5 steps to teach concepts ... in math

Last year, I shared a blog post with friends and colleagues on Facebook entitled "Are You Doing It Wrong? How to Introduce Telling Time" by Cindy Lee of Ainslee Labs (which has since been reposted here: Avoid the biggest mistake teachers make when teaching time ).

It caught the attention of my second grade colleagues and now together, we're trying to figure out how to put Ms. Lee's ideas into practice with students.

Traditionally, students are introduced to telling time by first learning time to the hour. Next, students learn to tell time to the half hour, then to the quarter hour, then to the nearest five minutes and finally to the exact minute.

Ms. Lee has experience that shows this way is confusing to kids (I've had that same experience!). We first teach kids to look wherever the short hand is pointing to to read the hour. But soon thereafter, this understanding becomes a misconception.

To address this confusion, Ms. Lee suggests using a clock graphic colorfully divided into twelfths, teaching the kids that each hour has an area or space that belongs to them.

Below, I've detailed how I used my five-step concept-based lesson framework to introduce this idea to the students.

1. Start with a concept.

Before the lesson, I decided on three related objectives to make sure that my lesson was three-dimensional (to learn more about 3D curriculum and instruction see anything by Lynn Erickson).

Concept: The students will understand that you can't achieve your goal if you use the right tool the wrong way.

Knowledge: The students will know that the hour has an area or space that belongs to them on the clock.

Skill: The students will be able to read the hour on a clock, regardless of how many minutes have passed.

Although I was specifically concerned with students gaining mathematical knowledge and skill during this lesson, I was equally interested in them learning the timeless, abstract, universal and transferable idea that even if you're using the right tool, you need to use it the right away to be able to reach your goals. Striking a balance between teaching and learning knowledge, skill and concept objectives is paramount, in concept-based teaching.

2. Pick a specific, concrete example of a person, place, situation, or thing that illustrates that concept & 3. Create an opportunity for students to explore that concrete example.

To introduce students to the concept of using the right tool the right way, I brought in a tool box, a hammer, a small board and my lanyard.

I told them I had a goal of making a lanyard hanger so that it wouldn't get tangled on my shelf at home. We first chatted about the right tool I'd need out of my toolbox. When we decided that a hammer was indeed the right tool, I tried to hammer in the nail with the end of the hammer. When that didn't work, a student suggested I use the top of the hammer. When that didn't work, someone suggested I used the circle part of the top.

As the nail went into the board, so too did the idea into the kids' brains that you need the right tool used the right way to get the job done!

Then I introduced them to another tool - a clock.

"When we use a clock as our tool, what is our goal?" I asked.

As one girl offered (and all the second graders agreed), we use clocks to read the time.

So, I asked the students to write on their whiteboards what the time was on each of the following clocks:

All the students were able to accurately tell me the time. We all agreed that to read the time on those clocks, you just needed to look where the hour-hand was pointing.

"Okay," I said, "now tell me what time it is."

We counted together that 55 minutes had passed and they all had to read the hour by themselves. Just like clockwork (pun completely intended) all but 1 or 2 students said it was 2:55.

Then, we had a conversation about how we had been using the right tool to tell the time, but we were using it in the wrong way.

Next, I showed them the clock face with the partitioned hours (thanks again to Ainslee Labs for the free clock face. Want to pay for her whole telling time kit?).

We used this clock face to practice telling the time with various times. Over and over again, the students had no trouble identifying the correct hour, even when the hour hand wasn't pointing directly to a particular number. Eventually I removed the gray numbers inside the clock and 100% of the students were able to read the hour on a clock, regardless of how many minutes have passed because they knew that the hour has an area or space that belongs to them on the clock.

4. Check for understanding by having them write a concept statement. 

(This step is very similar to the thinking routine Headlines found in Making Thinking Visible by Ritchhart, Church, and Morrison, p. 111)

Since I was confident that the students had reached the knowledge and skill objectives of the lesson, I asked, "What did you learn today about how you should use a tool to achieve your goal?".

I gave the students a chance to write down their individual responses on their whiteboards and then one-by-one we shared until everyone's ideas were represented in our statement. Below is the sequence of revisions that our conceptual statement when through. My prompts are highlighted in yellow.

"What do you have to remember about tools when trying to achieve a goal?" First child responds:

"Does anyone want to add anything?" Second child adds:

"Does anyone want to add anything?" Third child adds:

 "Is this true just when we're telling time?" Fourth child adds:

"Who needs to do all of this?" Fifth child responds:

"That doesn't sound right." Sixth child corrects: 

The students were comfortable that their conceptual statement encapsulated all the learning they had done up until that point, so we ended the lesson there.

Often times when crafting conceptual statements (sometimes called generalizations, central ideas, or enduring understandings) we create hard-to-understand statements that don't sound like how we normally talk. Concept statements should be written in language that is easy to understand to all who read it. One way to do this is let the students craft the statement, as we did together in this lesson.

