<story><title>An alarming trend in K-12 math education: a guest post and an open letter</title><url>https://scottaaronson.blog/?p=6146</url></story><parent_chain><item><author>heyitsguay</author><text>As someone who did a BA and PhD in math, went through high school geometry proofs, taught geometry proofs to math students and math education students:&lt;p&gt;Geometry proofs are a &lt;i&gt;terrible&lt;/i&gt; way of introducing rigorous mathematical proofs to students. Seriously. I cannot overstate how misleading they are. I remember my first abstract algebra class in college, after the first HW I got called in by the TA because I tried to structure my proofs like I learned in geometry class - a sequence of symbols and references to hard-coded lists of axioms and prior deductions. I thought that&amp;#x27;s what proofs are, but they are not, at least not as humans do higher math. High school geometry tries to distill this process down to a symbol manipulation game that students can memorize and regurgitate, and in the process loses the essence of the thing it was meant to capture in the first place.</text></item><item><author>cperciva</author><text>&lt;i&gt;they are cutting geometry a bit, which doesn&amp;#x27;t make as much sense today as it did 100 years ago when far more people grew up to be farmers or ranchers.&lt;/i&gt;&lt;p&gt;The point of high school Geometry was never to compute areas of fields.  Trigonometry is far more useful for that anyway.&lt;p&gt;The point of high school Geometry is that it is the first introduction to rigorous mathematical proof -- and has been, ever since Euclid&amp;#x27;s Elements.</text></item><item><author>humanistbot</author><text>The manufactured outrage over the California math recommendations keep getting posted on HN. Read the actual text of the plans here [1]. The FAQ is at [2] and directly responds to these characterizations. They are not banning gifted &amp;amp; talented programs or advanced students taking accelerated courses. They are not taking algebra out of the curriculum, although they are cutting geometry a bit, which doesn&amp;#x27;t make as much sense today as it did 100 years ago when far more people grew up to be farmers or ranchers. They are adding more statistics and probability, which I think are crucial in today&amp;#x27;s society.&lt;p&gt;What they are fundamentally doing is breaking up the classic U.S. staged path where you learn algebra for a year, then geometry for a year, then back to algebra &amp;#x2F; pre-calc for a year, then maybe take statistics or calculus as an elective, etc. Instead, all branches of math will be taught in an integrated approach focused around applied problems.&lt;p&gt;This is how a lot of European math courses are taught. In fact, I think HN would appreciate the shift from focusing on pure numbers and classic formulas to more applied uses of math, including algorithms, probability, data collected and analyzed in charts, etc. Students also forget a lot of algebra when they do a year of geometry by itself, then have to go back to algebra &amp;#x2F; pre-calc.&lt;p&gt;It also does mean that in the transition, it will be harder for students to &amp;quot;test out&amp;quot; of the classic algebra I&amp;#x2F;II, geometry, pre-calc sequence, because it will just be &amp;quot;year X integrated math.&amp;quot; But the framework does not forbid gifted and talented programs or anything like that. There will just be a few awkward years while the curriculum shifts.&lt;p&gt;Now, there are some on the left who advocated for the elimination of gifted and talented programs altogether, for equity reasons. They did not get what they wanted in the new California framework. That hasn&amp;#x27;t stopped a lot of people from looking at what California is doing and imagining it is actually some kind of Harrison Bergeron dystopia, when that is absolutely not the case.&lt;p&gt;[1] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;www.cde.ca.gov&amp;#x2F;ci&amp;#x2F;ma&amp;#x2F;cf&amp;#x2F;&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;www.cde.ca.gov&amp;#x2F;ci&amp;#x2F;ma&amp;#x2F;cf&amp;#x2F;&lt;/a&gt;&lt;p&gt;[2] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;www.cde.ca.gov&amp;#x2F;ci&amp;#x2F;ma&amp;#x2F;cf&amp;#x2F;mathfwfaqs.asp&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;www.cde.ca.gov&amp;#x2F;ci&amp;#x2F;ma&amp;#x2F;cf&amp;#x2F;mathfwfaqs.asp&lt;/a&gt;</text></item></parent_chain><comment><author>cperciva</author><text>I would agree that proofs as introduced in high school Geometry have more in common with formal axiomatic logic than the vast majority of &amp;quot;real-world&amp;quot; mathematical proofs; but a large part of that is just clumsy notation.&lt;p&gt;The fundamental concepts&lt;p&gt;1. We have axioms which are things we accept without proof because we all agree that they&amp;#x27;re obvious;&lt;p&gt;2. Everything else should follow logically from things which come earlier;&lt;p&gt;3. Don&amp;#x27;t skip steps!&lt;p&gt;are extremely important and apply regardless of the field you&amp;#x27;re in.</text></comment>
