{"id":15854,"date":"2019-05-19T20:00:32","date_gmt":"2019-05-20T00:00:32","guid":{"rendered":"http:\/\/advancedmathyoungstudents.com\/blog\/?p=15854"},"modified":"2019-05-19T20:00:41","modified_gmt":"2019-05-20T00:00:41","slug":"lessons-from-math-sat-prep-for-the-math-classroom","status":"publish","type":"post","link":"https:\/\/advancedmathyoungstudents.com\/blog\/?p=15854","title":{"rendered":"Lessons from Math SAT Prep for the Math Classroom"},"content":{"rendered":"\n

A\nbrief<\/s> rant<\/strong><\/p>\n\n\n\n

In light of the on-going controversy regarding admission to New York City\u2019s selective high schools, there has been some discussion of bringing test prep to the middle school classroom.\u00a0 At first, test prep in the math classroom may sound like an awful idea.\u00a0 Should we take what is frequently, sad to say, a joyless, stultifying slog and add in a large helping of stress-inducing rote drill to prepare for the SHSAT, PSAT, SAT, what have you?\u00a0 That can\u2019t be a good idea. And yet\u2026 <\/p>\n\n\n\n

Maybe we should work on fixing the joyless, stultifying part.\u00a0 I have taught (oh, lord) a ton of SAT prep.\u00a0 None of it involves rote drills.\u00a0 It does involve a number of ideas and methods that absolutely help make math more joyful.\u00a0 And many, many students have come into my SAT class announcing that they don\u2019t like math only to later report that they \u201ckind of like\u201d SAT math.\u00a0 \u00a0So what is happening in SAT class that isn\u2019t happening often enough in school?\u00a0 Here are some possible answers:<\/p>\n\n\n\n

1. SAT math is subversive.<\/em><\/strong>  And being subversive is fun.  You don\u2019t have to solve the problem the \u201cright\u201d\nway!  You can play, guess, draw pictures,\nmake lists\u2026and if it works, you get full credit!  School math is often presented like some\nderanged cooking class where everyone learns to cook the same dish in the exact\nsame way. No invention, no experimentation, no surprises. (And it\u2019s rarely\nanything anyone wants to eat anyway.)  We\nshould be encouraging mathematical improvisation and creativity \u2013 even when it\ndoesn\u2019t \u201cwork\u201d.  After all, how often do\nwe really care where the vertex of the parabola is?  Isn\u2019t the more valuable experience the\nsearch?  A math problem that only has one\nsolution path isn\u2019t really a \u201cproblem\u201d — it\u2019s more like a \u201cchore\u201d.  No one likes doing chores.<\/p>\n\n\n\n

2. SAT math rewards trial and error.<\/em><\/strong>  School math often treats trial and error as a\ntool for the weak.  But trial and error\nis a fabulous way to begin a problem when you don\u2019t see an algebraic path.  It is not merely an algebra-evasion technique\n(though on the SAT, it can certainly work that way). Trial and error can serve\nas an on-ramp to the algebraic solution. \nI have used trial and error this way with students in SAT class, physics,\nprecalc, calculus, and multivariable calc. \nTo keep this rant brief (well\u2026) I will expand on and demonstrate this\nidea in a separate post.  But to give you\nthe flavor: <\/p>\n\n\n\n

1.\nYou take a guess (or pick an answer if it is a multiple choice test)<\/em><\/p>\n\n\n\n

2.\nYou mess around with your guess, applying whatever line of reasoning you can\ncome up with to show that your guess is wrong (as it usually will be).<\/em><\/p>\n\n\n\n

3. You replace your guess with a variable and then use your earlier reasoning as a template to come up with the algebraic equation or equations that will lead to an answer.<\/em><\/p>\n\n\n\n

I have shown this method to students as young as 6th<\/sup> grade and to colleagues who have been teaching for decades.\u00a0 I can tell you that most of them got a kick out of this.\u00a0 Again, it feels subversive. And who doesn\u2019t like the feeling that you are getting away with something?\u00a0 But really, you’re not. You are just separating the problem into two phases, doing the reasoning before you attempt to apply the algebra.\u00a0 But it only works if you are willing to take a guess. \u201cCould this be the answer?\u201d is an amazingly helpful question, and not just on a standardized test.<\/p>\n\n\n\n

