Lets do homework!

And now for something completely different: mini break from Math Awareness Month for an update on my tutoring sessions yesterday. Don't worry, I'll put up today's MAM post later.

I worked with some of my fourth graders yesterday. There was another sub in the classroom, but this sub had things much more under control. He put free time on the table and asked for suggestions, but since the kids couldn't settle down he handed out homework and made them work quietly for a few minutes before they got free time.

I was one of those kids in school who liked to do as much of my homework as possible so I could go home and chill or do whatever. My two boys decided they would rather finish up as much as they could instead of play Uno with the rest of the class. One of the boys told me about his new BMX bike and how excited he was to go riding after school. I teased him and asked about the rain outside and how he could bike while holding an umbrella. They were so into their work, it was awesome! I reviewed long division with them to convert fractions into decimals. They picked it up really quickly, but when we tried it again (which they suggested for another problem) I feigned ignorance.

This is roughly the conversation we had:
Student - "Let's do division to get the fraction."
Me - "How do we get started? I don't know how to do it."
"What? Did you forget it? You just did it! I don't know how to do it!"
"Okok I can help you guys out..."
*teasing noise mehhhh~* "I thought you didn't know how to do it..."

I loled and we finished up with the math section of homework.

I had misjudged my other student. He has troubles focusing in the large classroom format and forgets to follow good student behaviors, but he was the one who mentioned he wanted to work with me and reminded the substitute. I was surprised when he focused through the math and almost quit halfway through. Positive peer pressure (or leadership as I've recently learned) from my other student "Lets get it done so we don't have homework!" and they started up on their reading comprehension handout.

They had to read through a passage and answer questions about roly polies. I was amazed at their writing skills. They rephrased the question "because you can't start a sentence with because" before answering and even underlined their evidence for their answers! They read it out loud, taking turns, and even asked me to read a section for them. I was just blown away at these two. My less focused student was really into the writing section, he took his time writing everything out and he really understood the passage.

It's unfortunate that when we're working with one student on one subject that we miss out on their other strengths. It's days like that when I wish I was a full time teacher, so I could have much more time with these students to see their full potential.



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Computer math games

Calculation Nation a website our ED sent to the office. It has eight math games that are pretty challenging. I'd say it's targeted towards middle schoolers because some games involve fractions, factors, multiplication, strategic thinking, and some luck. I've been trying to beat the computer on Ker-Splash, and I haven't even gotten close!

Good luck and may the math be with you!

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Greg Tang: Place Values

Part two of the workshop is place values and Greg making us compute in different bases. Place values boiled down are groupings of different size units. Although many students learn about place values in elementary school, we don't spend enough time teaching students to really understand place values enough so when they get to adding and most importantly subtracting, that they understand regrouping or borrowing and why they need to borrow from the next place value.

In the number 43, the 3 represents 3 ones, and the 4 represents 4 tens. Our number system is in base 10, which is also called the decimal system.

Base 10 means grouping place values by 10s. A group of 10 ones can be grouped into one group of tens. A group of 10 tens can be grouped into one group of hundreds, etc. Greg simplified it by saying, 10 is too many, so we group things into groups of ten.

We had to switch to a different base, where place values are different depending on different size groups. We did some exercises in base 5, and changing it back to base 10 where we understand the numbers better. Changing bases helped me understand place value a lot better and even helped me understand enough to be able to explain to my students.

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Even in "real life" math, we do deal with different bases, and a firm understanding of place values. For example, in dealing with time, we have to switch to base 60 to figure out elapsed time. Greg's example was finding the total elapsed time for:

1 hour 45 minutes and 30 seconds
1 hour 19 minutes and 50 seconds

When we're adding up the seconds, you get 80 seconds, but of course, no one would leave 80 seconds in the final answer, you have to regroup for a group of 60 seconds, or one minute, and have the remaining 20 seconds.

With our regrouped minute, we add to 45 minutes and 19 minutes, which equals 65 minutes. Again regrouping into 60 minutes (1 hour) and remaining 5 minutes, we add to the hours place value.

Ending up with:

3 hours 5 minutes and 20 seconds

instead of:

2 hours 64 minutes and 80 seconds

Which mathematically makes sense, but no one in the real world would be comfortable leaving an answer like such. It goes to show that anyone who understand time and time conversions have an understanding of base 60, but very few people talk about changing bases.

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Having a firm understanding of place values is incredibly important for doing fraction and decimal equations.

