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The world has changed and so Math has changed

Posted on by Carol Lloyd


I have heard people say the only things certain in life are death, taxes and change.  But then you also hear people say, especially in education, “some things never change.”  There is also a good amount of disagreement these days on the benefits of new state standards and the changes it is producing in education.  In my experience most of those opposed to these changes are parents, politicians or citizens without a true understanding of what the new curriculum is all about.  I have frequently heard:  “If it was good enough for me to learn it the old way why do we need to change it now?”  But what about life, society and the world today is the same as thirty years ago?

I remember when I started teaching at a high school in 1980 and they received the first computer to ever enter the building.  I had NO idea what to do with it, but within five years was teaching programming classes.  Today’s schools are becoming so technology dependent that students are encouraged to BYOD, Bring Your Own Device!  When I started teaching you could graduate from high school with General Math 1; now you need the equivalent of Algebra 1 and 2, Geometry and another math above those.  The world has changed, our expectations for students has changed and as a result, the way we teach has to change.

Several people have given me the link to the following video.

Why is Math Different Now from raj shah on Vimeo.

Dr. Raj Shah does a great job of explaining one way of answering this question about why the math curriculum has changed.  This answer is part of a big picture that many people outside education do not understand.  I also addressed this in an earlier blog.  Here is the way I explained it.


After taking advantage of the Buy 2, Get 3 Free chicken breast special, I told my daughter she could come by and pick up a pack.  She was having guests for dinner so was excited about the free meat (she lives on a 3rd year teacher salary).  When she came to pick it up, she looked at the pack which was labeled 1.8 pounds and said, “So how many ounces is that?”  Now this girl is 26 years old, scored high enough on her AP Statistics Exam her senior year to place out of math in college and made an A in high school chemistry under a very tough teacher.  Really? She was in a hurry and not in the mood to be coached to the answer so I answered:  “Well there are 16 ounces in a pound, 0.8 times 16 is 12.8 and 16 + 12. 8 would be 28.8 ounces.”  “Ok, so that’s more than enough for 4 people,” she replied.

The next day as I was spending 7 hours driving, I started thinking about that conversation.  Why is it so many people have trouble doing mental math, particularly like this case, multiplication.  My high school students were always amazed when I could multiply problems faster in my head than they could turn on their calculator and type it in.  After some pondering, I have a hypothesis.  Students don’t understand that all multiplication is simply an application of the distributive property.  I certainly didn’t understand that in elementary school.  I knew how to line up a multiplication problem and “move over” a space on the second line, but I really didn’t know why.  You see it’s that WHY that is our problem.  For example, why can’t students remember that to divide by a fraction means to multiply by the reciprocal? Because they don’t really understand the relationship between multiplication and division.  That’s not just the rule for dividing by a fraction, it’s the meaning of all division, to multiply by the reciprocal (the multiplicative inverse).

All this brings me back to the Common Core Standards for Mathematics.  It is this type of conceptual understanding that the Common Core Standards are demanding of students.  Students won’t memorize procedures that have no meaning to them and therefore will not be retained over time.  Students will need to develop a deep mathematical understanding of the structure of mathematics, from kindergarten through college.  If students grow up understanding that all multiplication of more than 1 digit times 1 digit is an application of the distributive property, then when they get to algebra they won’t need something called FOIL to tell them how to multiply (4x + 5)(2x – 3) and they won’t freak out when suddenly the problem changes to (3x + 4) (2x2 -5x + 2).

So teachers are you ready?  When a student asks you WHY they do a math problem a certain way, the answer should not be, “Because that’s the rule.”  I would love to hear some of those “WHY” questions you or your students struggle with answering.  Who knows, maybe together we can grow a generation of mathematically literate citizens!


What’s the difference between Personalization, Differentiation, and Individualization?

Posted on by Thinking Maps


With the changing role of education, we’re all learning and exploring a variety of ways to meet the needs of 21st-Century Learners. There are three types of instructional strategies that have been used interchangeably but actually refer to different things: Personalization, Differentiation, and Individualization. So what really is the difference between these three strategies? Personalize Learning created an awesome resource to clarify and we couldn’t resist mapping it. Enjoy!


