Response: 'Challenges Are a Natural Part of Mathematics'
(This is the first post in a two-part series)
The new "question-of-the-week" is:
What do math teachers view as their biggest challenges and how can they best respond to them?
All of us educators face challenges. This series will explore what specific ones face math teachers.
You might be interested in a related previous series on what science teachers view as their major challenges.
Today, Makeda Brome, Pia Hansen, Linda Gojak, Marian Small, Kenneth Baum and David Krulwich contribute their responses. You can listen to a 10-minute conversation I had with Makeda and Pia on my BAM! Radio Show. You can also find a list of, and links to, previous shows here.
Response From Makeda Brome
Makeda Brome is a high school math teacher and department chair at Lincoln Park Academy in Ft. Pierce, Fla. She is also a Florida Teacher Leader Fellow through the University of Florida and in partnership with the Center for Teaching Quality:
Our subject is about productive struggle, solving everyday problems, and seeing patterns in the world around us. Challenges are a natural part of mathematics. As teachers, we have to make it our goal to overcome classroom challenges so that our students learn.While there are many challenges that math teachers view as their biggest, there are three that we can best respond to. Time is a challenge that we cannot change, so what do I see as the three biggest challenges other than time? They are prerequisite skills, student mindsets, and resources.
Most math teachers would agree that student's come in lacking and/or not remembering prerequisite skills for their math courses. The "summer slide" is a contributor - not practicing math skills for over 2 months in between a previous course and and a new course is not helpful to a student. Most curriculum starts as if the student finished their last course the previous day. So what can teachers do? We can offer opportunities for parents and students to not fall down that slide during the summer by 1) giving a list of prerequisite skills that should be accomplished proficiently by students in order to start the next course on level. and 2) identify resources that parents and students can use to practice those skills (i.e. Khan Academy, websites, problem sets).
Student mindsets about math tend to be fixed. They either believe they were born with the ability to do math or they were not. When you combine this fixed mindset with the lack of and/or not remembering prerequisite skills, teaching math can become an even greater challenge. To combat this, we as teachers must first evaluate our own mindsets about mathematics. We have to have a growth mindset about mathematics. If we have a fixed mindset, then students will perceive that and our classroom becomes a mix of those that "can" and those that "cannot." There are many resources available about the growth mindset (Mindset, Carol Dweck). There are even specific resources about mathematical mindsets which address parents, students, and teachers. Stanford Professor and author, Jo Boaler has written the book Mathematical Mindsets and she even offers a free online course called "How to Learn Math: For Students." The course is split into two major parts: the Brain and Math Learning and Strategies for Success.
Last, but certainly not least, are resources. The implementation of the Common Core Standards nationally have caused teachers to change the way mathematics has been taught and presented for decades. Students are required to know more than procedural fluency or how to do a problem by just following the steps as quickly and efficiently as possible. Gone are the days where teachers just followed the text book. Common Core Math requires rigor, depth, coherence across classes, and application to real life circumstances. Most textbooks have not fully been adopted to these changes.
The newness of the standards, what they mean, and what they look like have not fully caught up to textbook makers. As of yet no textbook has been found to be 100 percent aligned to the standards. With this knowledge, teachers have to become seekers. Googling resources and specific standards can produce many resources, including activities, for the standards. EngageNY has lesson plans, videos, activities, etc. that teachers can use as a starting point. NCTM Illuminations, Illustrative Mathematics, and Learn Zillion are just a few more sites with great resources. I am a true believer that if you seek, you will find, but it may take some work to find what you need.
Response From Pia Hansen
Pia M. Hansen has been a classroom teacher for twenty-seven years, teaching pre-school to college level. She is the co-author of Performance Tasks and Rubrics for Early Elementary Mathematics: Meeting Rigorous Standards and Assessments with Charlotte Danielson, published by Routledge, and the 3rd grade curriculum Bridges in Mathematics, published by the Math Learning Center. Pia continues to work with teachers on best practice as the director of professional development for the Math Learning Center:
Math teachers face three major challenges: Their beliefs about teaching and learning, their content and pedagogy knowledge, and time for reflection.
1. Beliefs about teaching and learning math: Many researchers agree about the importance of changing teachers' beliefs and yet there is disagreement about what changes beliefs and practice. Some would argue that because beliefs influence one's perception of the world, beliefs must change before one can perceive the changes that must occur (Pajares). On the other hand, Guskey proposes an alternative model, arguing that "significant change in teachers' beliefs and attitudes is likely to take place only after changes in student learning outcomes are evidenced". After reviewing many studies about the relationship between changes in beliefs and changes in actions, Philipp (2007) argues for a dialectic approach to this apparent tension, indicating that the two work together to contribute to teachers' learning and growth. He writes, " determining which changes first is less important than supporting teachers to change their beliefs and practices in tandem and reflection is the critical factor for supporting teachers' changing beliefs and practices". Teachers might respond to this challenge by considering their beliefs personally. What do I believe about teaching, and teaching mathematics? What are the most effective models, strategies and practices to nurture all my students' mathematical thinking? How do the school/district/state/national initiatives reflect my own experiences in the classroom? What do I want to change in my practice?
