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# Elements of an Effective Math Lesson

(This is the last post in a two-part series.  You can see Part One here.)

The new question-of-the-week is:

What has been the best math lesson you have taught and why do you think it was so good?

In Part One, Beth Kobett, Jill Henry, Avery Zachery, Cindy Garcia, Molly Rawling, Catherine Murphy, Dr. Jennifer McBride-Donaldson, and Dennis Griffin Jr. share their choices. You can listen to a 10-minute conversation I had with Beth, Jill, and Avery on my BAM! Radio Show. You can also find a list of, and links to, previous shows here.

Today,  Erin Bolte, Laura Landau, Deven Ware, Lindsay Melia, and Douglas Reeves contribute their responses.

You'll also see the next question-of-the-week at the bottom of this column.

Holiday Budgets

Erin Bolte is entering her 18th year of teaching. She currently teaches 3rd grade in Manville, N.J.:

I am a 3rd grade teacher in a blue-collar public school district. Money is something our families earn through multiple jobs. Even with hard work, about half of our families qualify for some type of support to help them meet their monthly bills.

Many of our families are multi-generational in Manville. Their offspring have lived in town for centuries. Holidays like Thanksgiving are well-attended at their homes and a source of pride.

I introduced a Thanksgiving-themed math lesson right before the holiday.  Students were given a variety of fliers from different food stores and a \$200 budget to create their family Thanksgiving meal.  At first, my 3rd graders laughed. I remember hearing, "\$200 is SO much money for one meal. I'm buying everything!"

As the students identified items, looked at serving sizes, cost per unit, which stores sold the most affordable items, and who was attending their Thanksgiving meal, students began to form a deeper understanding for what their parents and/or guardians do in order to make the holiday special, successful, and affordable.  The comments soon changed from "I'm rich!" to "Oooh, this may not be enough money, Mrs. Bolte!"

The children realized that everything they use, eat, and see costs something—and it all adds up!  It was a great way for students to be engaged in a real-life situation and have more empathy for what they have—and what is provided for them.  Students made wise choices based on price, eliminated some items due to prioritizing, and utilized a variety of different skills in order to create a meal that fit the budget.

Teaching personal financial literacy through a holiday that means so much in our community was something that continues to resonate with my students for years after they've been in my class. I often will see my former students, and the lesson they most remember is the "Thanksgiving Math" one.

"Math Selfies"

Laura Landau has taught kindergarten through 3rd grade over her 20-year career. She also teaches 3rd grade in Manville, N.J. She comes from a family of educators:

One of the best math lessons I taught was at the end of last year in 3rd grade when I created a math scavenger hunt called "Math Selfies."

I have special education certification and work primarily with children who learn in nontraditional methods. As a lifelong learner, I appreciate creative and diverse approaches to teach subjects that can be looked at as traditionally lineary—as math is.

Students worked collaboratively in small groups to identify parallel lines, arrays, line segments, and other math concepts we focused on during the unit. They searched around the perimeter and area outside the school. When teams identified math concepts that matched what they noticed outside, students took "Math Selfies" with their Chromebooks.

Students needed to use their speaking and listening skills to articulately explain to their teammates why certain items matched math concepts/terms once found and then to their classmates when "Math Selfies" were shared between groups. Students debated, asking questions when unsure, and coming to a common understanding of why certain selfies were correct or not. Students learned from one another that many different items in our world can be viewed with a mathematical lens. Students and I loved this lesson because regardless of traditional math levels and/or abilities, students were able to demonstrate real-world mathematical understanding and were entirely engaged and excited during the process.

"A Warm-Up Puzzle"

Deven Ware is the math-curriculum lead for the online school at Art of Problem Solving (AoPS), a leader in math education for highly motivated students since 1993. Before joining AoPS, Deven taught math to motivated students at the Los Angeles Math Circle and MathPath summer camps. In his current role, Deven enjoys the opportunity to design math curriculum which excites and engages many of the top math students around the country:

The best lesson I ever taught startedand endedwith a warm-up puzzle. Two players are playing a game. The game starts with several stacks of random numbers of chips:

On a player's turn, they remove exactly one chip from any number of piles. So, for example, on your first move, you might remove one chip from every pile, leaving your opponent with this configuration:

Then your opponent might remove one chip from just the first pile, leaving you with this configuration:

The winner is the player that removes the final chip. A simple setup but a much more devious game than I imagined. I encourage anyone reading now to stop reading for a bit and think about the problem!

A big part of what makes a math lesson awesome is having meaty problems with good entry points for students to chew on. It's important for students to be able to make progress and figure things out themselves (with perhaps just a dash of guidance from the instructor). This problem fit that description perfectly for my class. It was just the right level of challengingthe students could make progress, but it didn't come easy. It was accessible to every student but also difficult. My original lesson plan went out the window, and we spent the entire time exploring this problem together as a group, like real mathematicians.

Perhaps the most important part of a good math lesson is having the students play the role of active participants, guiding the ship rather than following along with what the teacher has to say. My students were definitely driving as we went through this lesson. My main contribution was to be the person occasionally spouting gospel from my book of problem-solving-strategies: "Try something!" and "Try something simpler!" were two of my most helpful mantras. Because of my encouragement, my students considered each of the following simpler problems before jumping to the more complex problem.

