<p>Do you know what is in your Common Core curriculum? </p>
<ol>
<li><p>Our school district says that it will teach LESS concepts in math, but deeper (I don't believe). </p></li>
<li><p>Kids will learn to solve math problems in teams (it means that kids are supposed to teach other kids how to solve math ... yes, in elementary school).</p></li>
<li><p>"it will no longer be enough to know that 2 times 2 equals 4; students will need to understand why 2 times 2 is 4". ..... That is not school math! Honestly, it takes Math Analysis to understand why 2x2=4. Studies in the context of real and complex numbers and functions. You need to know calculus to understand it. </p></li>
</ol>
<p>Now I see why "white suburban moms" rebel against Common Core. According to our dear Education Secretary Arne Duncan. </p>
<p>Team problem solving is, all things being equal, a good thing. Math 55 kids at Harvard do most of their work in study groups, so why not? But the problem in public schools is you get stuck with a bunch of snotnosed rat kids with no interest in math or anything else and the whole exercise becomes a waste.</p>
<p>I honestly have no recollection of what was taught in math between 5th and 7th grades. We finished learning our times tables in 4 grade and did algebra in 8th. In between it was a whole lot of spinning of wheels. So maybe fewer concepts per year would be better? I can’t imagine how the concepts could get any lower really.</p>
<p>“students will need to understand why 2 times 2 is 4” - obviously they won’t be teaching real analysis or group theory to 4th graders. From what I have seen this seems to mean drilling kids intensively on the associative, distributive and commutative properties. That seems to be the main conceptual/theoretical content that distinguishes common core math from what I grew up with. Which is a different thing from really knowing why 2 times 2 is 4, but still important.</p>
<p>I am conflicted on common core. On the one hand, I like the ability to assess schools by the same standard. Frankly, some states have very low standards and those kids are being done a disservice when they must compete against kids from states with high standards. That being said, I am having an issue with the methodology that common core seems to push.</p>
<p>My own educational background was extremely traditional until high school - the three Rs, rote learning of times tables, etc. Went to boarding school that was heavy in the Socratic method. After boarding school, college was actually quite easy.</p>
<p>Without those fundamentals that had been drilled into my head early on, I would not have been able to take those and figure out ‘why’ 2 times 2 equals 4. If I started off learning math by learning the theory behind it, I think I would not have had the same measure of success.</p>
<p>My take is that, until a certain level, memorization and basic fundamentals that are drilled into these kids heads year after year is a good method. Let them become a bit older before introducing abstract theories.</p>
<p>You’ve used a very bad example of the why concept. The 2x2=4 example is actually one where kids have always been taught the why. Imagine how difficult it would be to memorize 3x7 = 21 if you didn’t understand that you were taking 3 7’s and adding them together. It would be like memorizing apple x banana = dog. It is the understanding of the why that allows kids to memorize their multiplication tables. </p>
<p>Unfortunately, when the concepts get harder, too many teachers don’t insist that the kids understand what is going on. </p>
<p>A better example would be that you need to know why, when dividing fractions you multiply the fraction in the numerator by the reciprocal of the fraction in the denominator.</p>
<p>The reason that so many kids have trouble with math is that they are taught to solve specific problems by rote and as soon as you change the problem in the slightest they are unable to apply what they should know because they have only been taught to solve the specific problem. We have been doing our kids a disservice by turning them into direction followers instead of thinkers. Step 1, step 2, step 3 is not math!</p>
<p>This concept goes hand in hand with my comment about harder testing in a different thread. An easy test allows everyone to memorize the steps to solving a problem and then everyone can easily get a good score and everyone feels like they have accomplished something. A proper test forces kids to apply what they have learned to problems of varying similarity to what was taught. Perhaps two questions can be solved via the memorized technique. Another requires slight adjustment and the fourth even further insight. </p>
<p>Only then can you actually see what the kids understand, and where they have weaknesses.</p>
<p>My son had the “why” figured out at age 4, probably because he was in a Montessori environment working with manipulatives. The “why” of multiplication is very easy to demonstrate. It’s just a matter of understanding counting and grouping. When my d. had difficulty with the concept, I ripped open a package of dried beans and we played with those until she got it. </p>
<p>The problem that is being addressed by the statement about understanding the “why” of it is that children are often taught merely to memorize their multiplication tables without having any clue as to what multiplication is. They often memorize them by rote, even using songs and rhymes to help with the memorization – so they develop a language-based memory to draw upon without understanding the process behind the numbers. It doesn’t require the ability to do a proof or math analysis — I think you are definitely over thinking the problem.</p>