5. Reflect on their thinking and decide next steps.

Moving forward, I would want these students to continue to practice reading the hour using this partitioned clock. Eventually, I would want them to independently be able to read the hours and the minutes first with the partitions and then after slowly taking away that support.

Conceptually, I would want to explore other tools and discuss the correct way to use them and contrast that with the wrong way to use them. This conversation would fit in wonderfully with any beginning-of-the-year guided discoveries that students experience as they are exploring the learning tools of the classroom.

After reading how I used a clock lesson to teach a timeless, abstract, universal and transferable concept to second graders, how do you teach these BIG IDEAS to your students?

Student Engagement: more than being busy

Educators in our school district are focused on increasing student engagement. We intuitively know that if students are going to learn, they must be captivated by what they're learning. But being engaged is more than just being busy. So how do we define student engagement?

The Science House at the Science Museum of Minnesota uses this formula to define engagement:

Talking on Task + Manipulating Materials = Student Engagement

They surmise that if students are engaged, students will learn.

Our Teacher Growth, Development, and Evaluation System plan relies on the following components of the Charlotte Danielson's 2013 The Framework for Teaching to define and measure student engagement:

• Domain 2 
• Component 2A: Creating an Environment of Respect and Rapport 
• Component 2C: Managing Classroom Procedures 
• Component 2D: Managing Student Behavior 
• Domain 3 
• Component 3B: Using Questioning and Discussion Techniques 
• Component 3C: Engaging Students in Learning 
• Component 3D: Using Assessment in Instruction 

These components give educators helpful parameters and strategies to engage students in "discussion, debate, answering 'what if?' questions, discovering patterns and the like," (p. 69). I acknowledge that if teachers implement such strategies, specifically Visible Thinking Routines that I have relentlessly promoted on this blog, students will be more engaged and more will be learned. However, if teachers want to truly engage students, it will take more than teaching strategies and techniques aimed at captivating student attention. For students to be authentically engaged, curriculum must be relevant, challenging, and significant.

Currently, in our the elementary school, students are often asked to learn literacy and mathematical ideas outside the contexts that these ideas are really for. Getting all students to engage at high levels proves to be extremely difficult, no matter the engagement strategy, when we teach these complex ideas in abstract, decontextualized terms.

Ron Ritchhart, in his book Intellectual Character: What It Is, Why It Matters, and How to Get It, tells the story of teachers who connect their course activity to big ideas to enhance the purpose and meaning of the work for students. He writes that by doing this, "students become clear about the larger purpose of the class and what the teacher wants them to understand and learn more about. This makes it easier to engage students in thinking because the work itself demands thinking and active exploration." He continues, "the thinking demanded of students is authentic in that it serves particular ends, unlike certain one-size-fits-all thinking-skills programs that might introduce a thinking skill in a discrete context disconnected from anything else students might be doing," (p. 150-1).

Ritchhart thus is advocating for curriculum that is based on students constructing understanding of relevant, challenging, and significant ideas, not delivering insignificant curriculum using engaging activities.

So, if we can agree that to authentically engage students, we need the right combination of engagement strategies and relevant and significant curriculum that is based on understanding big ideas, what are these ideas that are worth knowing about?

In the International Baccalaureate's (IB) Primary Years Program (PYP), the answer is to design a concept-driven curriculum; a curriculum where the understanding of significant ideas is more important than the memorization of isolated facts. H. Lynn Erickson writes in her book Concept-Based Curriculum and Instruction for the Thinking Classroom that concepts are timeless, abstract, universal, and transferrable (p. 31). But there are many big ideas worth knowing that fit this description. How are we to know which ones to teach?

Grant Wiggins and Jay McTighe suggest a list of transferable concepts in their 2005 book Understanding by Design (p. 74). They also give tips for finding big ideas within the state academic standards. Erickson echoes this idea by suggesting using the stem, "The student understands that ..." to identify concepts in state standards. She notes that this lead-in phrase "sets up the structure for a generalization sentence - two or more concepts stated in a relationship," (p. 52).

The more I work with teachers at identifying concepts - that is, ideas worth teaching - the more I think that another way to "test" to see if the idea you're considering is a concept is to consider how important the idea is to adults. If adults are continuously wrestling with the particular idea, chances are you're working with a concept.

The bottom line is this: as long as our elementary curriculum is exclusively delivered separately in reading and math blocks with little time dedicated to learning literacy and mathematical processes and strategies in the context of constructing understanding of key and related concepts within the PYP units of inquiry, our students will never truly be engaged with any of it and standardized test scores will continue to lag.

Engagement is more than students paying attention and being busy. In order for students to internalize literacy and math processes and knowledge concepts, they must be taught in the context of learning real-world big ideas; ideas that are timeless, abstract, universal, transferable, and still being discussed by adults.

When we begin to deliver this kind of engaging, authentic curriculum, coupled with employing engagement strategies designed at getting students actively and meaningfully thinking, talking, and working, we will start to see increases in levels of authentic student engagement and increases in standardized test scores soon will follow.