F2016%2F20160624PEMDAS.pdf&usg=AFQjCNHbNMIwtEuLjoc8bixwHapldUxHjQ&sig2=lzu3DKaIMf3V8pOPgSwTPg&bvm=bv.141320020,d.ZGg

http://www.mathematicalgemstones.com/gemstones/amber/pemdas-is-broken-and-how-we-can-fix-it/

 

I would way rather my students understand when and how we can ignore the “left to right” rules and the arbitrary order (and in fact that we can ignore the order of operations in certain cases and get the same answer because different parts of the expression don’t affect each other!), than for them to robotically repeatedly simplify expression after expression after expression…