On Friday the 13th of March, all K-2 teachers worked with Lisa Demick and me to review and assess how things are progressing this year, to unpack upcoming units, and to look ahead into next year. 'Explaining your thinking' has been a major theme with the new EM4 program, and it was a theme on our day of professional development too.
There were some challenges at the start of the year. Students struggled with the self-explaining problems, often not knowing what to say and (especially) what to write. Teachers were faced with the difficulty of scoring the self-explaining problems on assessments, and finding ways to support students who were having difficulties. And of course parents were asking questions about the self-explaining problems and wondering why their children were suddenly "developing" when last year they were "achieving" or "exceeding." These challenges were discussed during Friday's professional development, and most if not all teachers shared these experiences between early in the year and now.
The good news is teachers also reported seeing some real improvement in students' ability to explain their thinking, both orally and in writing. It was also noted that students are gaining confidence in their mathematical thinking! This was refreshing to hear, as it might be the most important intended outcome of this instructional emphasis being placed on self-explaining problems. The more opportunities we provide to share their thinking, to record their strategies, to collaborate and communicate with each other about their problem solving, the more confident our students become with their mathematical thinking.
The theme of "explaining your thinking" and communicating mathematical thinking is emphasized in the Common Core Standards for Mathematical Practice. On Friday morning teachers reviewed the curriculum materials and recorded some of the specific expectations for each grade level at different points of the year. Here are some of the results.
The above image displays some of the ways Kindergarten students are given opportunities to share their mathematical thinking at about halfway through the year. One example is "students are asked to draw, describe and compare shapes and vertices." Another is "Students are asked to describe shape names and use positional words."
Here (above) are some examples of ways students are communicating their math thinking in early 1st grade. "Partners discuss their responses and self-evaluate," and students are asked, "Why is it important to name your unit of measurement?"
Halfway through 2nd grade, students are "encouraged to use each other's strategies," and explain how more than one strategy can work. See above.
By the time students are in the 3rd grade, students continue to make sense of others' strategies and practice communicating their mathematical thinking with stems such as, "I noticed..., I wonder..., How did you..., and, Why did you..." Students are expected to communicate their thinking in writing more fluidly and are given many opportunities to practice.
Self-explaining problems are the meat of the new Everyday Math program, and are hugely important in building understanding and confidence in mathematics. That is why it is emphasized so much in the Standards for Mathematical Thinking (SMP's).
Everyday Math 4 has done us a big favor by taking the Common Core SMP's and breaking each SMP into several Goals for Mathematical Practice (GMP's). Below is a list of all 8 of the Standards for Mathematical Practice and their accompanying Everyday Math Goals for Mathematical Practice. These practice standards are key to the instructional shifts we speak of:
SMP1: Make sense of problems and persevere in solving them
GMP1.1 Make sense of your problem
GMP1.2 Reflect on your thinking as you solve your problem
GMP1.3 Keep trying when your problem is hard
GMP1.4 Check whether your answer makes sense
GMP1.5 Solve problems in more than one way
GMP1.6 Compare the strategies you and others use
SMP2: Reason abstractly and quantitatively
GMP2.1 Create mathematical representations using numbers, words, pictures, symbols, gestures, tables, graphs, and concrete objects
GMP2.2 Make sense of the representations you and others use
GMP2.3 Make connections between representations
SMP3: Construct viable arguments and critique the reasoning of others
GMP3.1 Make mathematical conjectures and arguments
GMP3.2 Make sense of others’ mathematical thinking
SMP4: Model with mathematics
GMP4.1 Model real-world situations using graphs, drawings, tables, symbols, numbers, diagrams, and other representations
GMP4.2 Use mathematical models to solve problems and answer questions
SMP5: Use appropriate tools strategically
GMP5.1 Choose appropriate tools
GMP5.2 Use tools effectively and make sense of your results
SMP6: Attend to precision
GMP6.1 Explain your mathematical thinking clearly and precisely
GMP6.2 Use an appropriate level of precision for your problem
GMP6.3 Use clear labels, units, and mathematical language
GMP6.4 Think about accuracy and efficiency when you count, measure, and calculate
SMP7: Look for and make use of structure
GMP7.1 Look for mathematical structures such as categories, patterns and properties
GMP7.2 Use structures to solve problems and answer questions
SMP8: Look for and express regularity in repeated reasoning
GMP8.1 Create and justify rules, shortcuts, and generalizations