The difference between instructional strategies

Educational Leadership: Best & Worst of Times

Posted on by Ken McGuire

“It was the best of times, it was the worst of times.” You probably recognize these lines as the opening of Charles Dickens’ A Tale of Two Cities. They are also lines borrowed by Dr. David Sousa to introduce the first chapter of his book, The Leadership Brain. He explains that it is the best of times because “never have we known so much about how the human brain develops, grows, and learns… It is the worst of times in that never before have the public schools been asked to do so much for society.” He continues that schools not only teach children, they raise them. While teachers are charged with presenting their curriculum, they are also asked to counsel on drugs, sex, family problems, and personal relationships. The point that gets driven home is that in these times, managing is not enough, educational leaders become the key to helping school staffs balance their responsibilities and priorities successfully.
I would advance this scenario from the educational leaders position considering that it is the best of times because never before have we had the depth and selection of outstanding professional growth and development opportunities. Many of these, like Thinking Maps, are linked to current brain research and offer better knowledge and instructional strategies that can ultimately lead to student success. It is the worst of times because we work on an ever-changing landscape where the demand for increased student performance data is accompanied by decreasing financial resources.
Consequently, educational leaders are compelled to serve as instructional leaders working with staff and community to ensure relevant and rigorous learning opportunities for their students. Schools cannot focus solely on dispensing knowledge, but on developing students who can learn and adapt to the knowledge and skills they will need to succeed in the 21st century. At the same time ,leaders must function as full-fledged business managers overseeing the prudent application of available resources at a time when those resources continue to diminish.
In summary,  let’s change the saying “managers tend to do things right while leaders do the right thing” to reflect the situation educational leaders find themselves in and say “managers tend to do things right while educational leaders do the right things right.” More than ever, our schools need strong leaders.

Who has been a strong leader in your life?

Mapping Gratitude

Posted on by Chris Yeager

I love magazines during the holiday season. I start collecting them about the middle of October and pour over them nightly. I don’t want to read them on my iPad; I want to tear out recipes, cut out sayings, and put them on my refrigerator, turn down the corners of pages I want to show someone, and put tabs on pages with possible gift ideas. I try to recreate clever decorating ideas that generally look pretty sad but still I love the engagement.
So when I was flipping through the November Good Housekeeping and saw the title “A Husband Who Does the Dishes,” I couldn’t resist reading. The Multi-Flow Map below summarizes the key points of the article. I printed out this map and have it posted on my refrigerator as a reminder.

While I have a million things to be grateful for in my personal life, I am also very thankful for a professional life as an educational consultant that I love. So here are a few things I am truly grateful for:


  1. Having the privilege to work with teachers who have devoted their lives to helping young people.
  2. Sharing, creating, and thinking with all of the fantastic educators in our company who are truly focused on contributing to the profession we all love.
  3. A job that allows me to stay connected to teaching and to share the critical and creative power of Thinking Maps.
  4. Any day that I can spend with children, hearing about their maps, and seeing them think.


Have you started your list? Create your Circle Map brainstorming these things, or if you are really motivated, create a Tree Map classifying all your blessings into categories. Can’t wait to see your thinking.

The “AHA” moments from the Kagan & Thinking Maps PLC scavenger hunt

Posted on by Thought Leaders


Bringing new programs to enhance learning to a school is sometimes a difficult undertaking. This year my school decided to take on the integration of not one but two new programs; Kagan strategies and Thinking Maps. Teachers crave meaningful trainings that they can feel are easy enough to implement immediately and not anything additional they feel like they have to do. Let’s face it, teachers are very busy people.

I have worked with the Kagan trainer at our school and we created a variety of ways that we can easily incorporate the two programs while involving the teachers. By doing this they are practicing both programs and seeing how the two can be easily integrated into one lesson.

This past week we had a faculty PLC Thinking Map scavenger hunt. I asked each grade level for a list of maps we could find if we “searched” their rooms. I then created a list for each grade level that required them to visit the grade level below them and the grade level above them. They had to take pictures of the maps they found and write down the teacher’s room number they found it in. Although we all work in the same place, how often do teachers actually have the time to visit anyone else’s room? Never!

By doing this we accomplished many things. First, teachers were able to see vertical movement between grade levels. Second, teachers were able to gain multiple ideas that they could incorporate into their own classroom without meeting with individual teachers. Third, teachers that were initially hesitant saw value in what was happening in other classrooms and realized if they didn’t jump on board their students would soon be left behind. Finally, as a trainer it helped me to see where we still had come confusion, and what my next follow-up training should be.

Once all the teachers returned to the media center we incorporated our Kagan strategy. We used mix, pair, share and rotated three times. Teachers had out their phones, were sharing pictures, and “aha” moments they had during their hunt. It was such an easy way for them to share what they had seen and the entire process was completed in the time it takes to hold a regular PLC meeting.

