This is a guest post by Mark Webster, continuing his tradition of guest posting for me during Blogathon. He is a graduate of Purdue University from the School of Science in the field of Mathematics Education and is currently a High School Mathematics Teacher in Indiana.
Please note that the author is representing general trends and personal experiences of a trained educator combined with popular evidence-based practices. By no means is this exhaustive. Please do not be butthurt. If you have evidence-based practices that conflict with anything I have said, please feel free to leave a comment.
There has been a lot of discussion on the problems in education, in general, but never do you hear a bigger cry for change in any other subject than you do in Mathematics. Perhaps it is time to analyze the problem and line up some solutions.*
Who are the problems in Math Education?
Teachers are certainly not the only problem, but when deciding to figure out the problem, it’s always best to start inward. In this post, I will be looking at what teachers need to focus on.
In our schools, there is still a lot of passive learning going on. What is passive learning, you may ask? Let me answer your question with another question:
When you imagine a math class, what do you think of?
Probably something like this:
**
A teacher, facing the board, not interacting with his students. Many of us, myself included, have had experiences like this. No teacher-student interaction. No checks for understanding. No eye contact. Perhaps, even more pernicious, there may not even be an analysis of the learning and long term progress of students.
Direct lecture-style math classrooms create an environment of passive learning. The teacher says a bunch of words at the front of the board; maybe, if he is a more dynamic teacher, waves his arms around a little bit; and then throws quizzes and tests at you. (Multiple, if you’re lucky—on the college level, there is rarely regular assessment…but that is neither here nor there.)
Even worse, the math classroom suffers from a lack of student metacognition and critical thinking—an ailment in a math classroom that baffles me to no end, particularly because that is, more often than not, the go-to excuse that teachers trot out when a student asks them “When are we going to use this?”
Rarely is the question asked, “Is our children learning?”, the answer to which is, more often than not, either “No.” or “Not well enough.” Now it is time to move beyond that question to “How can we help them to learn?”
How will the Common Core Standards help?
One of the country-wide initiatives that will be taking hold in the coming years, the transition to which will be complete in 2015, is called the Common Core Standards. While there are standards for Math and English, for obvious reasons (Hint: I’m a math teacher) I will be only talking about the Math standards.
1. Stricter Math standards for the USA.
If you look on the national report cards, you might notice grades for the states are changing. Of the 50 states, only six of them have not bought into the common core standards. If you or your children are students in any state besides Washington, Alaska, Texas, North Dakota, Nebraska, or Virginia, you will have (hopefully) heard of the change. A common set of standards across the country will mean that students are learning with the same level of rigor and relevance in Indiana as they are in New York or Mississippi.
2. Increased focus on Critical Thinking
In my professional development and my own personal research on the PARCC exams, I’ve come across the same thing over and over: The standardized tests of the past will not go away, but they will be refocused. Instead of fifty “Math Problems” on a test, the student may be given five or six “Math Tasks.” These tasks may involve, on the elementary school end, explaining the purpose of a step within a problem that they have completed for you, finding errors in simplifying a problem down and explaining what the error is, or even taking a newspaper article and analyzing it to take a position, using evidence. Instead of focusing on finding an answer, the test will be concerned with how they can apply math to that answer. How wild!
3. Broader, more targeted learning objectives
Most state standards and assessments have, in the past, been far more focused on students simply demonstrating their knowledge of a process, i.e. “Plot these two points and determine whether the slope is positive or negative.” Common Core standards are far more broad and far more targeted. Much like the National Counsel of Teachers of Mathematics standards, the Common Core standards concern themselves with specific domains of learning and applying them to metacognitive tasks.
Now, instead of plotting those points on the exam; they may, instead, be given a question that asks them to analyze, display, and track profit margins for a company and take a position on whether or not they will be able to afford to stay open in five years.
The accountability is changing from teaching mathematical processes to teaching thought processes. I approve of this, but does this really address the problems that we have seen above? I don’t think so.
So…How Do We Do This?
There are many champions of new processes within the Math Education community. Many of you have heard of Salman Khan and his famous Khan Academy, fewer of you have probably heard of Dan Meyer***—some Math teachers even haven’t (A fact which breaks my heart.), and I’m sure even less of you know about the conflict of pedagogical styles that lies between them.
