We can all agree that metacognitive thinking is obviously beneficial. The environment in which a student feels comfortable enough to be a self assessor can be tricky. What does that look like? What is the goal?
Getting away from the time-honored question, “Is this the right answer?!”
As former students, this is probably the question that motivated most adults as young learners. The response of, “Yes, you are correct!” or “No, you need to work harder.” always seemed to hold a finality of the lesson.
That is where we all decided that we were “great at math” or that we would be lifelong math strugglers. We all have those memories.
So how do we as teachers change that narrative? How do we facilitate students in becoming their own teachers? How do we give them a voice and the confidence to speak to their own capabilities and shortcomings and how is that beneficial?
Learning Goals and Scales
Reflecting back to John Hattie’s book, “Visual Learning”, there are questions that guide a student through the self-assessing process.
Relating to the content each student should ask themselves:
Where am I going?
How am I going to get there
Where to next?
Keeping those guiding questions in mind each student is given a Learning Goals and Scales chart for each chapter studied.
The goals and scales are based on a combination of Common Core Standards and State of Florida Item Descriptors.
Seen within the items are leveled abilities. This is important to guide the learners in their self-evaluation. It is also very helpful for the students to keep the chart throughout the chapter.
This is not a summative assessment or checklist to use at the end of the chapter. The students should have access to help guide their thinking and goals. The goals and scales are written in “kid-friendly” language as “I can” statements.
This is done to make ownership of the goals easily relatable to elementary-age students. In reading each statement the student is then able to designate the color that best fits their comfort level for that item.
As it relates to their feelings about a topic:
Red indicates that the student feels that they are not confident in this skill. They feel that they require more work in this area.
Yellow indicates that the student feels somewhat comfortable with the statement. They can be effective when leading in whole group scenarios, or with partner support, but may still have questions.
Green indicates that the student feels very comfortable with this skill. Not only could they complete the tasks at hand, they could also explain them to a classmate.
Support through Discussion
The students’ responses to their Scales and Goals can be seen below. Those are then used to guide the students through self-assessment by applying the goals to different examples that have been either created in their Math Journal or through concrete examples that the students can demonstrate and explain.
Examples below show the students reflecting on their goals for their first chapter in their mathematics classroom this year.
The students were allowed to use their goals and scales to keep track of their thoughts and their own accountability and then relate those directly to examples.
These discussions are truly the cornerstone of metacognition for the students. Being told that their thinking is more valued than a simple score is the first step in supporting a student in seeing themselves as not just as a spectator in their education but as their own teacher.
Initial Teacher Takeaways
Having a specific routine for this method of accountability and reflection is crucial. There must be a procedure put in place that helps the student guide themselves independently. If these steps are not taken students become distracted in the implementation of the task and the quality of the actual reflection is lost.
Providing the students with the Scales and Goals (I can statements) is the first step, but the teacher contribution in creating those goals must be instructionally sound and purposeful.
Something as simple as incorporating the math topic in the journal headings allows students to organize their thoughts for their accountability discussions with their teacher.
For example, if the first three “I Can” statements relate to place value and the student feels that they can prove that they are comfortable with that topic, looking back in their journal headings that address Place Value will give them a place to reflect and prove their thinking.
It should also be said that in some of the discussions and pictures seen below you can see that through the interactions the students actually change their ability levels from their initial thinking. That might be the most profound takeaway with these strategies early in the school year.
Precision teaching is the idea that relates to the book, Visible Learning by John Hattie. The premise behind Hattie’s research is the correlation between surface, deep, and transfer learning.
The findings go on to address the influence that different aspects of the education system have on our students and the math classroom. Basically, the idea of getting the “most bang for your buck”.
Every school district, school, and classroom teacher wants a magic formula for success. With limited time and funding, how do we ensure our academic goals are being met in the math classroom?
As math teachers, how do we know if our students are learning?
Setting routines are one of the cornerstones of a
successful classroom. It is important for teachers to understand the difference
in a compliant classroom and an environment conducive to learning.
When beginning the school year it is crucial to
ask ourselves a few questions when considering routines and procedures.
Do your students know what they are supposed to be learning? And more importantly why?
Are your students benefitting from your expertise?
Are your students capable of connecting their learning to what they’ve already learned and are they able to see where this learning goal will lead?
Are your students able to manage their own learning?
At the end of the lesson, are your students able to hold themselves accountable for what they have learned? (leading into self-assessment)
Self-assessment has one of the largest effect sizes for student learning (Hattie, 2009). The process of self-assessment requires several different elements implemented in the classroom.