2. Shallow content and pedagogy knowledge: More than half of our high school graduates end up taking a remedial math class their first year in college. Some young adults choose to teach because they don't believe they can do math. They may become teachers without the necessary mathematical content knowledge to teach for conceptual understanding. They may know rote procedures and hang on to rules they memorized rather than exploring mathematical relationships and celebrating multiple models and strategies. Their challenge is to deal with a very real phobia and learn more about the mathematics at least 2 years ahead of where they are teaching, that will allow them to effectively differentiate instruction for their students.
Other students may have been fast at math drills and good at remembering algorithms. They decided to teach math because it came easy to them. They have the content knowledge, perhaps without the pedagogy. Their challenge is developing a growth mind-set, a belief that all children can do important mathematics. Visualization, reasoning and justification, problem solving and cooperative group work are the hallmarks of 21st century mathematics classrooms. These practices can be missing from a classroom where speed and memorizations is valued.
Some recommendations for increasing both content and pedagogy are participating in a book study related to practices or models and strategies, creating a study group with colleagues for an upcoming unit, or taking online or local university courses. These experiences will increase student engagement and achievement and create a culture where the challenge to work hard on what matters most is accepted.
3. Time for reflection: In the words of John Dewey, "It's not the doing that matters; it's the thinking about the doing ". Professional Learning Communities, lesson studies, formal and informal mentoring and coaching relationships can provide that reflection. As teachers observe other teachers and their students, share multiple perspectives and build a conceptual understanding from discussions of students' ideas about math, they build competence and self-confidence. Almost every teacher-evaluation model includes some element of self-reflection, and yet few teachers have and make the time, for reflection. Beyond knowledge about the students, the math standards and curriculum, the content and pedagogy, the teachers must know themselves.
Response From Linda Gojak
Linda Gojak is a past president of the National Council of Teachers of Mathematics and the National Council of Supervisors of Mathematics. She spent 28 years as an elementary mathematics specialist, teaching all grades from 5-8 and working with the K-8 teachers in her school. For the past 15 years, she has provided professional learning opportunities around the country for K-8 mathematics teachers. She is the author of The Common Core Companion K-2 and 3-5, What's Your Math Problem? and Paths to Problem Solving:
I would like to speak to K-8 mathematics instruction since that includes the grade levels of my experience and work.
This is a great question and I often hear the following concern from the teachers with whom I work.
a) Mathematics content knowledge, not only at one's teaching level but across grade levels, is critical information as a teacher plans instruction for his or her grade. It is a concern of many teachers. Because content knowledge is a prime issue, many teachers do not feel comfortable teaching mathematics. Teachers need to know where their students are coming from and where they are going in math. Too often, too much valuable instruction time is spent reteaching a concept from the previous (or several previous) grade level. That means that all teachers have to deeply understand the content (and so many teachers have had only 1or 2 courses to prepare them to more deeply understand. This is hardly adequate.). How do teachers respond? Attending professional learning workshops that focus on content and pedagogy that are offered by credible facilitators or organizations as well as encouraging the administration in their districts to provide such opportunities that focus on mathematics content and pedagogy. Schools that have math coaches should be sure that teachers can use their expertise to help them more deeply understanding the content and analyze student thinking.
b) While I believe that new standards in mathematics are making a difference in instruction, some teachers do not understand what they entail. Our standards are no longer a checklist of skills that students need to "do" by the end of a grade level; rather, they require a deep understanding of what rigor looks like in the classroom, that is, a balance of conceptual understanding, procedural skills and opportunities to apply mathematics to a variety of situations. Additionally the practices or processes describe how a student who is proficient in mathematics thinks about and does mathematics. Using math coaches to support classroom teachers, especially teachers who are responsible for teaching four or more different subjects a day, is one way to help teachers better understand and teach mathematics around their state standards. Professional learning, including book studies and providing dedicated time for teachers to work together to discuss, plan and examine student work will help all teachers to better understand the depth and progressions of their standards.
Response From Marian Small
Marian Small is the former dean of education of the University of New Brunswick in Canada and have written about 100 resources for teachers and students on K-12 mathematics. Much of the work she does is on differentiating instruction and teacher questioning:
There was a time, not that long ago, when a math textbook modeled how to correctly complete every sort of task to be learned. A teacher could share those models that she or he did not even have to create, and ask students to answer a slew of questions, where the answers could be checked in the back. There was a lot of security for the teacher, with very few decisions to make.
