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Response: Math Can Be a 'Hard Sell'

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

The new "question-of-the-week" is:

What do math teachers view as their biggest challenges and how can they best respond to them?

In Part One, 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.

Today's suggestions come from  Mary Mueller, Cathy L. Seeley, Daniel R. Venables, Nancy Villalta, Erik M. Francis, and Rik Rowe.

Response From Mary Mueller

Mary Mueller, Ed.D, is an associate professor of elementary and special education at Seton Hall University who specializes in pre-service and in-service teacher preparation and development in mathematics.  Her research focuses on the development of mathematical reasoning, teacher preparation in mathematics, and the collaboration between teachers and families in supporting children's literacy and mathematics learning. She is co-author of Nurturing Your Child's Math and Literacy in Pre-K - Fifth Grade: The Family Connection (Rowman & Littlefield, 2016):

There are many challenges associated with teaching math.  Math, more than any other subject, builds on prior learning.  Some teachers say that children do not retain what they learned the year before and others will argue that if they don't retain it, they never learned it.  Either way, math builds on the knowledge and skills that children learn each year. Fractions are a perfect example.  Children begin learning fractions in third grade.  Or do they?  Depending on how you define learning—children are shown how to perform operations with fractions in the elementary grades but often they don't conceptually learn what fractions are and how they work.  Many children get to middle school not understanding the difference between 5/4 and 4/5.  When presented with solving 3/5x=9 they are lost.

Homework presents an additional challenge.  Math needs to be practiced. Anyone who took an instrument as a child knows the need for practicing - you can have a lesson each week, but if you don't practice in between you will never advance past Twinkle, Twinkle Little Star.  Math is the same way.  After (hopefully) learning the lesson during the day, children need to practice and reinforce it that night because most likely the next day will build on the prior lesson so understanding of this material is essential!  Unfortunately, children often don't do homework or do it but just go through the motions (maybe checking answers in the back of the book).

All people learn at different rates so a math teacher is constantly wondering whether she is going too fast or too slow. Teachers might be tempted to teach to the middle but them they are only reaching a small portion of their class.   This is especially difficult given the tightly packed curriculum that teachers are expected to follow.

These issues are worsened by the fact that people often think they are either good at math or not.  Many believe in the math gene and are sure they didn't get it. These negative views of their own ability are difficult to change and thus children enter into a failure cycle.  This failure cycle often leads to math anxiety, which follows many through college.

What can math teachers do?

•             Don't assume that kids have the necessary previous knowledge. Pre-assess and then add what is missing into the lesson.  In other words, review and reinforce the necessary prior knowledge in the mini-lesson while still teaching new material.

•             Assess continuously—little "exit slips" at the end of each lesson to check understanding—this shows what needs to be reinforced the next day

•             Instead of large tests, give shorter quizzes. This gives student's little successes.

•             Make homework doable. Instead of assigning 20 problems, assign five carefully selected problems.  Make doing homework a challenge by monitoring the class completion and offering a small incentive.

•             Get parents involved!  Stress how important math homework is and ask that they work with you to ensure 100 percent completion.

•             Differentiate your instruction.  You might do this by giving a short mini-lesson at the beginning of class (including a review of prior learning) and then assigning groups to work on problems. You can then work with small groups who need more support.

•             Connect math to everyday life so that children see its value and pertinence. This is done by engaging children in problem solving and reasoning and inviting them to discuss their ideas with their classmates. Invite them to act as mathematicians would.

•             Finally, build them up!  Take each child from where he or she is and build them up by giving them small successes which boosts their confidence.

