Response: 'Nix the Tricks' in Math Instruction
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 "kicked 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.
In Part Two, Sunil Singh, Laney Sammons, Abby Shink, Cathy Seeley, and Shannon Jones shared their thoughts.
The series is wrapped up today with answers from Dr. Hilary Kreisberg, Richard Robinson, Rachael Gabriel, Tamera Musiowsky, Dr. Fuchang Liu, Bonnie Tripp, Bill Wilmot, and Bradley Witzel, Ph.D.
Response From Dr. Hilary Kreisberg
Dr. Hilary Kreisberg is the director of the Center for Mathematics Achievement at Lesley University, an assistant professor in math education, the president of the Boston Area Math Specialists organization, and an author. She holds her bachelor's degree in mathematics with a minor in elementary education, a Master's degree in teaching and special education, and a doctorate in educational leadership and curriculum development:
First, I want to start by saying how many great things are happening in math instruction today. Teachers are much more versed than ever before on using and incorporating hands-on, concrete manipulatives to guide beginning instruction of concepts. In addition, many math teachers today are invested in engaging students through meaningful math instruction by presenting math concepts through multiple representations and modalities for easier access. However, with all this greatness, there still exists gaps in math instruction. It's hard to quantify the "greatest" mistakes, so instead I will identify two things I wish all math teachers would stop doing immediately! I know just saying "stop" isn't enough, so I will also include rationale for why these instructional routines are bad practice and offer alternative practices and resources to improve instruction.
1. Nix the tricks.
WHY STOP: In my experience, math teachers resort to tricks when they don't understand the mathematics deeply enough themselves to teach the conceptual version. If one looks up the word "trick'"in a dictionary, one will find that "trick" can be defined in many ways, based on its part of speech and usage. Spoiler, one will never see "conceptual understanding" in the definition. It boggles my mind that some might associate the word "trick" with "learning," especially after seeing the various definitions. One interpretation of a trick is an illusion (or a false idea). Another definition is "liable to fail." This is exactly why we need to stop teaching children tricks in mathematics. The tricks are liable to fail once they've "expired," and they create false perceptions that they always work.
RESOURCES: If you haven't read 13 Rules that Expire by Karen S. Karp, Sarah B. Bush, and Barbara J. Dougherty (Elementary), 12 Math Rules that Expire in the Middle Grades by Karen S. Karp, Sarah B. Bush, and Barbara J. Dougherty (Middle School), and/or Nix the Tricks by Tina Cardone and MtBoS (Elem/Middle/High School), then that's the first step.
HOW TO CHANGE: Start thinking about math as a subject that isn't "magic," but rather a science. It is in fact a natural science. Though like in literacy, there are some conventions we just adhere to without definitive reason. One suggestion is to become more familiar with the properties of numbers. Common examples of these are the commutative, associative, and distributive properties. By developing a deep understanding of the way in which numbers work, one will be more inclined to rely on the mathematical facts, rather than tricks, to help students understand the math. Another suggestion is to ask yourself, "Does this always work?" before spewing out a generalization. I can't tell you how many 3rd grade mathematics teachers I've heard say that multiplication makes things bigger or that it is repeated addition. This becomes problematic in 5th grade when students learn that multiplying by a fraction or decimal less than 1 results in a smaller product (e.g., 4 x ½ = 2). Rather than generalize, talk about multiplication as scaling or resizing. We can resize a quantity (enlarge or shrink) by multiplying.
2. Stop timed fluency tests.
WHY STOP: Timed tests have been linked to math anxiety, which can ultimately lead to a negative mathematical mindset or belief that one can't do math or isn't "good" at it. I was one of those students who suffered from anxiety growing up and especially during those nasty "Mad Minutes," where it was me versus the clock to see if I could maintain my status as "good at math." I realized early on that I had to spend an average of one second per problem to be able to get the page done in one minute or under. If it took me three seconds to spit out the fact for one problem, then I literally felt like my life was over. The frustration, anxiety, and stress would actually prevent me from being successful ... and it was not because I didn't know my facts. This is precisely why teachers need to stop timed fluency tests. What is the goal? If the goal is to see if a child is fluent or automatic with their facts, then timed tests are not the way to do it.
RESOURCES: Dr. Jo Boaler, professor of mathematics education at Stanford University, is the pioneer of the fight against timed tests in math classrooms, and her reasoning is based on evidence. If you haven't read her article in the NCTM journal, then start there: Timed Tests and Math Anxiety. If watching a video is more your modality, then check out Jo Boaler as she talks about the research here.