4 and 5/7ths + 3 and 5/7ths = 7 and 10/7ths, but improper fractions are usually frowned upon, so students have to understand that in this case, 7 is too much and you have to regroup to get a proper fraction. Regrouping the 10/7ths into 1 whole and 3/7ths gets us to the "proper" answer:

4 and 5/7ths + 3 and 5/7ths = 8 and 3/7ths

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Where this comes most handy in most Math Rules! math is subtraction and fractions & decimals. Helping students with regrouping in subtraction can be easier for this problem:

800
-347

Instead of regrouping twice for both zeros, thinking of 80 tens, and 0 ones and regrouping once to get 79 tens and 10 ones. Which is much easier and doesn't involve regrouping twice.

79 tens | 10 ones
- 34 tens | 7 ones

It also works for bigger numbers, to see that
1000
-458

Can also be seen as:

100 tens | 0 ones
-45 tens | 8 ones

And regrouped once to get:

90 tens | 10 ones
-45 tens | 8 ones

Making this problem a lot easier to solve with less work.

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I'm looking forward to the third workshop with Greg because like I said yesterday, it's a revolutionary way (for me) of looking at math problems that will help me help my students. Not only do I feel better at math, I have a firm grasp of number sense and Greg's two main concepts of partial sums and place values which will be useful for working with 3rd - 5th graders.

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We like fractions!

Yesterday's tutoring involved an introduction to fractions in my 4th grade class. I think it's cool to see the intrinsic nature of understanding fractions. For students, doing math computations with fractions quickly becomes difficult. Even as adults, we shy away from fractions, decimals, percentages, and ratios, which all express the same ideas. It's hard to conceptually think of a quarter of a third or to add up 1/6 a teaspoon to 3/4 of a teaspoon and even more so to compute complex fraction equations. Personally, I can't verbally explain what happens when you divide a fraction by another. Completely out of my realm.

However, being able to divide something into fractions and the visual understanding of equal parts is something kids pick up quickly. Our class was dividing up 4x6 arrays into fractions, like a quarter, an eighth, thirds, and sixths. The arrays have 24 smaller units inside, but my students did very well with dividing up squares and rectangles into thirds then sixths. I pushed them further and asked what 1/3 or 1/6 of the whole represents in units, which they quickly got as well.

One of my students came up after we had finished and declared, "I like fractions!" which I thought was nice. I just hope she continues to like fractions when you start adding different denominator fractions...

Mr. Steven Strogatz's newest article is also about the existence of fractions and the ensuing chaos of irrational numbers.

More cartoons of fractions.

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Too much candy?

The kids were a little hopped up on an entire day of sugar last Friday. It took a lot more effort for me to keep them focused on their work. They had a substitute so we had a review worksheet the kids had to get through. Adding and subtracting decimals and then comparing decimals ( < , > , = ) that sort of busy work. I managed to convince them that the comparing decimals side was much easier, and they whipped through it. The other side was a lot more tedious and boring for them. I got a lot of "can I go get a drink?" and "can I go sharpen my pencil/can I go get another pencil?" today. I put up with it and let them go as needed, but compromised with their work.

One of my girls is extremely studious and finished the worksheet by herself before the other kids had a chance to finish just one side. One of my other students is a very sharp student but had a lot of trouble focusing today, so I bargained with her. She could finish the rest of the addition problems if she could make up a fraction of what she had finished. At first she balked and said that was too hard. Fortunately for me, I pushed her a little more and worked through it with her. We talked about numerators and denominators and came up with a reduced fraction representing the addition problems she wanted to do. I also used this trick with the boys on Friday because neither one was very intent on finishing a review worksheet.

In summation, it was tough to keep all four students focused on their work, but I managed to trick them into doing "less work" with a challenge question that actually required more work than simple addition and subtraction.

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Steven's third installment of his math column discusses negative numbers.


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New Year

It's been a while since I last updated and I apologize for the hiatus. Happy New Year! Here's to hoping that 2010 will be a great year for everyone out there.

I missed out on volunteering last week and didn't get a chance to catch up with my teachers. Unfortunately, my schedule will be changing in the near future and I need to figure out some logistics. I hope this means I'll have more time to blog about different experiences.

I met with one of my groups this week and we went over a fraction and decimal packet. The students had a hard time focusing this week because the lesson was set up differently than usual. All students in the class were rotating between the teacher, and two volunteers. My kids stuck with me in the group and we managed to get through the entire packet with some time to check over answers.

I've been trying to help my small group with multiple choice questions and figuring out the best way to get to the right answer. It's tough because most of the students I'm working with don't want to check their answers, they'd rather get the work done and go on to do other things. It might just be a kid thing, but I'm trying to get them to check their answers against each other when we're all done with problems.