<story><title>An alarming trend in K-12 math education: a guest post and an open letter</title><url>https://scottaaronson.blog/?p=6146</url></story><parent_chain><item><author>heyitsguay</author><text>As someone who did a BA and PhD in math, went through high school geometry proofs, taught geometry proofs to math students and math education students:&lt;p&gt;Geometry proofs are a &lt;i&gt;terrible&lt;/i&gt; way of introducing rigorous mathematical proofs to students. Seriously. I cannot overstate how misleading they are. I remember my first abstract algebra class in college, after the first HW I got called in by the TA because I tried to structure my proofs like I learned in geometry class - a sequence of symbols and references to hard-coded lists of axioms and prior deductions. I thought that&amp;#x27;s what proofs are, but they are not, at least not as humans do higher math. High school geometry tries to distill this process down to a symbol manipulation game that students can memorize and regurgitate, and in the process loses the essence of the thing it was meant to capture in the first place.</text></item><item><author>cperciva</author><text>&lt;i&gt;they are cutting geometry a bit, which doesn&amp;#x27;t make as much sense today as it did 100 years ago when far more people grew up to be farmers or ranchers.&lt;/i&gt;&lt;p&gt;The point of high school Geometry was never to compute areas of fields.  Trigonometry is far more useful for that anyway.&lt;p&gt;The point of high school Geometry is that it is the first introduction to rigorous mathematical proof -- and has been, ever since Euclid&amp;#x27;s Elements.</text></item><item><author>humanistbot</author><text>The manufactured outrage over the California math recommendations keep getting posted on HN. Read the actual text of the plans here [1]. The FAQ is at [2] and directly responds to these characterizations. They are not banning gifted &amp;amp; talented programs or advanced students taking accelerated courses. They are not taking algebra out of the curriculum, although they are cutting geometry a bit, which doesn&amp;#x27;t make as much sense today as it did 100 years ago when far more people grew up to be farmers or ranchers. They are adding more statistics and probability, which I think are crucial in today&amp;#x27;s society.&lt;p&gt;What they are fundamentally doing is breaking up the classic U.S. staged path where you learn algebra for a year, then geometry for a year, then back to algebra &amp;#x2F; pre-calc for a year, then maybe take statistics or calculus as an elective, etc. Instead, all branches of math will be taught in an integrated approach focused around applied problems.&lt;p&gt;This is how a lot of European math courses are taught. In fact, I think HN would appreciate the shift from focusing on pure numbers and classic formulas to more applied uses of math, including algorithms, probability, data collected and analyzed in charts, etc. Students also forget a lot of algebra when they do a year of geometry by itself, then have to go back to algebra &amp;#x2F; pre-calc.&lt;p&gt;It also does mean that in the transition, it will be harder for students to &amp;quot;test out&amp;quot; of the classic algebra I&amp;#x2F;II, geometry, pre-calc sequence, because it will just be &amp;quot;year X integrated math.&amp;quot; But the framework does not forbid gifted and talented programs or anything like that. There will just be a few awkward years while the curriculum shifts.&lt;p&gt;Now, there are some on the left who advocated for the elimination of gifted and talented programs altogether, for equity reasons. They did not get what they wanted in the new California framework. That hasn&amp;#x27;t stopped a lot of people from looking at what California is doing and imagining it is actually some kind of Harrison Bergeron dystopia, when that is absolutely not the case.&lt;p&gt;[1] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;www.cde.ca.gov&amp;#x2F;ci&amp;#x2F;ma&amp;#x2F;cf&amp;#x2F;&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;www.cde.ca.gov&amp;#x2F;ci&amp;#x2F;ma&amp;#x2F;cf&amp;#x2F;&lt;/a&gt;&lt;p&gt;[2] &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;www.cde.ca.gov&amp;#x2F;ci&amp;#x2F;ma&amp;#x2F;cf&amp;#x2F;mathfwfaqs.asp&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;www.cde.ca.gov&amp;#x2F;ci&amp;#x2F;ma&amp;#x2F;cf&amp;#x2F;mathfwfaqs.asp&lt;/a&gt;</text></item></parent_chain><comment><author>jacobmartin</author><text>It is interesting to me that you didn&amp;#x27;t encounter problems with the rote geometric method before abstract algebra. I went to a so-so public school in the American South, so I don&amp;#x27;t think I exactly had a stellar introduction to math or even school&amp;#x2F;academics in general, but beginning in trigonometry, we had to structure our proofs with more words and &amp;#x27;connective tissue,&amp;#x27; and less explicit axiomatization&amp;#x2F;symbolization. Calculus continued this trend, and by the time I took linear algebra (at a local university while I was still in high school), the proofs were exactly like the proofs I would write throughout college. But there were several &amp;#x27;stepping stones&amp;#x27; away from the original geometric proof.&lt;p&gt;Though admittedly I only got an SB in Math, I think geometric proofs were an okay &lt;i&gt;introduction&lt;/i&gt; to the idea of the proof and of logic. Of course they are not representative of professional math. There are many things we learn as an introduction which don&amp;#x27;t turn out to bear that much resemblance to the more advanced form. I don&amp;#x27;t write 5-paragraph essays with clear thesis sentences anymore either. But it helped me to learn to write that way as a way of clarifying that I was stating my arguments effectively.&lt;p&gt;Sometimes, for my homework, I tried to state very explicitly in my proofs which axiomata I was using, both as an intellectual exercise and to see, e.g., where I might have gotten my logic backwards or where I used the axiom of choice (which constantly surprised me). I wouldn&amp;#x27;t turn these overexplicit proofs, but it helped to clarify that I was doing things correctly. If anything, when I was in college, the trend was &lt;i&gt;away&lt;/i&gt; from &amp;#x27;informal&amp;#x27; language. When I took, e.g., Discrete Math, for more complicated proofs, we were encouraged to write out a DAG for the dependencies of the proofs&amp;#x2F;lemmata so that it was exceedingly clear what depended on what and that we had in fact proven what we set out to prove.</text></comment>