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.
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.
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!
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?
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:
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.
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.
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).
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.
At a risk of stepping on the toes of some of my favorite lower elementary teachers, I wanted to address the shapes used in Thinking Maps, more specifically, the Bubble Map. Do you ever walk away from a conversation where a question stumped you and days later you think, “Oh, I should have said ____.”? This describes what prompted this blog, only it’s about two years overdue.
When I first recieve Thinking Maps training, I learned to stay true to the shapes of the Maps. I’m a rule follower, so when I came back as a trainer and shared the Maps with the staff at Saranac Elementary School, I reiterated that Bubble Maps are circles, Flow Maps are boxes, etc. Mrs. Cooper, a thoughtful and highly creative first grade teacher asked if, fitting the theme in her room, the Bubble Map could be drawn as hearts, clouds, snowflakes, etc. My unsupported, but clear reply was “No! They have to be circles.” However, the reason behind it wasn’t really solid until recently.
Of course we all know that Maps provide consistent patterns for students. Our brain recognizes (actually seeks) patterns to help new learning occur. Consider how a student fluent in Thinking Maps might process the Bubble Map below.
It might look something like this:
The automaticity the student has learned through knowing the pattern in the Maps tells the student immediately it is a familiar Map, it is a Bubble, and it’s used to describe something.
But if the teacher used the classroom theme for February to make the Bubble Map:
The thought process might be a little more like this:
If our goal is to create a pattern in thinking, so students have automaticity based on the visual, what is the purpose of changing the visual? Is there additional learning that occurs for the student or are we doing it because we are following the theme we picked from Pinterest?
Let’s look at it a little differently. Even if you’re not fluent in Korean, can you read this sign? Of course you can. It’s a STOP sign. You know this because it’s a red octagon. If Koreans decided they love it when you stop at an intersection because it keeps everyone safe, so they decided to make all the stop signs shaped like hearts, would you be able to interpret the sign as easily?
Therefore, my theme based friends, decorate your room to the hilt, but stay true to the Maps. Bubble Maps need to be created with circles, not the shape of the month. It provides a consistent tool for thinking, not a just a cute poster.
Connie Hamilton is a K-12 curriculum director and elementary principal at Saranac Elementary School in West Michigan. As a consultant for both Thinking Maps and classroom questioning, Connie shares her experiences with teachers and districts to help students think more deeply on curriculum.
Of all the slides we have shared with educators during our trainings over the past decade and a half, the Behavior Reflections slide has been by far the most popular! Every time this slide has been displayed on the screen during training sessions, most participants have excitedly voiced their appreciation of this Multi-Flow Map example and nearly everyone has asked for a copy.
As I have visited schools throughout the country, I have seen stacks of copies of this slide on teachers’ desks and in administrative offices. More numerous stories than I can remember have been reported to me about how educators have enjoyed the benefits of using this Map. I often wondered what it was about this particular Map example that resonated so deeply with training attendees. Perhaps this Map example triggered an epiphany in these thousands of educators because it provided for them such a clear example of how Thinking Maps can effectively enhance students’ social development as well as their intellectual development.
As an assistant principal for several years (before I knew about Thinking Maps), I addressed hundreds of instances of student misbehavior. When students are sent to the administrative office they may be inclined to focus on rationalizing their poor choice or seek to blame other individuals or situations. I often found students beginning the conversation by explaining WHY they did WHAT they did. Instead, I had to redirect them and ask them to begin by telling me WHAT THEY did. Once students accepted responsibility for their actions, I would then give them the opportunity to offer explanation. After listening, I generally responded by telling them their reasoning served as an EXPLANATION, not as an EXCUSE for their behavior.
After acknowledging their inappropriate choice, we discussed more appropriate responses to the types of experiences that tended to trigger their poor choices. In essence, I was working with them to teach and build new behavioral habits and patterns of response. Of course, had I known then about the Behavior Reflections piece, I would have used it to make their thinking more concrete. Additionally, I would have encouraged students to keep the visual reminder of the Multi-Flow Map and any other Maps they created to plan for more appropriate responses in the future.
