Archive for the ‘MathET’ Category

History of Numeration Systems


I just stumbled upon this great little video about Ancient Numeration Systems.  It does not go into depth on any particular system, but it wanders through the following:

  • Tally marks
  • Sumerian symbols
  • Babylonian symbols
  • Egyptian symbols
  • Roman symbols and modifications of it
  • Number systems based on the body (Zulu)
  • Commerce-based number systems (Yoruba in Nigeria)
  • Number systems involving knots and string (Persians, Incans)
  • Numerals 0-9 (invented in India)
  • Place value
  • Fractions as a solution for “fair-share” situations in culture
  • Unit Fractions (Egyptians)
  • Fractions with base-60 (Sumerians and Babylonians), still used for time measurements today
  • Abacus (Chinese)
  • Use of the “bar notation” in modern-day fractions
  • Computation by the double-half method (Russian)
  • Computation by a doubling procedure (Egyptian)
  • Computation by an abacus (Europe and Asia), the “handheld calculator of its day”
  • Introduction of Arabic Numerals in Europe
  • Importance of mental math algorithms to check for reasonableness

This would be a great introduction video to a unit that involves Numeration Systems.

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Collection of Math Games


The page of digital and non-digital games has grown too long and unwieldy, so I’ve finally taken the time to reorganize the content by topic area. I’ve also added all the new “Block” games on various topics in Trigonometry, Rational Exponents, and Logarithms.

If you’ve bookmarked the old Games page, you’ll see that it now just tells you how to find the new sub-pages.

Direct links to the new game pages are below:

I’ve also decided to collect your suggestions for other digital and/or paper games, puzzles, and manipulatives  using a Google Form, but before you submit a game for me to review, PLEASE check it against my criteria for Lame Games.

Submit your suggestions here.

 

 

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Signed Numbers: Colored Counters in a “Sea of Zeros”


The “colored counter” method is an old tried-and-true method for teaching the concept of adding signed numbers.  However, to show subtraction with the colored counter method has always seemed painful to me … that is, until I altered the method slightly.

Now all problems are demonstrated within a “Sea of Zeros” and when you need to take away counters, you can simply borrow from the infinite sea.  Voila!  Here’s a short video to demonstrate addition and subtraction of integers using the “Sea of Zeros” method.  You can print some Colored Counter Paper here.

Video: Colored Counters in a Sea of Zeros

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Math about the Electoral College


This was a surprisingly good video about the math of the U.S. Electoral College system.  At first I kept saying “but wait a minute…” but all my concerns were addressed in the video, and then some.  I was surprised by the revelation (towards the end of the video) that it is theoretically possible (although not likely) to win the seat of President of the United States with less than 23% of the popular vote.  Wow.

There is some great math of ratios and percents here.  You can find data and other pertinent information about the Electoral College here.

You might also enjoy playing the Redistricting Game with your students, where you can “recast” who wins an election based on how you draw the boundaries on a map.

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Nature by Numbers


If you teach Math for Elementary Teachers or Math for Liberal Arts, you just have to see this Nature by Numbers video by Eterea Studios.

The Nature by Numbers website provides background information about the mathematics in the movie.

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Numenko: Math Game for Arithmetic


Although I can’t see using Numenko for many college classes (except possibly for MathET), it would be a great game to help adult students practice their arithmetic skills in Developmental Math courses.  If you run a tutoring lab or a math help lab, I could see having a copy of Numenko handy.

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NYT Opinionator Series about Math


For a few months now, the NYT Opinionator Blog has been hosting a series of pieces that do a phenomenally good job of explaining mathematics in layman’s terms.

The latest article is about Calculus (with a promise of more to come): Change We Can Believe In is written by Steven Strogatz, an Applied Mathematician at Cornell University.

There are several other articles in this series, and if you haven’t been reading them, you really should go check them out.  Assign them.  Discuss them in your classes.

Given the discussions we’ve been having about teaching Series and Series approximations lately on Facebook, Twitter, and LinkedIn, I wonder if he’d consider writing an article explaining “Why Series?” to students.

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Prime Number Manipulatives


tcm_blog_button2For those of you who are curious what we were actually doing in class in yesterday’s post (Record with a Document Camera and a Flip), we were “fingerprinting” the factors of composite numbers with their prime compositions. The wooden “prime number tiles” are created using the backs of Scrabble tiles and the white tiles you see on the board are the student version of the prime number tiles (each of them cut out a sheet of their own prime number tile manipulatives at the beginning of class).  Just for the record, I would like to confess to destroying more than 5 Scrabble games in the last 6 months.

I tell students that the prime numbers are like the building blocks of the whole numbers, in the same way that the nucleotide bases (adenine, guanine, cytosine, and thymine) are the building blocks of DNA. It was ironic, then, that at the end of our prime factorization fingerprinting, the board looked (to me) the way an agarose gel electrophoresis looks.

