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?

Zoom In

Last year, the G1 teachers with whom I work conceptually reviewed a unit of inquiry that is largely based on Minnesota K-12 Academic Standards in Science. In the unit, the teachers want students:

• to know the different attributes used to describe rocks (knowledge-based benchmark),
• to be able to group and classify rocks by those attributes (skill-based benchmark),
• to be able to identify similarities and differences between soil and rocks (skill-based benchmark), and
• to be able to identify and describe large and small objects made of Earth Materials (skill-based benchmark).

As part of the review process, the teachers identified the big ideas (called concepts) that they want their students to understand:

• Form
• Change
• Causation
• Observation
• Description
• Comparison
• Classification
• Material

Ultimately, the teachers want their students to understand that "observing the environment leads to discoveries." By wording this enduring understanding in this way, teachers can easily transfer this idea to other areas of the content, not just science.

To kick off this unit, I was invited to lead a class of G1 students through a provocation.

While planning for this provocation, I used similar steps I have used previously to teach conceptually (until it becomes more natural for me to teach with these BIG IDEAS in mind). However, I've recently seen a need to make a couple of changes:

1. Start with a concept.
2. Pick a specific, concrete example of a person, place, situation, or thing that illustrates that concept and 3. create an opportunity for students to explore that concrete example. 
4. 3. Check for understanding by having them write a concept statement.

5. 4. Reflect on their thinking and decide next steps.

1. Start with a concept.

I want the G1 students to understand that you have to make careful and close observations to really learn about something.

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

To give students the opportunity to think about observing carefully and closely in order to really learn about something, I decided to use the thinking routine "Zoom In" from Making Thinking Visible by Ritchhart, Church, & Morrison.

Although it has nothing to do with rocks (or perhaps because it has nothing to do with rocks) I wanted to use an image from a book by Carmen Lomas Garza, "a Chicana narrative artist who creates images about the everyday events in the lives of Mexican Americans based on her memories and experiences in Texas and California," (from Garza's website). The specific image I used was from the story "Birthday Party" from the book Family Pictures / Cuadros de Familia.

At first, I showed the students a small bit of the image and asked them, "What do you observe?" and "What is your hypothesis of what is happening?" Students recorded their observations and hypotheses on a blank sheet of paper (samples of these observation recording sheets follow).

Students observed leaves, a tree, branches, and sky but they weren't really able to form a hypothesis of what was happening. We discussed that is the case because they didn't have a lot of information, so I revealed a bit more of the image.

With this new revelation, I asked them, "What new things do you observe?" and "How has your hypothesis of what is happening changed?"

Students noticed the flowers, the fence, and the woman with the crutches. They thought she might be planting a garden, but still weren't very sure what was happening because how can a woman with crutches successfully plant a garden? I pointed out to the students that their observations were actually leading to more questions.

We repeated this process three more times.

Students observed the woman's friend and thought she had invited her over for lunch.

Students observed more people and the man with the piñata. They then hypothesized that they were at a birthday party, which I challenged them on. I asked, "what is typically at a birthday party?" to which they responded "cake and presents." Because they didn't see any of that in the image, we re-hypothesized that this was an image of a party, but that we couldn't confirm what kind of party it was.

After revealing the entire image to the kids, they did observe the cake and the presents and therefore were able to confirm their original hypothesis that this was indeed a birthday party.

Next, in the routine I asked the students if they had any lingering questions. Some asked:

• How old is the girl?
• Why are there so many people there? Babies, children, and adults all at the same party?!
• Are her cousins there?
• Is the girl hitting the piñata the birthday girl?
• What game are they playing in the corner?

Below, are some samples of the students' observation recording sheets.

I observe a leaf and tree branches and sky. The new observation: a person with a crutch. The new observation: She invited her sister. She had a party for her grandson as a fiesta. Its called a fiesta. She had a big big big big fiesta party.

I observe I see leaves and sky. There is a lady and a fence and she broke her leg. Her friend is visiting her. They are having a party. A girl is hitting the piñanta and there are cake and presents.

I observe a little of a tree branch and the tree is a sky. A fence and a lady in crutches and a flower in a garden. I think she's reaching for something. There's her sister, I think. A piñata I think and kids and more people and I think a dog and it might be a party and it might be a birthday. I see a cake and presents and even more people and a girl with a stick and blindfold.

Its getting dark. Sky. I think that someone is under playing. A women is under the tree right behind the fence and she has a broken leg? There's a little grass. I think she's looking for something. I think she is talking to someone and she doesn't like it and she is told her, "Would you like some pie?" They're having a party and I see someone doing something. I think the lady is setting up the piñata. They're gathering around the girl because she is going to hit that piñata first. See someone saying look she is going to hit the piñata and they are watching that girl hit the piñata.

leaves. branch. I wonder what the person is doing. cups. bag. having a dinner. a drink. flowers. ropes. presents. cake. sky. fence. nest. plants. adults. person. piñata. cat or a dog. birthday party. kids.

I observe smiling (?) Sky! I observe that it is a party. I observe planting. I observe her mom is helping. Maybe they're having a party.

Leaves branches sky. Girl little grass someone hurt flowers fence crutch. Girl carrying a bag. Kids kids talking adults adults talking kids playing. a lot of people piñata cups cake presents.

A tree and sky. they're going to cut down a tree. A girl broke her arm and forest. Her mom is probably mad? A lot of kids! And a piñata. A cake and drinks and presents.

3. Check for understanding.

Next, I told the students to forget about the specific picture and think about the BIG IDEA of observation. I asked them to finish the phrase, "I have to make careful and close observations because ..."