<p>I think the example of 2 times 2 is 4 is an oversimplification of the issue of teaching math based on theory first or basic memorization first. I am likely a minority in that I believe just flat out rote memory exercises first is beneficial. This, however, must be reinforced again and again. Once that child has those ‘basics’ memorized, they can delve into the whys and hows. </p>
<p>My daughter struggled with some math courses for this reason. Delving too deeply into theory was a mistake for her at first. Honestly, sometimes it’s about the plug and chug method so they can at least start to see the connections.</p>
<p>I think my point is that there can be a balance. Going too far one way or the other will likely lead to many students being frustrated and lost. My fear, based on the way I’ve seen math education moving in the past ten years, is that we are in real danger of slipping too far over into theoretical teaching as the primary method.</p>
<p>One of my degrees is in math and I’ve spent quite a bit of time tutoring. I find when I break it down to apply this formula to this type of problem and THEN explain the whys, the students click and get it quite a bit faster.</p>
<p>I guess one thing that’s always confused me is how other counties are so far ahead in early math. I lived in England and moved to America before 4th grade. In 5th grade I took Algebra, and in 11th I took multi variable calculus. </p>
<p>What are US schools doing that it takes until high school for some kids to take algebra?</p>
My personal opinion? Too much busy work. If you look at the K-8 and what the students are actually learning, there’s not much being added each year. I’m fine with redundancy in math because it helps to cement skills but honestly, some of the work my daughter brought home was nothing more than busy work. Perhaps it’s the result of teaching to the lowest common denominator in the classroom - something further cemented with some of the educational standards schemes we’ve seen presented as an effort to ‘reform’ education.</p>
<p>I am only familiar with the common core for math, K-3. I like it. I agree with Dreadpirit on this one. Children can certainly understand number at the simplest levels when it is presented at a developmentally appropriate way. The problem is that our teachers don’t know how to teach these concepts, but I would hope that the curricula that are developed to accompany the core concepts will help in that regard.</p>
<p>And I definitely see the value teaching fewer concepts but at a deeper level and with more understanding. I know it’s possible because when I taught first grade math in a small private school in Princeton, New Jersey in the 80s, I used a program that would have fit in with these standards perfectly. We introduced the number line, inequalities, the concept of tens, ones, and hundreds, we had students creating their own word problems around addition and subtraction. We even introduced negative numbers. These are not difficult concepts for six-year-olds if they are introduced and practiced using manipulative materials, although the lessons are not “easier” than what I had taught in the public schools previously, but they were better. They helped the students develop their mathematical understanding so that when they moved on to the next level, they were solid.</p>
<p>I also don’t see the problem with working in teams. The standards aren’t requiring that everything be done in teams- it’s just one method that can be used when appropriate. Again, well-designed curriculum materials can give teachers examples of activities where teamwork would advance the learning. Part of learning in the early years involves engaging the students, and building number patterns, playing number games using number lines or building patterns with blocks are all appropriate activities for primary students that could easily be team activities. </p>
<p>The core standards also don’t discount memorization entirely, either. I noticed that they included knowing their facts -addition, subtraction, multiplication and division- by the end of the appropriate grade levels. </p>
<p>I haven’t studied the core for the language arts, so I won’t comment, but if they are as thoughtfully put together as the early math core concepts, there should be a lot to like.</p>
<p>Gosh our kids learned why 2x6 = 12 by looking at egg cartons, or other arrays. They spent maybe a week talking about different ways to think about multiplication before they set into memorizing tables. What’s the big deal? I think it’s very helpful to know what multiplication is for.</p>
<p>I think the basic multiplication is too simplistic of an example. That’s easy to see and understand. I can remember trying to teach derivatives, for example. If you base that solely on theory as a teaching method, kids have a really hard time understanding it. Of course, taking the derivative is just about the easiest math problem out there and yes, they need to understand the why but they also need to understand how to get the answer too.</p>
<p>Children can struggle with the theoretical approach because you are learning more “stuff” at once. The tricky part is getting kids to understand that struggling is ok because when they get to the ah-ha moment, they will have more than they would have if they had simply memorized rules.</p>
<p>As an over simplification, if a child memorizes 2x3 = 6 it doesn’t help them when they encounter 2x6. On the other hand, if you teach that you are taking two 3’s and adding them together, then they will understand how to approach 2x6 = 12. Once they have that understanding then the memorization has context and is needed to make life easier and faster later.</p>
