Response: Mistakes That Math Teachers Make
(This is the first post in a three-part series)
The new question-of-the-week is:
What are the biggest mistakes made in math instruction and what should teachers do instead?
We all make mistakes, and this series will explore which ones might be particularly unique to math teachers!
This series will "kick off" with responses from Bobson Wong, Elissa Scillieri, Ed.D, Beth Brady, and Beth Kobett, Ed.D. You can listen to a 10-minute conversation I had with Bobson, Elissa, and Beth on my BAM! Radio Show. You can also find a list of, and links to, previous shows here.
Response From Bobson Wong
Bobson Wong (@bobsonwong) has taught high school math in New York City public schools since 2005. He is a three-time recipient of the Math for America Master Teacher fellowship, a recipient of the New York State Master Teacher Fellowship, and a member of the advisory board for the National Museum of Mathematics. As an educational specialist for New York state, he writes and edits questions for state high school tests:
When I look back at unsuccessful lessons that I've taught, I find that they have many things in common. The list below contains some of the biggest mistakes I've made in my math instruction over the years and how they can be avoided.
- Forget about why students need to know the math. "Why do we need to know this?" is a question that students ask all the time. Many lessons fail because they don't adequately address this basic but important question. Answering it properly provides a solid foundation for the rest of the lesson. If the topic has a real-world application, then coming up with the right motivation is easy. However, a good motivation in a math lesson can often be as simple as asking students to find a pattern or extend their knowledge to unfamiliar situations. This technique is especially useful for topics like geometric proofs or multiplying polynomials that are too abstract to connect directly to the real world. Sites like Ask Dr. Math are excellent resources for giving clear explanations of mathematical questions, like why we can't divide by zero.
- Ignore the relationships between mathematical ideas. English and social studies teachers know that students need to organize their thoughts in an outline before writing an essay. Creating an outline helps students see the big picture by ordering information logically and seeing connections between ideas. We can use the same technique when we teach math by looking at each lesson as part of a larger unit and thinking about how each lesson relates to concepts that students already learned. For example, abstract ideas such as multiplying polynomials like (2m + 1)(3m + 2) can be easier to understand if we relate them to familiar ideas like multiplying 21 x 32.
- Don't customize your lesson for your students. With so many math education resources available in textbooks and the internet, copying someone else's lesson is tempting. However, it can lead to disaster in the classroom. A motivation that may work for one class may make no sense to another. Some classes may not understand the vocabulary or explanations used in someone else's lesson. In addition, finding the right combination of problems for your students is critical. If the problems are too difficult, students will get frustrated. If the problems are too easy or repetitive, they will get bored and start misbehaving. If they can't relate to the problems, they will feel lost. Problems should get progressively more challenging but at a rate where students have enough confidence doing easier problems that they can advance to harder ones. Writing a good lesson requires knowing your students' abilities and comfort levels so that you can customize it for them.
- Use sloppy language. Good math instruction isn't just about correct calculations or procedures. To convey mathematical ideas clearly, we need to model proper mathematical language for our students, many of whom are so accustomed to auto-correct features on electronic devices that they lack precision. We should avoid ambiguous phrases like "cancel" or "move to the other side." We also have to teach students how to concisely and precisely explain mathematical concepts so that they don't resort to saying, "You know what I mean." Using language techniques familiar to English teachers (such as word walls, visual aids, graphic organizers, and student dictionaries) can improve student understanding of math vocabulary.
In short, good math instruction helps students see how ideas connect logically, how math relates to the real world, and how these ideas can be clearly expressed.
Response From Elissa Scillieri, Ed.D
Elissa Scillieri, Ed.D., is a math supervisor from New Jersey who considers herself a math missionary. She previously taught all subjects in the elementary grades before being given the opportunity to share her enthusiasm for math with all ages. Follow her on Twitter: @EScillieri:
Mathematics is a challenging subject to teach. While the content itself is difficult enough, overcoming the mindsets connected to math is often a steeper climb. For many teachers, especially those at the elementary level, math conjures feelings of uncertainty and inadequacy; however, by avoiding a few pitfalls, teachers can overcome their math anxiety and build their confidence while improving their math instruction.
Pitfall 1: Telling students about math rather than allowing them to explore and discover
Remember how as preservice teachers it was frustrating to take a class about teaching instead of actually teaching? Teaching and learning became abstract with someone telling us that a certain type of lesson would engage students. It was not until student-teaching that those pieces of advice came to life, and with that experience, we understood and learned firsthand why certain lessons were more successful than others.