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You can also use the prime number tiles to line up the prime factorizations of two numbers (leaving spaces when the prime is not in both numbers) and read the GCF as the intersection of the two lines of tiles and the LCM as the union of the two lines of tiles.

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I’ve tried explaining these concepts in writing before (circling common factors and using colors or highlighting to show the steps), but the ability to easily rearrange the prime numbers after they’ve been found makes all the difference.  Students find the prime factorization using a factor tree or short division, then place out all the tiles for each number.  At this point, they can rearrange each row into least-to-greatest order.  Then line up the primes, bumping the whole row when there is no overlap in primes between the composite numbers.

We also were able to look at more interesting problems now that students could “see” the inner workings of the factorizations better.  For example, find a pair of numbers with a GCF of 42 and an LCM of 4620 (you may not use the “trivial” answer of 42 and 4620).  By constructing a “board” of prime number tiles, starting with the factorization of the GCF and the LCM, students begin to see which primes have to be in the two numbers and which can only be in one of the two numbers.  Note that there will be more than one answer here … which is another interesting discussion to have!

I’ve never seen “prime number tiles” in any math manipulative kit … they were just something I created last semester.  I think this method of using Prime number tiles would also be helpful with explaining how to find the GCF in the factoring unit of algebra, and with explaining how to find the LCD in fractions.

Feel free to print the prime number tiles and use them in your own classes if you’d like.

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How to Grade a Student Blog


tlda_blog_buttonLast semester I began using learning blogs as one of the assignments for Math for Elementary Teachers.  It was the first time I have ever used blogs as a graded student learning assessment, and I didn’t really know what to expect out of the students.  Would they all have created blogs before? [no]  Would they understand intuitively how to make hyperlinks, load in images, and embed videos? [no, no, and no] Would they write naturally in a conversational tone (in the style of most blogs)? [yes]  Would they make their blog posts two or three times a week (as directed) or would they cram them all in during the last couple days? [some of both]

Overall, I was thrilled with the results.  The students reflected on their learning, both in class and out of class.  They found and shared games, videos, articles, and vocabulary sites that they found on the web.  Some of them acted as a class reporter, summarizing what was covered in class each day (with their own personalities coming through).  Before you read the rest of this post, you might want to browse a few of their blogs to get an idea of the variety or writing and styles.

So let’s just say that this first time using blogs was a learning experience for both my students and for me.  I drafted a rubric for grading the blogs, and stuck to it all semester.  However, I realized that both the clarity of the assignment and the specificity of the rubric needed to be improved for “Round Two” (starting next week).

During the last round of blog grading, I revised my old rubric to try and tighten up the quality of the results.  Here are the specifics of the assignment now.

Set up a blog using Blogger or WordPress.  You should make at least six blog posts of at least two paragraphs each, using appropriate spelling and grammar.  The mathematics in your posts should be correct.  Blog posts should focus on what you have learned, what you’ve struggled with, or what you’ve found to help you learn.  Posts can discuss learning in class or out of class, but must relate to the current topics we are covering in the unit.  You should not refer to specific chapter or section numbers in your blog posts, and if you mention an activity from class, please use enough detail that a 3rd party reader would understand it.  Here are some specific details:

  • Blog posts should be spaced apart (not all at the last minute).
  • Your blog should include an appropriate  title (not just Maria’s Blog)
  • Your blog should include a profile (picture and brief bio). This can be fictional if need be.
  • Your blog should contain a “blogroll” with five of your favorite educational blogs.
  • Your blog should contain a list of tagged topics or categories.
  • Your blog should contain four images (or embedded videos) and should contain at least six links to web resources that you’ve found yourself.
  • Links to web resources should be properly “clickable” within the text of the post (not just a pasted URL).
  • Each post should be tagged with appropriate keywords.
  • You should make at least six comments on the blog posts of other students.

I think that the nature of the blog (what to write about) needs to stay as open as possible, but the fine detail of the assignments is difficult to assess if the quality of blogs varies wildly.   If you choose to try an assignment like this, I highly recommend a table-style rubric (like the one below) to keep track of where you are assigning points.

learn_via_blog_rubricI also found it helpful to use a screen-capture program (I used Jing and SnagIt) to make grading comments about specific blog posts (because, of course, you should not comment those in on a public blog site).

One last tip:  About halfway to the deadline, I give every student feedback on how they are doing so far.  I gently remind them about details that they might have forgotten so that they have time to correct or regroup.  I’ve found this results in immediate improvement in the blogs and is well worth the effort.  I use quick 1-3 minute Jing videos to give the feedback most of the time.

Note: You can see the rest of the learning projects and a “big picture” idea of how I fit all this in (timewise) by reading Transforming Math for Elementary Ed.

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Can we Teach Students to Understand Math Tests?


A few weeks ago, I gave a test where the grades were less than stellar. Whenever this happens, I try to sit down and reflect on whether the poor test grades were a result of something I did differently in class, a poorly written test, or a result of poor studying habits. After careful reflection and analysis of my own, I was pretty sure that this was the result of lack of studying (a theory which was verified … later in this blog post).