HOW TO CHANGE: Distinguish the difference between fluency, automaticity, and speed. These words are not interchangeable, and despite their common usage referring to fastness, the goal is not to create "quick" math thinkers but rather problem-solvers with number sense. In discussing fluency, the focus should be on efficiency and students' abilities to reason (how did they think strategically?). For example, to solve 5x6, a fluent math thinker may use 5x5 as an anchor and add one more group of 5 (25+5). This is called using a derived fact. Alternatively, another fluent thinker may know that 5x6 is equivalent to 10 x 3 because the multiplicative inverse property states that a person can double one factor and halve the other and not change the product. In discussing automaticity, the focus should be on memory retrieval. If one can recall a fact automatically, then it should be effortless on the student's part. Yes, this means the student most likely will produce the result quickly, but the emphasis is on accuracy of fact retrieval without a conscious effort, not speed. Speed is exactly what we want to steer clear of during math instruction. It is absolutely important for students to learn their basic facts and be able to use them fluently and with automaticity, but being "fast" in math class isn't the goal.
Response From Richard Robinson & Rachael Gabriel
Richard Robinson is assistant professor of mathematics at The Citadel. Rachael Gabriel is associate professor of literacy education at the University of Connecticut:
If we want students to develop an interest in mathematics and to develop a sense of themselves as people who engage with mathematics for applied or theoretical purposes, we need to design instruction that engages with authentic mathematical practices. As teachers unpack the Common Core State Standards Mathematical Practices Standards, they may notice that several common instructional habits often go against the thinking, habits of mind, and dispositions that support deep understanding.
What are students doing if they are engaged in truly mathematical practices?
A. Showing more detail.
In mathematics, writing is a form of reasoning. It is communicating with oneself just as much as it is communicating with someone else. By asking students to show more detail in their writing, students are forced to slow down their thinking, making their mathematical moves more intentional. More detail not only includes showing every step of a mathematical process, it can/should also include brief comments to oneself and the reader as to what one "sees" mathematically at any given moment. By asking students to provide a mathematical play-by-play in the margins of their work, students engage more meaningfully in the processes of mathematics. Disciplinary practice: writing as a form of reasoning.
B. Doing fewer problems.
The mathematics, and how it reveals itself to the mathematician, is not responsive to a preset schedule that says you can explain an entire problem set in a single class period. In some sense, the mathematics itself sets the pace—with the very nature of the problem dictating whether it should be solved by hand, using a calculator, or using a fast or slow mental process. Thus, classrooms in which a teacher flies through examples without giving students a moment to reflect, process, and truly wrestle with the mathematics at hand, promote practices antithetical to the discipline. It is common to increase the amount of practice when students are struggling with a concept in a classroom setting; however, in a work setting, when a problem isn't well understood, people do not do 20 more just like it. Instead, they might slow down, reframe, and work into the details of one problem at a time. Disciplinary practice: letting the mathematics set the pace.
C. Doing better problems.
Some people say that a teacher is only as good as the questions they ask. We believe that a teacher is only as good as the way they engage students in asking and answering important questions. When we assign large problem sets, we often assume that students will be able to compare across multiple examples and pull out the essence of a concept. Instead, a few well-chosen example questions compared explicitly by the teacher can help students attend to the heart of the mathematical matter. If chosen purposefully, each question/example should do real pedagogical work, moving students closer to clarity of understanding. One might provide an example where X applies (example) paired with an example where X does not apply (nonexample), clearly highlighting how an expert knows the difference. Promoting this sort of professional vision helps students "see," i.e., attend to, what's important in a mathematical situation. In this way, it may only take a set of three questions to do the pedagogical work of 23. Disciplinary practice: attending to what's mathematically important.
Response From Tamera Musiowsky
Tamera Musiowsky is an international educator and adviser who has taught in Singapore, New York City, and Edmonton, Canada. She is an active member of ASCD and is the president of the Emerging Leaders Alumni Affiliate. Her previous roles include elementary teacher, teacher leader, instructional coordinator, and student-action coordinator. She currently resides and teaches in Singapore:
Teaching Math: Learning From Your Limitations and Mistakes
Teaching mathematics is a scary experience for many teachers. This tends to be an area in which teachers are afraid to take risks in teaching because they may not be comfortable with the math themselves and they fear making mistakes. But I would argue that math is the perfect place to let go of some control, take risks, and learn from mistakes together!
