We recently had a heated debate on the blog about math instruction. There were more “sides” than I can count: pro-Common Core, anti-Common Core, pro-constructivism, anti-constructivism, pro-memorization, anti-memorization. I may have missed a few camps.
Here at last are some straightforward answers about how to improve mathematics education. Why wonder, why wander, why flail around when the answers are right here in this post.
Diane Ravitch's blog 
My Favorite Made Me LOL:
Buddy up Alabama with Massachusetts
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A few random thoughts:
Certified elementary math specialist at K to 6.
End Convoluted Common Core math. Over-complicating math is doing all harm and no good.
Create a developmentally appropriate scope and sequence K to 10.
KISS! Stop over-complicating simple ideas with ridiculous numbers and un-real applications. Use the simplest numbers in all computations and problems while concentrating on the concepts.
Eliminate all current textbooks – way to busy for kids
Mandatory industrial arts classes. Designing and building real objects that require measuring and calculating.
Students should rarely write a math answer without a unit label
Math educators need a strong background in science, technology, and basic engineering. Very few math teachers have any capacity for explaining or describing real world applications.
Adopt a lab driven approach similar to science.
Offer theoretical mathematics as an elective
Require statistics in high school
Do not grade elementary math students
STOP requiring students to explain in writing how and why they arrived at an answer.
Showing their work answers this appropriately.
Create new ways of developing number sense
Students must memorize common multiplication/division facts, common adding/subtraction facts, common decimal equivalencies, and common squares, and cubes.
Stop ignoring the simplest and most practical algebra: solving by proportion
Stop identifying/labeling so-called “accelerated” math students in 6th grade
Stop making math the be-all and end-all in K to 12. Millions upon millions of adults who were not successful in math are leading happy, fulfilling, and productive lives despite their numeric illiteracy.
Accept the fact that we now have on-line calculators that can instantly solve any math, engineering, or physics problem – including any formula – ever presented.
Make metric measurement an integral part of the math curriculum
STOP Common Core math NOW! Good standards would never have so many “implementation” problems.
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I’ve read that in Chinese elementary schools, most subjects –including math –are taught by subject-area specialists.
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I agree with many of your points, but am confused by others.
Why shouldn’t students be labeled accelerated? I guess I’m biased because I was labeled and was able to finish through Differential Equations and Linear Algebra while still in high school. I was also gifted in chemistry and finished Gen Chem 1, 2, and Organic Chemistry 1 and 2 while still in high school. This allowed to to graduate college 1.5 years early. I think if there are students with this capacity, let them shine (and there are many).
Math isn’t the end-all, be-all in k-12 education, nor is it in life.
We do have online calculators, and in engineering, we have a lot of software that calculates information we need; however, this doesn’t mean we shouldn’t learn the actual concepts in depth, or do formal mathematics. People who design these softwares/calculators need to understand mathematics…also, people who are reading the results should know what the results are saying (and understand it at a deep level) to explain it to others who may not.
Eliminate current textbooks? I disagree. I think they should seriously be revamped, however 🙂
“use simple numbers in computations while focusing on the concepts”…I agree with this, but there’s nothing wrong with giving problems with “harder” numbers to work with as the student progresses through the problem set. Once they have the concept down, they know how to work with the “harder” numbers…put it together.
I do agree with making math more developmentally appropriate and some of your other points.
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Good list, totally agree.
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I think you’ve summed up my whole experience with test-based school reforms in that one statement: Good standard would never have so many “implementation” problems!
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“Students should rarely write a math answer without a unit label.”
Which contradicts your requirement for simplicity. One of the problems with CC is exactly the insistence on lots of word problems.
As we discussed in another thread, the utilitarian education of Victorian England you seem to be favoring for math was already criticized by Dickens 200 years ago.
Math is coming from the world around us, yes, and it can be applied to the world around us, yes, but math is not the world around us. It’s simpler, cleaner. Kids instinctively understand this when they ignore unnecessary fluff like units and just deal with numbers and geometrical objects. Let them do that.
Similarly, let kids paint, draw pictures, make sculptures or wooden boxes without worrying about the real life applications of their art in business and economy.
I am not saying, kids shouldn’t see how their math applies in practical situations. No, it’s actually one of the striking things about math is that one can draw conclusions, make predictions about the real world just sitting at home and using paper and pencil and brain cells. But bombarding kids with word problems as if solving them was the main justification for math is crazy. As crazy as suggesting that math should always be taught by specialists.