Using the Kagan strategy to share gave teachers an additional idea for their classroom. How easy would it be to have students work on maps individually and then share what they discovered in a five minute mix, pair, share!

In the upcoming weeks we are incorporating Kagan into every Thinking Maps training we have including a timed, pair, share, a picking stickies (all write round robin), and line ups. Teachers left the training saying things like “this was a great idea”, or “I never thought I could use a Thinking Map and Kagan together”. Every time we leave feeling accomplished and that our time was not wasted. It’s an amazing experience to provide beneficial trainings to teachers that love to learn!

The attached photos are an example of a circle map exercise that was found in a first grade classroom, and a working picture of the teachers doing the mix, pair, share Kagan strategy to share what they found.

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My name is Christian Luciano and I teach second grade at Three Oaks Elementary in Fort Myers, Florida. I am passionate about sharing my love of learning with others. I love Thinking Maps! I honestly dream ideas on how to extend thinking in my classroom through the use of Thinking Maps.

Metacognition: Adding weights to an exercise

Posted on by Thought Leaders
By Florence McEachern


As is the case with many of us, I have struggled with maintaining a healthy weight and a healthy lifestyle. In pursuing this goal, I embraced every new weight loss strategy that has come along. I realized that I needed a functional workout plan.  My routine lacked consistency and I didn’t have the knowledge or ability to contribute to my own success.  Out of sheer desperation, I hired a trainer.

Fortunately for me, she had an educator’s mentality.  She not only taught me the “how,” but she also taught me the “why.”  She focused on making me independent so I wouldn’t always have to rely on her to tell me what to do.  Here’s a little secret that she shared with me: adding weights to a workout increases rigor and improves results by burning calories throughout the day.

Being a Thinking Maps trainer for more than a decade, I immediately saw the correlation between this and educating students.  Students not only need consistency but they need a workable strategy that allows them to contribute to their own success.  What my fitness trainer taught me is what we need to teach all our students.  I learned to focus on how I knew the things that I knew about exercise and to identify and examine my sources of information.  That metacognitive ability built my independence toward maintaining a healthy lifestyle.  I started asking myself meaningful questions such as, was the information that I relied on from my own perspective and needs or was it someone else’s opinion or point of view? Was the information biased? More importantly, what did I learn overall and why was it important to know?

The Frame of Reference is like adding weights in exercise. Reflecting on your thinking adds rigor in the classroom.  It is perhaps one of the most impactful components of Thinking Maps because it allows students to reflect on their thinking.  We know from Dewey that “We don’t learn from experience. We learn from reflecting on experience.”  When students are able to visualize and reflect on their thinking, it makes it possible for them to use metacognition to increase their individual chances for success. By actually taking a step back and thinking about their thinking, students also become more critical and independent learners.

As teachers, our greatest gauge for success should be observing our students thinking and facilitating their learning as if we’re not even there. If they can do that, they will continue to reap the benefits even when they’ve left the classroom. This is the true definition of “lifelong learners.” Ultimately, our goal as educators is to produce students capable of functioning independently and effectively in today’s global society.  After all, is there really any better reason to teach?

Why knowing what bad professional development is matters

Posted on by Hortencia Piña


Who is in the driver’s seat when it comes to student learning success?

Teachers? Students? Parents? Administrators?

If you guessed student, pat yourself on the back because you’re right. As teachers, we can hope that students listen, engage, and learn. We can create exciting lesson plans that encourage curiosity and ignite passionate learning but at the end of the day, the students themselves are in control of what they do learn. All we can do, is become more effective teachers, making professional development one of our top priorities for schools and educators.

So how do you prepare yourself and your colleagues to be as effective as possible when inspiring student learning?

A big question deserves a big answer but we can’t always arrive at the big answer with the first step. So knowing what not to do is a great place to start. After years of providing professional development trainings as a Consultant, I can attest that these “10 good ways to ensure bad professional development” are accurate. And you’ve probably experienced these as well.

10 good ways to ensure bad professional development” via Learning Forward

Beyond these 10, having an implementation plan with SMART goals will reassure teachers that this training is not just a “drive-by training” and will provide the necessary scaffolding and support for implementing a new system.

What would you add to this list? How has quality professional development impacted your teaching?