Unfortunately, this "telling" approach is one that many teachers employ with their math students. For example, in elementary school, we tell students that they have to find a common denominator in order to add fractions. Consider instead giving students the opportunity to play with math and discover its beautiful patterns. Before teaching them how to add fractions, let students play with pattern blocks or fraction strips and put together ½ + ¼ . Ask students what they notice. Encourage them to explain and then have them test out their ideas by trying another problem, such as ⅔ + ¼ . The more students play with hands-on manipulatives as they explore math, the more they will remember and make deep connections between those patterns and the procedure that they will later learn.
Pitfall 2: Teaching students tricks to memorize in place of developing a deep understanding
There is a poem that is sometimes used to teach rounding involving five or more and going next door to add one more—or maybe it is less than four makes one more since that rhymes, too. (Seriously, search images for "rounding poems" and take a look at all of the different versions that appear!) While it's entertaining to memorize rhymes and poems, depending on them for precision in math class or anything close to actual learning is far from ideal. Instead, spend time developing students' understanding of numbers. Show them how to find 136 on a number line and have students examine whether it is closer to 130 or 140. Eventually, students will realize that they can also use this same strategy for rounding to the nearest hundred. Dig a little deeper while questioning students to uncover their misconceptions and engage in that uncomfortable discussion about how 15 is exactly between 10 and 20 and not actually closer to 20. Rather than hiding behind tricks, teachers can give students confidence by teaching them the concepts behind the mathematics. (For a comprehensive list of math "tricks" to avoid at all levels, check out www.nixthetricks.com. )
Pitfall 3: Trying to hide our own mistakes instead of acknowledging that we (teachers) are all still learning
While teachers may be confident in confessing that they do not know a historical fact, many teachers are uneasy telling students they don't know an answer in math class. Or perhaps they are horrified to have a student point out a math mistake on the board. There is no need to save face during math class. Students need to see adults making mistakes and learning from them, especially in math class. They need to know that it is acceptable to attempt a problem and not always come out with the correct answer. This is not something that needs to be hidden; rather, it is an experience to demonstrate that true math learning is an ongoing process of trial-and-error, with an emphasis on the latter. The opportunity to see that even teachers struggle in math gives students the confidence to take risks and publicly make mistakes without suddenly feeling as though they no longer are "math people."
Response From Beth Brady
Beth Brady has taught in the Northampton public schools in Massachusetts for 26 years, initially as a 1st grade teacher, then 2nd grade, and currently as a math interventionist and Math Recovery Champion for the district. She graduated in July 2018 with her second master's from Mt. Holyoke College in the Master of Arts in Mathematics Teaching:
- Not doing the math ahead of time.
Be prepared. Do the math ahead of time. When you take time to think about your students and how they might approach a task, then you know what to look for. If you brainstorm on one sticky note for each strategy, you can order them from least to most sophisticated. It's even better with a grade level colleague.
Think about the math understandings needed for each strategy. When you put them in a hierarchy like this, it's thinking about the mathematical progression. It makes it easier to think about questions that might help students to get them to the next level of understanding. It becomes more clear for yourself what you are looking for when students are engaged in the math task and you can predict which of your students would use which strategy.
Then it makes pairing kids up so much easier. You can pair kids up who are using similar strategies, or you can pair up kids who are one "step" away from each other. When they share strategies, they can look for connections between the two different ways. Often, when there are mistakes, students fix their mistakes when they're explaining their thinking. You have to be intentional about who you are pairing up because sometimes you might want kids at very different levels of thinking to work together ... you have to think about why you're pairing the students and what each child would benefit from with the pairing.
You need to establish a climate in your classroom to be one in which there is a growth mindset and respect for each other's think time. The class needs to understand that think time is a gift that we need to give each other and blurting out answers or doing work for each other doesn't help grow math minds. Also the quote, "Give a person a fish, and they'll eat for a day. Teach the person to fish, and they'll eat for a lifetime." has a lot of power when you work through what it means and have the class live it. Responsive Classroom has a lot of resources to help you think about classroom climate.
- Talking through the independent work so much that they have no thinking to do when they sit down.
How do you introduce a worksheet? Try this: Have the kids think about it together. Help them work through the worksheet without giving away any answers, yet focusing on lifelong skills like being able to read and comprehend and problem-solve. If you don't have a document camera to project the worksheet, you could give the worksheet to two or three kids to have them think about it together. It's important to establish a protocol. Make it really clear that they are thinking about how to do the sheet, not working through the math yet. Have them discuss the directions on the page and how they might go about thinking about it. They should talk about what math tools they might need to help them solve the problems. They should talk about how to start. This promotes important self-talk that people need to be able to do to be independent learners.