3. SAT math rewards making up numbers for variables.<\/em><\/strong>  This one is an SAT classic that should really\nbe a staple of school math as well.  Many\nstudents see an algebraic equation as a sequence of letters and symbols,\nprobably including an \u201c=\u201d somewhere in there.  We want them to see it as a statement, a\nsentence written in the language of algebra, one that might be true if the\nvariables have the \u201cright\u201d values.  <\/p>\n\n\n\n

This gives us an efficient way of finding\nwhether two expressions are equivalent. \nMake up numbers (ideally, slightly \u201cunusual\u201d ones), put them into both\nexpressions, and see if they come out equal. \nAnd yes, this method can generate false positives. But it never\ngenerates false negatives \u2013 if you get two different numeric answers, then your\ntwo algebraic expressions were definitely not equivalent.<\/p>\n\n\n\n

This idea has many uses.  First of all, we can teach students that\narithmetic is a testing bench for algebra. \nAny time you are not certain whether something is allowed, try it with\nnumbers!  We all have taught plenty of\nstudents who believe that (x + y)2<\/sup> = x2<\/sup> + y2<\/sup>.  So let x =4 and y = 6 and see what happens!<\/p>\n\n\n\n

Secondly, it inoculates students from the classic standardized test question: \u201cwhich of the following expressions is equivalent to\u2026\u201d , a type of question that deserves to be subverted.<\/p>\n\n\n\n

And thirdly, once again it gives us a way\nto separate the reasoning from the algebra. \nYou can make up numbers for the variables, work out an answer, and then\nput your variables back into the answer choices.  It\u2019s probably the SAT method that seems most\ndevious. Even in physics class, often my beginning students are flummoxed by\nquestions like:<\/p>\n\n\n\n

A\ncar has mass m and speed v.  A second car\nhas half the mass and three times the speed. Find the ratio of the second car\u2019s\nkinetic energy to the first car\u2019s kinetic energy. <\/em><\/p>\n\n\n\n

The problem is not that these students\ndon\u2019t know the formula for kinetic energy. It\u2019s that they have not yet\ndeveloped the ability to think about the ratios and proportions involved.  When they are fluent, they will be able to\nlook at the formula, K = \u00bd mv2<\/sup> and think through the problem with\nease.  But while they are still\ndeveloping, isn\u2019t it nice to have an alternative scaffold available?  <\/p>\n\n\n\n

Make\nup numbers for <\/em>m\nand <\/em>v and calculate the kinetic\nenergy.<\/em><\/p>\n\n\n\n

Cut\nthe <\/em>m\nin half  and triple the <\/em>v and recalculate the kinetic energy.<\/em><\/p>\n\n\n\n

Find\nthe ratio of your two answers.<\/em><\/p>\n\n\n\n

Beginners are often surprised that this\nworks, and even skeptical, thinking that there was something special about the\nnumbers they made up. So let them make up a new set of numbers.    It\nseems like magic. And again, it serves as an on-ramp to the algebraic discussion.<\/p>\n\n\n\n

Two more points and I\u2019ll call it a rant<\/em><\/strong><\/p>\n\n\n\n

1. None of what I am recommending will\ntake time away from your curriculum. These are attitudes and strategies that\ncan overlay whatever you are currently teaching. Still, take a hard look: it\nisn\u2019t all joyful now and it isn\u2019t all as essential as you may think.  But we can argue about that another time.  I would teach these methods even if the SAT\nand its ilk were to vanish.  It\u2019s a more\nfun way to do math.<\/p>\n\n\n\n

2. Test prep in middle school is not\ngoing to magically bring diversity to the elite high schools. It might help,\nbut I would guess that it won\u2019t be nearly enough.  We are going to have to do more.<\/p>\n","protected":false},"excerpt":{"rendered":"

A brief rant In light of the on-going controversy regarding admission to New York City\u2019s selective high schools, there has been some discussion of bringing test prep to the middle school classroom.\u00a0 At first, test prep in the math classroom … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4uvY7-47I","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/15854"}],"collection":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=15854"}],"version-history":[{"count":5,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/15854\/revisions"}],"predecessor-version":[{"id":15859,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/15854\/revisions\/15859"}],"wp:attachment":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=15854"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=15854"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=15854"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}