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Response From Cathy L. Seeley

Cathy L. Seeley has worked as a teacher, district mathematics coordinator, Texas state mathematics director for grades K-12, and is a former president of the National Council of Teachers of Mathematics. Her books include Making Sense of Math: How to Help Every Student Become a Mathematical Thinker and Problem Solver (ASCD) and Building a Math-Positive Culture: How to Support Great Math Teaching in Your School (ASCD):

Some of the challenges that math teachers today face are similar to those their colleagues in other disciplines face, especially colleagues in subjects that are heavily tested, such as English and reading. Many teachers face unreasonable pressures to cover content, even at the expense of student understanding, and often in superficial ways reflected in many large-scale tests. Some teachers are expected to document that all the material likely to be included on the high-stakes accountability test has been covered. Yet, often teachers know their students would benefit from taking more time to dig deeper into a problem, topic or concept, in the process building a stronger foundation for lasting learning than by just covering content.

The overemphasis on testing is one example of the many political influences on teachers. In state after state, policy makers are doing the worst thing we do in education—"yo-yo decision making." The on-again-off-again support for the Common Core Standards is the most visible example in recent years, putting math teachers (and their colleagues in English) in the middle of a risky and fragile situation. In too many states, teachers were rallied around the adoption of these standards a few years ago, with extensive investments in professional development and new curriculum materials.

Within a year or two, many leaders began to completely change direction, decrying the very standards they had asked for and supported just a few years before. The result has been demoralizing for teachers, with little or no time to realize the gains that any significant investment in a sound set of standards might yield. How unnecessary and irresponsible! Even though the new math standards represented, at most, minor shifts from what most state math standards had called for over the previous 10 to 20 years, policy makers backed away from following through, failing to capitalize on the investment of both teacher time and taxpayers' dollars. In the process, both teachers and their students have suffered.

The end result of all of this political chaos is that teachers are placed in the untenable position of knowing their students could benefit from more time engaging with interesting math problems and challenging content. Yet, they feel compelled to cover what will appear on the test—often the most mechanical elements of their state's standards.

Some states have made progress and affirmed in their standards the importance of thinking, reasoning, and problem solving. But these elements are difficult to test and expensive to administer and score if tested appropriately. Thus, many states have either held onto (or regressed back to) limited forms of testing, such as bubble-in multiple-choice tests, simply because they can be quickly and economically scored by machine. Good teachers who see the negative influence of such tests on teaching and learning are frustrated. Many feel they can't teach in the ways they know will help students, as they are pressured instead to cover content.

Let's put an end to yo-yo decision making. The challenge for policy makers and leaders is to be thoughtful and careful in deciding to replace any program, test, or standards. Determine first whether a wholesale change is called for or whether making modifications to the existing program will accomplish the intended goal. If a new program or initiative is implemented, invest not only upfront in professional development and day-to-day support, but make a commitment to stay the course and allow teachers to implement the program with common sense and professional judgment. Encourage and trust them to be the professionals they can be, helping students increase their learning in meaningful ways.  

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Response From Daniel R. Venables

Daniel R. Venables is founding director of the Center for Authentic PLCs and author of How Teachers Can Turn Data Into Action (ASCD, 2014), The Practice of Authentic PLCs:  A Guide to Effective Teacher Teams (Corwin, 2011), and Facilitating Authentic PLCs:  The Human Side of Leading Teacher Teams (ASCD, forthcoming).  He can be contacted at [email protected]:

With no disrespect to other tough-to-teach subjects, math is a hard sell. And while a number of things contribute to this fact, I will address what I perceive as one of the biggest and persistent challenges facing math teachers.

Challenge: Teaching students to understand mathematical concepts is much more difficult than teaching them procedures. It is a pedagogical temptation that lurks behind every concept.  When they just don't seem to get it, just give them the cookbook.  My own high school Algebra 2 teacher gave us a series of steps to follow in order to solve nearly every textbook problem. For each problem type, she offered a five or six step recipe which, when followed, produced the correct answer. This focus on the procedure left us not understanding the underlying concepts.

