Making the Most of Math Manipulatives

                Manipulatives in the classroom are nothing new. As a primary/junior educator, I see them in classrooms everywhere. Oftentimes, they are displayed in labeled buckets and the students readily use them; unfortunately, this is not always the case. In one of my practicum placements, manipulative tubs were pulled out during indoor recess to be used as toys. This certainly saddened me.

I have seen first-hand how this amazing tool can help students learn by providing hands-on learning and support to students. Not only would this be beneficial in the primary and junior years, but manipulatives can support student learning in the intermediate and senior years as well, so it is essential for educators to fit them into their instruction in order to improve math proficiency. Manipulatives take something abstract and turn it into something concrete; thereby allowing students to become engaged and take ownership of their own learning.

Marilyn Burns, is a math specialist who has been teaching for over thirty years. She is a highly respected educator and continually creates professional development resources for educators. Marilyn is an avid supporter of using math manipulatives and backs the idea that these tools help students get a grasp of math concepts. She is such a strong believer that she created a Scholastic resource on how parents and educators can use manipulatives to help with math education. It is definitely worth a read, especially if you’re on the fence about whether or not to use them in your classroom.

How Can We Make Math Meaningful?

My earliest memory of math is from way back in grade three, which is over thirty years ago. We had to memorize our multiplication tables. The class would chant as a collective, while the teacher pointed to where we were on a chart. I knew my multiplication facts inside and out. Unfortunately, I did not really have a grasp of what multiplication was or meant. Luckily, that did not seem to matter to my teacher. The class would often play a multiplication “racing” game. We would form two lines, side-by-side, on one end of the classroom. The teacher would stand somewhere in the middle of the room facing us, and she would hold up multiplication flash cards. The first student to answer the question correctly would move forward one step. The first to the finish line was the winner. I was very competitive and was very often the winner, thanks to my diligent work done at home, repeating the times tables over and over each night.

In grade three, this was fun for me; unfortunately, the fun in math did not last and things went downhill from there. Not my grades, though—my math grades remained exceptional. I was always good in math, and by good I meant that I was able to memorize formulae and follow prescribed algorithms to arrive at the correct answer, in the exact way the teacher expected. Unfortunately, as the years passed, math was no longer fun for me. Despite my grades, I did not really feel like I completely understood it. I did not know why the steps involved were important to follow, and, just like with multiplication, I did not understand what many concepts meant. Math class always started with taking up the previous night’s homework. We would then move on to definitions, formulae, algorithms, and procedures. Next, it was time to watch the teacher put an example up on the board. Finally, homework would be assigned. This was the same, day in and day out—a never-ending cycle.

As an educator in the 21^st century, I feel that it is essential to move away from this model of teaching. Rather than the student being a “blank slate” and the teacher the giver of knowledge, we must make math meaningful. Kyle Pearce and Jon Orr are two math educators who are revolutionizing the mathematics classroom. They strive to “make math moments that matter” . They share their strategies on how to create a classroom where students want to keep exploring. The main idea is that the way I learned math is not how it should be taught today because that format rarely leads to students becoming interested or curious about math. They suggest that we start with some kind of a hook activity to spark the students’ curiosity. Then we let the students explore the topic. Let them come up with strategies and answers of their own on how to solve the problem. After they have made sense of it all on their own, or at least tried, then we come together as a group to consolidate our learning. Looking at this model from a young student’s perspective, I can see how this would certainly make the learning more meaningful, less boring, and highly engaging.

