Thursday, October 29, 2015

Can you remember more than 7 digits?

The other day, I came across this website that tests your ability to remember digits.

I thought it was interesting that, according to the website, the average person can remember 7 numbers at once. I've heard this before. This is supposedly the reason why telephone numbers are 7 digits long. At this point, I'm sure you're wondering if you are an "average" person. So, go try it...

Did you do it? I did it a few times myself and the farthest I got was 12 digits (my worst was 10). This probably means that I'm a superhuman or I have evolved past the rest of you. I'm sorry, but your days are numbered. (Numbered! Get it? No, of course you don't.)

I was still curious about this 7 digit claim, so I posed the problem to my students. Can the average person really only remember 7 numbers?

I had all of my students load the website and play along. After everyone was finished, I recorded the results and made a line plot with the data.

I asked the students to talk to their neighbors about whether or not this data confirms that the average person can remember 7 digits. Overwhelmingly, they felt pretty good about it, especially since the median of the data was 7. (I should note that sixth grade standards are all about analyzing distributions.) They were also able to see that more than half of the students were able to remember at least 7 digits, but less than half could remember 8 or more. Another reason to believe the claim that the average person could remember 7 digits.

We then discussed strategies for memorizing the numbers. Some students mentioned that they chunked the data...remembering 62 as "sixty-two" instead of "six-two". Some of them would practice typing them to build the motor memory. 

I also shared a couple of my own strategies...sometimes I could associate a number with something. For instance, once I saw a 53 and, for whatever bizarre reason, I remember that as Bobby Abreu's jersey number. Once I had that image of Bobby Abreu in my head, I stopped worrying about remembering 53. For the longer sets of digits, I would repeat the second half of digits over and over again while staring at the first half of digits. This way, I was relying on both my visual and auditory memory.

Now that the students had some new strategies, I gave them another chance to increase their digits.

As you can see, the data changed, but there really wasn't much improvement. Many students did worse while a few did marginally better. We couldn't make much sense of it, though we suspected that some of these strategies need to be practiced before we could see some results.

At this point, it would have been nice to keep practicing to see if we could improve, but my period is only 37 minutes long. I also had a couple of situations where students figured out they could copy and paste their answers. Cheating would be difficult to monitor.

Side note: Some of my students with IEPs could only remember three digits. This was consistent each time they made an attempt. This was eye-opening for me...when short-term memory is so weak, learning anything must be a huge struggle.

Saturday, October 24, 2015

Why, Common Core? Why?

The other day, I was checking students' work on mean, median, and mode. One of the problems involved finding out what grade you would need to get on a fourth test to have an average of 85 for the class. It's basically a mean problem in reverse, and for students who have never solved this problem, it can be challenging.

One of my students was struggling with this and wrote in her notebook, "WHY COMMON CORE WHY". I laughed and assured her that this problem has been around a lot longer than Common Core. What I really found amusing was that, in terms of content, this sixth grader really hasn't been exposed to some of the more unique things about Common Core. Most of that is happening in elementary school and Pennsylvania only switched over last year, when she was in fifth grade.

In all likelihood, this girl's hatred towards Common Core probably stems from something she overheard her parents say. And now, every time I present her with a challenge, a little voice in the back of her head is going to tell her that this problem is Common Core and it's not really important for her to figure it out. And that's all she needs...another reason to give up.

Tuesday, September 29, 2015

Warm-Ups with a Purpose

Warm-ups last year:

I would display four or five review problems on the Smartboard for students to work through as I took attendance. I would then walk around the classroom to see how students were progressing, but would often struggle to help very many of them, nor would I have a good sense of how the class did as a whole. We would then review every problem which was time consuming and not always helpful. The next day, I would create a few more warm-up exercises but I never had a clear picture of what my students were still struggling with or why.

Warm-ups this year:

I was asked to move into a new classroom where every student would have his or her own computer. Over the summer, I looked at several websites that would help me use formative assessment on a daily basis. I was happy to find Socrative (which is FREE!) and I use it everyday for my warm-ups. Students can quickly log in and start working on the exercises. I can create multiple choice, true/false, or short answer questions, and as students are answering them, I can see their responses live! It looks something like this...