Let’s break them down:
Dan Meyer’s pedagogical philosophy, a project that began as WCYDWT? or What Can You Do With This? and has slowly morphed into something he calls Any Questions? is designed as a student-centered, inquiry-based, generally collaborative effort to force students to lead the discussion and gain ownership of the material by creating the payoff in the medium used to teach the material; and, instead of spoon feeding them concepts, force them to push through a mathematical task and create the demand for the material.
Salman Khan’s pedagogical philosophy, a project that began as a series of youtube videos, has become a website, and some might say a school, of its own. The Khan Academy allows students to develop on their own, facilitates continuous improvement for the students at their own pace and on their own terms, and allows for constant pedagogical moderation.
Math teachers and parents alike have raised concerns about the methodologies that these two men have created. Dan Meyer critics fight him because they believe his philosophy creates a lack of rigor. Salman Khan critics fight him because his videos do just as much lecturing as one might see in a classroom, and do nothing to enrich and create ownership of the material.
Even with the miles of space between their two philosophies, it is worth the time to compare them and see a similarity:
These philosophies depend on a consistent foundation of what has come before. The genius of Dan Meyer’s method lies in the students being able to work through the tasks because they are absolutely prepared to tackle it. The utility of Sal Khan’s method is that because students get to a topic on their own terms, they are prepared to see it and they can meet it head on.
Without constant analysis and moderation of our students’ learning, we cannot teach to our fullest potential. If we leave students by the wayside because we don’t know where they are, we have put a student in a hole that they may not be able to escape from.
*I’m by no means the first nor am I the most qualified person to look at this. We’ve been overhauling since before I started teaching, but one more eye on the problem can’t hurt. Even if I’m not saying anything new, informing new people can’t hurt. Right?
**http://www.canyonville.net/wp-content/uploads/2011/07/Math-Class-620×270.jpg Disclaimer: This picture is not necessarily an indictment of this teacher. It is a photo that I found on the internet with a teacher that had his back to the classroom. The fact that it took me about five minutes to find one is encouraging to me.
***Full Disclosure: I worship the ground upon which this man walks. The classroom ideal he has created is my personal mission.
This is post 28 of 49 of Blogathon. Donate to the Secular Student Alliance here.
I have often heard math professors bemoan the fact that students aren’t learning, but it always seems the focus is on the method, rather than what, is being taught.
I am thoroughly convinced that the reason students are being lost is that they are, in fact, getting lost. Take Calc 1 for example. Often, the professor will begin with limit theorem before getting to derivatives, and onwards towards integrals, without a single concrete example that connects it to the real world. Give the students a picture, something tactile they can hold, and I feel you will get a better response. If, instead, we started Calc 1 looking at the simple stone dropped from a cliff example, they have a basis from which to connect ideas.
Now, I want to distinguish that I am not suggesting a truly experimental approach; more that we actually learn from the College of Education, and try to build on our students’ intuitive knowledge rather than give them several ideas which appear disjoint to them, and expect regurgitation without connection. I think, in short, most have forgotten what it was to be a beginner, and to sit on the opposite side of the desk.
There’s nothing in this guy’s brief bio to indicate he has a degree in math. He also seems unable to express simple ideas or more complex ones in simple terms. He speaks in jargon inappropriate to his intended audience.
He’s a high school math teacher, yet didn’t say a word about high school math; his only examples were about elementary school and university. He never says what happens to the bright kid who’s most likely bored to death with the ever-so-slow group-think approach. Nor does he tell us about the not-so-bright or passive kid who just follows along letting others do the exploring.
There’s zero discussion of what he thinks the end result of high school math is supposed to be, or whether any of the methodology he discusses actually achieves it. It’s on and on about how to teach but close to zero on what needs to be taught.
Moreover, all this stuff about practical applications in math class makes me wonder if this guy admits the existence of other high school classes. Hey, long ago, when I was but a mere slip of a lass, math teachers taught the concepts and the mechanics while the applications came in general science, business, physics, chem, etc. I mean, teachers of those various subjects actually talked to each other and even coordinated the overall learning process. This guy seems to have missed that boat.
Richard Feynmann would have torn this guy to shreds!
I remain convinced that far too many high school math teachers only more or less know the math they’re teaching, know little about where and how the math is used, and that damn few of them understand the material they’re teaching. They’re thus unable to generate understanding and enthusiasm in students.