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A bit too concrete for me but I like many of your points, especially ditching Common Core. One of my math profs constantly bemoaned the weak skills college math and engineering students have with abstraction, proofs, and logic. It remains my challenge that I am not as good as I’d like with math theories. Math teaches a way of thinking and there are many ways of doing this. Numbers and calculations are secondary. One complaint I hear from business, besides lack of soft skills in employees, is the need for independent problem solvers.
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Word problems. Math existed before words, so in its purest form, math should be done with as little talking as possible.
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I think the smudge on the isosceles triangle transparency is probably the main culprit.
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I love the straightforward answers on the onion – LOL!
I have two mathematics degrees (bachelor and master), taught as a high school mathematics teacher in the inner city for 4 years (worked with at-risk youth before that, teaching mathematics and chemistry – chemistry is my undergrad. minor), and am now an engineer. I don’t plead to know everything, but I do have my experience and my personal beliefs.
1. Our current mathematics curriculum is much too “wide” and not “deep” is a true statement. Our books have many chapters of “spiral review”, and plenty of “review” material within chapters of new content. Get rid of the spiral review and have review topics as a separate, supplemental aid during those lessons students need review. This will allow more time to learn new topics more deeply and allows more time for mathematics labs (which can be so fun for students), exploration activities, USEFUL projects, and in-depth discussions.
2a. Related to the first topic, our mainstream mathematics textbooks are very watered down. I’ve seen two things: (a) books that claim to require less problems for mastery that are “rigorous” and promote mathematical thinking, and (b) books claiming the model “the more practice, the better”, and therefore have many problem sets for each lesson, and even extra in the back of the book. With case (a), the few problems available are not enough, and don’t promote “mathematical thinking” at the level our students can reach. Also, mathematics needs to be practiced. I don’t believe in the 7 problems of homework each night to learn mathematics (refer to my background); however, I don’t believe in 50 problems each night, either. In case (b), these quantity of problems available for practice is fine, but the quality isn’t. With these books, the difficulty of problems may get increasingly more difficult, but the problems tend to only mirror the examples and don’t require students to think deeper and connect other material. Scaffolding the problem sets is great. These problems are like example 1, example 2, etc., but then other problems requiring more than just “this lesson’s” material is the point of mathematics, problem solving, etc. My thought: With less of a spiral review, students can focus more of their energy on quality problem sets with many problems of increasing difficulty to choose from, as well as problems that challenge their true mathematical problem solving skills, and true REAL-world applications.
2b. Provide many high quality example problems in the textbooks. Currently, there may only be a few examples, but don’t necessarily scaffold learning enough for students/parents. We need to see more high quality, scaffolded examples for reference (and even for teachers who may be teaching mathematics, but may not have the background to teach it). Of course, teachers can bring this to their own students (as I did), but the book is a great resources, so it may as well be great!
3. Let’s talk about the real-world applications. I’ve seen many teachers not teach them (perhaps it’s the time, quality,…, who knows?). I think some these real-world examples in the current texts are fine and I wouldn’t get rid of them (I would refine others, though…some of them are very bad). I would also add more real-world and applicable “word problems” to these texts, as it’s fun for students to see applicability, but we should also keep in mind that it’s not always about being applicable to a student’s life; there’s a glory in learning abstract material in both abstract and concrete senses…this is learning.
4. If students aren’t ready for a course, don’t promote them. I’ve seen this all too often at the old high school I worked for (and even the three high schools I, myself, attended). Students with grades of “D”s and “F”s were promoted to the next course, say, Algebra 2, who weren’t successful in Algebra 1. This is setting our kids up for failure. I have also seen many times students come from Algebra 1 courses that were so watered down (these students had decent grades, too) that they were ill-prepared for their next assigned mathematics courses. This happens more than I think we would like to admit (especially in urban schools) and it sets our kids up for failure and sets low expectations.
5. Stop inflating grades. I recently was speaking with a teacher who had 50% of their grade go to homework, 10% toward tests, and 40% toward projects. She explained that her principal was tired of the high failure rate in mathematics and mandated this grading policy to reduce the failures. This mentality is not helping students. Instead, let’s offer other supports (such as lab courses and tutoring, etc.). I know money is a factor, but let me dream here!