When students visited the office due to misbehavior, I often felt as if there was an expectation of taking punitive measures designed to extinguish the misbehavior that caused them to be sent to me in the first place. Of course I always followed the Student Code of Conduct and administered the identified consequence for the misbehavior in which the student engaged. Still, I intuitively realized that while such consequences may influence future choices, in order to really change behavior, students needed to be taught more appropriate behavioral responses. Interestingly, the form that accompanied students to the office was called a Discipline Record. One day, after receiving a Discipline Record for the same student several times that week, I decided to look up the word discipline in the dictionary. I discovered that the root word for discipline is disciple and to disciple means to instruct or teach. From that moment forward I worked with my teachers to develop systems and strategies for modeling and teaching desired behaviors.
I guess then, it is no wonder that the Behavior Reflections slide has resounded so deeply with so many educators. Dedicated educators recognize the importance of addressing the needs of the whole child. Thinking Maps provide the patterns and strategies needed to effectively do so! Please take some time to share how you are using Thinking Maps for behavior management and to address the social/emotional development of the students you serve.
Before I even begin this blog I need to make sure you understand that I am NOT a math teacher. I taught high school English for 20 years. As a student, I took all of the high level math courses and did just fine in them, but I think that had more to do with knowing how to be a student than truly understanding math.
It has only been in the last 4 or 5 years that I have even dared to think of trying to understand what it means to Think Like a Mathematician, and I really took this on so that I could do a better job of making the connection between math standards and specific Thinking Maps. Carol Lloyd, consultant and writer for Thinking Maps and consultant for Art Costa’s Habits of Mind, is the person who challenged me to believe that even I could, and indeed should, focus on learning how to think mathematically.
She issued this challenge by calling me on the carpet to stand in front of teachers and say, “I don’t really understand math, but here is an example.” Teachers would almost always laugh with me as we all acknowledged our lack of math skills. Carol later asked me, “What if you had said I can’t really read… Would teachers laugh with you and acknowledge their lack of reading skills?” Should it be acceptable to empathize and support our inability to think mathematically while it’s unacceptable for reading? She is often the voice in my head and one of my go-to resources as I work to apply Thinking Maps to specific math standards and math practices.
With the rise of rigorous college and career readiness standards, there is a renewed emphasis on teaching students to not just to follow a set of steps to solve a problem but to focus on thinking about the problem before rushing to solve it. Sounds like a life lesson, doesn’t it?
Regardless of the standards that these eight math practices come from, at their foundation they are all about meta-cognition; with the exception of “modeling with mathematics,” they are math practices that can be applied to all content areas, making them critical thinking practices.
I’ve taken those math practices and I have started to align them with the Guiding Questions in the Frame of Reference for problem solving. Here is my thinking so far and I would love to get some input from you about these new ideas and applications:
Guiding Question #1: How do you know what you know?
In reading applications, students identify their sources and evidence from a text to justify or support the information in their maps. For math, students should explain what math skills, vocabulary and concepts they brought to this problem. In other words, students would say “I need to know or use my understanding of ——- in order to solve this problem.” This guiding question (what we call a Green question) addresses almost all of the eight math standards.
Guiding Question #2: What is influencing your thinking?
Again, in reading applications this question asks students to analyze perspective or point of view. In math, students should reflect on what strategies they used to solve a problem. They might say, “To solve this problem, I _______.” This reflection helps students understand that there are lots of ways to approach a problem and their way is valuable. Giving all students the same problem and then having them share all of the different ways they approached the problem is what will truly help them Think Like Mathematicians, not just follow a set number of steps. This guiding question (what we call a Blue question) addresses the third math practice.
Guiding Question #3: So what do you now understand? And so why is this concept or practice important?
Students construct “so what” statements in reading to summarize a main idea. They then write “so why” statements to help them connect this new knowledge to their own world. In math, students should develop two “so what” statements: So what do I understand about this math standard and so what do I understand about the math practice I used. ”So why” statements (heard when students so often ask ‘so why do I need to know this?’) really should ask students to connect the math standard and math practice to real life situations.
I’m going to present these ideas in my upcoming training and will let you know what teachers think. I would love to hear from you as I continue my journey to Think Like a Mathematician.
As a neophyte Consultant for Thinking Maps in 2004, I had the privilege of shadowing Chris Yeager, our Director of Consulting, for a four-day Thinking Maps TOT. At the end of the second day, Chris was approached by an 8th grade social studies teacher who needed inspiration for his opening the unit on the Constitution. He explained that he wanted to focus on the first ten Amendments by engaging students in both critical and creative thinking. Chris not only agreed to help, but she actually taught the lesson to his students! The students were at the edge of their seats actively thinking for the entire 60 minute presentation. I want to share this lesson with you.
