Two additional videos that capture the nature of the Colorado Academic Standards and reform efforts in mathematics education, supporting conceptual understanding, procedural skill & fluency, and applications with equal intensity. It’s not an “either/or” situation, it’s “Yes, and…!”
NCTM’s Principles to Actions describes effective teaching practices, conditions, and structures to ensure mathematics learning environments that benefit all learners. A simple assessment of whether or not a mathematics classroom is learner-centered is to ask the following questions: Who is carrying the cognitive load in your classroom during mathematics lessons? Who is doing the mathematics? Who is doing the thinking and talking?
These curated resources, produced by reputable experts in the field, may help you create a dynamic learning environment that promotes mathematical understanding, thinking, and student voice.
I. Creating Cultures of Thinking: The 8 Forces We Must Master to Truly Transform Our Schools (Ron Ritchhart)
II. Mathematical Tasks
- Mathematical Tasks as a Framework for Reflection: From Research to Practice (Smith & Stein)
- Selecting and Creating Mathematical Tasks: From Research to Practice (Smith & Stein)
- Thinking Through a Lesson: Successfully Implementing High-Level Tasks (NCTM paid article)
- The October 2016 issue of Educational Leadership offers other ideas on general lesson planning with some direct applications to mathematics classrooms! (Some articles are paid access)
III. Promoting Student Discourse
- How to Get Students Talking! Generating Math Talk That Supports Learning
- 5 Practices for Orchestrating Productive Math Discussions (summary)
- Never Say Anything a Kid Can Say! (NCTM paid article)
- Connecting Mathematical Ideas (book with classroom videos)
IV. Using Questioning to Elicit Student Thinking & Promote Discourse
- PBS TeacherLine Tips for Developing Mathematical Thinking
- The Art of Questioning in Mathematics (Robert Kaplinsky)
Some thoughts on questioning in the classroom:
- If you want to know what a student is thinking, ask a question you do not know the answer to. This avoids the “guess what’s in my head” type of questioning which can typically be lower-level recall.
- Ask a questions that require students to respond with a second sentence (or more). How many of your questions require students to answer with a second sentence (or even a single sentence)?
- What questioning sentence starters could you post in your classroom to encourage student-to-student questioning and explanation of thinking? How can these questioning sentence starters become a classroom norm among teacher-student, student-teacher, and (ultimately) student-student interactions?
V. Classroom Norms
So often, it is teachers that design the learning environment and set up the classroom rules and expectations. But do students have a say in designing their learning environment that meets their needs? Some questions to ponder:
- Do students have choice and voice in your classroom?
- What is “tight” and what is “loose” in the classroom? Who decides?
- How often do students have the opportunity to provide you with feedback on how the class is running and what should be improved?
Some teachers have even adapted the Seven Norms of Collaboration from Adaptive Schools (or had students adapt these norms with their language) to build common collaborative school norms for students and adults alike.
I recently came across an article on Twitter, “Nine simple math tricks you’ll wish you had always known.” My immediate response to was suggest renaming the article to “Nine simple tricks you’ll wish you had always known,” since these tricks are isolated in their utility and provide little-to-no understanding of mathematics and how operations work. To me, the most troubling part of this article was the opening line, “It’s not entirely your fault if you’re terrible at math — maybe you just didn’t know the tricks to make any math problem a piece of cake!” While these types of “tricks” can be introduced to students with positive intentions of helping them attain procedural fluency, do they actually help? Do they promote a growth mindset in learning and understanding mathematics? Do they build conceptual understanding to support procedural fluency and transparency behind algorithms? And finally, are these tricks flexible – can they be applied in a variety of situations, or are they strictly dependent on the digits that appear?
Even though most of these tricks are elementary-based, they exist within secondary mathematics, too. Ever heard of “copy dot flip” for division of fractions or “cross multiply and set equal” to solve a proportion? Even the “F.O.I.L. method” for multiplying binomials attempts to illustrate the distributive property, but this trick fails when multiplying a binomial and a trinomial together. (See Phil Daro – Against “Answer-Getting” video and a recent blog post by Dan Meyer for additional examples.)
With all of this said, is there value in teaching these kinds of math tricks to students? is there a way they can become “mathematics” and not just “tricks?”
Following Unit Plans and adopted instructional resources are necessary to help our students master standards, but we must also be mindful of the updated standardized assessments students will take each spring. Whether we’re referring to PARCC, PSAT, or SAT expectations, we must ask ourselves, “Do our assessments and instructional tasks hit the right target of expectations?”
If we’re loyal to The Teaching & Learning Cycle, we start with looking at the standards (i.e. Unit Plans) to determine what our students are expected to know, understand, and be able to do. But it’s the next stage that typically get short-changed: Creating the post-assessment that will determine what/if students have learned. Instead of focusing on developing the end-of-unit assessment and matching instructional lessons & tasks to the rigor of that assessment, we typically fall into the pattern and habit of marching through the adopted instructional resources (with little-to-no modification at times) and using the publisher-created assessments that mirror the format and rigor of the lesson examples and homework problems.
To see if these publisher-created assessments are hitting the right target of expectation, let’s compare assessment items and tasks to the PARCC/PSAT/SAT released items. If they are falling short (which they most likely will), how can we use these released and practice items (and other test specification resources provided to us) to raise the expectations for classroom assessments? Paraphrasing a colleague of mine, if PARCC/PSAT/SAT is the hardest assessment students take all year, then our classroom assessments do not match the appropriate expectations. Once the end-of-unit assessment is intentionally designed, only then can we ensure our lesson design and choice of instructional tasks will help students prepare for the assessment. It may require adaptation and modification of the adopted instructional resources and tweaking tasks slightly. Ideally, our lessons and tasks would regularly include and incorporate the Standards for Mathematical Practice & Colorado 21st Century Skills and Readiness Competencies in Mathematics where students are doing the thinking, talking, and the mathematics, not the teacher. (Mark Chubb has some nice blog posts around this – check out A Few Simple Beliefs, Focus on Relational Understanding, and Aiming for Mastery? Dan Meyer’s April 2016 keynote “Beyond Relevance & Real World: Stronger Strategies for Student Engagement” is a good reference, too.)