6a. Give students options. Some schools may fare well with a more “core” approach, where Algebra, Geometry, and Prob/Stats are combined within each years’ curriculum in grades 9 – 12. In other schools, perhaps requiring Algebra, Geometry, and Algebra 2 (as they do in Michigan) separately is the best policy, but also requiring, say, 1 semester of Prob/Stats, etc. There’s too much overlap in these courses, but the material is presented extremely disconnected. For example, in Algebra 1, students have an extensive spiral review of middle school topics, then learn important Algebra topics, and then given prob/stats at the end of the book. WHY? Remove these extra topics, beef up the problem sets and examples, and refer to #1, 2, and 3 above for the rest.
6b. If students have to repeat a mathematics course due to failing, poor performance, etc., don’t enroll them in an online version of the course (especially if the school’s mathematics courses taught by teachers are very good). Depending on the course and the company it’s offered through, they tend to be very inflated, watered down, spiral, etc. I have known many students who were very successful in these courses who, truly, shouldn’t have been. Don’t kill me for saying this, but we all know what I’m talking about. We’re trying to increase graduation numbers, which is fine, but not at the expense of learning. BTW: one student I’m referring to gave drugs to another student to take this course for him. Win, win?
7. This is just my $0.02. Again, I don’t know the answers, but I do know this is how I want my own children to be taught mathematics and I won’t settle for less 🙂
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I went the other way from engineering into math instruction (our state is decidedly low tech – the largest employer is Walmart and I have family obligations here right now). But math, science, and engineering are still a hobby. I think I do a good job. I’ve had kids entering the class hating everything about me and ending the year with a thank you, handshake, or hug (is that allowed?). But our state has made it impossible to teach math with the anti-teacher test and punish, so it is likely I’ll be transitioning into something else soon. Ohio doesn’t want good teachers, just cheap, pliable teachers.
1. I do like separating spiral review. Spiral bores the kids that already understand and confuses those that do not. We need extra support and aides to keep all kids on track or gaps in understanding do develop, causing failure to snowball.
2. A guy at Penn State (Riccomini) had spoke to us about an interleave strategy – examples, problems, examples, etc. I have applied his suggestions to my classroom and it works reasonably well. Here’s where I do think HIGH QUALITY online instruction can supplement (not supplant) the classroom as homework. Lots of practice but with feedback. And not the abused “flipped” classroom concept, either.
3. Real world applications are good and follow Bloom’s, but they can seem contrived. But we are trying to teach thousands of years of math knowledge in 12 years. This remains a challenge. Often, really interesting real world problems don’t present themselves until after analysis or group/ring/field study. But it is our job to try to connect.
4 and 5. I agree, but this is not up to math teachers anymore, especially if lacking due process. Parents understandably see grades as college cash plus have, at times, unrealistic expectations. Many schools pressure teachers to inflate grades or set quotas.
6. Too true. It makes no sense for a music major to go through agony over pre-calculus anymore than forcing a future engineer to take orchestration. The goal of math instruction is to teach a way of thinking and problem solving. That can be accomplished many different ways.
7. More like $10 worth! I appreciated you comments as part of a discussion. I encourage you to support math teachers and vote for candidates that support schools, teachers, and learning. Then you do not have to settle at all.
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MathVale,
I always enjoy reading your comments on Diane’s blog. Thanks for the response to my comment! Totally agree!
Honestly, Michigan (we’re close) doesn’t want great teachers either. Just like in Ohio, we want the cheap, pliable ones too. Thank goodness there are a few higher paying, good suburban districts in the Metro Detroit area. Unfortunately, these districts, too, are under pressure from our test-and-punish legislation and are declining in quality.
I love your comment for #2. Thanks for pointing out the HIGH QUALITY SUPPLEMENTAL (not supplant) online instruction note. So true!
For #3, they are TOTALLY contrived! It’s funny that the kids notice it too…when the kids spoke up about them once in a while, we usually made our own real-world problem up (in my past life as a teacher, of course). So true about Abstract Algebra 🙂
Thanks for your comments on #7! My full support for this profession goes without saying :). I would leave engineering in a second and go back!
