(This is the first post in a two-part series)
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?
We’ve all taught some great lessons and we’ve also all taught some pretty bad ones. This two-part series will feature math educators sharing more of the former ...
Today, 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.
You might also be interested in past columns appearing here on Math Instruction.
“My ... Students Were Solving Problems Interesting to Them”
Beth Kobett, Ed.D., is a mathematics educator and an associate professor of education at Stevenson University in Baltimore. She is a current member of the National Council of Teachers of Mathematics board of directors and co-author of Formative Five: Everyday Assessment Techniques for Every Math Classroom and The Mathematics Lesson Planning Handbook: Your Blueprint for Success:
I have asked this question hundreds, perhaps thousands, of times in my career. As a practice, I never ask a question I won’t answer myself. Still, I squirm a bit as I write this. As I reflect on the best mathematics lessons that I have taught, I consider one of my very first “best” mathematics lessons and, 30-plus years later, one of my most recent “best” mathematics lessons. While one lesson occurred in a 5th grade classroom and the other lesson took place in a college classroom, there are several common themes.
First, I planned both lessons with depth. Planning means I know the goals for the lesson, what I am doing, where I am going in the lesson, and how I will respond to student understanding, including misconceptions and advanced thinking. I do this all the time, but when it is a “best” lesson, I actually get pretty close to figuring out how my students will respond to the learning experience. One thing I know is that if I don’t plan deeply, I won’t get even moderately close. When I anticipate what my students will do and say, I can respond thoughtfully with feedback that values who they are and advances their thinking in meaningful ways.
Second, the lessons involved a lot of mathematical discourse. Of course, I didn’t call it discourse 30 years ago, but that is what it was. Students talked to each other enthusiastically about the mathematics they were doing and learning. Back then, I stood in my room in complete awe while students made discoveries about squares, square roots, primes, and composite numbers as they made concrete multiplication models while shouting their ideas with glee and enthusiasm that I couldn’t quite believe but wholeheartedly loved. More recently, my college students formed debate teams and argued their ideas about the relationship between area and perimeter and why it was an important idea in the elementary classroom. This time, I took a seat in the back of the classroom and let my students shine. Oh, and there was a little bit of good-natured shouting once again. Now that I think about it, in both lessons, someone curiously peeked in from the hallway to see what was going on.
Finally, but most importantly, my elementary and college students were solving problems that were interesting and compelling to them. They were focused on proving their ideas, constructing arguments that others would find convincing, and showcasing their incredible mathematical ideas in thoughtful and creative ways. These mathematics problems are worth solving because they show my students how much I respect and care about who they are as mathematical learners and people.
My “best” can always be better. I talked too much then and find I still do. I fire off questions because I am excited, and my students scramble to figure out which question to answer. I am still working on how to get out of the way when my students are thinking, analyzing, and problem-solving, and step in at just the right moment to pose a question that will prompt a new strategy or idea. More recently, I have turned my focus to recognizing and leveraging my students’ innumerable strengths in every mathematics lesson. This focus is a work in progress but has the potential to increase access to mathematics and provide a more equitable learning environment for each and every one of my students. Right now, my effort to thoroughly plan, provide multiple student-discourse opportunities, situate my lessons within a problem-solving context, and leverage students’ strengths is my hope for my next “best” lesson. If I can get these components to align, perhaps my next lesson will be my “best” lesson.
Student Engagement Is “The Barometer for a Good Math Lesson”
Jill Henry teaches problem-based Algebra 2 at Flintridge Preparatory School, a 7-12 independent school in the Los Angeles area. Jill holds a master’s degree in secondary mathematics education with a specialization in constructivist learning and has served as a curriculum consultant for several national mathematics education grants:
I’ve always considered student engagement to be the barometer for a good math lesson. The best ones are those that grab students’ attention by getting them invested in the content. Ones that prompt students to ask thoughtful questions with enthusiasm. Ones that result in a collaborative learning environment where the whole room is engrossed in a discussion together. Year after year, I consistently find that my students display their highest levels of engagement when we talk about college-student loans in my Algebra 2 class.