Building Them Up Through Literacy in Math

Math Literacy: An Integral Strand While Weaving the Basket of Learning

It wasn't so long ago that teachers were seen as the givers of knowledge who were meant to fill the empty vessel: the student. A great paradigm shift has taken place in education and the goal of the teacher is now known to be to build up the student with the many fabrics (i.e. facets) of learning in the model of weaving a basket. A facet of learning in the Math classroom that must not be overlooked in the regular incorporation of literacy in Math. You may ask yourself, "Did she mean to use the term 'numeracy' since she is referring to a Math classroom?" To that, my answer is an emphatic, "No!" Literacy in the Math classroom starts by making the concepts in Math accessible and meaningful to all of your students through the use of language, text, graphics and questions that put students in a position to ask meaningful questions and access information themselves. Chad Broome is an American middle school teacher who uses YouTube videocasts to showcase how Disciplinary Literacy in Mathematics can be used to encourage literacy in the Math classroom. Broome begins a new concept by showing the class a graphic or piece of text and then poses the questions, "What do you notice and what do you wonder?" By opening up the discussion in a conversational manner, students are encouraged to introduce their own thoughts and ideas based on their own observations and curiosities on the topic at hand. Through this approach Broome encourages neighbouring students to collaborate and identify trends and patterns in data presented and to formulate problem solving strategies based on their own reasoning and collaborative input from their peers. Broome encourages literacy in the classroom and uses it as a key strand in the fabric of higher order thinking that allows students to problem solve and ask important and meaningful questions of themselves and of each other. Students are able of asking and answering the 'big questions' if only we give them the opportunities and encouragement to do so!

Rhetorical or Meaningful; Are We Just Talking to be Heard?

We have all had that experience of knowing someone, whether it be in our personal or professional lives, who seems to talk and ask questions just to be heard. As educators, we should strive to make the direct questions we ask of our students effective and meaningful. There is great value in the use of open-ended questions where students are asked to dig deep and make their own connections to the topic at hand, however now I would like to consider strategies used to as effective and direct questions of your class. In their Literacy and Numeracy initiative, the Ontario Ministry of Education, published the Asking Effective Questions in Mathematics which can be accessed through the EduGAINS site.  A number of helpful guides to higher level question-asking for the Math classroom are featured and one example of such is pictured below:

Posing such questions allows for the students to make connections and allows them to make inferences and draw conclusions based on their observations and their reasoning skills. The use of prompts provides the opportunity for the teacher to give the student the floor to use their own voices to work out a problem verbally, justify and support their thought process, and essentially, to think out loud. If we want to welcome literacy into the Math classroom a great start would be to keep encouraging our students by leading them in a way that they become comfortable thinking out loud. 

Knowing Thy Impact

Knowing Thy Impact
John Hattie's work on Visible Learning is applied to Mathematics in his book titled Visible Learning For Mathematics: What Works Best to Optimize Student Learning.  In this book, the authors examine the plethora of "research-based" instructional practices in an effort to provide recommendations for best-fit teaching.  In a seminar in Toronto this summer, John Hattie recommended that educators resist getting seduced into a single form of instruction and opt for an understanding of the impact of various types of instruction to produce significant results and maximize student achievement.  He labelled it the Goldie Locks approach to instruction:"Not too hard and not too boring". This reminds me of Vygotsky's Zone of Proximal Development.  Matching instructional practices and strategies with the student's readiness to learn.  In sum, Hattie et al.'s framework asks the teacher to consider when particular teaching strategies are most beneficial, and where a student is in his or her understanding of a concept.

Making Algebraic Thinking Visible

Making Algebraic Thinking Visible

This clever problem would be an excellent way to lead student's into a dialogue about using different methods to solve a problem.  Further, it has an accessible entry where visually a student may be able to deduce the height of the table and it allows other student's to reason algebraically with multiple variables.  This could be used in an elementary class as an intro the algebraic inquiry, yet it could also be used as a "minds on" exploration in the senior grades before demonstrating how to solve equations with more than one variable.  Overall, it is a gripping challenge and would allow the students to engage in mathematical dialogic investigation.

Watch the video below by Presh Talwalker at Mind Your Decisions to see two methods to solve this problem.  It could also be solved using bar modelling from Singapore's Primary Mathematics program. 

Geometry Snacks

Geometry Snacks

I was intrigued by a geometry question I saw on  Twitter which credited authors, Ed Southwell and Vincent Pantaloni.  Geometry Snacks is a great mathematical puzzle book with many geometrical figures that will lead the student through several complex challenges.  The problems are fun, increase in complexity and allow students to see how many approaches one can use to solve a problem.  The first questions could be used to challenge elementary and middle school students, whereas the latter challenges involve the application of higher level mathematics and logic.  Once a student has solved the problem, the authors also provide further challenges. The rich geometric vocabulary makes this little pocket book, an excellent resource for teachers of Mathematics.


A Sample Question for Geometry Snacks as presented by blogger and author Presh Talwalker.