I won't totally spoil the problem for anyone out there that hasn't solved it yet, but this type of reasoning is crucial to solving difficult problems. My students were trying special cases and making discoveries on their own. They would make partial progress and share it with the class. When someone figured out the one-pile case, they were excited to share their progress with the rest of the class, which then moved on to trying two piles (and quickly discovered it was a mess and simplified that to one chip in the second pile).

What sets this lesson apart from other lessons in my mind is how student-led it was. Not only were the students making discoveries, but they were also asking questions that led to those discoveries. The students were able to be creative and work together as a group to make progress. Nowadays, I'm always looking for more problems like this that get a group of students thinking and excited. It doesn't really matter what the problem is, as long as the students grow their problem-solving chops working to solve it.

"Construct the Concept"

Lindsay Melia received a Multiple Subject Teaching Credential from California State University, Long Beach, with a Subject Matter Authorization in Mathematics.  After teaching for two years, Ms. Melia pursued her longtime dream of education reform and studied education statistics at the University of Arkansas. Her time was split between researching and teaching statistics to incoming freshmen and teachers pursuing graduate degrees. During this time, Ms. Melia's passion for teaching math continued to grow, and she decided she could make greater impacts through classroom teaching. Ms. Melia returned to California and started teaching the middle and high school at TVT Community Day School.  She now teaches Beast Academy math to 2nd through 5th grade students at TVT:

One of my favorite 3rd grade lessons is a Beast Academy math lesson on the concept of finding perfect squares for numbers ending in 5. This is not your typical 3rd grade math skill, but taught with the right resources, many 3rd graders can easily grasp this mental math trick.

My first two years teaching this lesson, I created a set of guided notes that were pasted into the student notebooks. I included visuals, guided problems, and independent problems. It was your typical "I do. We do. You do." lesson. The third year I prepared to teach this lesson, I decided I would also use it as my sample lesson for a parent math night. As I reviewed my guided notes, I thought the parents would likely have no idea what we were doing or why. The calculation was clear, but the notes were not engaging. In the past, my students were able to grasp the concept because they were familiar with our note-taking routine, and I'm lucky enough to work with kids who are almost always up for a math challenge. I worried that I wouldn't have that same buy-in with parents who may or may not enjoy math or have a background in it, so I started cutting up paper and trying to construct the concept:

I cut a perfect square and deconstructed it into smaller parts until I had a hands-on model of the calculation. The problem became a puzzle that allowed students to reconstruct the square into a rectangle with an extra little 5-by-5 square to add in at the end. For instance, a square with side length of 35cm could be reconstructed into a 30cm-by-40cm rectangle with a little 5cm-by-5cm square left over. Instead of using the multiplication algorithm to calculate 35 squared, this model demonstrated how to simply multiply 30 by 40 and add 25, thus making the perfect square of 35 equal 1225. It wasn't just a random calculation to be memorized once the model was available.

The parents, and later my students, would see how we created the rectangle from the original square and where the little 5-by-5 square came from at the end. Some of the parents were concerned that they wouldn't master this mental math trick, but everyone understood it through the model. A week or so later, I taught this revised lesson to my students, and they had much more fun using the physical models of deconstructed squares than taking the guided notes I had used in prior years. Each pair of students worked with a different square ending in 5.  The partners had so much fun that they started passing around the puzzles to each other to try out different numbers. In no time, they were able to simply look at the number and provide the perfect square in mere seconds.

My students know that it's not the speed of the answer I value but rather the mental dexterity and ability to visualize and explain the concept. However, the excitement and enthusiasm that comes with mastering a new mental math trick is contagious. When students see math as a puzzle with the goal of finding a pattern, they are so much more engaged and excited. Letting the students discover the pattern instead of simply teaching it through direct instruction allows the kids to make deeper meaning that they will retain longer. I love this lesson because it was the reminder I needed to step back and let kids (and parents) explore before giving away the main concept.

Douglas Reeves is the author of more than 30 books and 100 articles on educational leadership, teaching, and student achievement. His videos and articles are all free downloads at CreativeLeadership.net. Doug Tweets @DouglasReeves and can be reached at [email protected]:

"Design Your Ideal School"  works well with students at every level.  It incorporates everything from number operations and the properties of  polygons for younger students to three-dimensional modeling and physical science properties for older students.  Best of all, it allows students to express their understanding of the relationship between their physical  environment and learning.

I typically have four separate tasks, including a written description of the school, a drawing of the school, a two-dimensional scale drawing, and a three-dimensional scale  model.  While reinforcing essential mathematical skills, this project  engages students in a real-world application of everything we have learned to a subject that is directly relevant to them—their learning environment.  I've seen students consider ways to make the school environment more conducive to collaboration and more welcoming to students with limited mobility.  It is wonderfully creative while maintaining a rigorous focus on essential mathematical knowledge, skills, and applications.

Next Question!

Instead of publishing separate posts announcing the next "question-of-the-week," I'm going to start publishing it at the end of the last post responding to the previous question.

The next question-of-the-week is:

In what ways can writing support reading instruction?

Thanks to Erin, Laura, Deven, Lindsay, and Douglas for their contributions.

Consider contributing a question to be answered in a future post. You can send one to me at [email protected]. When you send it in, let me know if I can use your real name if it's selected or if you'd prefer remaining anonymous and have a pseudonym in mind.

You can also contact me on Twitter at @Larryferlazzo.

Education Week has published a collection of posts from this blog, along with new material, in an e-book form. It's titled Classroom Management Q&As: Expert Strategies for Teaching.

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