Monday, July 18, 2011


Sometimes I find myself looking for some sort of a “cheat sheet" (for lack of a better term because it is not being used for cheating) that will sum up all I need to know in one place. This holds true when trying to organizing my brain around what I should be thinking about when I teach math, especially from the angle of teaching it effectively and making it relevant to my students.  I believe I may have found a good summary, thanks to Jennifer Suh, assistant professor of mathematics at George Mason University in Fairfax, Virginia and her article “Tying it all Together: Classroom Practices that Promote Mathematical Proficiency for all Students”.  

Here is a link to the article.

I find this article very resourceful for thinking about teaching mathematics because it outlines the National Research Council’s “Five Strands of Mathematical Proficiency” and goes on to discuss what the author coins “Modeling Math Meaningfully”.  This modeling is illustrated with a great graphic known as “Lesh’s Translation Model”.  When I look at these three things, I feel like I have gathered a "cheat sheet" for my future teaching.  This article sums up so much of what I have been learning about how to teach math in a way that is meaningful and purposeful. 
Of course, "cheat sheets" are not meant to take the place of a real understanding of the content, which I feel I do have.  The Five Strands of Mathematical Proficiency include conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.  Basically, if you read the article, you will notice how these strands involve the learning of mathematical ideas, the skills in carrying out math procedures, the ability to apply those skills to solving math problems, the ability to make sense of what is learned via reflection/explanation/justification, and the attitude that math is personally useful/valuable/learnable.  I can see the value of setting up my classroom and designing my instruction in such a way that each of these strands are hit upon.  

Also of value to my future teaching is Suh’s “Modeling Math Meaningfully” activity.  This is an activity where students model their math understanding in each of five modes.  You will see these modes illustrated in the article via Lesh’s Translation Model.  They include (not in any set order, for their uses can be intertwined) using real life situations, pictures, verbal symbols, written symbols, and manipulatives.   In order to demonstrate understanding students explain their math through writing numbers, verbal explanations, drawing pictures, using manipulatives, and writing real-life stories or situations where the problem can be applied.

The article also has an example of a very useful rubric and graphic organizer.

Out of curiosity, and as a side note (because I always like to add side notes), I decided to Google “math cheat sheets for students".  Of course I would never let students rely on these without understanding the deeper concepts behind the math, but perhaps they could be useful at the right time (maybe in developing fluency) in their learning.  Here are some links.

Scroll down to “Math” in this link:

Click on “Basic Math and Pre-Algebra” categories in this link:

Check out the "Pre-algebra" section in this link:

P.S.  If any bloggers out there can come up with a better term than "cheat sheet" (because it is not really cheating) I am open to suggestions.  Resource Sheet?  Summary Sheet?  Any creative ideas out there?

Thursday, July 14, 2011


I was thinking about manipulatives. 

As I was thinking about manipulatives, I began to wonder whether or not they would be helpful for math beyond the elementary and middle school years.  Does it not make sense that, even with more advanced math, students would benefit from moving from the concrete, to the pictorial, to the abstract?  I decided to Google “using manipulatives in high school math” and found the article, “Manipulatives: The Missing Link in High School Math”, by Marilyn Curtain- Phillips, M. Ed.

In the article, the author notes that children in the U.S. scored high among nations in fourth grade math assessments.  However, by the time they graduated from high school, they were last.  Hmmm...I wonder...  Could the use (or the absence of use) of manipulatives come into play here?   The author also notes  that the National Council of Teachers of Mathematics has long pointed to the value of using manipulatives in all grades.  In spite of this, the use of manipulatives in high school math is rare.

The article goes on to give a few reasons as to why the use of manipulatives in high school math is rare.  It also goes on to talk about the value of manipulatives in bringing abstract concepts to life, and in engaging students by addressing multiple learning styles.  Let’s face it, math is just as abstract (perhaps even MORE abstract) in the later grades.  Manipulatives seem like the perfect tool for learners of all levels. learning styles, and math genres.  Here is a link to the article.  It is definitely worth reading.

Manipulatives: The Missing Link in High School Math

Here is another link I found in my Google search.  This paper talks about the effectiveness and natural use of manipulatives in elementary school, middle school, high-school and (Yes! Even ADULTS use manipulatives in the "real world"!) beyond.

Tuesday, July 5, 2011


This is one way to tackle the "word problem" problem.

So is this.

One of the philosophies and pedagogical approaches to Singapore Math is to use a problem scenario that gives students a familiar context for contemplating an abstract concept. Basically, the aim is to give the student a word problem that they can relate to so they are more interested in taking the steps necessary to find an answer. Another philosophy and pedagogical approach to Singapore Math is having students move along in their learning by first using concrete manipulitives, then moving on to pictorial representations, and finally delving into abstract calculations.  Both of these practices make a lot of sense to me.  In fact, they are similar in philosophy to Montessori Math, a topic I am interested in learning more about, and one which I briefly touched upon in my previous post.

These approaches are also integrated into Thinking Blocks, a math system where students use special manipulatives to solve word problems. Thinking Blocks were developed by Colleen King in 2003. Prior to creating Thinking Blocks, Colleen was using elements of Singapore Math for advanced students to help them with their algebraic thinking. Thinking Blocks evolved from her desired to create a math method for students of all math abilities.  This is a compelling reason to use Thinking Blocks in the classroom, which will no doubt contain students with a diverse set of math skills.

I spent some time working with Thinking Blocks and am excited to have my son try some problems to see if the modeling aspect helps him better understand how to decipher a word problem into a math equation. Word problems come much closer to real-life scenarios (or, better yet, can be designed to emulate actual real-life situations) so finding ways to help students master the process seems like worthy goal.

I think the Thinking Block manipulatives are very useful for working with word problems (and other math problems) because they not only provide students with a visual representation, but they help students break apart and organize the elements within the math problem.  This organizational process requires the student to identify the missing components or numbers in the problem, which is not an easy thing to do by just reading the scenario.  The hands-on, visual nature of Thinking Blocks give students "extra information" that aids in comprehension of the task at hand.  I think that these manipulitives would benefit students in all the elementary grade levels and even middle school.  The games on the Thinking Blocks website contain math that is learned or reviewed in all of these grades and the level of difficulty has a broad enough range to accommodate many ability levels.  I would definitely use Thinking Blocks in my math classroom because I believe the modeling leads to a greater understanding of what is being asked and how to go about answering it.

Here are links to the Singapore Math and Thinking Blocks websites. 

The following link has a few statistics about Singapore Math - just as a side note - because of its connection with Thinking Blocks.