MTMS Bloghttp://www.nctm.org/rss/rssfeeds.aspx?fid=599536 Bringing in the Real Worldhttp://www.nctm.org/publications/blog/blog.aspx?id=42692&blogid=599536<p>I have been writing about how to use real world contexts in the mathematics classroom. What are some practical ways to do this? Let’s consider three examples that explore how to integrate diversity, connect to students’ lives, and analyze social</p>nctm@nctm.org (Customer Service)Thu, 17 Jul 2014 15:10:28 GMTMathematics and the Real Worldhttp://www.nctm.org/publications/blog/blog.aspx?id=42609&blogid=599536<p>In my previous post, I argued that in addition to teaching mathematics for its own sake, we should also teach mathematics so that students learn to value diversity, see mathematics in their lives and cultural backgrounds, and analyze and critique</p>nctm@nctm.org (Customer Service)Mon, 30 Jun 2014 10:27:19 GMTWhy Teach Mathematics?http://www.nctm.org/publications/blog/blog.aspx?id=42561&blogid=599536<p>NCTM’s new Principles to Actions Ensuring Mathematical Success for All aims to ensure high quality mathematics education for all students. But what does high quality mathematics education look like? Another way to come at this question is to ask, “Why</p>nctm@nctm.org (Customer Service)Thu, 12 Jun 2014 12:30:55 GMTTeaching Students about Functions with Dynagraphshttp://www.nctm.org/publications/blog/blog.aspx?id=42504&blogid=599536<p>Introduction Function is a fundamental concept in mathematics it is one that students explore repeatedly, at increasing levels of sophistication, throughout the early and middle grades (and beyond) (Steketee and Scher 2011). In the early grades, students may encounter functions</p>nctm@nctm.org (Customer Service)Tue, 03 Jun 2014 11:08:45 GMTA Critical Look at Movieshttp://www.nctm.org/publications/blog/blog.aspx?id=42440&blogid=599536<p>Several weekends ago, my family suggested that we go to the local theater and watch a movie. As most any parent of a teenager knows, finding a movie that everyone can agree on is no small task. My wife and</p>nctm@nctm.org (Customer Service)Mon, 19 May 2014 11:51:06 GMTCreative Problem Posing in the Middle-Grades Classroomhttp://www.nctm.org/publications/blog/blog.aspx?id=42399&blogid=599536<p> Earlier this year, I was introduced to Paul Lockhart’s remarkable text, A Mathematician’s Lament How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form. Lockhart persuasively advocates for mathematics teaching and a mathematics curriculum that provides students</p>nctm@nctm.org (Customer Service)Wed, 07 May 2014 10:48:56 GMTTowering Mathematicshttp://www.nctm.org/publications/blog/blog.aspx?id=42282&blogid=599536<p>Towering Mathematics In a recent visit to a seventh grade mathematics classroom, I was delighted to see students working collaboratively on the Skeleton Tower task as shown in figure 1. MTMS_blog_2014 04 28_Art01.png Fig. 1 How many</p>nctm@nctm.org (Customer Service)Thu, 24 Apr 2014 15:24:24 GMTPeace through Constructionshttp://www.nctm.org/publications/blog/blog.aspx?id=41979&blogid=599536<p>Compass and straightedge constructions have been in the geometry curriculum for a loooonng time. These constructions were around before Euclid included them in his textbook, The Elements, 2300 years ago. The painting shown here is an illumination from a translation</p>nctm@nctm.org (Customer Service)Mon, 07 Apr 2014 11:31:14 GMTBootstrapping Revisited: A Regular 3 1/2-gon?http://www.nctm.org/publications/blog/blog.aspx?id=41601&blogid=599536<p> This is another weird example of the bootstrapping I mentioned in my first post. We were brainstorming a geometry class a few months ago when my friend Ian asked me, “Can you draw a</p>nctm@nctm.org (Customer Service)Thu, 20 Mar 2014 11:35:10 GMTApples and Remaindershttp://www.nctm.org/publications/blog/blog.aspx?id=41325&blogid=599536<p> <strong>Here’s a classic
question:</strong>
</p>
<p>Out of the 6000 apples we harvested, every third
apple was too small, every fourth apple was spotted, and every tenth apple was
bruised. The remaining apples were good. How many good apples were there?
</p>
<p>What is the purpose of constructing a viable
mathematical argument or proof? Many people would say that the purpose is to
convince another person or yourself that the position being taken is true,
beyond a shadow of a doubt. But I think we have focused too much on conviction
as the purpose of mathematical argumentation. I think the primary benefit of
constructing a viable argument is that it uncovers the way mathematical ideas
relate to one another. When you make a viable argument for a claim, it reveals
what mathematics the claim relies on. Here is an example:
</p>
<p>Students usually start by saying, “Subtract from
6000 the 2000 apples that are too small, the 1500 that are spotted, and the 600
that are bruised to get 1900 good ones left.” This explanation isn’t correct,
because the students have subtracted some of the apples twice (e.g., the ones
that are spotted <em>and</em> bruised).
</p>nctm@nctm.org (Customer Service)Tue, 04 Mar 2014 11:52:53 GMTThe 2° th posthttp://www.nctm.org/publications/blog/blog.aspx?id=40375&blogid=599536<p>Welcome to the Blogarithm! We’re exploring ideas raised by the Standards for Mathematical Practice in the Common Core State Standards for the middle grades. </p>
<p>I noticed something while watching an eighth-grade class. The teacher was getting the students to notice the rule about adding exponents, asking them to figure out 2<sup>4 </sup>x 2<sup>3</sup> and other similar cases. When they saw it was 2<sup>7</sup>, and that you could, in general, just add the exponents with the same base, she asked them why it is always true. One student said, “Because you are just multiplying 2 times itself 3 times and then 5 more times, so it’s 8 times total.” They then did it for division (e.g., 2<sup>7 </sup>÷ 2<sup>3</sup> = 2<sup>4</sup>), which was just a rearrangement of the same thing.
</p>
<p>Later, the teacher was talking about 2<sup>0</sup> and negative exponents like 2<sup>-</sup><sup>3</sup>. A student asked why 2<sup>0</sup> is 1 because shouldn’t 2 times itself 0 times be 0? Hmm. The teacher fielded this pretty well by asking the class what 2<sup>3 </sup>÷ 2<sup>3</sup> was, based on the subtraction rule. They said that it was 2<sup>0</sup> and noticed that it also had to be 1 because it was 8 ÷ 8. So it makes sense to say 2<sup>0</sup> is 1. Crisis averted.
</p>nctm@nctm.org (Customer Service)Fri, 03 Jan 2014 10:33:54 GMT