As we work through our measurement chapter, students are learning to be more precise by watching out for gaps and not overlapping. These are skills that will sharpen with time and exposure. Allowing students to work together to tackle how they will measure and what they will measure with, also provides invaluable skills.

When students are given the opportunity to work together on tasks, they are forced to use mathematical language and reasoning skills to explain their thinking with their partner/group.

In the pictures below students were placed in groups to measure different paths. They had to agree as a group which tool they would use (paper clips, tiles, straws, string or paint stirrers). As they traveled from path to path, their measurements became more precise because they collaborated and discussed their strategies for measuring. The lesson to be learned here is that with time, collaboration and the teacher stepping to the side, students can figure out everything we want them to. Know what the key ideas allow the learning to happen!

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.

In these initial phases, we are teaching kids how to reflect. To look over their week and explain certain images where they learned, wondered, or were confused about. Many have never been asked to “reflect”, especially in math. As we begin this year, we are trying to introduce this and make it a habit. The hope is that the mathematical language becomes stronger and the thinking becomes deeper and meaningful to understanding concepts better.

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.

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?

Counting Activities should include matching objects to pictures and different representation of a number.
Games like “Go Fish”, “Concentration”, and “War”, where student have to make a match are great ways to practice the early stages or recognizing numbers and counting.