November 2012
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This document discusses the importance of developing students' mental math and estimation skills. It notes that the Saskatchewan curriculum emphasizes teaching both computational fluency and helping students understand the mathematical concepts underlying procedures. While practice is important for developing fluency, it should not come at the expense of exploring concepts. Finding the right balance is key. The document provides examples of formative assessment strategies like "commit and toss" and "fist to five" that teachers can use to check students' understanding without putting them on the spot. It also shares online resources for virtual manipulatives and interactive applets that can engage students.
November 2012
- 1. Mathematical Process of the Month: Mental Math Our Saskatchewan curriculum is based on the NCTM (National and Estimation ME Council of Teachers of Mathematics) framework. The teaching of The Saskatchewan Curriculum describes this as calculating mentally and mathematics is guided by Content Standards (what we teach) and reasoning about the approximate size of quantities without calculators or pencil Process Standards (how we teach). and paper. It is not only estimation skill, but also computational fluency that The process standards as described in develops efficiency and accuracy. NCTM further describes the need for students the Saskatchewan Curriculum are to develop procedural fluency. It is essential to success in mathematics. Communication (C), Connections (CN), Mental Math and Estimation The renewed SaskatchewanCurriculum is clear about the need to teach for (ME), Problem Solving (PS), deeper understanding. Students are to be given the opportunity to understand Reasoning (R), Visualization (V) and the mathematics that underlie procedures. We provide students opportunities Technology (T). The process to construct meaning for themselves, explore relationships through inquiry, and standards are not topics we teach, but things we teach through. We teach to represent and verbalize their understanding. Though we may fear that taking through technology, using whatever time to allow students to create meaning around math concepts comes at the tools we have to enhance instruction. expense of developing procedural fluency, this is not the case; rather, the two Similarly we teach through problem are intertwined. So as long as we are providing opportunities for students to solving; it’s not a unit, it is a process that calls into action the skills we are discover relationships and explore the meaning behind the math, we can helping our students develop. The confidently provide practice and expect students to develop mental recall for process standards appear in the facts and procedures. Rather than taking away from concept exploration and curriculum guide as bold letters at the deeper learning, procedural fluency enhances learning of new concepts because bottom of each outcome, as a reminder of processes we can use to procedures become routine and automatic, allowing the student to focus on address each outcome. We must mathematical relationships and developing new skills. Developing personal consider incorporating these strategies is encouraged, but sharing and reflecting is important to help students processes into our instruction as we select strategies that are efficient and accurate. plan. Each month will feature and examine The amount of practice required to develop procedural fluency seems to be a one mathematical process. This subject of much debate. This is a matter left to our professional discretion, month the focus is Mental Math and understanding that procedure without context is meaningless, and the amount Estimation of practice may not be the same for every student. Our job is to find the balance; -Florence Glanfield, (2007). not so much practice that it becomes meaningless and contributes to a negative Building Capacity in Teaching perception of mathematics, but certainly enough practice to allow students to and Learning. Reflections on process quickly so their thinking can be focussed on new learning and Research in Mathematics. understanding. Pearson Education Canada We can give students information, but we cannot give them understanding. Upcoming Events: Middle Year Sciematics: The Changing Face SUM conference: May 3-4, Math Workshop, Dr. Brass of Education. Saskatoon, May Saskatoon. Featuring Dan School. Date TBA 9-11, 2012, College of Meyer and Marian Small. Agriculture and Biosciences, U http://www.smts.ca/sum- PreCalculus 30 Collaboration of S. conference Workshop, YRHS, Nov 26 5:00- 7:00 pm. http://www.sciematics.com/
- 2. Formative Assessment Feature Formative Assessment Commit and Toss: This is a technique for eliciting anonymous student responses. It is a fun Sometimes called “Assessment and safe way for students to express their ideas. Students are given a “probe” question, for Learning” the primary preferably one that generates some debate; an example would be “Do you agree with the purpose is to promote student statement ‘two negatives make a positive’? Why or why not?” or, a forced choice question learning (Hodgen & William, where students have to commit to one answer and justify their reasoning (such as selecting 2006). It does this by helping the correct changed volume of a cylinder if the radius is halved, and then explaining why students monitor their own they chose the answer). Students toss the papers around the room till the teacher says learning in order to develop stop (or into a centre pile or box, or changed to ‘commit, fold, and pass’, whatever suits the self-reflective learners, as well climate in the classroom). Students then share the answer and explanation of the paper as to inform instruction. they end up with, and they present only that idea and not their own