George Lucas Educational Foundation
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Three young girls are standing beside each other wearing hats that say, "I heart math."

The ultimate goals of mathematics instruction are students understanding the material presented, applying the skills, and recalling the concepts in the future. There's little benefit in students recalling a formula or procedure to prepare for an assessment tomorrow only to forget the core concept by next week. It's imperative for teachers to focus on making sure that the students understand the material and not just memorize the procedures.

Here are six ways to teach for understanding in the mathematics classroom:

1. Create an effective class opener.

The first five minutes of the class period set the tone for the entire lesson. Ideally, teachers would start by sharing the agenda for the class period so that students will know the expectations for what will be occurring. Next, teachers could post and articulate the learning objective or essential question to the class so that students know the purpose and, at the end of the lesson, can self-assess whether the objective has been met for them. Finally, the opener might include one or more warm-up problems as a way to review and assess students' prior knowledge in preparation for exposure to the new material. This video shows a class opener for a seventh-grade lesson on rectangular prisms:

2. Introduce topics using multiple representations.

The more types of representations that you can present to students addressing their different learning styles, the more likely they will truly understand the concept being presented. Different representations could include using manipulatives, showing a picture, drawing out the problem, and offering a symbolic representation. For example, when presenting linear relationships with one unknown, illustrate to students the same problem as an equation, on a number line, in words, and with pictures. Students who are exposed to and can recognize the same relationship posed in the different representational modes are more likely to have conceptual understanding of the relationship and perform better on assessments (PDF).

3. Solve the problems many ways.

In the best classroom environment, the teacher is able to show different ways to solve the same problem and encourage the students to come up with their own creative ways to solve them. The more strategies and approaches that students are exposed to, the deeper their conceptual understanding of the topic becomes. Empowering students to create their own problem-solving methods can make the teacher nervous. What if we don't follow their logic? What if they're incorrect? However, it's worth the risk to have them explore. After an individual, pair, or small group of students finish solving the class problem using a single method, encourage them to look for alternate ways to come up with the same correct solution. Having students develop their own methods and then share the correct steps with the class is a very powerful learning experience. The video below shows how a teacher encourages students more than one way to solve the same problem on rectangular prisms:

4. Show the application.

In a perfect world, we would always be able to demonstrate how every concept can be applied to the real world -- and when that's possible, it helps improve the students' understanding. When a concept cannot be applied in that manner, we can still share how it might be applied within mathematics or another subject area. Another option is showing how the concept was developed through the history of math. Consider taking a minute out of each lesson to show your students where or how the math can be seen or used in life outside of the classroom.

5. Have students communicate their reasoning.

Students need to explain their reasoning when solving problems. In order for a teacher to determine if every student truly understands the objective for the class period, it's necessary for each student to communicate both orally and in writing. By giving the class ten minutes to discuss their reasoning with each other while exploring multiple ways of solving the problems, you'll promote excellent engagement and learning. It's not always easy to get students talking in class, but there are ways to encourage them (PDF).

6. Finish class with a summary.

Everyone can get lost in the class period, and it's easy to lose track of time until the bell rings and class is over. The final seven minutes might be the most critical in making sure that students have understood the day's learning objective. You can use this time to accomplish three very important things:

  • A quick formative assessment to determine how much was learned, such as students self-rating their comfort with the concept on a 1-5 scale
  • Reviewing the objective for the class period and brief discussion as to where the lesson will go next time
  • Previewing the homework together to avoid any confusion

These are just some of the activities for the close of a lesson. There are at least 22 additionally powerful closure activities. This video shows the summary phase of the same lesson:

In the comments section below, please share your own tips and tricks for helping students understand math.

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Rapid Leaks's picture

Great article, thanks for sharing. Gave me some new tech tool ideas to try in my class.

(1)
Shira's picture
Shira
a New Teacher in progress...

Thank you! When I began teaching (as an emergency entry) in 2001, I was told NOT to explain how to solve math problems in more than one way, as it confused most students. But I felt intuitively that there must be a way to effectively show students multiple problem solving methodologies. I feel that I could still use work on refining that skill.
Best Regards,
Shira

(1)
Dr. Matthew Beyranevand's picture
Dr. Matthew Beyranevand
Author of book "Teach Math Like This, Not Like That"

Hi Shira, Thanks for the comments. If you are showing multiple ways to solve the same problem, it might be a good idea to demonstrate a new way in different class periods to avoid confusion. If you are having the students create their own methods during the class period, it can be all occur during the same class period. Best, Matthew

(1)
Shira's picture
Shira
a New Teacher in progress...

Ah, ok, now that makes more sense. Thank you Matthew,
Best,
Shira

Dennis Baker's picture

I agree with both Dr Matthew & Shira- students need to see different ways to solve problems to gain different perspectives., but confusion also needs to be avoided. I think the key to effective learning is as Dr Matthew writes "encourage students to come up with their own creative ways to solve them." Learning does not necessarily follow from a student being shown a solution, but rather through a student grasping the solution by either independently solving a problem, or assimilating a solution that they've been shown & like.

lincco's picture

Thanks for all your tips.

It is not always easy to show students how every concept can be applied to the real world. But helping students to see them through the lens of history is intriguing for me. Will definitely look into that. This would certainly enrich not only my students but my own learning too.

Dennis Baker's picture

Hi lincco!
Please explain about "helping students to see them through the lens of history." How would this help students with understanding & what is the best way to use this method in a maths lesson?

Lisa_MCcoy's picture
Lisa_MCcoy
Parent. Teacher. Budding Writer

Explaining concepts with real life applications go a long way in developing the understanding of students. This is extremely important in a subject like Maths where students struggle to understand how it would help them in real life scenarios.

Mark Sandel's picture

Understanding math is vital for students to solve complex problems and it is impossible to catch a new topic without comprehending (better mastering) previous ones. You know it works the same with writing too. At first, you know nothing and just learn the letter (numbers), that you realize how to manipulate those items. And finally, you master every skill getting professional intuitions on what do you have to do and why. Those who do not want to do this, may use calculator or checking tools (http://www.paragraphchecker.com/ this one for example). They are effective enough for people who do not have professional skills. However, it is always better to rely on own skills, not on programming code.

michele's picture

Reading your strategies, and watching the short video really helped to focus me on what I want to do and show my students on the first day. Thank you.
I was stymied about how to introduce group work as well as a math topic on the first day. Although, I don't believe that I will truly realize the goal of my topic, I believe that introducing the students to the "creative class opener" (with each step being timed) is a great beginning. I will let you know how it worked out. My first objective (other than teaching procedures and expectations for group work) is adding integers with the same sign.
Wishing all a good first day, and a great and productive school year.

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