First Chris began with a (1) Circle Map to determine what prior knowledge the students had about the Constitution: “Tell me all the things you know about the Constitution, and in your Frame of Reference tell me where you learned that from, whether it was from your personal experience, from your mom, or dad, or school.”
Next, students used the (2) Brace Map to identify the parts of the Constitution.
She then had students transfer the information from the Brace to the (3) Bridge Map. Students pulled information from their resources to identify the function of each part of the Constitution while also including the page numbers within their Frame. Citing your sources is a key habit to build in all students, especially in this digital age where we can easily borrow ideas from each other.
Using a (4) Tree Map, a copy of the Constitution, a US History textbook, and reliable websites, the students were placed in heterogeneous cooperative learning groups and assigned an Amendment. They were instructed to define their Amendment using their resources and include an illustration. As each group completed their assignment, they presented their Amendment to the entire class and posted their work on the large Tree Map.
This first part of the lesson is all critical thinking: Defining a concept with a Circle Map, dissecting the parts of a whole with a Brace, using resources to bridge the relationship between the parts, and classifying the Amendments with a Tree Map employs multiple levels of understanding and analytical skills, especially as you move from one thinking exercise to another. The second part of the lesson transitions students from thinking critically about related concepts to thinking creatively by connecting those concepts to their own value systems.
Chris asked students to work individually to reorganize the Amendments in order of importance from their point of view by using a (5) Flow Map.
Then she asked students to make predictions with a one-sided (6) Multi-Flow Map.
I always like to have students take their thinking “off the Map” so I conclude the lesson with this last step in both critical and creative thinking: I ask students to present an argument in an essay format for why the Amendments are an important part of the Constitution. Students used the (7) So What/ So Why Frame of Reference questions for their opening and closing and the information from the Maps to provide the supporting details for their argument.
These rights, as defined by the various Map activities, are guaranteed to all citizens of the United States.I will leave you with this one sentence, which for me captures the essence of the amendments also known as The Bill of Rights:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof’; or abridging the freedom of speech, or of the press, or the right of the people peacefully to assemble, and to petition the Government for a redress of grievances.”
Do you love or hate technology? Most of the time the answer to this question is complicated. Other times, it’s easy to say “I love it” or “I hate it.” On one end, technology can enable students to access a depth and breadth of knowledge at the touch of a button. On the other end, it can also distract them while driving and eat away at their grammar. At the end of the day the most important thing to remember is that technology is a tool. It’s an extension of the human body that you can manipulate and hold. How you choose to use it is what makes it good or evil.
As you read the following collection, consider whether the critique is of the practice or of the technology itself.
Give students a thought and they’ll learn for a day. Teach students to think and they’ll learn for a lifetime.
Teaching is one of the most important professions of all time. As it changes to reflect what students need in the 21st century, we stick to our foundations: equipping learning communities with the tools for critical and creative thinking. With that frame of reference in mind, here are seven lessons by Chris Yeager on the practice of teaching critical thinking.
Chris Yeager is the Director of Consulting for Thinking Maps, Inc, and co-author of Thinking Maps®: A Language for Learning and Thinking Maps®: A Language for Leadership, 2nd Ed. She joined the company in 1995, following 20 years of work as a high school English teacher, Instructional Curriculum Facilitator, and assistant principal with Cumberland County Schools in North Carolina. She blends her training in critical thinking, mastery learning, cooperative learning, elements of instruction, and cognitive coaching with her work in Thinking Maps. Chris sees Thinking Maps as a realistic way to apply the principles of brain research in the everyday process of teaching and learning.
The activity I am sharing with you is one I conducted with my staff when I first served a principal. I hope you find it to be useful.
As you prepare to begin a new school year, there are certain to be memories of previously-faced challenges that are threatening to cloud your sunny view of the coming year. I encourage you to begin by acknowledging them by creating a Circle Map to brainstorm and record these challenges.
When you have noted the challenges you previously faced, select the one about which you would most like to consider the resulting lessons learned. Now, create a one-sided MultiFlow Map to identify as many positive outcomes of this challenge as possible.
Once you have identified them, take some time to reflect on these lessons learned and select the one that has been most valuable for you.
Following this selection, please tear a strip of aluminum foil and write this valuable lesson learned on the aluminum foil strip using a permanent black marker. Place this strip in a prominent place in your classroom or office as a shiny, tangible reminder of the silver linings that result from even our most difficult challenges. This silver lining should serve to foster a growth mindset and inspire hope throughout the coming school year.