Instead...Question. Listen. Watch. The kids are supposed to be thinking, talking, and developing their own understandings ... not the teacher's understandings!
- Not having a climate in the classroom that values using math manipulatives.
Math manipulatives or tools help bring out the ideas that are inside children's heads when they are thinking about the math. Some teachers complain that children end up playing with the manipulatives instead of using them as math tools. That has to do more with how the teachers are setting expectations of the tools than with the presence of the tools themselves. Also, the tasks that are being presented may be too difficult for the children who end up playing with the manipulatives. If kids don't understand the point of showing their thinking, and if the math is not within their Zone of Proximal Development, then you're going to get "behavior problems."
When kids have access to materials and they use them to show others what they were thinking, then kids can make connections between the models. They can better understand what others are thinking.
- No mental math.
Mental math must be emphasized. Children need to solve problems in their heads and pay attention to their thinking so that when they record their thinking, it's a true record of their thinking. When kids are given pencil and paper, they are using the paper as a record for their thinking and a lot of the actual thinking is lost. Kids need to practice explaining their thinking, and at first, teachers need to scribe their thinking so that they can see how they might do it on their own.
Response From Beth Kobett, Ed.D.
Beth Kobett, Ed.D., is a mathematics educator and an associate professor of education at Stevenson University in Baltimore. She is past-president of the Maryland Association of Mathematics Teacher Educators and a current member of the National Council of Teachers of Mathematics board of directors:
Most teachers teach with the very best intentions. Most of these missteps, rather than mistakes, are made out of concern for their students. Teachers desperately want their students to succeed, and sometimes these teaching steps have unintended consequences. When this occurs, students exit the learning opportunity, whether it's one lesson or a full week of instruction, with incomplete understandings and potential misconceptions. Three changes can empower and engage student learning in mathematics.
Making Math Accessible
With the goal of making mathematics accessible, teachers often break math down into parts and deliver small pieces of mathematics content to their students. Parsing mathematics into bite-sized chunks may work in the short term because students can produce answers, but when teachers do this, students don't see how the small piece of mathematics that they are learning at that moment is connected to more comprehensive conceptual ideas. Instead, teachers should select mathematics tasks that encompass bigger ideas, which, in turn, provide more entry points for students to access the mathematics, develop mathematical language, and connect to prior learning. One way teachers can avoid this "bite-sized" trap is to present a meaty problem to students, facilitate student learning by asking probing questions, and then use the student work to unpack the mathematics in a meaningful, connected, and visible way. For example, a teacher who wants to teach multiplication of fractions may present a problem like the following:
"After the school picnic, you noticed that ¾ of the pan of brownies were left over. The principal asked you if she could take 1/3 of the pan to take to the school secretary to eat later. You get to take what is left home. How much of the full pan of brownies do you get to take home?"
Instead of teaching the algorithm for multiplication of fractions first, students solve the problem by constructing representations and engaging in a productive discussion about the problem. Then, the teacher can use student solutions to make the mathematics visible and connect their representations to the procedures for multiplying fractions. Students will then know why and how procedures work and are more likely to understand and remember the mathematics.
Planning Improves Teaching
Even when teachers have taught for many years, taking the time to plan lessons collaboratively makes a big difference in student learning. First, teachers can select the most appropriate activity for their learners, considering their learners' unique learning strengths and challenges. Second, they can anticipate how students will respond to the task or activity they have selected and map out different instructional paths they might take depending on the students' responses. This kind of strategic planning translates into effective teaching practices because teachers regularly anticipate how to use formative assessment throughout the lesson and are therefore prepared to respond in ways that promote student understanding. Without such focused, learner-centered planning, teachers may fall into a "corrective feedback" loop in which teachers' prompt students to give and get answers without understanding. Explicit, targeted, and thoughtful feedback makes a difference!
Supporting Students' Mathematical Identity
Teachers make many instructional decisions. They may be about grouping, task selection, what materials to use in a lesson, questions they will pose, and so much more. Grouping students by perceived ability sends clear messages to students about their capability to do mathematics. Varying these grouping practices to promote students' strengths and collaboration skills builds positive dispositions about math. Teachers can support all learners by drawing upon students' real-world experiences to construct tasks that are relevant to them and invite different perspectives. Students who have their ideas recognized and valued develop positive beliefs about themselves as capable and empowered mathematical learners.
Thanks to Bobson, Elissa, Beth and Beth for their contributions!
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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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Look for Part Two in a few days.