Distinguishing Between Factors & Multiples

Posted on by James Dean


While reading through the new state standards, I was reminded of another frequent question I am asked by math educators: How to teach the difference between factors and multiples? The fourth grade standard indicates “gain familiarity with factors and multiples.” For years this puzzled me as well, but the real key is to understand that factors are a Whole-Part relationship, and multiples are about Sequencing. Hmm. . . .do you hear Thinking Maps on their way? I certainly do!

A Brace Map shows students to clearly SEE what Factors are. Products are comprised of Factors.

The product of 24 has several factors.

After building several Brace Maps to show the products of a number, we then add a Frame of Reference and ask students to write a clarifying statement to demonstrate their understanding.

Brace Map Math Factors and Products

When I teach multiples to students they need to see this as a sequence. It is almost like skip counting, if you will. What are the multiples of 4? What are the multiples of 24?

What are the Multiples of 4?

Once students have a firm understanding of factors and multiples, it is easy to see how a Compare/Contrast Map is in order to help solidify the concepts in their mind. A wise Thinking Maps consultant once told me that the real key to comparing/contrasting concepts and ideas is to determine ahead of time if the concepts are considered more alike or different in the students’ minds. Based upon that decision, determine where to focus their attention on the Map. Most math educators tell me that this concept is confusing for students (and sometimes educators) because they think they are the same. Let’s focus on the differences to help clarify concepts.

Through careful thought, planning, and mapping we can help ALL students to SEE and understand Factors and Multiples.

Share some feedback. What has worked for you?

How does writing help you make connections?

Posted on by Thinking Maps


At the heart of Connected Educator Month is a strong desire to empower educators by bringing all of us a little closer together. Through collaboration we can grow as a community and learn the best practices that will increase student achievement and inspire student learning!

Today is the National Day on Writing. To celebrate, educators are answering the question, “How does writing help you make connections?”

Share your thoughts here and then share the Google Doc with others so that we can connect with as many people as possible!

Do you speak educationese?

Posted on by Chris Yeager


If you have been a teacher for more than a month, then you have learned that education has a language of its own.  And you have probably learned how important it is to sprinkle these special terms throughout any conversation about quality teaching and learning.  Any of the following words should be a mainstay in your vocabulary:

21st Century Skills, College and Career Ready, Differentiation, Formative Assessment, Authentic Assessment, Real Life Problem Solving, Data-Driven Decision Making, Scaffolding, Collaboration, Critical and Creative Thinking, Rigor…

I’m sure you could add a few more to this list.  The concepts behind these terms are essential to student success, but do educators really understand the concepts, the definitions, what each should look like and sound like in a classroom designed to meet the needs of all learners?

Let’s focus on just one term:  Rigor.  In his blog, “Using Webb’s Depth of Knowledge to Increase Rigor,” Gerald Aungst says that the reason we struggle with rigor is that “we have adopted the jargon without a clear understanding of what we really mean.”

He provides the definition of Webb’s Depth of Knowledge Levels.  Instead of rushing to focus on the list of verbs, we should begin our discussions by thinking about the title of each level.  We should analyze our student tasks to see if they are asking students to:

  • Recall and Reproduce what they have been taught?
  • Apply Skills and Concepts?
  • Think Strategically?
  • Extend Their Thinking?


The author suggests that teachers collect student tasks they have designed, classify them according to levels (Tree Map) and look for patterns (Frame of Reference reflection statements).


During the discussion that will take place during this classification (often seen as Level 2 thinking), teachers will ultimately be engaged in both strategic and extended thinking (Levels 3 and 4).  Aungst makes 4 great points that should be a part of this discussion:

  1. Don’t let a list of verbs lock you into a specific level of thinking.   Defining can be done at all levels of thinking; Comparing can require strategic and extended thinking; etc.
  2. Remember that time alone does not make a task more rigorous.  I’ve seen students spend way too much time on low level thinking and have witnessed strategic and extended thinking taking place in relatively short, focused tasks.
  3. Understand that these DoK levels are not sequential.  Thinking strategically (Level 4) about a word problem in math helps students strengthen their understanding of mathematical skills and concepts (Level 2).
  4. DoK Levels are NOT developmental.  “All students, including the youngest preschoolers, are capable of strategic and extended learning tasks.”


While Aungst advocates for using Webb’s Depth of Knowledge as a framework, he emphasizes that “Regardless of how you define ‘rigor,’ the important thing is that (all) students are thinking deeply on a daily basis.”


I agree.  Focusing on thinking and cognitive depth is what Thinking Maps is all about.