One example of self-assessments in the math classroom is the use of Math Journals. Beyond a journal prompt containing the “content goal” teachers have an amazing opportunity to go beyond a right/wrong answer by providing students with a voice for self-reflection.
Below is an example of a pre, mid, and post year self-assessment used in a fourth-grade classroom to gauge confidence and attitude toward mathematics.
This same questioning is also used in this classroom in the aforementioned Math Journal activity. The example seen below addressed the math topic of Place Value. The students were asked to describe the value of the 8 in the number 88,888.
They could use pictures and words. The prompt was actually shared at the beginning of class as an informal discussion. Students were asked to read the prompt and think about what they already knew about place value, what they remember from previous lessons that would help them with today’s goal, if they felt like the topic was going to be difficult and if so, why?
They were given time to think about their responses and write them in their journal, then after the lesson, they returned to their journal to work on the prompt and reflect on their thoughts from the beginning of class. Once journals were complete, they were allowed to share the journals with their group.
The outcome of this part of the activity is to foster a safe space for math discourse, more importantly, mathematical talk is proven to lead to the ultimate level of transfer of learning and metacognitive awareness.
If self-assessment is the goal and math journaling is one tool, what is the payoff? How do these techniques help spark success in our math students? Metacognition is the ability to think about our thinking (almost sounds too simple).
While it may sound simple it is the cornerstone of all actual and true learning. Metacognition happens when students practice self-reflection on their level of thinking. In addition, when students can relate this to a target it becomes powerful enough to increase understanding and motivation.
This knowledge promoted in the classroom is invaluable. As teachers, we all dream of that day when our students are intrinsically motivated. A moment in the classroom when the value of the content rises above a state standard or an arbitrary notion but becomes a self-fulfilling desire to learn and grow. But students need guidance and tools to develop their metacognitive awareness and become confident in the ability to self-question.
Formative evaluation is the process of gathering evidence to inform instruction. In simpler terms, it is the process for teachers and students to communicate about the learning that is taking place and the direction that instruction should go. Formative evaluation should drive instruction decisions, gathering real-time data is crucial in guiding how the teacher will proceed with the delivery of the lesson.
In the book, Visual Learning, Hattie speaks of several internal questions
that drive learners:
Where am I going? What are my goals?
How am I going there? What progress is being made towards my goal?
Where to next? What activities need to be undertaken next to make better progress?
Consequently, these are the same three questions that teachers must ask about instruction as they make adjustments based on the data they gather from students. Some examples shown below, include response cards (whiteboards), and exit tickets, journal entries may also serve as a way to inform instruction.
Building community beginning on the first days helps to create new relationships and strong bonds that will last throughout the year. Creating a shared vision of the expectations, developing a common understanding of classroom limits, and fostering a love of learning are only a few of the characteristics you might have in mind as desired outcomes. Ultimately, achieving mutual respect and a spirit of collaboration creates an ideal working environment for the classroom.
When community exists, each child feels valued. A sense of shared purpose unites the group and working together to accomplish goals becomes a priority. Our goals are BIG and require the effort of all of our members. The uniqueness that each student provides as a member of the community must be valued and each individual strength will make the community stronger and better. As children develop a sense of duty to the community, self-discipline is likely to emerge more naturally and from the child’s (intrinsic) motivation rather than from external or reward-based methods (extrinsic).
Early in the year, creating purpose in the child’s movement and activity is desired and we balance the freedoms offered within the environment, the needs of the young child to move, and the constraints of the environment. Providing structures and routines will help to create order as well as ensure a safe environment for your children. A strong sense of community is one of the most effective ways to teach how to use individual freedoms.
How do we build community?
We play games and have fun together. We share lunch and work with each other, mixing-up our groups with an emphasis on getting to know new friends. We interview and find out more about each other by sharing experiences, stories, traditions, and the accomplishments we are proud to have achieved. We make time to appreciate each other and learn how to recognize others, as well as ourselves.
In our community, we learn to problem solve, developing the skills necessary to take care of ourselves and others. When solutions are found and conflicts resolved with little or no direction or intervention by an adult, students feel great pride! Creating a class agreed-upon list of rights and responsibilities with the students allows them to partner in holding others accountable and enforcing your shared vision of community.
