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.

Tuesday, June 28, 2011


Why do so many students struggle with math?  Sometimes I think we are grasping at straws trying to come up with a way to get students to grasp math.  Yes, I know that students should have a variety of strategies to solve math problems, but is it not possible that this variety adds to math confusion for some?  Perhaps we are (however rightfully constructivist it is) giving students the opportunity to construct their own methods for solving math problems without making sure they have the solid foundational number sense that will aid them in the process.  Perhaps, as an example, offering up strategies like the lattice method is premature for many. 

Does the lattice method really promote understanding?  My son, who struggles with math, uses the lattice method to multiply multi-digit numbers.  If someone were to ask me if I thought the lattice method connects my son to a real understanding of multiplication, I would also have to say “no.”  It is just his preferred technique because it has brought him the most success, in terms of getting correct answers.

Where am I going with all of this?  Two words...Core Strength.  I think something core is missing.  I  believe that students who struggle with math missed something foundational in their early math learning.  The problem with current math education, in my opinion, is that we RUSH so many of our children into math arena that they simply are not ready for.  Yes, there are some kids who are brilliant with numbers from a very early age.  These kids should be able to be in math arenas that challenge them and cater to their exceptional math abilities, but that is another discussion altogether.
What is the solution? I suppose defining what deep math understanding entails is a good first step.  This has been done, in the form of a document titled “The Common Core Standards for Mathematics,” written and published by the Common Core State Standards Initiative.  Here are the links to both.
The Common Core Standards for Mathematics is written as a guide to bring deep mathematical understanding to students in the U.S.  Many states have adopted these standards.  The guide is broken down into Standards for Mathematical Practices and Standards for Mathematical Content.  The Standards for Mathematical Practices list eight areas in which students should exhibit mathematical ability.  The Standards for Mathematical Content break down what students should learn in each grade, K-12.
As a teacher, I will find the breakdown helpful in determining what is to be taught to whatever grade I end up teaching.  As a teacher, I will find the practices helpful as a means of assessing student progress and ability.  I sincerely hope that, as time goes on, these guidelines will succeed in providing foundational number sense so important to successful math progression.

Finally, in my quest to find math curriculum that is both effective and relevant to students, I have been looking into Montessori math.  I am very drawn to the Montessori philosophy and approach to learning, how they connect math to everyday life, and how they combine hands-on engagement with simultaneous integration of number operations and facts memorization.  I am wondering if Montessori methods might be the answer (or at least part of the answer) I am looking for.  I plan to do more research.  In the mean time, this link and video are worth checking out.

Introduction to Montessori Math video

Wednesday, June 15, 2011

PICTURE THIS: Cooperative Learning

Picture One

Cooperative learning may look like noise.  The classroom is buzzing with activity.  Students are talking, actively discussing and sharing ideas.  There may be some debates and some differences of opinion.  There is a lot of “doing” as groups implement ideas in hands-on and engaging ways.  With all of this going on there is bound to be noise, but it is a joyful noise - or at least it should be.  How, then, does one (especially the teacher) discern whether or not this noise is indeed ‘joyful”?

Picture Two

Working toward group goals and incorporating individual accountability is key.  If these things are happening, then most likely the “noise” IS a joyful noise.  One of our class readings (which you don't absolutely need to have read to understand and contribute to this blog) points to these two main goals being necessary for effective cooperative learning to take place.  A group should be working towards a common goal through the sharing of ideas and the implementation of the steps needed to meet the goals.  In addition, each person in the group should have a role in meeting that goal.  If students are doing this, while treating each other respectfully, then they are moving forward in a positive fashion.  The teacher’s role, then, is one of facilitator, where he or she provides time and space for groups to work together and provides guidance and feedback where necessary.

Activities that students might be engaged in can range from simple “one day” activities like when a group gathers to solve one particular math problem, to a “multi-day” activity where students gather numerous times to complete a task or project.  Students might be engaged in an activity where one group works together to become “experts” in a particular aspect of a subject while other groups become “experts” in other aspects of that subject.  Each group eventually teaches the other groups what they have learned.  In addition, they learn from the expertise of the other groups. This cooperative effort can effectively enlighten an entire class on a particular subject.  Students could also be engaged in a group investigation activity of their own choosing.  This cooperative learning activity is student directed.  Students decide what they want to learn about and students define roles and responsibilities within the group.

Math is just one of the subjects that can be taught through cooperative learning.  Here is a video that discusses the subject.  Much of the focus is on teaching the process of cooperative learning but there is some commentary on math as well.  The video also touches upon the power cooperative learning has in making math relevant and meaningful to students, perhaps the most significant reason being that cooperative learning is student driven, meaning students are active learners and even teachers themselves in some cases.  It is so interesting to watch how engaged the students in this video are. 

Monday, June 13, 2011


In our class text book (which you don't absolutely need to have read to understand or contribute to this blog) the author describes three basic uses for subtraction, taking away, separating, and comparing.  The author goes on to illustrate and model six practical uses for subtraction.  These are 1) taking away a subset from a set, 2) separating a set into two disjointed sets, 3) comparing two sets, 4) taking away part of a length, 5) separating an area into two parts, and 6) comparing two lengths.

I really like how the uses are broken down and defined for me.  It allows me to simply move on to thinking about how I could help students make connections to subtraction concepts.  The modeling and illustration in the book points to how a teacher would want to model these concepts for students so, as a teacher, I would set up manipulatives at stations for students to explore each of these uses and get practice writing down the equation or equations that would go with their particular problem/solution.  I would then move on to having the students, as the book suggests, come up with their own subtraction-based story problem and solution.

After that, as a means of reinforcing the concepts, I would try to implement some sort of project-based learning math activity, the level of involvement being appropriate to the developmental level of the students.  I mainly go off on this tangent because, from what I know, project-based learning (also known as inquiry-based learning) is a way of learning that give students ownership over their learning.  I found a few websites that are helpful for getting my brain around the concept of project-based learning in general, and project-based learning for math specifically.  Here are the links.

I know that all kind of branches off into the realm of what would be learned in a method class so, for discussion purposes, I am wondering what kinds of practical uses for subtraction would students find meaningful and interesting?  In other words, why might an elementary student want to take a subset away from a set, or separate a set into two disjointed sets, or compare two sets, or take away part of a length, or separate an area into two parts, or compare two lengths?

Also, I found this video interesting.  It presents a philosophy and a few ideas on how to make math relevant for elementary students.  One of its main ideas focuses on how elementary students are not ready to connect math with "real-life" scenarios as an adult would define "real-life" scenarios.  How does that idea fit with the above question?