{"id":646,"date":"2021-03-30T16:33:12","date_gmt":"2021-03-30T16:33:12","guid":{"rendered":"https:\/\/www.terc.edu\/empower\/?page_id=646"},"modified":"2021-04-06T15:47:24","modified_gmt":"2021-04-06T15:47:24","slug":"how-empower-addresses-the-mathematical-practices","status":"publish","type":"page","link":"https:\/\/www.terc.edu\/empower\/how-is-empower-different\/how-empower-addresses-the-mathematical-practices\/","title":{"rendered":"Addressing the Mathematical Practices"},"content":{"rendered":"\n<p>Pam Meader is a long-time adult educator, <em>EMPower<\/em> user and contributor, and Adult Numeracy Network board member. She selected the following <em>EMPower<\/em> materials as exemplars of activities that address the eight CCR Standards for Mathematical Practice.<\/p>\n\n\n<div class=\"accordion\" id=\"accordion-100\"><div class=\"card\"><div class=\"card-header\" id=\"headingfaq-451\">\n\t\t\t\t        <h5 class=\"mb-0\">\n\t\t\t\t          <button class=\"btn btn-link\" type=\"button\" data-toggle=\"collapse\" data-target=\"#collapsefaq-451\" aria-expanded=\"false\" aria-controls=\"collapsefaq-451\">\n\t\t\t\t            Mathematical Practice 1: Make sense of problems and persevere in solving them\n\t\t\t\t          <\/button>\n\t\t\t\t        <\/h5>\n\t\t\t\t      <\/div>\n\t\t\t\t      <div id=\"collapsefaq-451\" class=\"collapse\" aria-labelledby=\"headingfaq-451\" data-parent=\"#accordion-100\">\n\t\t\t\t        <div class=\"card-body\"><p><div class=\"ezcol ezcol-two-fifth\"><strong>Sample lesson recommendation:<\/strong><\/div> <div class=\"ezcol ezcol-three-fifth ezcol-last\">\n<p><em>Many Points Make a Point, Closing the Unit,<br \/>\nActivity 1: \u201cStock Picks\u201d<\/div><div class=\"ezcol-divider\"><\/div><\/em><\/p>\n<p>Problem solving is everywhere and everyday, but a good problem to grapple with really gets at the core of perseverance. I like \u201cStock Picks\u201d, as it involves several choices by the students.<\/p>\n<\/div>\n\t\t\t\t      <\/div>\n\t\t\t\t    <\/div><div class=\"card\"><div class=\"card-header\" id=\"headingfaq-854\">\n\t\t\t\t        <h5 class=\"mb-0\">\n\t\t\t\t          <button class=\"btn btn-link\" type=\"button\" data-toggle=\"collapse\" data-target=\"#collapsefaq-854\" aria-expanded=\"false\" aria-controls=\"collapsefaq-854\">\n\t\t\t\t            Mathematical Practice 2: Reason abstractly and quantitatively\n\t\t\t\t          <\/button>\n\t\t\t\t        <\/h5>\n\t\t\t\t      <\/div>\n\t\t\t\t      <div id=\"collapsefaq-854\" class=\"collapse\" aria-labelledby=\"headingfaq-854\" data-parent=\"#accordion-100\">\n\t\t\t\t        <div class=\"card-body\"><p style=\"text-align: left;\"><div class=\"ezcol ezcol-two-fifth\"><strong>Sample lesson recommendation:<\/strong><\/div> <div class=\"ezcol ezcol-three-fifth ezcol-last\"><em>Split It Up, Lesson 9, Activity 1:<br \/>\n\u201cWhat Is the Story?\u201d<\/em><\/div><div class=\"ezcol-divider\"><\/div><\/p>\n<p>Reasoning abstractly and quantitatively has much to do with contextualizing and decontextualizing. An elementary math teacher explained it well: <strong><em>\u201c<\/em><\/strong><em>If students have a problem, they should be able to break it apart and show it symbolically, with pictures, or in any way other than the standard algorithm. Conversely, if students are working a problem, they should be able to apply the \u201cmath work\u201d to the situation.\u201c<\/em> (Everette, 2013). <em>EMPower<\/em> gives students many opportunities to draw representations of problems and to show situations with manipulatives. The point is to see how quantities, situations, or spaces are related.<\/p>\n<p>There are a variety of activities to choose from in the <em>EMPower <\/em>series. In \u201cWhat Is the Story?