- Let go of the "lesson." Instead of trying to control every part of a math "lesson," turn it into a learning experience for students, and for you. Some of the best learning comes from watching students struggle with a concept, discussing their thinking, and coaching them through their thoughts. Naturally there are parts of the math block that you will always include such as an introduction to the concept or skills and sharing time, but what you do with the meat of math time really rests in what you want students to experience. Having an "open structure" to your math time will allow you to learn a lot from your students and yourself as a teacher. Be patient and give it a try.
- Quality versus quantity. Instead of giving students a bunch of quick practice questions at the end of a lesson that add little value to the learning experience they just had, think about shifting the independent work and sharing time. Experiment with giving one complex problem that requires students to use their vast bank of skills to solve problems. Engage them by challenging them to use their listening, speaking, time management, collaboration, and planning skills. Let them dig in and be creative and keep in mind that one great question can take longer than one class period to solve, and that is OK!
- Mixing learners. Instead of tracking students or pigeon-holing them in an ability group, mix up the groups that work together. This type of grouping provides equity in the learning experience by giving all learners a chance to contribute their strengths and skills, voice their thinking, reason out solutions, and present their solving processes.
- Make it relevant. Instead of presenting problems to students that have no relevance to their lives, let them create the problems! Students know what kinds of problems need solving because they have lives outside of school! Opening the door to problem creation gets students authentically involved in recognizing all the kinds of real-life math problems that need a solution.
It takes time to let go of some of the fear of teaching math, but it is possible to unlearn some of the limiting behaviors that have been instilled in how we teach. Think about an area in which you are willing to take the first step in changing the math learning experience for you and your students.
Response From Dr. Fuchang Liu
Dr. Fuchang Liu has been an educator for more than three decades. Currently, he is associate professor of math education at Wichita State University, working with pre- and in-service elementary teachers on a daily basis. He is the author of Common Mistakes in Teaching Elementary Math - And How to Avoid Them:
The Order of Operations Is Never PEMDAS
The single biggest mistake in math instruction, in my opinion, is the mnemonic concerning the order of operations: "Please Excuse My Dear Aunt Sally," PEMDAS for short, standing for parenthesis, exponent, multiplication, division, addition, and subtraction. This mnemonic supposedly tells students the order by which a math problem containing two or more operations should be carried out. Due to its popularity, it has been causing tens of thousands of students, including college students, to make mistakes on a problem as simple as 7 − 2 + 3.
You can easily test this out. For a group of students (elementary, high school, or even college students), pass to everyone a pencil and a piece of paper. Then write the above-mentioned problem on the board: 7 − 2 + 3 = and ask them to solve it without using a calculator or consulting with each other. It is very likely that some students will get it wrong. If you ask those who got it wrong how they solved it, you will invariably hear the mention of "Please Excuse My Dear Aunt Sally" or "PEMDAS", such as "By PEMDAS, I did 2 + 3 first, and got 5. Then I did 7 − 5, so my final answer was 2."
This is precisely why PEMDAS is so misleading. When students are presented with this "order of operations" in a linear, sequential manner, they will naturally interpret it as meaning that any operation takes precedence over those following it. More specifically, students interpret PEMDAS as meaning that multiplication comes before division and that addition comes before subtraction, because the mnemonic says "M-D-A-S".
In reality, however, the rule regarding the order of operations is that addition and subtraction are on the same level and must be executed from left to right. Neither operation has precedence over the other. Thus, for doing the problem at issue, the correct procedure is: 7 − 2 + 3 = 5 + 3 = 8. And the same is true for multiplication and division. These two operations are on the same level and must be performed from left to right. For example, for 18 ÷ 3 × 2, the right procedure is 18 ÷ 3 × 2 = 6 × 2 = 12.
As to how to correct this mistake, it is simple: Stop teaching "Please Excuse My Dear Aunt Sally" immediately. Never say the order of operations is PEMDAS. It's extremely misleading and utterly untrue. If it helps you to remember, just think Aunt Sally is evil and you should never excuse her.
Instead, teach the rule regarding the order of operations the way it should be:
- Multiplication and division are higher than and must be performed before either addition or subtraction.
- Multiplication and division, in themselves, are on the same level and must be carried out from left to right. So are addition and subtraction, which also must be carried out from left to right.
Response From Bonnie Tripp
Bonnie Tripp has been a high school math teacher and department chair at Lake Highland Preparatory School in Orlando, Fla., for the past 33 years:
I will be discussing two of what I believe are the biggest mistakes made in mathematics instruction today.