Grace and courtesy work also play a role in learning how to act in a community. A firm handshake and smile in the morning set a respectful tone for the day. Allowing students to have the role of a “class greeter” is a great way to have students serve in a leadership role as they create personal and inviting welcomes to the community. Practicing how to greet visitors with a cup of tea and a special chair or preparing a class snack are other ways students can assume responsibility. Modeling ways to ask for help, challenge other student’s ideas and even how to say “no thank you” respectfully are tools your students will need to have in order to work effectively in their community.
A natural extension of building community within our classrooms is to reach outward. The work that starts within our classroom might find opportunities in other areas within the school. Participating in the work of the larger community helps the students feel proud and invested. Students experience, on a small scale but in a real way, that they can create change. We can act individually or as a group – and we DO make a difference!
Specific Ideas to try at the beginning of the year might
Toss a ball in a group to help learn names
Learn a favorite food of a new friend
Create a scavenger hunt in the room to learn a new environment
Share with a friend something you like about yourself
Work together to line up without talking
Offer lessons on classroom jobs
Provide lessons and model grace and courtesy
Make a list of “Classroom Rights and Responsibilities” WITH your students and have them initial or sign
Have a procedure or place in the classroom for resolving conflicts –create a “Peace Table” or “Peace Corner”
Follow our Journey. Lesson Study with a Focus on Visualization.
Jugyou kenkyuu, a Japanese phrase gives us the term “Lesson Study”. Introduced in the U. S. in the late 1990s, interest in Japanese lesson study remains strong in the education world throughout the United States. Our Lesson Study this year will focus on visualization and metacognition.
Lesson Study & Mathematics
Lesson study works well across education and in particular, in improving mathematics education. We will wrap up professional summer reading on visualization in September with a look into the routines we create in classrooms that promote visualization. During “Introduction to Lesson Study” in October, we will explore what lesson study is, how it works, how to use it, and best practices with a focus on creating metacognition in students.
Pre-Lesson Study Questions
We engaged our focus group from St. Edward School in Vero Beach by asking the following questions:
What attracted you to this Lesson Study?
Overall the participants felt this lesson study would improve their ability to use visualization strategies in their own classrooms. They felt the experience would allow them to “dig deeper” into learning the best way to improve their teaching skills to build visualization.
What do you hope to learn from this Lesson Study
Participants generally responded similarly, wanting a deeper understanding of the science behind visualization, learning how to integrate visualization into their daily teaching, and using visualization to help students see concepts in a different way.
What is visualization to you?
It is creating a picture in your mind, being able to ‘see’ what you are hearing or reading to help you better understand the lesson, and it brings life to situations, assisting a student in understanding the concepts being taught.
What do you feel you already know about visualization? (before reading)
The response to this question was consistent with all participants. All felt that visualization was a way of seeing something in your mind to better or fully understand it and using it in math as well would bring life to situations and assist students in better understanding the concepts being taught.
Ideas on how to get kids to visualize math?
Using various concrete and pictorial models
Incorporating color in our board witing to connect ideas
Relating ideas especially in the operations
Have children create a short movie in their minds with each math concept so they can ‘see’ the process and verbalize it before computing
What questions do you have before we start the lesson study?
Can all students visualize?
How are other teachers using visualization?
Does the brain have any physical limitations with visualizing?
How do we teach visualization to students so they use it seamlessly when seeing a math problem?
What forces the brain to want/have to visualize?
We will be holding a private Lesson Study at St. Edward’s School, Vero Beach, FL in September.
Follow us through this Lesson Study.
We’ll be at Oak Hill High School in Nashville, TN, October 2, 2019 – October 4, 2019. Seats still open!
Click here to register for this event and for details on this Lesson Study.
I am dying with my 5th grade…they are struggling with fractions (the chapter is hard) and with not a strong Singapore foundation from last year…well…we are on our 4th week! Should I do the 4th grade fractions chapter?? I honestly am at a loss.
I am currently in the unit of Parts to Whole, teaching bar modeling. Looking ahead, it seems to me that bar modeling is a big part of not only this unit but upcoming units as well.
My wondering and question: I have students who are looking at the problems and can figure them out by stacking numbers then using the renaming strategy which was what they learned in lessons before bar modeling. Should I be encouraging all of my students to draw the bar model before stacking the numbers to solve the problem? How critical is it for them to bar model? Please know that I am not saying that I don’t want to teach it at all, but I don’t know how much I should be “pushing” those that either don’t understand it or those that feel like they can work the problems using other strategies.