<p>The theoretical concepts are often more difficult (that is exactly why so many teachers go the plug and chug route). This is teaching to the test in the abstract. They are being taught how to solve specific problems, but as soon as you change the situation at all they get lost.</p>
<p>Take a slightly more complex example. If you teach that you find distance by mechanically multiplying speed by time, they can find distance. Then you could repeat the process two more times to show them how to find speed or time. Or you can teach that d = r x t on a more conceptual level and let them discover that they can find the missing one of the three. This is probably still a trivial difference in this example, but I think it better illustrates the issue.</p>
<p>Perhaps your experience is much different from ours. I have seen zero movement into the direction of theoretical. My kids are essentially done with high school and I’ve been “fixing” their math education since they were in elementary school. In 14 years with two kids we have had exactly one math teacher who wasn’t all about plug-and-chug.</p>
<p>Dreadpirit - It’s interesting because my daughter experienced the opposite. Her math education was so steeped in the abstract that she was genuinely lost and confused half of the time. It wasn’t until I showed her the plug and chug and THEN how that connected to the abstract that she ‘got it’. So, I am a bit gun shy about teaching in the abstract.</p>
<p>I am an enormous fan of taking the math they learn and then showing them the usage of that math via word problems or to a higher level, physics. I think they can take what is seemingly a bunch of different equations and concepts and then figure out how to actually use them. We perhaps are not completely on different pages. I think it’s just methodology that we may differ. I favor learning the basic equations/theorems and then connecting it to the abstract after.</p>
<p>southbel - I think we are pretty much in agreement. My experiences were such that I couldn’t ever fathom a teacher being that theoretical, but as you show, anything is possible.</p>
<p>Dreadpirit - Oh it was terrible going through it with my daughter. The math curriculum in her case was one of those ‘hey I’ve got this great idea’ things from an administrator somewhere in the district office. Honestly, I wonder how other kids managed without having a mathematician mother. I know lots and lots of parents in her school employed tutors - well I was one of those tutors. It did get better once she hit high school but really, there needs to be a balance and I am leery of any program that is abstract based for this reason.</p>
<p>Best lesson I ever taught my tutored kids and my daughter was how to read word problems and figure out how to make an equation from it. I’ve literally had kids thank me for that later on because, for some reason, this skill is not taught well at school. I think they believe the kids will just figure it out on their own. Pshaw. It needs to be taught.</p>
<p>I like the Common Core. Really, it is not all that different from our previous mandated state benchmarks. Knowing “why” and being able to explain has been part of our state benchmarks for years. </p>
<p>It’s like anything else…it’s relatively new. Change takes time.</p>
<p>Plug and chug kids can’t translate a word problem either if it doesn’t match one of the n identical word problems they covered in class. </p>
<p>Maybe I’m using misleading terminology when I talk of teaching the theory.</p>
<p>Something the common core stresses is the ability to “think” mathematically. I think that your example is exactly where our kids are getting short changed. It is an over simplification to look at it as a continuum between plug and chug and theoretical.</p>
<p>In the end, no matter the curriculum, teaching math requires people who understand math and understand that teaching kids to think involves a lot more than teaching them how to solve a prescribed set of problems.</p>
<p>As an elementary math coach I am very excited by the possibilities if the Common Core math standards. I am concerned that numbers of teachers aren’t yet accessing the mathematical pedagogy. They are used to teaching algorithms, not multiple approaches like use of decomposition, area modeling, and number lines that show understanding of properties. Lots of training is needed to pull off this shift in instruction.</p>
<p>^I thought our school’s math programs (and there were many!) did a pretty good job of getting kids to think mathematically as well as to actually memorize the facts. That said, sometimes by showing so many ways of thinking about something relatively simple they actually confused kids. For example - they showed kids how to do lattice multiplication. I think the only kid who thought it was cool was mine, a first grader in a third grade class, who really liked different ways of looking at numbers and who read math theory books for pleasure. Learning how to do word problems (at least the ones that have some applicability in real life is enormously important.) How many gallons of paint do I need if a can will paint 500 square feet and the room is 12’ x 10’ x 8’ high? If the storage rental is $20 the first month and $10 a month after that at one place and $30 for the first month, but $5 a month after that when should I choose place number two? Not so much if Joe is twice as old as Mary is and in ten years he’ll be three times older sorts of problems.</p>
<p>I think, in the end, we may agree Dreadpirit. However, it will, as it usually is, fail or succeed in the execution of the idea. </p>
<p>Like I said before, I do not have an issue with Common Core. Our state has been using the same basic thing for years now. I am just worried about the execution of it. I’ve seen firsthand when it is bad, it is very bad.</p>