For St. Vrain teachers, we’ve curated some other resources to assist with designing classroom assessments that meet the expectations of student mastery of the Colorado Academic Standards. Check out Illustrative Mathematics and our assessment tasks linked to applicable Unit Plans for ideas and prompts. The i-Ready Standards Mastery component can also be a great tool for PARCC-like items in a digital format.
Summary points to remember:
Have fidelity to the standards. These define and describe what our students must know, understand, and be able to do. There is room for prioritization here, identifying the conceptual “big ideas” and focusing instructional time accordingly.
Assessment communicates expectations. Are your classroom assessments communicating the right expectations?
Adopted instructional resources are tools. One of the Standards for Mathematical Practice is “Use appropriate tools strategically.” You are the highly-qualified teacher that builds relationships with your students, builds a classroom culture of mathematical discourse, and designs dynamic lessons to actively engage students in the content. Don’t outsource yourself to the script of a textbook or go on “autopilot” by blindly covering lessons without knowing the assessment expectations! Check out Elementary Mathematics Achievement by Design or Secondary Mathematics Achievement by Design for more ideas around structuring your math class/block.
Here are two examples of how conceptual understanding that focuses on number sense helps make meaning of traditional algorithms with computation, based on videos that have been recently shared with me. An additional bonus is these are visual models that appeal to a wide range of learners. It is an end goal for students to use standard algorithms for computation that are the most efficient, flexible, and accurate. The Colorado Academic Standards for Mathematics (i.e. Common Core State Standards) nicely scaffold this learning progression across the grades. The change from past practices (of which many of us experienced as learners in school) is the use of conceptual strategies and methods first to build an intuitive foundation before using standard algorithms that can be abstract in nature. We do not expect students to use these conceptual methods beyond elementary school, yet they can easily be referred back to and used if the steps associated with the standard algorithms are forgotten or if the standard algorithm is not the most efficient for the given quantities.
Multi-Digit Subtraction Using a Number Line (2nd grade standard)
Using the number line defines subtraction as a finding the difference between two quantities. The abstraction of the standard algorithm, especially when ungrouping is required, can be challenging for students. Of course, as the place values of the quantities being subtracted increase, the number line may lead to more room for error and inefficiency.
Multi-Digit Multiplication Using an Area (or Array) Model (4th grade standard)
The area (or array) model of multiplication is also applicable to the conceptual development of multiplying fractions and polynomials. This is an example of coherence across the curriculum, providing a common representation for all multiplication. While the standard algorithm for multi-digit multiplication, the algorithm of “multiply straight across the numerators and denominators” for fraction multiplication, and using F.O.I.L. to multiply binomials are the most efficient, flexible, and accurate, they themselves do not promote any mathematical understanding. Only representations such as the area (or array) model provide understanding and transparency to these standard algorithms.
I received an interesting e-mail the morning of August 26, 2016:
“I would like to personally congratulate you as your blog St. Vrain K-12 Mathematics has been selected by our panelist as one of the Top 100 Math Blogs on the web … I personally give you a high-five and want to thank you for your contribution to this world. This is the most comprehensive list of Top 100 Math Blogs on the internet and I’m honored to have you as part of this! Also, you have the honor of displaying the following badge on your blog.”
This site serves as a Frequently Asked Questions (FAQ) resource for the St. Vrain community. Even though it is more of a website vs. a true blog, it’s good to know this space is providing relevant resources and information for mathematics education. Please contact me on how this space can be continuously improved.
Folks will oftentimes use the line, “But the district says we must do it this way.” The question really being asked is, “What is ‘tight’ and what is ‘loose?’ Where do I have flexibility and professional discretion?” With respect to mathematics Tier 1 instruction and lesson planning, here is a summary:
Question: What is required to create, support, and sustain a culture of access and equity in the teaching and learning of mathematics?
The National Council of Teachers of Mathematics (NCTM) has resources available to foster conversations around access and equity in mathematics for all students.
- NCTM Position: Access and Equity in Mathematics Education
- Equitable Pedagogy (from Principles to Actions toolkit)
- Using Identity and Agency to Frame Access and Equity (from Principles to Actions toolkit)
Here are some resources, gathered from various sources over the years, to help establish a classroom culture that promotes the Standards for Mathematical Practice, growth mindset, and students’ understanding of mathematics. How can we begin to shift the cultures of our mathematics classrooms to ensure students are doing the mathematical thinking, talking, and “heavy lifting,” and not solely their teachers?
Looking for other ideas? Check out these math classroom norms from youcubed.org (Jo Boaler).
From Education Week blog: “A new paper from the Institute for Education Sciences lists 28 ways federally funded research has changed what we know about how to teach whole numbers, fractions, algebra, and other math topics.
IES, part of the U.S. Department of Education, funded more than 200 studies about math instruction between 2002 and 2013. A synthesis of that research, published this month, lays out some of the contributions these studies have made to the field.”
View the research paper to see the IES-funded research contributions (28) on Whole Numbers, Operations, and Word Problem Solving in Elementary School, Fractions and Algebra in Middle School, and Development and Evaluation of Teacher Professional Development Approaches.