More, we didn't imagine that the concepts were even important; we could do well on tests by memorizing the recipes. It turns out, as discouraging as it sounds, this practice of teaching math students steps or recipes to specific problem types is much more widespread than math educators want to believe. Fighting the good fight is teaching kids the concepts that, when fully understood, lead to student success in solving not only specific problem types but generalize to a wide range of problem types related to the concept.  The NCTM (National Council of Teachers of Mathematics) has been advocating for this kind of conceptual teaching since its first Standards document in 1989, echoed decades later by the "CCSS Standards for Mathematical Practice."

One solution that has worked for me in addressing this challenge is what I call the 80/20 Principle. It works like this: 80 percent of the test items on any given test are just like the problems that kids practiced from the book. They are essentially lifted from the book problems with different numbers. The other 20 percent of the test items are non-routine problems. These questions assess the students' understanding of the concepts. There are problems the likes of which students have not seen before.  They tend to be "inside-out" versions of familiar problems, and they tend to require authentic understanding and higher order thinking to solve. For examples,

FAMILIAR PROBLEM:  Find the area of a rectangle whose sides are 3" and 5".

NON-ROUTINE PROBLEM:           Draw and label the sides of a rectangle whose area is 12 square feet. 

FAMILIAR PROBLEM:  The heights of the players on the high school basketball team are 72", 73", 70", 69", 74".  Find the average height.

NON-ROUTINE PROBLEM:           The heights of three players on the high school basketball team are 72", 75", 70".  Find possible heights for the remaining two players if the team's average height is 73". 

FAMILIAR PROBLEM:  (Students are asked to read a given graph and answer questions from the graph.)

NON-ROUTINE PROBLEM:           Without using any numbers, sketch a possible graph showing the number of popped kernels v. time when making popcorn.

It's important to keep in mind that non-routine problems must be unfamiliar to students.  If they've seen or practiced one like it, it's not non-routine.  When teachers assess students this way, they tend to teach this way and push students' thinking during instruction and not just on tests.  The point is not to trick students but to assess how deeply they understand the concepts (and not just the problems).  It was not uncommon for my students to try to anticipate a 20 percenter (as they called them) for concepts on a pending test.  Even if many students miss the 20 percent problem, going over it after the test was a huge teachable moment conceptually and practically—witnessing the solution left most students thinking "Oh, I could have gotten that one."  

Start small. Try one 20 percent problem on the first and second tests, two on the next, and 10 percent to 20 percent of the test thereafter. They will quickly catch on to the fact that true understanding is expected in the class and their questions during instruction will commonly reflect their pursuit of that understanding. 

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Response From Nancy Villalta

Nancy Villalta is a teacher at Moffett Elementary School in Lennox, Calif. and is a member of the Instructional Leadership Corps, a collaboration among the California Teachers Association, the Stanford Center for Opportunity Policy in Education (SCOPE), and the National Board Resource Center at Stanford:

The biggest challenge I face is the pressure to follow a fast pacing plan so that I can provide data in the form of summative assessments. The number of lengthy assessments severely cut into time that could be spent teaching and the pacing plan is unrealistic. Common Core standards are fewer than in the past, but more in depth. They require that students understand the material and that only happens if they construct the knowledge on their own through discovery tasks and experience. Those experiences take time that is not accounted for on the pacing plan and is sidetracked by the summative assessments.

Students should be solving interesting, relevant problems, having discussions about the solutions, looking for patterns, making conjectures, and experiencing number talks. The time invested in these learning opportunities pays off. I have found that my students are better able to take on unfamiliar tasks by the end of the year than students in classes that have covered all of the material quickly using a Direct Instruction model. Since they have spent the entire school year tackling difficult problems, choosing appropriate tools, testing different strategies and persevering through them, their ability to think critically increases. They have more confidence in their own ability and they are not put off by the difficulty of summative assessments.

I use informal, formative assessment to inform my instruction. As the students are working in groups, I interview them, observe their interactions and listen to their reasoning. In this way, I am able to determine what my next steps should be and what misconceptions my students may have. This gives me far more information than summative assessments and does not cut into teaching time.