\u201d students use drawings, situations, and equations to make sense of mathematical relationships.<\/p>\n<\/div>\n\t\t\t\t      <\/div>\n\t\t\t\t    <\/div><div class=\"card\"><div class=\"card-header\" id=\"headingfaq-693\">\n\t\t\t\t        <h5 class=\"mb-0\">\n\t\t\t\t          <button class=\"btn btn-link\" type=\"button\" data-toggle=\"collapse\" data-target=\"#collapsefaq-693\" aria-expanded=\"false\" aria-controls=\"collapsefaq-693\">\n\t\t\t\t            Mathematical Practice 3: Construct viable arguments and critique the reasoning of others\n\t\t\t\t          <\/button>\n\t\t\t\t        <\/h5>\n\t\t\t\t      <\/div>\n\t\t\t\t      <div id=\"collapsefaq-693\" class=\"collapse\" aria-labelledby=\"headingfaq-693\" data-parent=\"#accordion-100\">\n\t\t\t\t        <div class=\"card-body\"><p><div class=\"ezcol ezcol-two-fifth\"><strong>Sample lesson recommendation:<\/strong><\/div> <div class=\"ezcol ezcol-three-fifth ezcol-last\">Construct viable arguments and critique the reasoning of others<\/div><div class=\"ezcol-divider\"><\/div><\/p>\n<p>I believe this practice is meant to ensure that students have the opportunity to verbally critique mathematical reasoning or articulate their own mathematical reasoning. As teachers we can ask ourselves: Are students using mathematical language, to <strong>support<\/strong> or <strong>oppose<\/strong> the work of others.\u201d<em>\u00a0<\/em><\/p>\n<p>The <em>EMPower<\/em> series contains many examples. Consider \u201cWatch Out!\u201d as well as \u201cReasoning It Out\u201d (found in <em>Using Benchmarks<\/em>). In both, the students have to figure out what is going on and explain whether the thinking is right or wrong. So first they are critiquing the work but second this could develop into a rich discussion in the classroom on how students go about explaining the reasoning for their critique.<\/p>\n<\/div>\n\t\t\t\t      <\/div>\n\t\t\t\t    <\/div><div class=\"card\"><div class=\"card-header\" id=\"headingfaq-556\">\n\t\t\t\t        <h5 class=\"mb-0\">\n\t\t\t\t          <button class=\"btn btn-link\" type=\"button\" data-toggle=\"collapse\" data-target=\"#collapsefaq-556\" aria-expanded=\"false\" aria-controls=\"collapsefaq-556\">\n\t\t\t\t            Mathematical Practice 4: Model with mathematics\n\t\t\t\t          <\/button>\n\t\t\t\t        <\/h5>\n\t\t\t\t      <\/div>\n\t\t\t\t      <div id=\"collapsefaq-556\" class=\"collapse\" aria-labelledby=\"headingfaq-556\" data-parent=\"#accordion-100\">\n\t\t\t\t        <div class=\"card-body\"><p><div class=\"ezcol ezcol-two-fifth\"><strong>Sample lesson recommendation:<\/strong><\/div> <div class=\"ezcol ezcol-three-fifth ezcol-last\"><em>Seeking Patterns Building Rules, Lesson 8, Activity 8: \u201cJob Offers\u201d<\/em><\/div><div class=\"ezcol-divider\"><\/div><\/p>\n<p class=\"Default\">Mathematical Practice 4 emphasizes that \u201cmathematically proficient students can apply the mathematics they know to solve problems arising in everyday life \u2026 They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions\u201d (U.S. Department of Education, 2013).<\/p>\n<p>Students exploring \u201cJob Offers\u201d do just this: They create a graph and equation to model the financial implications of two job offers over time and then make a decision based on their model.