This is kind of a big deal. As soon as a student gets something right or wrong, I know. And there's a lot I can do with that information. During those exercises, you'll routinely hear me say things like...

"Mary, awesome job on that last one. Everyone's having trouble with it."

"Almost everybody's getting #1 wrong. Make sure you read it carefully!"

"Sheri, that last are you supposed to set up an addition problem with decimals?"

"Fawn, you seem to be having trouble with greatest common factor. Can I see your work for that last problem?"

"Hey, Andrew. Where's your notebook? Stop trying to do the work in your head. You're not Rain Man!"

After the students finish the exercises, I share the results with them and I let them tell me which ones we need to review (and which ones we don't). We look at commonly selected wrong answers and think about what mistakes students were making.

At the end of the day, I can throw this data onto a spreadsheet (shown below) and decide which topics/skills students have a firm grasp and which need further review. I can see how students progress in some skills over time and share that as a model of learning.

I love that students are getting instant feedback. I love that I have evidence of their growth. I love that we can review results as a class and, rather than students only focusing on their own mistakes, we can ask ourselves, what are we, as a class, doing wrong? What are we, as a class, doing right?

Sunday, September 20, 2015

Quizzes without Grades

A few weeks ago, I blogged about how I was going to stop putting grades on quizzes. This decision was heavily influenced by Dylan Wiliam's ideas from his book, Embedded Formative Assessment. I also need to mention that Ashli Black has very helpful explaining how she does comments-only grading and pushing me to design a system of grading that works.

This past week, I was finally able to test-run this idea after the students took a quiz on the Order of Operations. I explained my reasoning to the students and, for the most part, they seemed to be okay with it. I told them that this creates a better working environment where students can feel less embarrassed about their performance and work together to identify and correct their mistakes, no matter how well they did. I marked the quizzes by circling the problem number for every wrong solution and then color-coding three problems that I wanted the student to correct. If a problem had a pink mark, they had to identify their error. If there was a purple mark, they had to rework the problem. If a student did not get anything wrong, I gave them a more challenging problem to solve. Finally, while grades were not written on the quizzes, they were calculated and recorded into the online gradebook so parents and students could see them at home.

Overall, I thought it went really well. The students had about 10 minutes to work alone or together on their mistakes and handed the quizzes back to me. Those who did not finish had extra time overnight to do so.
The next day, I used socrative (an online quizzing tool) to ask my students how they felt about my "no grade" policy. The good news is that 70% of my students either liked it or didn't care. More students liked it than didn't like it. However, there is still 30% of my students that didn't like it. While it was not obvious in their responses, I believe that this frustration comes from not having that instant gratification of knowing what your grade is. This impatience isn't unexpected. Many times students will ask me if I graded their quiz ten minutes after handing it in.

In the end, I think the benefit of students revisiting their work and working together to fix mistakes outweighs the annoyance of not getting their grades right away. I'm hoping that, over time, students will begin to also see that benefit.

As a side note, I should say that I'm not really doing "comments-only grading". I had considered writing out comments, but it occurred to me that most of what I'd be writing could later be discovered by the student upon more reflection or figured out with help from a classmate. I believe that writing comments on every wrong answer would have been extremely time consuming and would have deprived my students from discovering their own mistakes.

Thursday, August 20, 2015

Motivated by Stature

Many people measure their success by comparing themselves to others. If they are at the top of that food chain, they will feel a sense of superiority and will likely continue to do well with as much effort to preserve that. If they are at the middle or the bottom, they will likely withdraw over time as they’ll never be able to surpass those at the top.

Anything we do in our classrooms that feeds into that culture will ultimately harm all of our students. What is needed is a belief system where people are not defined by their class rank but where everyone, including those at the top, has potential to improve.

I made the decision this summer to switch to comments-only grading which I believe will help instill this belief. Students will no longer be able to compare their grades with other students to determine where they fit in this hierarchy. All students will be asked to extend their thinking, including my highest-performing students. However, quiz feedback is just one small aspect of everything I do in a classroom. I can’t help but wonder how many of the other interactions I have with students might imply that I value ability over effort.

