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Response: 'It's Time to Slow Down and Smell the Mathematical Roses!'

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

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.

Today, Sunil Singh, Laney Sammons, Abby Shink, Cathy Seeley, and Shannon Jones share their thoughts.

Response From Sunil Singh

Sunil Singh is the author of Pi of Life: The Hidden Happiness of Mathematics (2017) and Math Recess (2019), and works as a mathematics learning specialist at Scolab, a digital education company in Montreal. He was also a regular contributor to The New York Times Numberplay blog:

There are many mistakes in math instruction that have been entrenched in the classroom for many decades. However, many of these missteps in students learning and exploring mathematics stem from how we view mathematics. Namely, that it is a body of knowledge of procedures and facts to be learned for practical and useful purposes. There is an unintentional dehumanization of mathematics. As such, the element of time is used in ways that reflect production and productivity. Everything requires measurement and accountability.

Mathematics is not a race. It never has been. Perhaps the most beautiful articulation of its heart and soul is the one below.

"Mathematics is one of the most essential emanations of the human spirit, a thing to be valued in and for itself, like art or poetry."

Oscar Veblen

 

So, the biggest mistake we make—which in turn creates so many related mistakes—is that we do not honor the most critical element in the natural, historical narrative of mathematics. We don't do things slowly. The Slow Movement has been occurring the last few decades in many aspects of our lives. Regrettably, that philosophy has not filtered down to our math classes, where things are still learned with the clock and on the clock.

Slowing down the learning of mathematics creates a better environment for learning and teaching mathematics. But, more than that, it does the opposite of when time becomes an unhealthy metric in math education. Learning with the element of speed creates and enables math anxiety among our students. Learning without it creates more time and space for not only mathematical connections but deeper, human ones.

And really, the improvement of math instruction is really about improving the relationships that we have with this most beautiful subject and with each other.

It's time to slow down and smell the mathematical roses!

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Response From Laney Sammons

Laney Sammons (@LaneySammons) is the author of Teaching Students to Communicate Mathematically (ASCD 2018). She was a classroom teacher and instructional coach for 21 years before becoming an author and educational consultant. She has written numerous books including her most recent work, Teaching Students to Communicate Mathematically (ASCD 2018), Guided Math: A Framework for Mathematics Instruction, and Building Mathematical Comprehension: Using Literacy Strategies to Make Meaning:

With the adoption of increasingly demanding mathematics standards, our expectations of what students need to know and be able to do have evolved. Yet, our instructional methods remain very much the same.

Here are two common mistakes that greatly impact students' mathematical achievement.

1.Telling, Instead of Teaching

For centuries, teachers have followed the advice of early math textbooks: State a rule, demonstrate the rule with examples, and finally have students practice the rule (Larson & Kanold, 2016). How do we improve on this approach? I suggest that we heed the wisdom of the poet Mark van Doren (n.d.), "The art of teaching is the art of assisting discovery."

Turn the traditional instructional sequence upside down by first assigning an exploratory problem to teach a new concept or skill. Challenge students to draw upon what they already know to solve unfamiliar problems, and, in the process, discover new mathematical ideas. In this context, the newly learned content is more meaningful, and in searching for solutions, students engage in vital mathematical practices, e.g., making connections, justifying mathematical reasoning, and identifying patterns and structures.

Allow students to engage in productive struggle. Although it is tempting to step in and tell students what they need to know and to do, grappling with challenging problems teaches them to draw upon their own resources and persevere when solutions don't come easily. Through struggle, meaningful mathematical learning occurs.

Promote student reflection. Encourage them to look back at their experience and then talk or write about the learning process. Ask them: What happened? What have you learned? Why is it important? How can you apply it? Don't expect students to do this well immediately. Most have had little practice. Give them plenty of opportunities for reflection.

2.Expecting Students to Communicate Mathematically Without Teaching Them How

Today's math standards require students to precisely express their mathematical thinking. So, teachers are asking students to explain their thinking when solving problems—and often are disappointed with the results. How do we improve the quality of students' mathematical communication?

Explore the criteria for quality mathematical communication, both oral and written, with students. There is little chance that students will measure up to expectations if they don't know what the expectations are. Consult colleagues to develop consistent expectations. Share these criteria with your students.

Borrow teaching strategies from literacy teachers. Teach students how to communicate about math beginning in kindergarten. Model how you speak and write about your own mathematical thinking. In primary grades, lead shared-writing activities in which students jointly compose a piece of mathematical writing as you act as scribe. In mini-lessons, teach strategies for effective math talk and mathematical writing.

