EDUC 4334 – J/I Mathematics Weekly Commentary Week 1: Short, non-routine activities: o Geometric application—name plates (triangular prisms) Extensions: What other quick applications can you think of? Building 3-D shapes, and applying them to ‗realistic‘ situations o Number application—Guess my (whole) number activity with post-it notes Simple but key rules of the game Active, social, mobile A problem solving situation (they don‘t have to be pencil & paper problems!) Logic, (deductive) reasoning Application of number concepts, properties (e.g., order, divisibility, factors, odd/even, etc). o Emphasize communication aspects Extensions: Once the children are up and have placed themselves in order, what activities can you now have them engage in? o Cross-curricular application—Reading Math Curse This particular book—very popular with children, teachers Variety of situations in which you might read the book Variety of uses you might make of book after reading In math class, to build links to other subjects (e.g. language arts); in language arts class, build links to mathematics. Extensions: Explore other children‘s literature with a math ‗angle‘ Some focus heavily on math; others have a math ‗component‘ Lots of opportunities for ‗communication‘—discussing, reading, writing, etc. o Hands-on History with Rope (String!)—‗Egyptian‘ 12 interval rope (13 knots) Build triangles: What types of triangles can you build; what types can‘t you build? The 3-4-5 triangle—what‘s special about this triangle? Measurement tool (length) also Hands-on, kinesthetic. Volunteers; or invitation! Extension: Read up on, and provide a brief history of early measuring tools to the students while doing such activities. o Visual tools to help provide alternative images of concepts being learned [Use of yarn to represent the five math strands—visual metaphor in this case.] Engage children with a variety of approaches, representations to support their learning. The five math strands of the Ontario grades 1-8 curriculum o Become familiar with each of these strands. You DO NOT need to memorize the expectations! But do have a general sense about what each strand involves Become familiar with the content (as you prepare to teach it). Use the Van de Walle & Folk text liberally for both content and pedagogical purposes! o Become familiar with the seven mathematical processes that you are expected to have students engage in as they ‗do mathematics.‘ (And which you have to evaluate.) Week 2 Main Focus: Problem Solving Teaching through Problem Solving and Teaching for Problem Solving o See the ―Two Approaches to Teaching‖ PowerPoint on Resources. A geometric context: o Polygon activity—properties, relationships, organization, display (TTPS) Note (again) that mathematical problem solving does not have to involve pencil and paper word problems! Encourage investigation, inquiry, problems (situations) that may be solved/approached in a variety of ways. o The teacher-directed lesson—A review lesson on angles (TFPS) The Polya Model—Four stages o A major model for teaching PS (see the Ontario curriculum document, see the hand out) o As the teacher, you need to expose children to a variety of problem solving strategies over time. It will take children time (years) to become fluent with strategies to solve mathematics problems Read Van de Walle, chapter 4. Support for TTPS. o Note the three part lesson—In Ontario, the 3-part lesson has been a significant focus, certainly in math at any event. o A comment on Van de Walle generally: You may not have time to read or even skim all the chapters, but my suggestion is at least to look over the ―Big Ideas‖ found at the beginning of chapters, and try to understand what those statements mean. Geometer’s Sketchpad o This is Ministry-licensed software, and should be in all schools. o The Math 1-8 curriculum explicitly refers to ―dynamic geometry software‖; in Ontario that has meant Geometer’s Sketchpad. o Continue to explore its features on your own. Week 3 Assignment 1—strand games and activities Ontario Ministry of Education mathematics resources o Leading Math Success materials, TIPS4RM, etc Measurement and Geometry activity (an example) o Visual, hands-on, concrete representations Area—grid paper Perimeter—coloured yarn Display a critical element to make the point and create discussion o Problem Investigation—relationship between area and perimeter! Start with scenario Encourage estimation, prediction to start. Then test. TTPS or TFPS? Week 4 (after Oct practicum) The Fall theme activity—Apples o Important that you think of this as an example of what you might do around a topic or theme, rather than as an isolated set of measurement activities to do with an apple. There are many possibilities—just be creative, while working within your contextual constraints (but maybe even pushing against those!). Give the students a voice. o See also the discussion of the activity