I've been teaching math for many years, and what is amazing about that is that I'm continually learning new and better ways to impart mathematical understanding and skill.
As I think about math education today, I believe that it is in our best interest at the intermediate grades to think of math as two threads. The first thread is computation and the second thread is concept. Obviously there will be regular overlap of the threads, but by creating two threads we build extra time into the schedule for math education, and greater time for attention to both areas.
During the computation thread students solidify a strong fact foundation, and work to become flexible and facile with the four operations: addition, subtraction, multiplication, and division. The second thread intersects with the first, yet focuses more heavily on concept development and problem solving. The two threads are taught simultaneously.
Online programs such as Xtra Math and That Quiz help students to solidify fact accuracy and speed. Lots of paper/pencil practice and review and response develops computation facility. Programs such as sumdog also help to strengthen students' fluency in this area. Checking subtraction with addition, and division with multiplication serve to strengthen students' computation too. The reasonableness and meaning of computation problem solutions are regularly discussed and noted.
Concept development is best when the Standards of Mathematical Practice (SMP) are embedded into the approach.
Using a similar roll-out for each standard is one way to embed the SMPs into students' concept and problem solving lessons with effect. The roll-out of each standard could include a process like this:
Note that the SMP's are highlighted in red.
1. Plan: Research and plan for each standard keeping in mind your students' needs, interests, and skills. To support that effort, I created a grade-level standards guide.
2. Explore: Give students time to play with and explore the tools related to each standard. At first I recommend that you don't guide the exploration with too-tight restrictions, instead make the time to observe students' interactions with the materials. Of course, different groups will demand different processes.
3. Vocabulary: After the tool exploration, make time to discuss student discovery, questions, and connections. Chart the results and highlight standards' vocabulary. Make a bulletin board online and/or offline of related vocabulary. Practice vocabulary throughout the standard's unit in multiple ways.
4. Model: Model the standard with vocabulary, mathematics, self-talk, and the strategic use of tools. Also model the use of structure such as charts/tables, graphs, number lines, patterns and more. Give students time to replicate the models while solving simple problems related to the standard. Encourage students to look for, create, and make use of structure as they practice. Practice sessions should always follow the 1-2-3 of differentiation: 1-Review, 2-Grade Level, 3-Enrichment--students are assigned the combination of practice that best fits their need.
5. Open Response Problem Solving: Once students have gained familiarity with the standard, provide opportunities for multi-step, real-world, open response (and yes, test-like) problem solving. Give students a chance to make sense of the problems and persevere. Let students know that "cognitive discomfort" is a path to learning, and allow them to struggle at a just right level. Also tell students to take the time to study the problem numbers, words, images, and diagrams and their relationship. Prompt them to bring to mind what they understand, and to make a solution plan. Encourage students to construct viable arguments in writing and speaking, share their thinking, and critique the reasoning of others. Teach children strategic problem solving strategies, and develop students' ability to utilize and share those strategies. Establish protocols and space for math talk and share. The key to this stage, as noted by my colleague, Mike O'Connor, is to use the best problems for each standards--problems that elicit dialogue, debate, multiple solutions and understanding.
6. Reflection: Gather to discuss the standard and look for and express regularity in repeated reasoning. What rules can we apply to this standard? When and how do we see this standard in our everyday life? Are there efficient ways that we can think about this standard?
7. Application: Design and implement collaborative, relevant, and meaningful project/problem base learning related to the standard that requires students to reason abstractly and quantitatively. Confer with students and foster attention to precision through teacher-students edits. Make time for student share in person or online with an audience such as classmates, schoolmates, family members, or an online community.
8. Final Assessment: Assess students' learning with a responsive tool or structure. Provide feedback to each student about his/her effort and learning, and also use the data to inform the roll-out of the next standard.
A strong blended math program embeds multiple tools, strategies, and emphases. The parallel paths of computation and standards based concept/problem solving can serve to create a strong path to mathematical understanding at the elementary school level. When presented with care, students embrace math learning and study joyfully. That positivity, in turn, creates enthusiastic and successful math students.
As you consider these thoughts and plan, please know that I'm open to your ideas and questions. I look forward to trying out these strategies in the math days ahead.
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