Provide ample feedback that is specific, descriptive, and timely. Based on established criteria, let them know specifically what they have done well and how they can improve. Give ample time to practice and revise.

Incorporate mathematical communication into math class every day. This provides many opportunities for students to hone these skills and sends a clear message to students that mathematical communication is an integral part of mathematical proficiency.

 

Changing one's instructional practices is never easy. Most teachers, however, thrive on challenges that improve the learning of their students.

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Response From Abby Shink

Abby Shink is a K-5 math interventionist at a small rural school in central Maine. She has been teaching for 10 years, focusing specifically on math for the last four. Abby is currently working on her master's in teaching mathematics K-8 at Mt. Holyoke College:

One of the biggest mistakes I see teachers making in math instruction happens before they walk into the classroom. Whether teachers are using a program, finding lessons on open-resourced sites, or using some other source, many teachers fail to do the math themselves before teaching the lesson in class. Before I teach any lesson, I start by doing the math myself. Furthermore, I don't just do it once, but I do it a couple of different ways, trying to anticipate ways my students might solve it. Anticipating ways that my students will solve the problem and solving that way myself help me to think about mistakes that they might make and scaffolds they will need in order to be successful when working on the problem. If you have the benefit of planning with other teachers, solving the math together and talking about your work and errors that your students might make will allow you to be even more prepared to do the lesson with your students. Inevitably, one of your colleagues will solve it in a way you had never thought of. Taking the time to listen and understand the different ways the problem could be solved can prepare you for understanding the ways that your students will solve the problem.

The hardest part about teaching is making those in-the-moment decisions during a lesson. Doing the math and anticipating possible student thinking help to make those decisions a little less "in the moment." You can never prepare for every possibility in the classroom; in fact, the students that surprise us with their thinking can be the most exciting thing that happens in our day. Being truly prepared to teach a lesson means that you understand the content and the possibilities of students' thinking and are ready to recognize important big ideas that students come across in the process of working.

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Response From Cathy Seeley

Cathy Seeley (@CathySeeley) is the author of Making Sense of Math: How to Help Every Student Become a Mathematical Thinker and Problem Solver (ASCD Arias 2016). Cathy has worked as a teacher, district mathematics coordinator, and Texas state mathematics director for grades K-12, and she is a sought-after speaker, having spoken in all 50 states and around the world. A former president of the National Council of Teachers of Mathematics, Cathy recently retired as a senior fellow with the Charles A. Dana Center at the University of Texas, where she worked on state and national policy and improvement efforts, with a focus on prekindergarten-grade 12 mathematics:

When I learned how to teach mathematics many years ago, the process seemed pretty straightforward: Prepare well, explain clearly, and present with enthusiasm(!) so that students would pay attention. Then work through some examples together with students before moving on to give them exercises to practice or problems to solve on their own. It was a familiar process—it was the way most of us learned math throughout our elementary, secondary, and college years.

Today, we realize the limitations of this 'sit-and-git' method of teaching (that's what we call it in Texas). The biggest mistake I think we make in teaching mathematics today is clinging to this belief that we need to tell students everything they will need to know in order to solve a particular type of problem before we give them a chance to actually solve that kind of problem. Instead, more and more educators are advocating a very different model of math teaching from the one that I learned in the last century, a model that was eloquently described by the National Council of Teachers of Mathematics more than 25 years ago in their Professional Standards for Teaching Mathematics and reinforced in their 2014 document Principles to Actions. Instead of telling students rules and procedures and then walking them through examples, we start with an engaging problem or task that they may not have already been taught how to solve. We give students the opportunity (and time) to wrestle with ideas and think about how they might approach the task instead of expecting them to remember what the teacher or textbook told them to do. And we purposefully structure classroom discussion around students' explanations of their thinking.

I call this model "upside-down teaching" (Seeley, 2017), referring to starting with a good problem instead of ending with one. Many educators over the years have recommended variations of a teaching model focused on student thinking and classroom discourse, and today, teachers across the grades are finding success with this kind of engaging teaching. The idea is that we should let students productively struggle with the underlying mathematics of a situation as they work toward developing some conceptual understanding of what is involved. Then we can use students' struggles as a basis for having them share their work in whole-class conversation that is structured and facilitated by the teacher. In the process, students learn the value of discussing not only correct approaches, but perhaps even more importantly, looking at incorrect answers and unproductive approaches as the teacher uses these "wrong turns" as a basis for students learning the intended mathematics of the lesson. In addition to structuring and facilitating the classroom discourse, the teacher's other critical role is to ensure that students connect their work and discussion to the mathematical goal of the lesson.