But now we are asking teachers to teach math so that it makes sense to ALL students, not just the high end student. That means teachers need to teach ideas to different students in different ways; that requires deciding which is the best way for different students. Oops- teachers have to make decisions now.
We also now want teachers to have students solve problems their own ways and have those students defend their processes. What if a student solves a problem in a way the teacher cannot follow? Where has all the teacher's security gone now?
In spite of all of this insecurity about what's correct or not and what decisions to make, teachers are feeling the pressure to ensure that each of their students does well on "standardized", often high stakes, tests. How can a teacher who learned math in a very conventional way be in a position to help students decide whether an argument is a good one or not in preparation for students judging an argument on a standardized test?
On top of all of this, we live in a world without much patience. While students used to be willing to quietly sit through relatively boring math "speeches", students in our world do not have the patience for this, and, on top of this, they expect their ways of thinking to be affirmed, never dismissed.
I think the best ways to respond are:
- Start learning more. We live in the age of the internet. Google is there to help teachers learn about things they didn't know before. So maybe the teacher has an obligation to become more aware of the background of the mathematics being taught and not just live in the foreground.
- Borrow from others. Again, because of the internet, there is a sharing culture that provides many resources from which teachers can choose, for free, to make their lessons more engaging and more meaningful. Teachers can just browse for what they want, and there it will be!
- Take risks. We ask students to take risks every time we give them a test or ask them a question. But too many teachers are risk averse. You can't be an effective teacher if you don't take risks. That means teachers need to try new approaches and strategies and give it their best shot.
- Listen hard. Teachers are used to doing the talking and not the listening. The best way to grow as a teacher is to put something out there and listen to what students have to say. But teachers need to listen hard, and with an open mind.
- Ask open-ended questions where LOTS of students can participate, from the weakest to the strongest and lots of opinions can be heard and valued; kids these days need to be heard and their interesting ideas, even if they are not typical, need to be appreciated.
Response From Kenneth Baum & David Krulwich
Kenneth Baum and David Krulwich are, respectively, the former and current principals of the Urban Assembly School for Applied Math and Science, a public school in the Bronx, New York serving grades 6 through 12. They are co-authors of the new book The Artisan Teaching Model for Instructional Leadership (ASCD 2016):
"What is this stuff good for?" This age-old question is the single biggest complaint that math teachers hear from their students and the source of much math teacher anxiety. Math educators have tried to address the issue most recently by stressing "real world math" in state standards, textbooks, and curricula. Glance at almost any textbook written within the past 10 years and you will see "real-world connection" icons displayed in almost every problem set. So why is it that with all of these resources, teachers still are having a hard time truly engaging students? There are four big things teachers can do about this.
First, avoid falling into the trap of following a text book in lieu of using lesson plans that feature higher-order thinking. When a teacher essentially uses the textbook as the backbone of a lesson, learning typically becomes rote because that's the way most textbooks—even in the common core era—are organized. The exercise sets have tons of easy "plug and chug" problems up front, with higher order thinking problems relegated to the back.
Taken to the classroom, if you give students 20 rote problems, say, about the Pythagorean Theorem and then the 21st is a word problem about a rocket, students not only get bored, but they also get saturated with essentially one technique. All pumped up with their one technique, students then attempt the word problem almost without regard to context, and entirely without need to select and an approach. That's not problem solving—that's conditioning. Although this approach will generate consistently deceptively "exit slip" data, it will not help students to become better problem solvers or better thinkers. Text books can be a valuable resource, but are not meant to replace well-written lesson plans.
Second, teachers need to look past the glitz of color photos of their resources and determine what kind of thinking kids need to exhibit to solve a problem or project. Specifically, teachers should "test" each curricular resource for how much it engenders and requires higher order thinking. Many "real-world" problems presented in expensive texts are actually not problems at all; that is, they are simple tasks "dressed up" with fancy pictures to look appealing, but are actually devoid of critical, contextual thinking.
Third, do not assume that just because a math problem has a "real-world" connection that the problem will hold "real interest" for kids. Far too often in textbooks and curricula, the "real-world" problems presented to students are boring.
EXAMPLE: A common application for exponential growth is to calculate retirement savings based on compound interest. To be sure, retirement savings is very real-world to a 35 year old teacher, but few 15 year olds can relate to retirement savings. The world of a teenager is now and a way more engaging application of exponential growth to the viral spread of a YouTube video
Fourth, co-plan your lessons and projects with another teacher in your school who teaches the exact same course to similar groups of students. If no such teacher exists at your school, take charge of your professional learning by advocating the value of having someone to authentically co-plan with.
Thanks to Makeda, Pia, Linda, Marian, Kenneth, and David for their contributions!
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