Math facts are simply the basics: addition, subtraction; multiplication; division. They are basic number combinations and calculations we do every day. So why is it that learning math facts creates such huge problems for teachers and students in our classrooms?
Common subtraction mistake when children are learning math facts.
Is it that some of us are just naturally better at math than others? Perhaps. The good news is you as a teacher can help anyone improve their math skills. You might be wondering how. Well, rote memorization alone won’t get us there.
Often we teach the meaning of equality as the equal sign indicating “put the answer.” To prepare students for future mathematics, we need to rethink the way we communicate this representation to students. The equal sign is too important to attach such a limited meaning, especially when students are moving to abstraction. When moving to abstraction, it is important to use this symbol as a “reader” versus a “speller.” What do I mean by that? Let me explain. First, a speller sounds foreign because the student is trying to make sense of abstract symbols that they don’t fully understand. A reader connects meaning that they can comprehend while connecting the story to the symbol.
For example, when teachers record the following equation,
Using a simple addition problem as a reader versus a speller.
A speller would say, “Four plus five equals nine”.
A reader would say, “Four apples and five apples is the same as saying ‘I have nine apples’” .
In the journal prompt below, we want students to see the statement as 5 and 4 is the same as 2 less than 11. If we focus too much as a speller, students see symbols as performing actions rather than relating ideas, which leads to misconceptions of the equal sign.
Less is more. Hesitate in thinking math becomes harder by creating larger numbers, rather provide depth early on using children’s intuitive understanding of basic number operations. A simple journal entry can help assess what students understand about operations and the meaning of equality. Once you find that a student understands the meaning of addition and subtraction, it is time to create tasks that dive deeper in understanding equality. These task can be as simple as asking questions such as, 4 + 5 = ____ + 6. Additional equality tasks can be found in the Algebraic Thinking and Reasoning with Numbers books (Groundwork Series) by Greenes, Carole, et al., (https://www.mheducation.com/prek-12/program/MKTSP-O8302M06.html).
In the book, “Thinking Mathematically,” Carpenter, Thomas P., et al. suggest some benchmarks to keep in mind while moving towards the conceptual understanding of equality.
Establish a starting point. What are students’ initial ideas regarding the equal sign?
Mix it up. Don’t always write equations in the form a + b = c, rather c = b + a.
See the equal sign as a relational symbol. Emphasize students proving that each side is the same as the other.
Compare sides without calculations. Encourage students to look for relationships without performing calculations.
Listen to students and be aware of where they are in the process. Encourage them to question if it is true. If so, why is it true? By listening and looking we may be pleasantly surprised at what we find out.
Below is student work demonstrating various levels of understanding within the concept of equality.
Spiky says that 5 + 4 = 11 – 2 is false.
Curly says that it is true.
Who do you agree with and why?
This student understands how to simplify expressions but lacks applying the meaning of equality.
This student is confused in understanding what it means to have expressions on each side of the equal sign.
This student understands the meaning of equality. What can we ask next?
This student demonstrates a shallow understanding of equality. What can we do next?
This student can solve for an answer but needs practicing in mixing up the form of an equation.
Often we share our students progress by sending home “Monday” folders with worksheets that were completed, test or quizzes taken the previous week, or maybe we have a site where parents can go and see the current grade in your class. Are we informing parents on their child’s progress? Does the grade or worksheet give an accurate picture of student growth?
I often challenge teachers to give feedback on an ongoing basis. We should not only assess from the papers turned in but from what we are seeing, hearing and collecting on a daily basis. Use journal entries or a table with your objectives (I can statements) to allow for a more productive discussion at a parent conference. The following guidelines can help your conversation whether it be in the form of report card comments, parent conferences or the chat in the pick-up line.
-What area is the child is doing well? (Have pieces of work that demonstrate growth.)
-What area does the child struggle? (Show specific concepts not just a broad topic.)
-What are you doing in the classroom to help?(The learning does not stop after a chapter test. It is your responsibility to help the child learn it!)
-What can they do at home to support you and their child? (Parents want guidance on how to help.)
Attached is an article from the Wall Street Journal – 10.18.17 addressing the idea of a student-run conference. This conference helps children communicate their progress while building the metacognitive process. Would love to see this happening more in our schools.
How can we differentiate instruction without putting more work on our plate?
Teachers do not have time to prepare multiple tasks and lessons to meet the needs of each student in the classroom. The key is giving an anchor task that all students can enter and knowing how to adjust the task based on our observations.
In this class, students were guided to use “Katell’s” method (aka-left to right strategy) to solve the problem 65 – 12. Notice how the student recorded their thinking process.