This lesson happens during the unit on geometric sequences and series and ties in nicely with any discussion about interest. For homework the night before the lesson, I give students three tasks. First, answer several research questions about loans: “What percent of college students carry student loans? What is the average amount borrowed? What are the typical interest rates for loans (federal and private)? What is the maximum loan amount (federal and private)? What is a typical loan length? How is the loan paid back, and how is the interest calculated?” Second, students are instructed to determine the full sticker price for four years (tuition plus room and board) at their “dream” school. Finally, if comfortable, they are encouraged to have a conversation with their parents about how they will be expected to pay for college. If they are uncomfortable having that conversation, and have no knowledge of any college savings that may exist for them, then they are told to assume a yearly family contribution of $5,000.
The next day in class, students are asked to determine how much money they would need to borrow to attend their dream school. This value is calculated with a simple difference: loan amount = total cost of school - total family contribution. They then break down their anticipated need into two separate loans, one federal and one private (if needed), as these have different maximum limits and interest rates. Finally, using Excel, students simulate the payback and calculate the most meaningful figure: the amount due per month.
The lesson typically ends with a brief discussion about that specific monthly loan payment, for which we brainstorm the cost of other standard “adult” expenses in order to give that figure some context.
There are many ways to extend this lesson. Some years, I’ve had students explore Roth IRA’s to illustrate the positive effects of interest. In others, I’ve had students research the starting salary of their dream job and create a budget for themselves, including their loan payment along with other life costs, to determine the amount of discretionary income, or “fun money,” they’d realistically have left over at the end of each month.
The personalized nature of this lesson is a large part of what I believe makes it so engaging. Students find it fun to both consider and research their future—where they might go to college, what they might do for work, and where they might live.
Taking out a student loan is often the first large financial decision that a person makes, and it’s of critical importance that high school students are educated about the ramifications of debt before they are faced with borrowing money for school. “Buried in Debt,” a national research study completed in 2018, illustrated the significant financial hardship and devastating domino effect that student-loan debt is having on the lives of graduates across the country. If you have a chance to review this report for yourself, I’m sure you’ll agree that the findings are very disheartening.
For many of my students, this lesson marks their first time looking at a college website, particularly at the cost page. The first time teenagers consider how they’re going to pay for college shouldn’t be during the college process itself, where emotions and stress are already running high for both student and their family. Talking about these things well in advance of college allows for a highly engaging but relatively low stress conversation, one that prepares students for real life in a way that few other math topics truly can.
“Understanding Fractions”
Avery Zachery graduated from Georgia State University with a bachelor’s degree in early-childhood education and a master’s degree in reading, language and literacy, and has an ESOL and mathematics endorsement. Zachery has been teaching for nine years, currently at Winston Elementary School in Douglas County, Ga., teaching math, science, and social studies in a team-teaching model:
As an elementary math teacher in Georgia, it’s important for me to be aware of the Georgia Milestones Assessment and particularly the fact that fractions account for 30 percent of the content. Without a firm understanding of fractions, students will struggle to perform well on their End of Grade Assessment.
With this in mind, the best and perhaps most important lesson I taught was “Lesson 17: Adding and Subtracting Mixed Numbers” in our Ready Georgia Mathematics curriculum. This is a pivotal lesson in giving students a firm understanding of fractions.
The lesson starts off with two problems featuring addition of mixed numbers. Both problems require the students to regroup fractions and rename them as whole numbers. Most teachers would start with adding like terms and not including regrouping in the first introduction of the concept. However, our curriculum builds in the rigor from the start of the lesson. The first two problems have real-world connections and are relatable to students. I really liked how the problem was taken step by step and presented in a variety of ways. The problems were modeled pictorially, on a number line, and numerically.
The second half of the lesson is on subtracting mixed numbers, which can be a major challenge for students. Ready Math introduces the concept with a problem that has a greater fraction in the subtrahend than in the minuend. When teaching this lesson, I was shocked at the depth of knowledge needed to complete this problem: 4 ¼ - 1 ¾. The problem starts with allowing the students to model it. Modeling the problem allowed the students to see how to subtract mixed numbers pictorially. They were able to show the regrouping within the model. The lesson also presented the concept of subtracting mixed numbers in multiple ways. This is a very difficult concept for students to comprehend, but the way it was taught here made it much easier for students to be successful.
