Response: Students Must 'Engage in Math Problem-Solving' & Not Just 'Follow Procedures'
(This is the final post in a two-part series. You can see Part One here.)
The new "question-of-the-week" is:
What are some best practices for teaching high school mathematics?
In Part One, David Wees, Jill Henry, Tammy L. Jones, Leslie A. Texas and Anne Collins shared their suggestions. You can listen to a 10-minute conversation I had with David and Jill on my BAM! Radio Show. You can also find a list of, and links to, previous shows here.
Today, Wendy Monroy, Jennifer Chang Wathall, Sunil Singh, and Dr. Matthew L. Beyranevand contribute their commentaries. I also include comments from readers.
Response From Wendy Monroy
Wendy Monroy is a Mathematics Coach for the Los Angeles Unified School District in Los Angeles, California and is a member of the Instructional Leadership Corps, a collaboration among the California Teachers Association, the Stanford Center for Opportunity Policy in Education, and the National Board Resource Center at Stanford:
The acquisition of best practices for teaching high school mathematics is necessary for student academic success. However, they will be discussed later. First and foremost, you must have as your guiding philosophical principle the belief that all students can learn. Unfortunately, once a student enters high school, they come with preconceived notions of their own capacity as math learners. Therefore, you must then adequately assess each student's prior knowledge and relationship with math. Then, you must have a clear vision for each student in which they are actively engaged in problem solving rather than just following procedures. This will unleash students' full potential as math learners. You must provide opportunities for them to fall in love with learning, since we are in an era in which anything is available with the simple touch of a button.
The Standards for Mathematical Practices (SMP) can be used as a guide since they describe the way students need to be engaged in mathematics. In the first weeks of school and throughout the year, I choose problems that are open-ended. This allows students to understand that math is more about being engaged with the process rather than just the search for the correct answers. Math is not supposed to be easy. If it is, it's because students are not being challenged enough.
In order for students to engage with the SMP's, you must choose a Higher-Level Demand Task. To ensure that the task is a Higher-Level Demand, ask yourself, does the task allow for the following?
- Does it allow students to explore and understand the nature of mathematical concepts, process, or relationships?
- Does it have multiple entry points?
- Are students engaged in conceptual understanding and, as a result, develop and connect it to procedural knowledge?
- Can the task be represented using visual diagrams, manipulatives, or symbols so that students can develop meaning of math concepts?
If the answer is yes, then students are engaged in doing mathematics. After selecting the task, you must consider the following:
- Solve the problem in as many different ways as possible. You must anticipate student responses and misconceptions. This will allow you time to prepare and set a goal for the task.
- While solving the task, jot down questions that will allow students to expand their thinking. Along the way you must provide guidance rather than the answer. Guide the students to find their own answers by questioning their thinking process.
- Acknowledge the process rather than the answers. Yes we want right answers but we want to value the process and encourage students to persevere through problem solving.
- If students find one strategy to solve the problem, ask for them to find a second strategy that will also allow them to prove their first initial answer.
- Mistakes are an important part of the learning process. I tell my students that we learn more from wrong answers than right answers. As teachers, we must cultivate a culture where students feel free to make mistakes and ask questions.
- Math, as Jo Boaler states, is not about speed.
- Student collaboration is the key to success.
- Provide meaningful feedback and allow for multiple opportunities to improve.
Response From Jennifer Chang Wathall
Jennifer Chang Wathall is an independent educational consultant specializing in concept-based curriculum and concept-based mathematics. She is author of "Concept-Based Mathematics: Teaching for Deep Understanding in Secondary Schools" and she travels the world facilitating professional development courses and delivering keynote addresses at conferences:
Often when I met people for the first time and tell them I am a math educator, the majority of the time I receive the reaction "Oh I hated math at school and I'm just not a math person!" This reaction greatly saddens me but is the product of years of being exposed to the drill and kill approach to math learning.