There was one question in particular where the kids were adamant that their answer was right. I said I agreed with the student who got the question correct and then asked him to explain to the others why. As soon as he finished explaining his reasoning the others quickly said "oh yeah, that's what I thought too." Having their peers explain is INCREDIBLY helpful for the other students to understand and it reinforces the concepts or problems they're explaining.

I've been introducing the elimination method for taking multiple choice questions. I ask them to X out any answers that don't make sense. I'm not too sure they're catching on though. We worked on a problem that asked for the number higher than the population of NH and less than Maine. So we went through and eliminated any choices/numbers that were more than both populations. However, when we went through the questions again to eliminate any choices/numbers that were less than both populations, some of the kids got confused and then frustrated. I think we'll go through this again as the year progresses.

Candy canes and fractions

Yesterday was even more miserable than in my previous post. I woke up to snow on the ground, which quickly progressed to blustery wind and freezing rain. I went and bought candy canes for my tutees because this is the last week I'll be working with them until the beginning of next semester. Walking to my schools, I think the candy canes comforted me more than my soaking socks and wet pants.


Working with my horseshoe group, we had a handout that reviewed lots of fraction work, converting to decimals and the like. I thought the session went incredibly well because my kids are warming up to me and we talk about other things in between problems. When I told them I wouldn't see them for a few weeks, one of my students got worried. I reassured them I would be coming back in January and she seemed ok with it.

Working through the worksheet, I showed them a shortcut for making fractions into decimals when the denominator is 5. They can just multiply the original fraction, both the top and bottom by two to get a fraction over 10, which is a lot easier to change into decimal form. After showing them the shortcut, I came up with problems not on the worksheet.
"But that's not on the worksheet!"
"Yeah, I know. I'm trying to challenge you."
And they really stepped it up and applied what they had just learned to the new problems. It was nice to know that they could adapt to new math rules and tricks.

When working with fractions, I find it really helpful to draw a fraction circle and ask them to color in sections. We also had a set of decimals they needed to change into percents which involves moving the decimal point. Once they understood the rule, they raced each other to finish and I 'quizzed' them on their answers afterwards.


I was so impressed and told them they were superstars (I even drew stars on their papers) because we finished early. I let them choose candy cane flavors and said bye for now. I'll miss them while I'm on vacation, but I'll have something to look forward to when I get back.

Connections

After a few weeks of volunteering, it seems like I'm slowly but surely building relationships with my students.

One of my groups has been touch and go because the math format changes from week to week. Sometimes we're working on reviewing for a test and playing a Jeopardy game, and other weeks we're doing group work on a new topic. This week we worked on arrays and played a game with different size arrays. Only two of my students were at school yesterday which was nice. I had more attention to give to both of them, and I felt like I got to know them a lot better because of the small size.

One of my students shared stories with me while we were playing the game, and it's really helpful to connect with them. I talked with her about the Nutcracker and how their class is going to the Urban Nutcracker performance in a few weeks. She also told me she practiced her multiplication tables in the morning before coming to school and the roles her parents play in getting ready. I'm glad she feels comfortable enough to share with me and to talk freely. My other student I hadn't had much of a chance to talk to, but he's an energetic and eager young man who knows his arrays and multiplication tables well. He also enforced taking turns "I got the game cards, you gotta spread them out now." Overall, it was a very productive session. I also let the kids pick arrays for me to do with them and they tried to stump me, but it didn't quite work. :)

With my Horseshoe group, we moved out of the classroom and working at a table in the hall. I got to chat with them as we were walking from the class and found out their ages and got to connect with one of my students who wrote a story about his favorite animal.

This session was a little more tough because we were working on fractions, changing to decimals and adding different denominator fractions. It's tough even for adults, so explaining to the kids was difficult. We started with decimals first and the kids seemed to know what they were doing when we went over it as a group, but when I asked them to do it by themselves they struggled quite a bit. I was a crutch for them in guiding them through the long division, but I couldn't get them to understand the long division steps on their own papers.


We managed to get through maybe half of the problems, then we switched to adding fractions. This worksheet seemed easier for the kids and we walked through those steps to get their answers. I skipped problems that would've taken more time to explain so that we could understand the process clearly.

One of my students took it upon himself to try the hard problem and got frustrated. He didn't quite finish the problem and seemed down, so as we were walking back to class, I helped him go through the rest of the problem and he did just fine. I wish there was more time to work with each student one on one because he wouldn't have gotten to that frustration point if I had been sitting right next to him. He's independent and didn't want to ask for help, but if he had had the help, he wouldn't have been so disappointed in himself. I tried to cheer him up and told him he tried a really hard problem and managed to get pretty far, which I think he appreciated, but I'll try to give him more attention next time.