This kind of teaching is hard to implement well, and it calls for considerable support in terms of professional learning and ongoing help through coaches or peer discussions. It may not be something a teacher uses every day, at least not at first. It takes more time to teach this way than teacher-telling with students passively receiving. And in order for a teacher to know how to deal with potentially unexpected student approaches, this approach depends on teachers' own mathematical understanding. The payoff for making such an investment into a different instructional model is huge, however. Students not only begin to develop a more positive opinion of what math is, they can also develop a confident view of themselves as mathematics learners and see themselves as mathematical doers. If our goal is to make mathematics accessible to every single student, regardless of demographics or prior success, then upside-down teaching is an essential tool.

Successfully implementing upside-down teaching relies on administrative support to understand what is actually going on in a classroom that might involve students talking to each other as they work through a task that might or might not look like the exercises we typically expect to find in math books. A colleague recently shared with me that he had visited a wonderful math class that was structured in this upside-down way. The class spent an entire class period digging into a problem, with students increasingly moving toward deep understanding of mathematics as they participated in well-structured classroom discussion about how to solve the problem and what the solution might be. The teacher beautifully connected the students' work to the intended outcome of the lesson. Unfortunately, the principal who was also visiting the class that day lamented afterwards that the class had moved so slowly that they only finished one problem. Clearly, we also need to help administrators understand the purposes and value of this kind of teaching as their teachers begin to shift their approach to teaching mathematics in ways that help students develop lasting learning and become proficient mathematical problem-solvers

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Response From Shannon Jones

Shannon Jones is a 12-year elementary school educator. She has taught 3rd-5th grades in high needs and Title I schools. She is currently the 4th and 5th grade math-focus teacher for her school. She lives in Maryland with her husband, who is also an educator, and their two little girls. She tweets at @purplstarfsh:

Mistake #1 A lack of precision with math language

● We don't want to simplify the language that we use in math class just to make it "easier" for students. Students can handle words like "congruent" at an early age if they become commonplace in the classroom. Use the correct math terminology from the very beginning and hold your students accountable for using these terms in their speaking and writing.

Mistake #2 Discontinuing the use of manipulatives in the upper grades

● Students of all ages can benefit from the use of manipulatives. Many students work at a concrete math level beyond elementary school. The use of balance scales or cups and counters can be particularly helpful to students in middle school algebra courses. Make manipulatives easily accessible and commonplace so that students don't feel embarrassed accessing them whenever they feel the need.

Mistake #3 Overgeneralizing math rules

● Telling students things like, "You can't subtract large numbers from small numbers," "Addition and multiplication make numbers larger," or "Altogether means to add" can cause confusion and create misconceptions that are hard to undo in the later grades. Instead, when students are problem-solving, encourage them to make estimations before solving and push them on why their estimation is reasonable for the situation. When they have found a solution, ask them what makes their solution reasonable. When answers are found within the context of a real-world problem, it is easier to discuss reasonableness.

Mistake #4 Thinking the "low" or ELL students can't access challenging math tasks

● Create and find math tasks that grab student interest, connect to the real world, and are relatable to students' lives and you will find that all of your students will be able to contribute to the math work being done in your classroom. Presenting a math task with the numbers initially blacked out or eliminated and asking students what they know, notice, or wonder about the task, is a conversation in which all students can participate.

Mistake #5 Choosing fun and cute over rigor 

● Look for tasks, websites, games, and centers that meet the standard being taught. We want to engage our students and create excitement but we want to do that at the same time that we are meeting the grade-level standard. If you find a fun and cute activity, try it out yourself first and make sure that it meets the standard. Ask yourself why you are choosing this activity for your students to be sure that you aren't just assigning it for the "fun factor."

Mistake #6 Paying attention to all of the mistakes students make.

● When grading student work, we don't need to address every mistake or omission we come across. Ask yourself, does the error or omission impact understanding? If it does, group student work by the type of mistake and address the mistake in small group. Don't meet with students who made simple calculation mistakes; their work shows that they understand the content. They aren't in need of small-group interventions or a whole-class reteaching. Take time to analyze your students' errors.

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Thanks to Sunil, Laney, Abby, Cathy, and Shannon for their contributions!

Please feel free to leave a comment with your reactions to the topic or directly to anything that has been said in this post.

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.

Just a reminder; you can subscribe and receive updates from this blog via email or RSS Reader. And if you missed any of the highlights from the first seven years of this blog, you can see a categorized list below. The list doesn't include ones from this current year, but you can find those by clicking on the "answers" category found in the sidebar.

This Year's Most Popular Q&A Posts

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I am also creating a Twitter list including all contributors to this column.

Look for Part Three in a few days.

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