The Power Tools of Mathematics

When we think of mathematics we often think of numbers, shapes, graphs, equations, adding, subtracting etc. We often see math as “representing” items, equations or situations. Take for example representing five. We could represent it with the word “five” the symbol or digit “5” and the quantity of five using manipulatives. (Manipulatives are any object that helps us represent our thinking for example: blocks, unit cubes, pencils, counters etc.) Have you ever stopped to think about if we are using manipulatives enough?  My answer is “no we are not using manipulatives enough.”

When we use manipulatives we are striving to make things clear for the students. Manipulatives provide that hands-on experience where students define and defend their own understandings of each situation.

Within some of our classes there is the misconception that if the teacher draws the representation then a student will gain an understanding. That's simply not always true. When a teacher is using the manipulatives, drawing the picture or representing the situation with base-ten blocks or counters then the teacher is the one in control of the learning. Students have to be able to construct their own meaning. They do this through representing their thinking. Students may be a drawing a graph or 3D shape. They could be using different blocks, using different tools, whatever works for them. It's not a one size fits all.

If you're wondering if a student really understands the concept that you're trying to instill in them ask them to represent their thinking. Then look at what they represented and better yet talk to them about how they represented it. What this representation means to them? Why did you choose to use these manipulatives/tools? Would these tools/representation work in every situation?

Using manipulatives in the classroom is not as easy as just getting them out and saying here some tools get to it and do it. As teachers we need to model what we expect. We need to allow students time to understand what each manipulative can and should be used for. There are times when base ten blocks are not the right tool to use maybe we need a ten frame and two sided counters.

We need to think about how our students to tackle the problem. We need to purposely plan and know our students. We need to ask ourselves more questions: How might my students struggle with this situation? What specifically are they going to struggle with? What ways can represent their thinking?  How can using manipulative in this situation clear up those struggles or clear up those misunderstandings? What are the possible manipulatives to use to allow students effectively and efficiently represent their thinking? Manipualtives are a way for ALL students at ALL levels to show us what they understand and how they understand it.

By taking the manipulatives off the shelves and showing your students how they can use them to represent their thinking will change what you know or think you know about your students understanding.

What to do over Break...MATH what else!

What do I want our students to do over holiday break?  Play games.  The power of numbers is often overlooked when playing games.  When students are able to apply their understanding of numbers to an application they build their number sense. Let’s explore a few traditional games and what math/number sense they help to strengthen.

• Dominos/Yahtzee – or any game that involves dice or dot patterns helps our students to be able
  
  to recognize dot patterns, refered to as Subitilizing, as a number and not just a group of dots. As students develop number sense they learn to recognize quantity. This is important to mathematics because we want students to move past counting every dot individually and just be able to recognize that the eight dots on a domino is eight or see it as two groups of four. This is the beginning of building our sense of number. For more about the power of Subitizing check out this article: http://gse.buffalo.edu/fas/clements/files/Subitizing.pdf

• Card games- any card game that involves students being able to sequence, develop pairs, three or four of a kind is helping students to develop similarity with number knowing that a two of hearts is the same amount as the two of spades. That two is always two no matter how it is represented. 
• When I was a child growing up we used cards to play the game of war. (Split the deck between the players, flip a card and the person with the highest card wins both cards) Why not put it into a math context? As you play have students add the two cards together, or subtract. Yes it will even work with multiplication. Have the face cards be tens or make the Jack eleven, the Queen twelve and the King thirteen.

• There is also some my family’s favorites: cribbage, Monopoly, Life and Clue to name a few. With the game of cribbage the power of knowing the combinations of fifteen and being able to problem solve which cards will provide me with the most points while not helping my opponent gain points. With Monopoly, Life and Clue all involve problem solving and strategies to help you progress toward the end of the game. With our younger students games like Sorry, Chutes and Ladders, etc. all help strengthen our counting skills, directional skills as well as strategic thinking. For example in Sorry I need to move four, do I move one of my pieces four or do I split the move. If I split it how do I split four, one and three or two and two?  This is important mathematics for students being able to conceptualize that numbers are made of other numbers.

Just a few thoughts for you to pass along to your students or as parents’ items you might consider as we spend a few quality family days together.  Wishing you all a merry and joyful holiday season.

Leadership involves WE not me...

With new opportunities comes new understanding. As many of us move ahead with our goal of providing a learning environment that follows a continuous growth model how do we tackle all the possible roadblocks along the way? There is so much to think about: is our curriculum viable, are we following best practice for each content area, are we meeting the needs of all our students? Wow, a lot to think about. It forces me to reflect on who am I a leader or a manager? I’ve come to the conclusion that a good manager is a good leader and great leaders are great managers. Yes I’m taking the middle ground. Not because I can’t make a decision but because there are times in all my professional “duties” that I have been asked to “manage” situations and also to “lead” people and I think that is the key.

We as the next generation of leaders need to understand and trust those around us. Provide everyone with the proper tools to be effective leaders within their classrooms, building and the district. We also need to manage situations so that there are opportunities for people to become leaders.  We need to remove or at least lower the barriers that might be in the way. If we are strong instructional leaders we should be always considering how learning is going to take place in each of our buildings. Are we working as a system? Not lock step with each other, rather all working systematically toward a common goal for our common clients with common work. So this gets back to the earlierquestion, which barrier I try to tear down first. All of them. However, I can’t do it alone. Working with the other leaders around me and empowering them to: model the way, to inspire a shared vision, to challenge the process to make it better, fostering collaboration in which enables others to act and finally celebrate the values and victories along the way.