Good luck in your transition! It’s a shame to lose another great teacher – I can totally tell in your writing and knowledge the passion you have 🙂
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As a non-math teacher, who has a kid who struggled in math, may I add one more suggestion? Stop treating all kids as if they’re going to major in math or engineering. My son agonized through all kinds of esoteric math that neither he or I could understand (and I have pretty good number sense). Calculus concepts introduced in 9th grade, high-level trigonometry, things that he will NEVER use. This year, he’s finally being able to take a Math of Personal Finance class, and he’s suddenly getting As, because it’s material he knows he will actually use. Ten years of being told he’s a “failure” in math–and what he needed was material that he could actually USE. I’ll bet a lot of other kids are in the same boat as my son.
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Threatened Out West,
I totally agree! That’s why I believe we should have more options in our schools. This is also why I believe we should have less breadth and more depth! This will help students who struggle keep up with unnecessarily fast paced courses! And it’ll give students more time to learn abstract mathematics and connect topics to the real world instead of rushing to learn rules and drill, drill, drill.
Like you said, not every student will be an engineer or math major. We need to reach everyone. This is one of the domino effects of test and punish – schools require students to take many pure math courses to do decently on the tests (and it’s not even working).
I think every student should be required a business/personal finance course!
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” Stop treating all kids as if they’re going to major in math or engineering. ”
Exactly. Math should be taught in a non-utilitarian way. teaching rigorous proofs is a mistake (which is what mathematicians suggest) and teaching only household and engineering applications is a mistake too.
Most of the trigonometry and calculus in current 8 lbs textbooks is used absolutely nowhere: not in math, not in engineering, not in physics. They are used to build expensive houses for their writers
http://www.wsj.com/articles/SB123872378357585295#project%3DSLIDESHOW08%26s%3DSB123869600484183257%26articleTabs%3Darticle
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Funny! Thanks for the chuckle. Why we don’t have competing standards for math teachers to synthesize an instructional approach is beyond me. I have faith math teachers could do it better than faceless bureaucrats. Math is beautiful, axiomatic, and elegant – sort of like Latin only with calculators.
Can I add a few?
– Have kids glue on 6 extra fingers and switch to hexadecimal.
– Make slide rules in neon colors and use product placement in Taylor Swift videos.
– Embed microchips into all adults that deliver a small shock upon detecting the phrase “I hate math”.
– All online math videos must first be tested to ensure they do not scare small children and pets.
– Any principal evaluating a math teacher must not be a former phys ed teacher unless they can prove Pythagorean Theorem Likewise, any coach must not be a math teacher unless they can name all past Superbowl MVPs in chronological order.
– One of the new Avengers must be a math teacher with really awesome super powers and look like Matt Damon or Jennifer Lawrence.
– Adopt friendlier math terms like “number buddies” for associative principle and “Math-libs” for two column proofs. Banish the term “latus rectum”.
– Contribute $1 to the early retirement fund of every math teacher who is asked “when am I ever going to use this?”.
– Pass a law that any politician must first calculate the exact value of pi before being allowed to dictate instructional policy in math classrooms.
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I love these! Thanks for adding them! LOL
For your last point, we’d hear, “I’m successful and haven’t ever had to calculate the exact value of pi…I don’t even remember what the numbers for pi were”
This is today’s mindset, no?
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Lol! Yes. I saw a great episode of “Person of Interest” where the one character was substituting in a math class. He drew pi on the board and told students “every book you have ever read or will ever read, every book written or will be written is in the number pi”. Talk about critical thinking skills…..
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Switch to base two, less numbers to learn.
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Can we get back to “borrowing the one”?
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Yes! And the old joke “there are 10 number of people in the world that understand binary, those that do and those that don’t”.
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Thanks, I liked that. Sent me to wiki, “When Noah sends his animals to go forth and multiply, a pair of snakes replies “We can’t multiply, we’re adders” — so Noah builds them a log table.”
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“Eliminate all current textbooks – way to busy for kids.”
Texts seem to be designed by mathematicians who lacked an understanding of literacy, intuition and counter intuition, or novice minds. I could not teach my students in resource how to read the text to find samples that would assist them in solving the problems the text provided.
I had never clearly understood what my algebra teacher was explaining. However, with the support of a readable text, I could “teach” myself when doing homework. Not so with current textbooks.
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Today’s textbooks are designed by marketing teams who through everything and the kitchen sink at a page. They have so much junk on each spread that it is next to impossible to follow a train of thought. (Mathematics involves following a train of thought.)
Our textbooks today look as though they were designed by gerbils on methamphetamine.