<\/p>\n<\/div>\n\t\t\t\t      <\/div>\n\t\t\t\t    <\/div><div class=\"card\"><div class=\"card-header\" id=\"headingfaq-909\">\n\t\t\t\t        <h5 class=\"mb-0\">\n\t\t\t\t          <button class=\"btn btn-link\" type=\"button\" data-toggle=\"collapse\" data-target=\"#collapsefaq-909\" aria-expanded=\"false\" aria-controls=\"collapsefaq-909\">\n\t\t\t\t            Mathematical Practice 5: Use appropriate tools strategically\n\t\t\t\t          <\/button>\n\t\t\t\t        <\/h5>\n\t\t\t\t      <\/div>\n\t\t\t\t      <div id=\"collapsefaq-909\" class=\"collapse\" aria-labelledby=\"headingfaq-909\" data-parent=\"#accordion-100\">\n\t\t\t\t        <div class=\"card-body\"><p><div class=\"ezcol ezcol-two-fifth\"><strong>Sample lesson recommendation:<\/strong><\/div> <div class=\"ezcol ezcol-three-fifth ezcol-last\"><em>Using Benchmarks: Lesson 6,\u00a0Activity 1:\u00a0\u201cFraction Strips and Rulers\u2014Tools to\u00a0Think With\u201d<\/em><\/div><div class=\"ezcol-divider\"><\/div><\/p>\n<p>The key point here is that the activity must be one where the student decides on his or her own what tool to use. Everette (2013) explains:<em> \u201cStudents can select the appropriate math tool to use and use it correctly to solve problems. In the real world, no one tells you that it is time to use the meter stick instead of the protractor.\u201d<\/em><\/p>\n<p>In this activity, students explore fraction equivalence using a variety of tools.<\/p>\n<\/div>\n\t\t\t\t      <\/div>\n\t\t\t\t    <\/div><div class=\"card\"><div class=\"card-header\" id=\"headingfaq-400\">\n\t\t\t\t        <h5 class=\"mb-0\">\n\t\t\t\t          <button class=\"btn btn-link\" type=\"button\" data-toggle=\"collapse\" data-target=\"#collapsefaq-400\" aria-expanded=\"false\" aria-controls=\"collapsefaq-400\">\n\t\t\t\t            Mathematical Practice 6: Attend to precision\n\t\t\t\t          <\/button>\n\t\t\t\t        <\/h5>\n\t\t\t\t      <\/div>\n\t\t\t\t      <div id=\"collapsefaq-400\" class=\"collapse\" aria-labelledby=\"headingfaq-400\" data-parent=\"#accordion-100\">\n\t\t\t\t        <div class=\"card-body\"><p><div class=\"ezcol ezcol-two-fifth\"><strong>Sample lesson recommendation:<\/strong><\/div> <div class=\"ezcol ezcol-three-fifth ezcol-last\"><em>Keeping Things in Proportion, Lesson 4, Activity 1: \u201cPart to Part versus Part to Whole\u201d<\/em><\/div><div class=\"ezcol-divider\"><\/div><\/p>\n<p>Precision enables problem solvers to use the language of math to explain themselves clearly. Recognizing and acting on the importance of labeling correctly are key to clear communication. Beyond that, precision allows problem solvers to choose strategies and solutions that speak to the authentic needs of a situation, for example, rounding time to hours or days, letting go of fractions of hours or seconds, depending on the situation. The \u201cprecise\u201d answer may not be the most useful answer. Fractions of cartons or rooms are not necessarily helpful in everyday contexts.<\/p>\n<\/div>\n\t\t\t\t      <\/div>\n\t\t\t\t    <\/div><div class=\"card\"><div class=\"card-header\" id=\"headingfaq-368\">\n\t\t\t\t        <h5 class=\"mb-0\">\n\t\t\t\t          <button class=\"btn btn-link\" type=\"button\" data-toggle=\"collapse\" data-target=\"#collapsefaq-368\" aria-expanded=\"false\" aria-controls=\"collapsefaq-368\">\n\t\t\t\t            Mathematical Practice 7: Look for and make use of structure\n\t\t\t\t          <\/button>\n\t\t\t\t        <\/h5>\n\t\t\t\t      <\/div>\n\t\t\t\t      <div id=\"collapsefaq-368\" class=\"collapse\" aria-labelledby=\"headingfaq-368\" data-parent=\"#accordion-100\">\n\t\t\t\t        <div class=\"card-body\"><p><div class=\"ezcol ezcol-two-fifth\"><strong>Sample lesson recommendation:<\/strong><\/div> <div class=\"ezcol ezcol-three-fifth ezcol-last\"><em>Everyday Number Sense, Lesson 10, Math Inspection:\u00a0\u201cRectangles, Arrays, Area, and the Distributive Property\u201d<\/em><\/div><div class=\"ezcol-divider\"><\/div><\/p>\n<p>One way to understand structure is as underlying principles that can be built upon over time. Structure speaks to how numbers and spaces are organized and put together as parts and wholes. I also think of structure when I think of the mathematical properties like commutativity, which allows certain changes within an equation. In particular, the Math Inspections in <em>Everyday Number Sense<\/em> get at these ideas.