I was dreading the task of passing back these tests and prepared myself for an onslaught of questions aimed at trying to discredit the test (or the teaching).  Then I got one of those great last-minute ideas that come to you right before you walk in to face the students.  Maybe I should let them “pick apart” the test BEFORE they see their own tests.  The class in question is Math for Elementary Teachers (MathET) and I figured that a detailed test analysis would not be an inappropriate topic for us to spend class time on.

First, I made some blank copies of the test (enough for each group to have one).  I also created a handout with every single learning objective and assignment that I had given the students for each of the sections on the test (these are all available in their Blackboard shell, but  I compiled these in a paper-based handout that was 3 pages single-spaced). Here is what one section looked like:

5.1 Integers
• Describe operations on signed numbers using number line models.
• Demonstrate operations on signed numbers using colored counter models.
• Explain why a negative times a negative is a positive.
• Add, subtract, multiply, and divide signed numbers (integers).
• Know the mathematical properties of integers (closure, identity, inverse, etc.)
• Complete #1, 3, 7, 9, 15, 17, 19, 21, 23, 25, 37
• Be able to model addition, subtraction, and multiplication of signed numbers using a number-line model
• Be able to model addition, multiplication, and division of signed numbers using a colored counter model (why not subtraction? because subtraction is really the addition of a negative – treat it so)
• Read the blog posts about why a negative times a negative is a positive and be able to paraphrase at least two arguments in your own words.
• Study for your Gateway on Signed Numbers (be able to add, subtract, multiply, or divide signed numbers)
• Play with Manipulative: NLVM Color Chips Addition
• Play with Manipulative: NLVM Circle Game

When we met in class, I counted the class into groups of 3 students each. Each group received a copy of the learning objectives & assignments (by section) and a blank copy of the test. All of these were un-stapled so that the group could share and divide up the pages as they wanted.

Task #1: Look at all the objectives and assigned tasks/problems. Determine where these objectives, tasks, and problems showed up on the exam. (20-30 min)

Objective: Make the connection between what I tell them they need to learn/do and what shows up on the test.

Task #2: What didn’t show up on the exam? We discussed why these objectives might have been left off (for example, maybe it was not something that I emphasized in class) (5 min)

Objective: Make the connection between what is likely to show up on an exam and what is not likely to show up (with the caveat that any of the learning objectives are really fair game).

Task #3: Make a chart that shows how the points for each test question were distributed between the sections that were covered on the exam. We then compared the results from each group and compared these results to my analysis of the point-distribution for the test. (5-10 min)

Objective: Clearly see that it is necessary to study ALL sections, not just a couple of them.

Task #4: I passed back the individual tests. Each student was given three questions to answer as they looked through their tests. 1. Where were the gaps in your knowledge? 2. What mistakes should you have caught before turning in your test? (read directions more carefully, do all the problems, etc.) 3. What can you do to better prepare for the NEXT test?  I collected these, made copies, and passed them back to the students the next class. (10 minutes)

Objective: Take responsibility for your own studying.

It was here that students surprised me by being honest on Question #3. Most of them confessed that they had not studied at all, but now realized that they needed to start studying. I cannot help but wonder whether I would have gotten the same result if I had simply passed back the tests with no analysis.

Task #5: At the same time the students were looking over their own exam, I passed around one more blank copy of the test and asked students to write their score  for each problem (no names) on that problem so that we could see what the score distributions looked like.  When this was done, I placed the pages on the document camera one-by-one so that they could see the scores problem by problem.

Objective: To show the students that for many questions, students either get the question almost completely right, or completely wrong (you know it or you don’t).

Task #6: We had already covered the first section of the next unit, so I had the students begin a set of “How to start the problem” flashcards. On the front of the flash card, they wrote a “test question” for the new unit. On the back of the card, they wrote some tips for starting the problem and details they might otherwise forget.

Objective: To begin to see tests from the perspective of a test-writer instead of a test-taker.

Take-home assignment: I told each student that they must come to the next test with at least 5 flash cards per section (35+ flash cards). I suggested that a great way to study would be to swap cards with each other and practice with someone else’s questions. I checked to see if they carried through (although I assigned no consequences if they didn’t). All but two carried through. Anyone want to guess how those two fared?

Results: I just finished grading the latest test (signed numbers, fractions, and decimals – not easy topics), and the test results were almost all A’s and B’s (instead of C’s and D’s on the previous test). Most students left the test with a smile on their face, and several finally got the “A” they had been trying for all semester.

Three questions for you, my busy and wise readers:
1. Will the students persist in better study habits now? For example, will they study like this for the next exam?
2. Would this detailed test analysis have had the same effect if the class hadn’t just crashed & burned on a test?  (i.e. is it necessary to “fail” first)
3. Was this a good use of class time? Would it be good use of class time in a different math course, like Calculus?

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