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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Group work pt II

So how does a Math Rules! volunteer manage a small group effectively? I found an article online that is pretty dense in content, and doesn't even address group work in schools. I still thought that many of the main ideas apply to working with a small group of elementary school students. The Personal Functions section is especially nice to note. The Managing conflict, Stopping conflict escalation, and Disruptive Behaviors sections are important for volunteers who have been having a grand ol' time trying to manage their small groups.


This article called Promoting Problem-Solving Skills in Elementary Mathematics is also quite useful for managing and provides best approaches to math group work. I have seen the teachers I work with use the Problem of the Day and Whole Group Learning techniques as the group lesson for the students and also for me to guide the group work afterwards. My teacher would write a problem on a flip chart, ask the students to copy down the problem in their notebooks and then to solve it. Afterwards, she asks for volunteers to explain what they got and how they got there. Directly from the article, the big group lesson for the day allows students to:

o identify the parts in the question
o find the best problem-solving strategy and explain why it is the best
o describe two different ways a problem could have been solved
o share student-generated questions
o ask other students to solve the problem and justify their answers

These skills and approaches to teaching are also useful in small groups. The article outlines that students benefit from being in mixed-ability groups (to encourage less motivated/focused/confident students to work with and learn from the more motivated/focused/confident students) as well as similar ability groups. Hopefully, Math Rules! groups are more similar ability small groups. I've also mentioned before on this blog that even in "similar ability" groups there is a lot of variation in specific abilities.


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Group work pt I

Before coming to Boston Partners in Education and becoming a Math Rules! volunteer, I had never worked with and helped tutor a small group of students on math before. My personal experience with math in elementary school and higher was doing individual math homework and discussing problems in class. I rarely worked with another student on class problems except in fifth grade. I was given an "advanced" math book and told to work with my good friend on the advanced math. I somehow got ahead of her on the homework, and ended up working mostly by myself with the occasional group work when helping my friend.

That ends my experiences with math group work. In other subjects, I dreaded group work. Group projects usually meant I would end up doing a significant portion of the work while other groups members slack off and still get credit for the work I did. I would later learn that this is called social loafing and happens in all kinds of group work activities. I think the main problem with group work is that teachers don't manage the small groups well enough. It's one thing to put students in a small group, but another to make sure all groups and individuals are contributing equally.


Math Rules! provides more structure to small groups, with an academic mentor, the group isn't supposed to exhibit social loafing. The way that Math Rules! is set up builds off the Investigations curriculum that Boston Public Schools uses for elementary schools. I honestly didn't know what Investigations was until I went to Math Rules! training. Even then, it wasn't until I started volunteering in an actual classroom to understand how the curriculum works. Investigations is a different approach to math education with more focus on the multiple approaches and perspectives to a math problem. Showing and explaining your process is more important than the final answer itself. The Investigations curriculum also encourages and works best in small group situations, where students work together to come up with possible solutions for a problem.

Many studies have shown that small group work is ideal for students. Although I couldn't get the full article, this study summarizes that "pupils in the fifth grade produced superior answers on questions requiring original contributions" and that group work creates "classroom conditions that foster positive social interaction and productive intellectual activity." The article also notes that small groups allow students to develop "investigation and problem-solving [skills], with pupils cooperating in seeking and interpreting knowledge from a variety of sources." On top of it all, this study was done over 30 years ago. Other studies and many years of research have shown that group work is most beneficial for students in a variety of subjects and across age groups. "Group work also helps students practice essential social, problem solving and communication skills needed for success in the workplace." Students who never get a chance to work with other students face difficulties in the future jobs that more often than not require working in groups and teams.

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Tutoring resources pt. I

Hello math tutors and supporters! This blog post contains resources and interesting things to think about when working on math with students:


Short and sweet article on tutoring tips to use with students

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TERC's library of free information about the Investigations math curriculum that the Boston Public Schools uses. Includes overviews of what grades cover in the year, sample sessions and even online teacher resources.

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I've also been reading through a San Diego tutor's blog with tips and advice on math tutoring, as well as other student related topics. I haven't read through the entire blog, but what I have read is pretty good advice that I will definitely use in the future.

Although many of his posts are about higher math (middle school, high school, and standardized tests), there are a few highlights from Mr. Casteneda's blog that talk about tutoring in general:

The importance of students knowning their multiplication tables.

Leading questions I use often to get students to explain their reasoning.

Comments on ADHD. Personally, I agree with Mr. Casteneda on ADHD. It came out of nowhere as a behavior problem for students and children and everyone started medicating and diagnosing children for it. Back in my day, ADHD didn't exist! Or more accurately, ADHD wasn't a 'condition' that had a list of symptoms. From the previous blog post: "I wish more parents would [take] the time to bust the "ADHD" label into pieces, and figure out what exactly is going on behind appearances in the particular case of their child." Although I don't know very much about ADHD or the medical procedures for diagnosing ADHD, I feel like ADHD is a label adults and children use to avoid figuring out and working to solve the root cause of attention deficits.

A blog post on encouraging postive thinking in students who don't feel like they're good at math.

And finally, a post on the unexpected skills tutors gain.

Math Education Funding

So I've been looking for math and tutoring blogs to read through and learn more about math and science education. I have a few resources and links that I will be putting up tomorrow, but today I'm going to talk about this article that I found at Wild About Math.

It's about a $200 million donation from Intel to support Math and Science programs and educators. A large portion of the money will be going to professional development and training for math and science teachers who are looking to expand their teaching practices. I hope some of that money will be going to support Boston Public School teachers. I'm hoping that Obama's "Educate to Innovate" campaign will be successful. In the press release, President Obama announced "new and innovative partnerships involving major companies, universities, foundations, non-profit organizations and government agencies" that will provide training programs for math and science teachers.

Video link to President Obama's speech on the partnerships

It's interesting also because President Obama's announcement today of a three-year government spending freeze threatens our education systems across the nation. BBC says "The spending freeze covers many domestic programmes and departments to which Congress allocates budgets each year, including agriculture, transportation and education...Security and defence spending, foreign aid, social security and spending on healthcare for the poor and retired would be exempt."

I'm not too sure what the implications of the spending freeze are or if it will actually affect education, but there's a good chance governmental funding towards education will be put on the backburner (very much like No Child Left Behind).

Math Rules! Blog

First week of volunteering! I met with one of our first new volunteers JD today at one of the schools. Our volunteers will be starting all of this week and next week, I'm trying to meet with most of the new volunteers so they know where to go and who to contact.

I'm going to start volunteering on Wednesdays, I'm both excited and nervous. Being the new kid in class is tough. What if the kids don't like me? What if the teacher doesn't like me? What should I wear? I hope I don't forget all my math! I'm also going to three different schools and four different teachers, a whole day of math tutoring. I'm looking forward to being out of the office for a while. It's a shame I'm going to miss the Math Rules! training at Boston Partners though.

As Math Rules! Coordinator at Boston Partners in Education, this is part of my AmeriCorps and Massachusetts Promise Fellowship year of service. The blog is meant to be a resource for the Math Rules! volunteers, as a forum to document the Math Rules! volunteer experience. I hope other volunteers will find it useful, but we'll see.







More updates to come!
-MN