<\/p>\n<\/div>\n\t\t\t\t      <\/div>\n\t\t\t\t    <\/div><div class=\"card\"><div class=\"card-header\" id=\"headingfaq-947\">\n\t\t\t\t        <h5 class=\"mb-0\">\n\t\t\t\t          <button class=\"btn btn-link\" type=\"button\" data-toggle=\"collapse\" data-target=\"#collapsefaq-947\" aria-expanded=\"false\" aria-controls=\"collapsefaq-947\">\n\t\t\t\t            Mathematical Practice 8: Look for and express regularity in repeated reasoning\n\t\t\t\t          <\/button>\n\t\t\t\t        <\/h5>\n\t\t\t\t      <\/div>\n\t\t\t\t      <div id=\"collapsefaq-947\" class=\"collapse\" aria-labelledby=\"headingfaq-947\" data-parent=\"#accordion-100\">\n\t\t\t\t        <div class=\"card-body\"><p><div class=\"ezcol ezcol-two-fifth\"><strong>Sample lesson recommendation:<\/strong><\/div> <div class=\"ezcol ezcol-three-fifth ezcol-last\"><em>Everyday Number Sense, Lesson 9, Activity 4:<br \/>\n\u201cShortcuts \u2014Multiples of 10\u201d<\/em><\/div><div class=\"ezcol-divider\"><\/div><\/p>\n<p><em>EMPower<\/em> lessons include several activities where the students look for patterns and then come up with a rule or method. Linda Gojak of the National Council of Teachers of Mathematics has commented on the importance of engaging students in this way:<em>\u201cOne of the things that sometimes students don\u2019t figure out is that math makes sense; it\u2019s supposed to make sense. There\u2019s not much you do in math that doesn\u2019t make sense. And in our traditional show-and-tell type of instruction, we don\u2019t allow students to actually have the opportunity to make sense \u2026 We need to give students time to make and refine observations.\u201d<\/em><\/p>\n<\/div>\n\t\t\t\t      <\/div>\n\t\t\t\t    <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>  <a href=\"https:\/\/www.terc.edu\/empower\/faqs\/how-emp-addresses-mp\/\">&nbsp;&raquo;&nbsp;Read more<\/a><\/p>\n","protected":false},"author":4,"featured_media":0,"parent":591,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"_relevanssi_hide_post":"","_relevanssi_hide_content":"","_relevanssi_pin_for_all":"","_relevanssi_pin_keywords":"","_relevanssi_unpin_keywords":"","_relevanssi_related_keywords":"","_relevanssi_related_include_ids":"","_relevanssi_related_exclude_ids":"","_relevanssi_related_no_append":"","_relevanssi_related_not_related":"","_relevanssi_related_posts":"","_relevanssi_noindex_reason":"","footnotes":""},"class_list":["post-646","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/www.terc.edu\/empower\/wp-json\/wp\/v2\/pages\/646","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.terc.edu\/empower\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.terc.edu\/empower\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.terc.edu\/empower\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/www.terc.edu\/empower\/wp-json\/wp\/v2\/comments?post=646"}],"version-history":[{"count":7,"href":"https:\/\/www.terc.edu\/empower\/wp-json\/wp\/v2\/pages\/646\/revisions"}],"predecessor-version":[{"id":859,"href":"https:\/\/www.terc.edu\/empower\/wp-json\/wp\/v2\/pages\/646\/revisions\/859"}],"up":[{"embeddable":true,"href":"https:\/\/www.terc.edu\/empower\/wp-json\/wp\/v2\/pages\/591"}],"wp:attachment":[{"href":"https:\/\/www.terc.edu\/empower\/wp-json\/wp\/v2\/media?parent=646"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}