Below are some representative samples of the students' thinking.

This student understands the basic idea: that we must make careful and close observations so we can learn.

During the lesson, I asked students who needs to look carefully and closely and students responded with dectectives, spies, and scientists. This student's response shows that she understands this BIG IDEA (concept) also transfers to artists.

This student knows that careful and close observations help us learn more things, but I really like that this student is also including that when we take more time and look closely, we "have more thinking".

Next, I had all the students read their responses to the class. If someone read a new idea, we added it to the class's concept statement (using a different colored marker to graphically show how much the statement changed with the addition of new ideas.

Finally, we rewrote the statement to which they can refer throughout the entire unit.

4. Reflect on their thinking and decide next steps.

I was impressed by the first graders' keen eye during the Zoom In routine, but I was more impressed by their ability to write down all their observations. Having every child write down their observations during the thinking routine assured that everyone was engaged and also gave them an authentic reason to write. The teacher could review their observation sheets and give mini-lessons on a number of things related to writing: spelling patterns, punctuation, spacing, capitalization, question formation, etc.

The students' concept statement right now includes some ideas that are not always true; for instance, we don't always have fun when we look carefully and closely. Moving forward, I would suggest to this teacher that she set up opportunities for looking closely and carefully that aren't necessarily fun (looking through the lost and found, for example). As the students explore more concrete examples where they must observe, describe, classify, hypothesize, and draw conclusions, I would recommend that they continually revisit this concept statement and make changes as necessary.

After reading about how I used the thinking routine Zoom In to get students to develop an understanding of the importance of observing carefully and closely, how could you or have you used thinking routines to develop students' understanding of life's BIG IDEAS?

Don't teach concepts to children

Yep! You read it right. Don't teach concepts TO children.

Instead, give your students opportunities to EXPLORE concepts.

Let them DISCOVER the complexities of concepts by INVESTIGATING concrete examples from a variety of times, places, and situations.

In doing this, children will CONSTRUCT UNDERSTANDING of these timeless, abstract, universal, and transferrable ideas that we call concepts.

Letting students construct their own understanding of life's BIG IDEAS is known as inquiry, which is more than question asking. Teaching in this way doesn't come naturally to me, so until it does, I've been trying to follow these five steps.

1. Start with a concept.
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. 
4. Check for understanding by having them write a concept statement.
5. Reflect on their thinking and decide next steps.

If you think using these steps could help your students construct their own understanding of life's BIG IDEAS, try them out and let me know how it goes. That is exactly what one second grade teacher did recently. Here is her story:


1. Start with a concept.

In grade 2, teachers collaboratively decided to have their students explore how people influence the world. During this unit of inquiry, students also explore the genre of biography, as they read about various heroes.

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.

During the last several weeks, students in second grade have read, listened to, and watched biographies and have learned a lot about specific heroes.

After all this learning about individual heroes, the teacher posed two questions to the students: "Who is a hero?" and "What is a hero?"

Students responded using the Chalk Talk thinking routine (for more on Chalk Talk, see p. 78 of Making Thinking Visible by Ritchhart, Church, and Morrison). In the last step of the thinking routine, when students were sharing the thinking, they identified common themes and ideas that emerged. Those were notated by the teacher and are written in black marker and boxed in the pictures below.

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)

The next day, the teacher simply gave students a blank piece of paper and prompted them to write all they understood about heroes. The following were the student responses:

They stand up for people.

(they) doing awesome

Then, the students worked collaboratively to create one, concise concept statement.

When each individual statement was read, the teacher confirmed with the class if it was true or not. Then, students had to decide if the statement had any new information to be added; if it said anything different.

If it did have some new and different information, students had to work together with their teacher to revise the existing concept statement. The images below show the progression the concept statement went through as the class collectively constructed their understanding of the concept of hero.

First, the students started with ...

Next, they added a couple of attributes and took out the pronouns.

2nd graders then added how heroes did this.

To fit better with the vocabulary of the IB Learner Profile, they decided to replace kind.

2nd graders realized that heroes don't just help people.

The students struggled with what is the "right" thing. That value-laden statement is based on one's perspective and is not an absolute truth, so they knew they had to get rid of it.

But the students felt something was missing when they took out that phrase, so they put the idea back in, while acknowledging that "right" is subjective, based on the perspective of the "hero".

At this point, the teacher invited me into the conversation to see what a fresh eye could add. To make sure that "hero" was front and center, I suggested moving the dependent clause to the end of the independent clause.

Then, I suggested to the students that they change some of the wording so that it sounded more like a second grader. Often times when crafting concept 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.

Finally, I had the students test the statement to see if it was true in any time and in any place. To do this, we briefly brought up "heroes" from different times and from different places that they had already learned about. We decided their definition of a hero held up when applied to all those heroes, so it was indeed timeless and universal.

The students were comfortable that their concept statement encapsulated all the learning they had done up until that point, so the teacher wrote it down on their whiteboard.

5. Reflect on their thinking and decide next steps.

The second graders' understanding of hero is certainly deep and complex. The next step now is to continue to explore the concept of hero: read more biographies, watch age-appropriate biopics, read age-appropriate news articles, interview heroes local to their community, reflect on how their own actions could lead others to believe they themselves are heroes, etc.

As they continue to learn about other "heroes", second graders should be encouraged to continually revisit and revise their definition of hero as needed.

After reading about how one second grade teacher set up opportunities for her students to EXPLORE the concept of hero, instead of directly teaching this big idea to her students, how could you or have you had your students construct understanding of life's BIG IDEAS, known as concepts?