Again where do we start?  The answer is with best practice. Below are some additional tools for your toolbox on mathematical teaching practices:

 Principles to Actions Ensuring Mathematical Success for All (2014) NCTM Executive Summary https://www.nctm.org/uploadedFiles/Standards_and_Focal_Points/Principles_to_Action/PtAExecutiveSummary.pdf Continuing its tradition of mathematics education leadership, NCTM has undertaken a major initiative to define and describe the principles and actions, including specific teaching practices, that are essential for a high-quality mathematics education for all students.

Strategies and Challenges Regarding the Mathematics Needs of Special Education Students

This research briefs synthesize the research on mathematics teaching for special needs students. It  discusses effective instructional strategies http://www.nctm.org/news/content.aspx?id=8452

Where to begin...at the end of course

As many of us begin our school year today or within the near future it is important to think about where we want to end up in May/June instead of where to begin. As a math teacher I want my students to “know math.” Ok so what does that mean, for me it is a focus on helping students to develop a mathematical understanding and be able to put that understanding into practice daily. I know you are saying ok Christine, really something that big takes time, years. Yes your right it does that is why it is so important for us as educators to work collaboratively.  Focusing on a mathematical instructional paradigm shift where teachers focus on HOW concepts are taught, not What is taught. 

We need to:

• Challenge students to make sense of what they are doing to solve mathematics problems
• Pose questions that stimulate student’s thinking, asking them to justify their conclusions, strategies, and procedures. 
•  Have students evaluate and explain the work of others, and compare and contrast different solution methods for the same problem
• Ask students to represent the same ideas in multiple ways (symbolically, pictorially or with manipulatives)

So how do we begin… with the end in mind? A small but mighty step is to allow students to explain their thinking. Having students explain their mathematical reasoning on how they came up with the answer is so huge in knowing what they know.  Also for those of us that work in a PLC (professional learning community) it is getting toward question #2 “How will we know…”  It is more than student’s having the right answer it is knowing how the students came up with the right answer.  What process did they use? Will that process work if the numbers are changed? Can they prove how they came up with their answer using tools, drawings or manipulatives? When students are asked to explain their thinking they have to be able to put their thoughts into some type of organization. “First I did, and then I did…” We often have students do this back at our desk one to one when they have gotten too many problems wrong. But what if we do this as a whole class or small group lessons on problems that students got correct? Think of the impact on the students, their self-confidence, their willingness to communicate/participate, the learning from each other not just the teacher just to name a few. No, this doesn’t happen overnight. No, there will be messy situations in which you the teacher are going to have to discuss with your math team what the child was attempting to do, but isn’t that ok, isn’t it about the learning both ours and our student’s? So as you start your math lesson planning try to work in some opportunities for students to share their math reasoning. It will open many doors to many great math opportunities in your classroom. To help you get going here are five practices from the NCTM publication 5 Practices for Orchestrating Productive Mathematic Discussions by Margaret Schwan Smith and Mary Kay Stein 2011 which help to have a meaningful discussion not just a show and tell of math ideas or procedures.

1. Anticipate student responses prior to the lesson 
2. Monitoring students’ work on and engagement with the tasks
3. Selecting particular students to present their mathematical work 
4. Sequencing students responses in a specific order for discussion.
5. Connecting different students’ responses and connecting the responses to key mathematical ideas.

Some additional resources on this subject are:

• Burns, M (2005). Looking at How Students Reason. Educational Leadership http://mathsolutions.com/wp-content/uploads/2005_How_Students_Reason.pdf

• Principles to Actions Ensuring Mathematical Success for All (2014) NCTM Executive Summary https://www.nctm.org/uploadedFiles/Standards_and_Focal_Points/Principles_to_Action/PtAExecutiveSummary.pdf Continuing its tradition of mathematics education leadership, NCTM has undertaken a major initiative to define and describe the principles and actions, including specific teaching practices, that are essential for a high-quality mathematics education for all students.

• Reinhart, S. (2000). Never say anything a kid can say. Mathematics Teaching in the Middle School, 5(8), 478–483.

• Smith, M. S., & Stein, M. K. (2011). 5 practices for orchestrating productive mathematics discussions. Reston, VA: National Council of Teachers of Mathematics.

A clean desk...what's next...

"Life is a journey"...full of many opportunities to stop or change direction. While the 2014-2015 school year will not find me at the same location that I had been for the last nine years, Area Education Agency 267, I will be still doing what I love, helping others be their best.
I am looking forward to traveling north each day to Albert Lea, MN where I will be the math specialist for the district. While my desk is still relatively clean, I thought I would dust off this blog and use it to provide resources to those I work with or anyone else who may find the information useful. I plan to post links to websites and resources in which anyone is welcome to share and use in their classroom. My only request is that you take what I post and make it better. I have found in my 20+ years in education that too often we are not willing to try new things, or we try something half way. Take what I post try it, learn it, understand it then mold it into something stronger and better keeping the integrity of it in place.  Remembering that it is not about how easy it is for us as adults to do what we do, but rather how successful our students are in building/growing their knowledge and understanding.