With the exponential growth in technology, knowledge has become a commodity. Our students are able to Google any answer or technique so education needs to focus on teaching students what to do with the knowledge and how to apply and transfer their knowledge to different situations. Math learning should be about developing conceptual understanding in order prepare our students for the 21st Century. Successful 21st Century math learning needs to foster 21st C skills such as communication, collaboration, critical thinking and problem solving.
Mathematics is a language of conceptual relationships. Traditional pedagogy has focused on rote memorization of facts and skills with little attention paid to the conceptual relationships involved. Facts and skills go hand in hand with conceptual understandings as illustrated by figure 1 below. "In order to develop intellect in our students and increase motivation for learning, curriculum and instruction must create a synergy between the lower (factual) and higher (conceptual) levels of thinking by utilizing the inquiry process continuum" (Wathall, 2014, p14).
Here are my tops tips for engaging your students in the math classroom with the goal of developing conceptual understanding and therefore enhancing learning:
Strategy One: Create a social learning environment that promotes team work and collaboration.
Strategy Two: Provide an open, secure environment to allow for mistakes as part of the learning process. Based on the work of Carol Dweck and Jo Boaler, this strategy includes fostering a growth mindset with your students and colleagues.
Strategy Three: Use the levels of inquiry (Table A) as described by the table below and employ inductive teaching approaches (Table B). Please see the side by side comparison of a deductive vs inductive teaching approach (Table C).
Strategy Four: Reduce whole class teacher talk time. Brain research shows secondary students can concentrate with direct instruction for around 15-20 minutes.
Strategy Five: Cater to all the needs of your students by differentiating for different backgrounds and interests. Carol Ann Tomlinson's work on differentiation discusses differentiating by: content, product, process and affect.
Strategy Six: Use Diagnostic, formative and summative assessment. I like to use visible thinking routines from Harvard University's Project Zero such as "I used to think... Now I think..."
Strategy Seven: Be purposeful when asking students to answer questions. I like to use the thumbs to chest strategy so students can discreetly show whether they have completed a question or would like to answer a question.
Strategy Eight: Flexible fronts and looking at the structure of your classroom. Traditional math classrooms have a "Sage on the Stage" layout which encourages a more didactic approach. Here is a diagram of how I structure my classroom to encourage more collaboration and discussion to promote the "Meddler in the Middle" type of teaching.
Response From Sunil Singh
Sunil Singh is the author of Pi of Life: The Hidden Happiness of Mathematics and a math consultant for Scolab, a digital education company in Montreal. He was also a regular contributor to The New York Times Numberplay blog:
One of the best practices for teaching high school mathematics is to give students problems that not only challenge them, but build patience and persistence in their maturing problem solving strategies. This means that students should be given ample time in and outside the classroom to work on these problems, so that the joyful struggle is independent of time--as has been the entire natural, historical development of mathematics.
One strategy I learned from my mentor, Peter Harrison, was to give students extremely difficult problems, but with the freedom to ask anyone--except himself--help in solving these questions. Not only could other teachers in the school be consulted, but even college professors outside the school. The endgame was to create rich mathematical dialogue that "left the building", to increase the radius of mathematical discourse. The notion of students given time and space to have their ideas marinate can extend to final assessment as well. On one final exam in a geometry and discrete course that we both taught, we picked 12 questions from the Challenging section of the textbook and told our classes that 5 of these questions will appear on the final exam in one month. We gave them class time to work on these problems as well. One of the questions was that a point P, inside a square, was 3, 4, and 5 units away from three of the corners. Find the length of the side of the square. The results of this question and the other four were spectacular, beyond the grades themselves. The solutions were dense and well-thought out--mathematical creations that could only have occurred if given time and encouragement to discuss and unpack ideas--in shared, equitable experiences.
Mathematical resilience is hopefully an idea that students have experienced even in elementary grades. In high school, it is imperative they explore problems that give a hint to the stamina and endurance that has been the lifeblood of mathematics for over 2000 years.