idea. Ideas and Instructional decisions such as solutions can be discussed. pacing, grouping, and Commit and toss allows students to see that there are different ideas in the room, not reinforcing are guided by how our students are responding to everyone has the same answer. Because the answers are shared anonymously students instruction. We could also may feel less threatened sharing their thinking. consider this “Assessment as Tips : Remind students to honour the anonymity. Do not overuse this activity or Learning”, since students often it loses its appeal. Establish a norm that no disparaging comments should ever be made grasp concepts through the about someone else’s answer or thinking. process. Formative assessment data is Fist to Five: This quick show of hands allows students to indicate their level of not used for grading, understanding of a concept or procedure. The teachers asks students to show their hand, accountability, or ranking. closed fist meaning “no idea!” , one finger “I barely understand”, two “I need help” thee “I However, data should still be understand most of it but can’t explain it” , four “I understand and can explain” five “I kept because it is useful in understand completely and can explain it well to someone else”. Students can raise their decision making and can be hands high, but if they don’t want to disclose their level of understanding you can ask them useful in discussions with to simply show their hands low, in front of themselves so that not everyone can see. You colleagues, administrators, and can also use “thumbs up, thumbs down, thumbs sideways” as a quick gauge of how well parents, as well as with students caught on. students themselves. -Keeley & Tobey, (2011), Mathematics Formative Assessment, Thousand Oaks CA: Corwin This is not a new initiative. Press and NCTM We’ve always checked for understanding and gauged student progress in a multitude If you haven’t checked out Michelle Morley’s collection of virtual manipulatives, the URL of ways. This forum will allow is http://gssdknowproblems.wikispaces.com/home us to exchange ideas for Use the menu on the left to select the strand, then the web applets are sorted by grade. formative assessment activities This list is extremely well organized to fit our curriculum outcomes, and the applets that are useful and engaging. make great demonstrations on SMARTboard, or can be student interactive. This site has tutorials on trigonometry topics, this particular page has a nice demo of trig function graphs. Click the “start here” button to view the creation of the graph http://www.analyzemath.com/unitcircle/unitcircle.html See the homepage at http://www.analyzemath.com/Trigonometry.html for more trig applets National Library of Virtual manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html For neat classroom ideas check out Pinterest. This URL takes you to the education Prototype Departmentals for WAM category and math related ideas http://pinterest.com/search/?q=math 30, Foundations 30 and PreCalc 30 are on line at Having students estimate the answer to a problem worked http://www.education.gov.sk.ca/pr out as a class group or teaching example increased ototypes engagement and gives learners a stake in the answer M. Burns, 2008
- 3. Mental Math and Estimation Teachers Need to: Students that struggle have Allows Students to: Provide daily practice of limitations in working memory. math skills and estimation Practice can help offset this by Quickly and efficiently recall skills. A few minutes of developing automaticity, which basic facts practice every day can make a difference! reduces the amount of Develop confidence in their Introduce strategies and information to keep in mind, math ability provide practice freeing up attention for new Judge if an answer is reasonable Help students understand learning. Become proficient problem the math behind strategies Computational fluency is solvers Model a variety of strategies necessary to prepare students -Nova Scotia Dept of Apply math in everyday Education for advanced mathematics. situations -Riccomini,P. 2012 “First, enactive mastery, defined as repeated performance accomplishments (Bandura, 1982), has been shown to enhance self-efficacy more than the other kinds of cues (Bandura, 1977a, 1982; Bandura, Adams, & Beyer, 1977). Mastery is facilitated when gradual accomplishments build the skills, coping abilities, and exposure needed for task performance.” Gist, M. E. (1987). Self-efficacy: Implications for organizational behavior and human resource management. Academy of management review, 12(3), 472-485. Mental math practice develops mathematical literacy and proficiency, and prepares students for participation in a technological society. Students can practice and build skills for short periods of time, but avoid timing practice, like “mad minutes”, because they contribute to math anxiety, lack of confidence, and diminished motivation. Practice does not have to mean worksheets! Students can practice in teams, peer teach, dialogue, do activities or games. Check out http://www.pedagonet.com/quickies/acingmaths.pdf for a collection of skill building card games. From Classroom Instruction That Works: Research-Based Strategies for Increasing Student Achievement By Robert J. Marzano, Debra Pickering, Jane E. Pollock ASCD 2001 Math Coach Please visit my blog at www.blogs.gssd.ca/csmith/ This site has useful resources, but it is a work in progress. Please email me if you have ideas or requests for this newsletter. Math Webinars. Nov. 5 ~ Pinterest & Math Resources – Michelle, Nov. 22 ~ Google Forms & Flubaroo – John, Dec. 10 ~ Kidspiration – Gary, Jan. 15 ~ SMART Math Tools – Gary, Jan. 23 ~ Screen Casting – Michelle, March 6 ~ Photo Story – John, April 17 ~ Building a Personal Learning Community - Michelle. These webinars are free. See Michelle Morley’s blog for log in info