“I Just Told Them We Were Going for a Walk”
Cindy Garcia has been a bilingual educator for 14 years and is currently the district instructional specialist for PK-6 bilingual/ESL mathematics in the Pasadena Independent school district (Texas). She is active onTwitter @CindyGarciaTX and on her blog:
One of the best math lessons that I taught was teaching perimeter to my 3rd grade students. It was a good lesson because it sparked my students’ curiosity, it prompted my students to be problem solvers, the task involved was a real-world task that made sense, they collaborated with each other, there was movement involved, they used real-world mathematics tools, and it was hands-on. I took my students on a walk along the fence that was around the perimeter of the field.
I didn’t tell them the real reason; I just told them we were going to go for a walk. As we walked, I asked them to share what they were noticing as we walked. They mentioned we were walking in a rectangle, walking by a fence, and we were in a straight line. I stopped the line four times and shared the measurement of the length we had walked and then we went inside. I handed each group a card that had a picture of a field with a rectangular fence with the same measurements we had just walked. I prompted them to determine the perimeter of the fence. They talked it out and determined that the perimeter must be the measurement of the fence surrounding the field and they figured out that they could add the four measurements to determine the perimeter. We talked about their solution process and confirmed the meaning of perimeter.
I then showed them our undecorated bulletin board and told them that we needed to staple on a border. I gave them border, a ruler, and scissors. Once again, they worked together to determine the perimeter of the bulletin board and how much border they would need. One of the main reasons this worked out so well was because there was time given for students to figure things out, draw their own conclusions, and test things out.
“Choral Counting”
Molly Rawding, K-5 mathematics coach, and Catherine Murphy, K-5 ELL specialist, have been working together for years to develop scaffolds in math for our shared ELLs in the building. They both teach at Fiske Elementary School in Lexington, Mass.
Our work started by going around the school teaching “row” and “column” to many classrooms in the building. These are crucial tier-2 words that are commonly misunderstood. To advance this work, we teamed up to co-teach a choral-counting routine that incorporates these terms in a meaningful way.
Choral counting is a fun, engaging routine in which students count aloud together, then notice patterns and make connections. We love this routine because all students are fully engaged in oral language in mathematics. This activity promotes authentic academic conversations that encourage students to make connections with each other and with the content.
In this routine, the children count aloud together by a given number—sometimes forward, sometimes backwards. The teacher selects a number for the students to skip count by, and on what number to start and end the count. As the Teacher Education By Design site says, “The goal of this activity is not just to practice rote counting but to engage children in reasoning, predicting, and justifying.”
Having the math coach come into the ELL pullout classroom and co-teach choral counting with the ELL teacher provides the scaffolds necessary for ELLs to engage in the routine. Classroom teachers lead the choral-counting routine in their rooms, and our goal is to give our ELLs practice using the scaffolds so that they can be more engaged with the native-language speakers in the classroom. While not all children may be counting up to the same range of numbers, it’s important for all students to hear the sequence, notice the rhythm, and make connections.
There are many benefits in using choral counting with ELLs:
- there is an entry point for all language levels
- the focus of the activity is on a visual with low language demands
- can be used for oral and written responses
- fosters great effective prompts for academic conversations i.e., " What do you notice?” We had lower WIDA level students paraphrasing each other to get valuable practice with talking about math.
- imbeds tier-2 words that students hear and need to know in their classrooms
We have integrated this 10-15 minute routine on a regular basis (every other week). Here’s what it looks like:
- Before class, the math coach and ELL teacher meet to plan the count: Determine the count (start and stop number) and how to record the numbers on chart paper.
- To begin, the teachers share what number the count will start with and by what number students are skip counting. (“Today, you are going to count starting at 5 and counting up by 5s and keep counting until 135.”)
- The goal is to have “one voice” so the students, similar to singing a song, all say the numbers at the same time. This takes practice, and it’s important for the teachers to stop if the voices are not in sync. Then start again.
- After the students count to 135, they count again often with a focus on making sure all voices are in the count.
- The teacher might ask, “What number comes next?” and “How do you know?”
Next, the teacher records as the students count. Based on the goal of the selected number sequence, the teacher records in a certain way (from the top or bottom—and a certain number in each row).
- After the count, students take a minute to silently look at the recording of the number to “notice” patterns and observations.
- Students then share, one at a time, describing their observations using a sentence frame such as, “The pattern in the row is _____________” or “I see a pattern that _____________” During this time, the teacher records the patterns on the chart.