My hope is that, throughout this coming school year, I will regularly reflect on how my interactions with students, which are often subtle, might help or hinder this outlook.

Monday, August 3, 2015

Spaced Practice and Repercussions for Teaching

I've been reading John Hattie's book, Visible Learning, in which he ranks the effect sizes of different strategies that help student achievement. One of the strategies that is pretty high on the list is that it is better to give students spaced (or distributed) practice as opposed to mass practice. In other words, rather than having a student practice something over and over again in one day, it is much better to spread that practice out over multiple days or weeks. (You can read one of these studies here.) The main benefit is that spaced practice helps with long-term retention.

While this research certainly gives some justification for providing students with multiple opportunities to revisit older topics, I am left to wonder if this should change how I structure my lessons and assessments. I, like many others, teach by units. My students might spend a month on fractions followed by a test. They then get a month of algebra followed by another test. We, as teachers, create this span of time when all learning about a particular topic must happen. We don't always give students the time to practice these ideas, particularly the more challenging ones that almost always happen at the end of the unit and right before the test.

Based on what I've read about spaced practice, I would propose that teachers shouldn't give tests at the end of a unit. Perhaps students need time to practice these skills over several weeks before you should assess them. This is something I'm going to explore this year with some of the concepts that were challenging for my students last year.

Note: This is probably not an original idea and I'm sure someone else out there has probably explored it. If you have any resources to share on the subject, I'd greatly appreciate it!

Another note: I do allow my students to retake quizzes which I had hoped would send the message that learning doesn't stop after the quiz is taken. However, very few of my students have taken advantage of this in the past. I am hoping to correct that this year with some ideas from Dylan Wiliam, Ashli Black, and others.

Update: Henri Piccioto has written about this and calls it "lagging homework". He also reinforces the idea that quizzing should happen much later then when the material was taught. Thanks to Mary Bourassa and Chris Robinson for helping me find his work!

Sunday, August 2, 2015

Movie Popcorn

I ordered a small popcorn at the movie theater and the cashier asked me if I'd like the large size for only $1 more. I knew that this had to be the better deal, so I took it. I mean, what if I had gotten the small popcorn and ran out during the movie? That would be unacceptable.

However, as I left the theater, I noticed that I didn't actually eat all of the popcorn. There was about two and a half inches of popcorn left at the bottom of the bucket. I could take it home with me, but stale popcorn doesn't sound too appetizing and I decide to throw it away. Did I just get ripped off? Should I have just bought the small popcorn?

There's a couple of ways of modifying this task to address the needs of different grade levels. It all depends on what information is given to the students. If you can just give the students the number of cups of popcorn in each bucket, then this is a fairly simple unit price problem. If you just give dimensions of the buckets, you will need to derive and use formulas. It would also be extremely helpful to use a spreadsheet.

6th Grade Version:

Info required...

Questions to explore...

What is the unit price for each size?
What is the percent change in size, price, unit price?
What is the least amount of popcorn from the large container (in cups) you would need to eat so that you don't get ripped off? (This is not as interesting a question as the 8th grade version because you can't usually tell how many cups of popcorn are left in a bucket.)

8th Grade (or beyond) Version:

Info required...

Volume of a truncated cone:

You will notice that there is a little bit of popcorn above the rim of each bucket. There is also a small gap on the bottom of each bucket. I assumed that the added and subtracted volumes of this popcorn would more or less cancel each other out. I could be wrong about this!!!

Questions to explore...

What is the capacity of each size?
What is the unit price for each size?
What is the percent change in size, price, unit price?
How many inches of popcorn would be left in the large bucket if you eat just as much as the small bucket?
What is the least amount of popcorn from the large container you would need to eat so that you don't get ripped off? In other words, how many inches of popcorn can I leave at the bottom of the bucket?

The answer....

I'm not leaving my full solution here because I'm curious to see how others might solve it. Basically, I used a spreadsheet to test different heights of popcorn eaten to determine where the unit price of the large matches the unit price of the small. If you think about it, this is further complicated because as you eat popcorn, the height AND top radius changes. You will have to come up with a formula that calculates the top radius based on the height.

I determined that you get ripped off if you leave more than two inches of popcorn at the bottom of the bucket.