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Response From Erik M. Francis

Erik M. Francis, M.Ed., M.S., is the author of Now THAT'S a Good Question! How to Promote Cognitive Rigor Through Classroom Questioning, published by ASCD.  He is also the owner and lead professional education specialist for Maverik Education LLC, providing professional development on teaching and learning that address the cognitive rigor of college and career ready standards:

The biggest challenge facing math teachers is that the instructional focus and assessment of teaching and learning in mathematics have changed.  No longer are students simply expected to demonstrate how to solve problems correctly.  Now teaching and learning for mathematics challenges students to think deeply and express and share how and why mathematical concepts, operations, and procedures can be transferred and used to attain and explain answers, outcomes, results, and problems.  It's about demonstrating and communicating mathematical knowledge and thinking.  

Just look at the standardized assessments.  There are hardly any equal signs on the test!  Students are asked and allowed to choose more than one response (if merited).  The word problems ask students to write and interpret the numerical expression that describes the rather than solve the problem itself.  Some questions even ask students to describe and explain subject-specific terms.  Students should still be taught and learn how to use math to solve mathematical and real world problems.  However, students are also expected to demonstrate and communicate—or show and tell—deeper and extensive mathematical learning of how and why math can be used in different contexts.

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Response From Rik Rowe

Rik Rowe is a connected educator with a passion for learning and a passion for empowering his Learners to engage in exciting learning opportunities. Rik has been a High School Math Teacher for the past 16 years. He previously had several roles in Systems Development in the corporate world when he realized his passion to become an educator with the hopes to ignite the learning in our classrooms. Rik is an avid reader, participates in a variety of Twitter chats and can often be found at EdCamps:

After conversing recently with several mathematical educators, it has become apparent that too much focus is being allocated to mere computation and not enough to learning to think and reason mathematically. We see too many Learners told to be 'compliant' and focus on busy, tedious and repetitive questions. They are being asked to solve questions for which we already know the solutions. I believe their minds and our time could be better spent thinking and wondering about situations that require mathematical thinking, but at a higher more abstract level. We could develop the fundamental skills on an as needed basis.

Some of my plans for the approaching school year include designing learning that presents situations to Learners to empower them to engage in more thinking, wondering and collaborating rather than immediately just solving a problem. I envision creating more 'situations' as opposed to problems. When we provide opportunities for our Learners to engage in collaborative questioning, we empower them to develop key skills for tomorrow. When our Learners develop an interdependence with their peers and teachers, we will be moving in a better direction.

Imagine when our Learners start to collaborate during and after school hours on real-world situations for which we cannot just Google an answer. I believe we need this shift in education from merely doing or performing the mathematics to actually collaborating, communicating and pondering over larger and more abstract mathematically related situations. As a Learner, I understand so much more and can apply my skills better when I relate and connect concepts. School activities have been more geared to performing skills based on a curriculum tied to a calendar. Much of this has been due to our Learners required to take and pass so many standardized tests. Most of these standardized tests do not prepare our Learners for a tomorrow based on uncertainty versus the more predictable past we've experienced.

I envision learning transitioning into situation and problem-solving that could in fact have a number of different solutions. Our Learners recently were given the opportunity to book a flight from Point A to Point B during a given week. They realized that there were many variables for which I had not stipulated any specific requirements. Through this learning opportunity, they learned about non-stop flights versus one or more stops, flying through various cities with a possible layover, and how being flexible with their dates might yield more attractive options. Several Learners shared how much more they enjoyed their 'Taking Flight' learning opportunity as opposed to just another worksheet with fifty or so math questions to solve.

Once we better understand the skills that today's employers are in search of, we can better prepare our Learners for more collaborative and engaging learning. Today is a very exciting time to be learning and especially math teachers need to shift our thinking from merely solving repetitive questions to creating and solving innovative situations.

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Thanks to Cathy, Mary, Daniel, Nancy, Erik, and Rik 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.

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