in my Resources, right next to the link to the activity handout itself. In this case, we focused mainly on measurement of various kinds (remember, measurement is fundamentally an act of comparison) o Inter-strand, and interdisciplinary activities, are important for students to experience. Beginning Number Sense and Numeration o First, what do we mean by number? What are they for? Why bother? Quantity, How much, How many, Counting, etc. o Base 10 number system (more technically, the decimal system) The most common of many positional number systems, based heavily on the position (or ―place‖) of the digits, 0-9. Thus Place Value is a HUGE concept for students (and you) to master (i.e., understand and apply) –over time! Base ten blocks: Concrete materials (manipulatives) with which to explore number, and operations with number. Proportional model—why is that? Another proportional model? A non- proportional model? o By the way—what do we mean by ―number sense‖? If you had a child in (say) your grade 6 class (and hopefully you would have several) that you felt was developing ―number sense,‖ what are some things that child might be doing or saying to lead you to that conclusion? Week 5: Continuing Number Sense and Numeration This week we focused mainly on operations (adding, subtracting, multiplying and dividing) of whole numbers. o We made extensive use of base ten materials as a manipulative with which to explore these operations—mainly adding and subtracting (with place value mats) and multiplying—several different models possible—arrays, repeated addition, area model (e.g., using base ten materials), for example. o Division: various ways of conceiving of division—as an operation I find the repeated subtraction model particularly appealing, but there are a various potential algorithms. o Remember that both Van de Walle AND the Ontario Math curriculum guideline call for having children engage in their own strategies or algorithms while investigating these operations. Remember what an algorithm is? Van de Walle refers to these as ―invented strategies‖ The Math curriculum document calls them ―student-generated algorithms‖ See also my PowerPoint that went with the Whole Number seminar sheet we followed. You do not have to discard those so-called ―tried and true‖ algorithms that you have long used to perform the operations in your teaching, but: You should understand why they work (as well as how they work), and You should be prepared for children to present you with other approaches o A reminder from earlier: Place Value is fundamental to understanding any operations algorithm. Understand the place value conceptual basis for the algorithms and you will go far in terms of your own operations understanding, and in understanding the new ideas and the difficulties the students present you with. Week 6—Numeration and Number Sense continued Fractions o Throughout the junior grades students spend considerable time learning about fractions, but almost no time on operations with fractions (according to the Ont. Curriculum). It‘s not until grades 7 & 8 in Ontario that the curriculum really calls for a focus on operations with fractions. Why do you think that might be? o Another hands-on model: fraction circles. A potentially compelling visual and concrete model that can be used for describing fractions, comparing fractions, adding & subtracting fractions, etc. Less effective for multiplying and dividing. We used the Fractions Seminar handout (Resources page) o There are many other concrete materials that you (and students!) can use to model fraction scenarios. Just a few are: Geoboards, Pattern Blocks, Cuisenaire rods, Fraction Bars, etc. o Division of fractions: Can you give a plausible description of what 8 ’ 2 might mean in a ‗real‘ example? I‘ll bet you can. Can you give a plausible description, then, of what 8 ’ ½ might mean in a similar ‗real‘ example? Out of which situation would you expect to get the larger quotient? Remember what ―quotient‖ means? Integers o Well, in some sections we did a little bit of integer introduction. o We used coloured chips (blue and red, old-fashioned BINGO chips) as cold and hot, -1 (negative one) and +1 (positive one) respectively to: Represent negative and positive integers Perform operations with integers—especially addition and subtraction Use them to derive the various integer ‗situations‘: like signs, unlike signs, etc It helps, also, to think conceptually about what these operations mean when applied to integers. And, connecting to real life contexts in which positive and negative numbers are found also helps. o Another model that we did not discuss but which is common, and can be quite powerful, is the number line—moving left and right (negatively and positively) along the line. o I regret that we did not discuss integers in all sections, and even when we did, it was brief.
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