{"id":347,"date":"2017-08-23T05:19:14","date_gmt":"2017-08-23T05:19:14","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/elementary-education\/?page_id=347"},"modified":"2017-08-29T03:58:54","modified_gmt":"2017-08-29T03:58:54","slug":"mathematics-instruction","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/elementary-education\/mathematics-instruction\/","title":{"rendered":"Mathematics Instruction"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/mathematical-reasoning-problem-solving\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/stats-data-probability-and-mathematical-reasoning\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/further-reading-9\">Further Reading \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Mathematics Instruction<\/h1>\n<h4>Objective<\/h4>\n<p>Now that we have learned all of these mathematical reasoning and problem-solving techniques, we can hopefully fulfill our goal of being able to teach these techniques to students. So let&#8217;s go over some strategies and tips for successful mathematical instruction.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>In the previous pages we looked at some simple problem solving techniques, as well as some common missteps in mathematical reasoning. We also brushed up on some basic types of proofs.<\/li>\n<\/ul>\n<section>\n<h3>Systematically Sequence Mathematics<\/h3>\n<p><strong>Example:<\/strong> Consider the situation where a teacher wants a student to learn how to multiply two fractions. How would this teacher systematically sequence instruction in order to achieve this goal?<\/p>\n<p>Assuming that a student is already familiar with basic mathematical operations (addition, subtraction, multiplication and division), the teacher would want to teach the students about fractions:<\/p>\n<p>A fraction is a rational number <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/stats7_1010.gif\" alt=\"a\/b\" \/>, where a is a real number called the numerator, and b is a real number not equal to zero called the denominator.<\/p>\n<p>Then the teacher would want to define the product of two fractions <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/stats7_1010.gif\" alt=\"a\/b\" \/> and <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/stats7_1012.gif\" alt=\"c\/d\" \/>:<\/p>\n<p class=\"center\">(<img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/stats7_1010.gif\" alt=\"a\/b\" \/>)(<img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/stats7_1012.gif\" alt=\"c\/d\" \/>) = (<img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/stats7_1013.gif\" alt=\"ac\/bd\" \/>)<\/p>\n<p>where the numerator of the product of the two fractions is equal to the product of the numerators of the two fractions, and the denominator of the product of the two fractions is equal to the product of the denominators of the two fractions.<\/p>\n<p>To end the sequence of instruction, the teacher would want to go over an example illustrating the concept of multiplying two fractions:<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/stats7_1014.gif\" alt=\"Completed example\" \/><\/center><\/p>\n<h3>Tailor Your Instruction to the Skill Level of the Student<\/h3>\n<p>For any given group of students, it is most likely the case that they are coming from very different backgrounds\u2014some of these differences being levels of knowledge, aptitude, and interest in the subject being taught.<\/p>\n<p>Therefore, it is important to structure lessons in order to accommodate all of these students. How would a teacher do this? Here is a possible course of action:<\/p>\n<ol>\n<li>Give the students a test at the beginning of the year or semester to determine their level and aptitude in mathematics based on material they have previously been taught.<\/li>\n<li>After determining the students&#8217; levels, create small groups according to skill levels.<\/li>\n<\/ol>\n<p>Of course, you probably won&#8217;t be able to teach all of these groups individually\u2014most teachers do not have the time or resources. One solution to this problem would be to give the entire class a lesson, and then assign each small group different problems to solve depending on their skill level. Ideally, the students would work on these problems together and help each other.<\/p>\n<p><strong>Example:<\/strong> If a teacher was giving a lesson on how to compute the area of simple geometric shapes, like squares and triangles, here are two different problems the teacher could give\u2014one for a less-advanced group and one for more-advanced group.<\/p>\n<p>Problem One (for the less-advanced group):<\/p>\n<p>What is the area of the triangle with a height of five inches and a base three inches?<\/p>\n<p>Now, what is the area of the square with one side six inches long?<\/p>\n<p>Problem Two (for the more advanced group):<\/p>\n<p>What is the area of the drawing of the house below, with height nine inches and base five inches wide?<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/7.1question3.jpg\" alt=\"House example question\" \/><\/center><\/p>\n<p>Why would the less-advanced group be assigned problem one? Because that problem simply requires computing the area of a triangle and then the area of a square. However, problem two requires a student to divide the house into a square and a triangle, compute the area of each, and then add these areas together to find the total area of the house. Thus, because problem two addresses more concepts and is not as straightforward as problem one, it would be appropriate for the more advanced group.<\/p>\n<h3>Be Able to Recognize and Correct Mistakes<\/h3>\n<p><strong>Example:<\/strong> When students have to learn how to add multiple-digit numbers, they commonly start with the left column. For example, when adding the two numbers 86 and 57, a student might start by adding the 8 and 5 to get 13.<\/p>\n<p>How might a teacher correct the student&#8217;s error?<\/p>\n<p>One method is to come up with a creative way for the student to remember the correct way to add two numbers. In order to add two numbers, the student needs to start by adding the numbers in the right column, so the teacher could say \u201cstarting with the left column is wrong, and starting with the right column is right! Remember: right is right!\u201d<\/p>\n<p>That way, the student would only have to remember the phrase \u201cright is right!\u201d when trying to add numbers with multiple digits, and thus be more likely to start by adding the numbers in the \u201cright\u201d column, which is the \u201cright\u201d way to do it.<\/p>\n<h3>Make It Relevant<\/h3>\n<p>Point out some of the practical uses of new math skills and knowledge by linking them to other subject matter, to careers, and to personal situations. This makes math more important and students will want to learn more when you can show them the power of mathematics and logic in their everyday lives.<\/p>\n<p>Not only is it important for a teacher to be able to successfully teach mathematics to his\/her students, but it is also important for a teacher to encourage students to be interested in mathematics and realize its importance and usefulness.<\/p>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/mathematical-reasoning-problem-solving\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/stats-data-probability-and-mathematical-reasoning\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/further-reading-9\">Further Reading \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Further Reading \u27a1 Mathematics Instruction Objective Now that we have learned all of these mathematical reasoning and problem-solving techniques, we can hopefully fulfill our goal of being able to teach these techniques to students. So let&#8217;s go over some strategies and tips for successful mathematical instruction. Previously Covered: In the previous pages [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-347","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/347","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/comments?post=347"}],"version-history":[{"count":6,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/347\/revisions"}],"predecessor-version":[{"id":1009,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/347\/revisions\/1009"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/media?parent=347"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}