Response From Dr. Matthew L. Beyranevand
Dr. Matthew L. Beyranevand is author of Teach Math Like This, Not Like That: Four Critical Areas to Improve Student Learning (Rowman & Littlefield, 2017), K-12 Mathematics and Science Department Coordinator, an ambassador for the Global Math Project, supporter for the With Math I Can campaign, and a member of the Massachusetts STEM Advisory Council. He also serves as an adjunct professor of mathematics and education at the University of Massachusetts at Lowell and Fitchburg State University:
There is a critical issue in mathematics education that does not devote sufficient time or emphasis to making sure that students know how to solve problems using different methods. This is most prominent in high school mathematics classes where the most knowledgeable teachers of mathematics reside, but often force one particular method onto their students.
The phrase "Work Smarter, Not Harder," coined by Allen F. Morgensen in the 1930s, is very appropriate here. The first step in teaching problem solving is to look for opportunities for students to have multiple entry points or strategies for solving a problem. Spend the extra time to allow them to explore their options. Take time to discuss strategic choices. For instance, when solving an equation that has parentheses, you can start with a simple problem:
5(x+3) = 20
Students will immediately use the distributive property. They learned to do this problem using their procedural knowledge in the eighth grade by using the distributive property. However, in Algebra I, the focus should be on conceptual understanding, so ask the students to find a "shortcut." (Students love it when you give them an opportunity to "do it in their heads" instead of showing work.) If they have good conceptual understanding, they will know that this is a multiplication problem that can be solved by using the inverse, division, first.
5(x+3) = 20
x + 3 = 4
x = 1
Then, ask them to find a flaw in this method. Eventually they will reach the understanding that not all problems are candidates for this method, such as:
3(x - 4) = 10
Why? Because we have to work with fractions too soon into the problem. In this case, the distributive property is the best method.
3x - 12 = 10
3x = 22
x = 22/3
Take an extended amount of time to examine a single problem and identify many different approaches for solving it. Some of the approaches can be shown by the teacher; however, giving students time alone, in pairs, or in small groups to find their own unique process for solving the same problem is one of the best ways to promote conceptual understanding.
For instance, ask the students to find three consecutive integers whose sum is 96. Most students will use a guess and check method and get the correct numbers. Then you can take the same problem and make it a little more difficult using consecutive even or consecutive odd integers. When students have developed a conceptual understanding of what numbers they are looking for you can introduce how to write an equation that will give them the numbers.
Another option would be to first have the students understand and apply a single approach demonstrated by the teacher. Once they attain proficiency, the teacher could ask students to individually attempt to solve the same problem creatively using a different approach.
If you want a student to use a particular strategy or method, you should show them why. You can either show them an instance in which their method won't work or give them a sneak peek into an objective that will require them to use the one you are currently touting. When students understand the purpose they are more willing to comply with your rules.
If you cannot provide a mathematical reason as to why they shouldn't use a method different from yours, then you need to open your mind to new and sometimes exciting approaches. A gifted student might work a problem very differently from the approach planned out by the teacher, but on inspection the teacher might see that the student had come up with a better approach. Good high school teachers allow that experience to mold how they approach that problem, or that type of problem, in future lessons.
Responses From Readers
Hands-on conceptual lessons in which students are allowed to grapple with the content and experience productive struggle rather than just being told a procedure that is regurgitated for a test, and is meaningless to students outside of the mathematics classroom.-- 4Awbe (@4Awbe) December 21, 2017
Explain WHY something is what it is (or does what it does). I just went through this, when teaching truth-trees for propositional logic:
When students merely try to emulate the rules of decomposition, they make mistakes;when students "understand 'the logic'" governing the rules, they're much more successful.
It is NOT telling the students to go watch a video...in lieu of one explaining it again. Not all students do well with flipped instruction (I do realize that some do).
Thanks to Wendy, Jennifer, Sunil and Matthew, and to readers, for their contributions!
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