- After two or three students share, students engage in academic conversations to paraphrase and elaborate on the patterns that they see.
- Students notice so many interesting patterns, and they build on each other’s ideas. We set a time limit of 15 minutes for this routine—and often revisit the chart later in the week.
This routine is one of our favorites because it is so engaging and students notice many interesting patterns and connections. This activity is highly beneficial for ELLs of all WIDA levels because it promotes development in all domains in mathematics.
For further thought:
→ We have also added a “clap” on the 10’s so students recognize groups of 10’s (10, 20, 30...), and when they say that number, they clap.
→ We have also connected choral counting to money as we count by nickels or dimes.
→ We audiotaped students during the choral count and replayed it for them so they could hear how they sounded. Students were excited to try again counting louder, clearer, and in sync.
More Resources:
Check out Choral Counting for more information, videos, and ideas.
Academic Conversation Recommendations
Teach academic conversation norms and develop a routine at the start of the year.
Check out Catherine’s website for more information on academic conversations.
Employ routines for paraphrasing and elaborating (so what you’re saying is).
Check out Jeff Zwiers for more information and ways to get started.
Developing “Student Understanding of Numeracy Vocabulary”
Dr. Jennifer McBride-Donaldson is a native of Augusta, Ga. She is in her 20th year as a teacher, currently as a kindergarten teacher at Richmond Hill K-8 STEM School. McBride-Donaldson gains the greatest pleasure by watching young learners use various technology devices to solve problems, explore, grow, and enhance their understanding of mathematics:
The math lesson I found to be the best and most memorable was one I taught with i-Ready, “Understanding Addition Level AA.” This lesson creates a classroom environment that develops student understanding of numeracy vocabulary. Understanding Addition is introduced through pictorial, concrete, and abstract problem-solving strategies. These strategies equip students with the skills and knowledge to know how to count and add. The lesson also provided me with data about the students’ strengths and weaknesses, which I used to form learning units and adjust classroom instruction to meet the needs of both succeeding and struggling students.
The Understanding Addition lesson provided students with practice, challenges, and fun. Math practice determines the students’ level of knowledge according to tailored math lessons and develops each student’s ownership of their learning. Challenging mathematics teaches students how to link math vocabulary and symbols when discussing concepts-steps that are needed to be used to solve problems. Promoting engagement with math―as opposed to drill and skill exercises―is a tactic that encourages students to use math to solve everyday problems. The following technology connections are within this lesson:
- Video
- Vocabulary
- Examples
- Guided Practice
- Independent Practice
- Assessment
All these components work together to improve student engagement in their learning process while personalizing lessons for students according to their strengths and weaknesses. Personalized mathematics-lesson data is used for charting data about student performance and progress. As a kindergarten teacher, I find that this lesson provides teachers with concrete data about student engagement, understanding, mastery, and needs for growth. A data-driven classroom is a classroom built for student success.
Cross-Curricular Teaching
Dennis Griffin Jr. serves as the principal of Brown Deer Elementary School in Wisconsin. He has seven years of experience as a middle school educator and is entering his sixth year as an administrator. He is currently pursuing his doctoral studies in educational leadership at Cardinal Stritch University. Dennis believes all students will be successful in school when they develop relationships with educators that value their gifts, cultures, and individuality:
The best math lesson that I have had the opportunity to teach was a collaborative effort with a language arts teacher. In math, the lesson was drawing and identifying units to scale in the month of February. While collaborating with the language arts teacher, we created a research project that would require our students to research and write a report about a prominent African American in history; the students would take a picture of the person they were researching and create an enlarged image of one of their pictures. Of course, our students were able to choose the person they would like to study and the picture they would enlarge.
The project took the better part of three weeks if my memory serves me correctly. We gave the students the opportunity to flexibly go between classrooms during this time. Each and every day the students were provided feedback from not just the educators but their peers as well. The students helped to create the rubric that they would be assessed on. Each student had to present their learning as a culminating project. We displayed their learning in the hallways. The students were proud of everything that they learned and were willing to put their learning on display for the school community. This is a lesson that I would love to teach again because it truly demonstrated the cross-curricular connections within our learning. I still have several of the finished projects in my possession.
Thanks to Beth, Jill, Avery, Cindy, Molly, Catherine, Jennifer, and Dennis for their contributions.
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Look for Part Two in a few days.