How can we differentiate with one task?
During the guided structure component of the lesson, teachers are observing and questioning students. This student demonstrates and can explain the left to right strategy. What comes next for this student? During our planning phase, we need to ask ourselves, “What can I do for the student that already knows the answer”? Looking at this student’s work, how can we deepen the mathematical understanding?
After listening to the student explain his thinking, the teacher commented, I can understand your verbal explanation, but your written work is confusing to me. I wonder why I am confused? The teacher then walked away, and the student grappled with the idea of equality. Differentiation is meeting the student where they are at and extending the learning.
Do not try to create more problems for students to practice. Plan your task thinking of what mistakes students typically make and how you can help extend the thinking. Use your observations to go deeper and differentiate!
What are other ways you could take the task 65 – 12 and extending the task? Share your thought below.
How can we help student refocus on understanding versus just an answer?
Too often when we pose a problem and students shout out an answer. We need to ask ourselves do students understand the concept or are they obtaining the correct answer by fitting the symbols and numbers into a structure they know?
In this example, the anchor task was to subtract 34 from 87. The teacher wanted to screen the children first wondering, Do students know what it means to subtract one number from another? To find out more the teacher posted the following problem, removing the numbers.
Do students have the conceptual understanding or just fit the numbers into a given structure?
Students were asked to set up the expression that could represent the situation. The student A on the far right was the only student in the class that seemed to understand. When asked to explain his thinking, many observing teachers felt he understood the concept and that he gave us a platform to generate a discussion.
Do students understand the part-whole relationship here?
Following this analysis, the numbers were inserted into the problem. Subtract 34 from 87. It was interesting to see student A’s work. Much to our surprise, Student A who seemed to understand the concept had a hard time transferring that knowledge to another situation. Notice his equation is not correct but he gets the final answer.
Can student A transfer knowledge from one setting to the next? Look and listen to help guide students to conceptual understanding. Looking at answers does not tell us the whole story.
Less is more. Spend more time on conceptual understanding and listening and watching students versus solving more problems. The answer does not help assess student reasoning or how we can extend or guide the learning process.
Participants were asked at the 2017 Fall Mathodology Institute if it was possible to use the digits 0-6 only once to make a true statement?
Is it possible? Participants worked and worked and did not find a way to solve the problem with the given conditions. The teacher allowed the task to move on without closure for a purpose. When students work to become confident problem solvers we want them stating, “this is not possible, I have exhausted every situation.” Listen to their reasoning, and look for a systematic way to prove it is not possible. All too often we focus on finding the solution, and we can deepen the process and allow for practice when we alter the task.
Try giving a task where it is not possible and see how your students handle it. Follow-up by asking, “if it is not possible, can we change one of the symbols to make it a true statement?”
When we do not provide immediate closure, we allow students to continue to explore. Many participants emailed after the institute because they would not quit until they figured it out. Below are a few of the pictures they sent.
Great perseverance by participants who refused to give up?
This is what we want from students…next question for Shelly…can I change any symbol to make it a true statement? If so is there only one way?
Ok…..I’ve worked literally for hours and I’ve come to the conclusion that this problem is NOT possible. If the lesson was to teach us to let kids struggle so they work harder like a ???? I get it. I’ve done at least 100 fraction problems over the weekend. If you would’ve given us the answer I would’ve never thought about this problem again……but you’ve let us struggle and now it’s just painful. Please tell me it’s not possible my sanity is at stake.
At 2017 Mathodology Fall Institute participants worked on a task from think!Mathematics which required them to use the given shapes to make a composite figure. Figure C posed a problem for many. Is it possible? Participants worked and worked and did not find a way to solve the problem with the given shapes. The teacher allowed the task to move on without closure for a purpose. When students work to become confident problem solvers we want them stating, “Figure C is not possible with the given shapes. I would need an additional… to complete this shape.” We have built perseverance and another way to assess them. Do they know what to ask? “I need two more pieces of one shape.”
When we do not provide immediate closure, we allow students to continue to explore. Many participants emailed after the institute because they would not quit until they figured it out. Below are a few of the pictures they sent.
Try giving a task where there is not enough information and see how your students handle it. Will they be confident to state this is not possible, and know what to ask?
Great perseverance by participants who refused to give up?
Students using the think!Mathematics program reinforce the count on strategy. Games/activities can be used during the guided practice part of a lesson. Students are encouraged to count on from the larger number.