Homework: What is the right amount of homework to assign in a high school math course? If I assign 10 problems of the exact same type, what am I really accomplishing? If students do not understand the concept, they will get all 10 incorrect and become discouraged. If they understand the concept, then they will get all of them correct and will become bored. We should assign what is necessary for them to understand the material. I also tell my students that they should be mature enough to decide if they should do more problems. Some students may just need a couple of examples, and they have it; others may need more. Homework should never be assigned as a punishment or busy work. Let your students take an active role in the number of problems they need to practice to understand the topic.
Partial credit: In my early years of teaching, I was guilty of not giving enough partial credit. Sure, it takes longer to grade if the teacher has to decide how much partial credit should be given. I tell my students all the time that I am looking for an understanding of the process, not simply the answer. If they make a careless mistake, they should not lose many points. If a student makes a mistake early on in a problem, but the rest of the work makes sense, the problem should still get a fair amount of partial credit. Students should be learning and understanding the process of a problem, not just memorizing. As I go through a problem with my students, I ask them on every step to explain what I did. Is that "legal" mathematically? Is my new step equivalent to the previous step? If not, they should stop and try to find the mistake. Maybe, even have the student verbally explain to you the process they went through to accomplish the task. Take the time and really look at what the student has written. Teaching our students to think is a tough process but well worth it in the end. We should be giving credit where credit is due.
Response From Bill Wilmot
Bill Wilmot was the founding head of school of the Tremont School, a project-based independent school in Massachusetts, designed around individuals rather than groups, and a math and science teacher in a variety of settings. Now, as the founding partner of Venture Educational Collaborative, he works with founders and schools on designing or redesigning new schools and programs to prepare students for an ever-evolving future:
The single biggest mistake in math instruction is to think of math learning as the delivery of information from an expert to a ready recipient. This idea seems particularly attractive when teaching math because there are clear right answers to the questions that we ask in math classes. However, our goal in giving everyone a mathematics education is for adults to use math in the world, not to answer questions in math class. Real-world math poses questions like: "How many bags of hot dog buns do I need?" Or, "How do I best model the growth of ice crystals on power lines?" Or, "How do I create a geometry with a different set of assumptions?" These questions have more than one answer, investigated respectively by the Memorial Day cookout shopper, a college student for her thesis, and Lobachevsky and Riemann. More important than the answer itself is a valid argument, no matter the complexity of the math.
Instead, we should think of learning math as a dialectical process, a process of questioning. Math benefits from being a deeply human subject, built on the foundations of how we perceive the world, how our systems of logic work, and how we account for discrete units. In some ways, it is helpful to think of math as a careful process of rational self-discovery. Holding certain propositions as true, we build entire logical, mathematical worlds. The teacher's job is to guide the student through this process of self-discovery by asking just the right questions at just the right time, building step by step on what they already know.
To nurture adults who use math, math teachers should ask questions. Math students should ask questions of each other and the teacher. Math students should present their work. And math teachers should ask themselves the simple question, "Who is doing the thinking, me or the student?" By thinking it through, by talking it through, our students can develop a deep conceptual understanding of math and become the future adults who use math in the world—hot dog buns, ice accretion, or the next Riemann.
Response From Bradley Witzel, Ph.D.
Bradley Witzel, Ph.D., is an award-winning teacher and researcher who works as a full professor and program director of the MEd in Intervention at Winthrop University. Dr. Witzel has authored 10 books and delivered nearly 500 presentations on strategies for students with academic needs:
Not allowing learning to occur on a continuum is the biggest mistake made in math instruction. Much debate surrounds the explicit versus implicit constructs of teaching mathematics. Truthfully, both are permissible and have decent research effect sizes (National Math Advisory Panel, 2008). However, when learning is new or difficult, explicit instruction should be used to initiate the instruction. As students gain competence, the teacher needs to fade input and corrective feedback while scaffolding supports in order to help the students gain competent independence. In other words, explicit instruction may have powerful effect sizes, but teachers must fade such principles and strategies as the students succeed.
The all or nothing, such as "I am a constructivist," is not necessarily beneficial to teaching math in that using only one approach ignores the needs of some learners, lacks the needed sophistication to meet the needs of multiple students, and is potentially detrimental to learning.
Thanks to Hilary, Richard, Rachael, Tamera, Fuchang, Bonnie, Bill, and Bradley for their contributions!
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