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Yikes. Typing too fast. “Who THROW in everything and the kitchen sink”
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“Texts seem to be designed by mathematicians who lacked an understanding of literacy, intuition and counter intuition, or novice minds.”
Lower level math textbooks are mostly written by people who want to make money (perhaps justifiably, since profs’ and teachers’ pay are getting worse and worse).
While your suggestion to write new kind of textbooks is welcome, since very few good ones are available in English, the task is much simpler than it appears, since there are great math textbooks in other languages.
In English, here is a Calculus book which is under $20, and hence is never used anywhere
I think it does the right thing: it makes math really intuitive by developing the concepts from the world around us, but doesn’t crowd and bore the reader with applications and millions of similar problems.
It’s thicker than it really needs to be, because it has all three semesters of college calculus in it. But it’s quite easy to extract 50 or so pages that could be taught to any high school student. Yes, any student.
It’s another matter, that I haven’t succeeded in finding any good Algebra I book in English.
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“Our textbooks today look as though they were designed by gerbils on methamphetamine.”
I’ve been saying that about foreign language textbooks put out by the major publisher for over twenty years. It’s almost impossible to follow any logic through the course of a chapter, unless you already know what is in it. More like shrews on psychedelics!
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“Our textbooks today look as though they were designed by gerbils on methamphetamine.”
Literature texts are like this as well. I finally ended up designing a custom made one for our 11th grade this past year. Just the literature is in it. Teachers can create their own assessments, graphic organizers, and so on, based on the composition of the class in front of them!
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So many superb ideas on this page! We are very, very far away from empowering teachers to make changes that make sense for their students, alas.
Here, the best thing I have ever read about math instruction, Paul Lockhart’s “A Mathematician’s Lament”:
Click to access LockhartsLament.pdf
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It really is great reading. If teachers would supplement it with concrete descriptions of how to break down feed discoveries in small enough steps so that all kids (not just the smart ones), we’d finally have a real math education not just “household calculations” and “fake engineering applications” courses.
Do you know where this aricle first appeared?
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Delete “break down” to make it comprehensible what I wrote.
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OK, so what I wanted to write was
If teachers would supplement the ideas in the article with concrete descriptions of how to feed mathematical discoveries in small enough steps so that all kids (not just the smart ones) could participate without frustration, we’d finally have a real math education not just “household calculations” and “fake engineering applications” courses.
This would require much more time on each topic than traditionally used. But then, there’s really much less material to teach than usually is assumed.
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Here’s a general observation form the article of Lockhart.
“Teaching is a messy human relationship;
it does not require a method. Or rather I should say, if you need a
method you’re probably not a very good teacher. If you don’t have
enough of a feeling for your subject to be able to talk about it in your
own voice, in a natural and spontaneous way, how well could you
understand it?”
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When do students learn how to calculate interest on a loan?
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They of course learn it in school. It’s just not a major issue on which kids should spend a lot of time on, solving dozens of home work problems on all possible variations of it.
The reason why they would learn to do this would not be because “it’s important to know when you grow up” but because the argument leading to the formula is common to many other fundamental laws in nature, finance, psychology, agriculture.
This philosophical difference is important: the purpose of math education is not to teach the basic applications of mathematics. It cannot possibly be done, plus what is important for you is unimportant for others. It’s more important that kids see where mathematical concepts are coming from, and they recognize some basic situations where these concepts appear.
The reason for this philosophy is twofold: first of all, what kids find interesting and important is very different from what adults think is important. Second, if kids learn the unifying nature of math, they will have no problems with recognizing and learning new applications of it.
If they learnt mathematics in a strictly utilitarian way—as it’s actually mostly done in the US—they’d be bored to death plus their problem solving abilities would be restricted to those problems they already covered in school. Which is the exact situation now.
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RageAgainstTheTestocracy stated:
“Do not grade elementary math students”
Just as “grading” doesn’t work for ration-logically determining/stating the quality of a school, district or teacher, it certainly doesn’t work well at all for stating/determining what a student is/has learned.
Grades are an abomination of education, a true educational malpractice that demonstrably harms many innocent children with a label of “failure” (that f#$k$%g “f” word again).
“Hi, I’m Timmy and I’m a 9 years old and am a f#$k$%g FAILURE, even though I tried my damnedest to do my math and have learned many things, just not enough to not be considered a f#$k$%g FAILURE! Screw math from here on out!”
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