Based on his article, I’m glad this guy isn’t teaching my kids.
>>There’s nothing in this guy’s brief bio to indicate he has a degree in math.
Why do I need a degree in general Mathematics?
>>He also seems unable to express simple ideas or more complex ones in simple terms. He speaks in jargon inappropriate to his intended audience.
What jargon did I use? Perhaps, the only acronym/insular word I used was PARCC which is the exam that the Common Core Standards are using. I figured that was clear? I apologize if it wasn’t.
>>He’s a high school math teacher, yet didn’t say a word about high school math; his only examples were about elementary school and university.
My statement about the level of the elementary exams was to show that, even in the elementary schools, they will be seeing that high level of metacognitive/critical thinking process required at the youngest of ages. The types of tasks that will be given there are equally applicable at all levels. The only distinct difference is that in the higher-level classes we might see, say, a log or exponential regression instead of a linear regression.
>>He never says what happens to the bright kid who’s most likely bored to death with the ever-so-slow group-think approach. Nor does he tell us about the not-so-bright or passive kid who just follows along letting others do the exploring.
The former is eliminated in both instances because every student, in *both* cases, meets the problem at their own level. The latter is eliminated through the use of monitoring of the classroom via consistent and frequent assessment. Were you paying attention? Are there any straws left for you to grasp at?
>>There’s zero discussion of what he thinks the end result of high school math is supposed to be, or whether any of the methodology he discusses actually achieves it. It’s on and on about how to teach but close to zero on what needs to be taught.
There was never any dispute as to WHAT needs to be taught. The whole ARTICLE was on *how* to teach! Apparently there were a few straws left. By presenting Meyer and Khan as two people who are making innovations in Math Education and by specifically telling my audience that there are parts of each that I think are important, I’m suggesting that this is the direction that Mathematics Education needs to be; and, indeed is, heading toward. Maybe I should have done an article on close reading instead.
>>Moreover, all this stuff about practical applications in math class makes me wonder if this guy admits the existence of other high school classes.
Remember that part where I said “I’m a math teacher. I’m going to be focused on the pedagogical aspects of math?”
>>Hey, long ago, when I was but a mere slip of a lass, math teachers taught the concepts and the mechanics while the applications came in general science, business, physics, chem, etc. I mean, teachers of those various subjects actually talked to each other and even coordinated the overall learning process. This guy seems to have missed that boat.
And now that teachers have actually talked to each other and have coordinated the overall learning process, they have realized that it is possible to supplement the learning of those various subjects *within* each others’ classes! How awesome it is that we can foster students that can see parallels between the thought processes in Math and English! How fantastic that they can utilize dimensional analysis to figure out that their calculations are correct without even needing to crunch a single number! Why would you be against this?
>>Richard Feynmann would have torn this guy to shreds!
Yes, well he’s dead now.
>>I remain convinced that far too many high school math teachers only more or less know the math they’re teaching, know little about where and how the math is used, and that damn few of them understand the material they’re teaching. They’re thus unable to generate understanding and enthusiasm in students.
Well I spent most of my college experience studying Topology, Group Theory, and Real Analysis…so you can be convinced all you want. Enjoy your delusions. I spend much of my time in class discussing things like the parallels between the Pythagorean Theorem and Fermat’s Last Theorem, showing them NBG and ZFC Set Theory, and take the time to foster that delight and curiosity in my students.
>>Based on his article, I’m glad this guy isn’t teaching my kids.
I’m sorry you feel that way. Enjoy your AIDS.
>>I have often heard math professors bemoan the fact that students aren’t learning, but it always seems the focus is on the method, rather than what, is being taught.
One of the big problems about how we teach the content is that we spend so much time repeating ourselves in class that we don’t have time to actually go in-depth and foster a *reason* for the material. One of my college professors was on the team that wrote the Connected Math Program. It does, in my opinion, a wonderful job of presenting tasks to students to develop the mathematics *once*, in a way that will stick, at the exact right time. More than anything, I think, at this point, it’s about timing.
>>I am thoroughly convinced that the reason students are being lost is that they are, in fact, getting lost.
Right on!
>>Take Calc 1 for example. Often, the professor will begin with limit theorem before getting to derivatives, and onwards towards integrals, without a single concrete example that connects it to the real world. Give the students a picture, something tactile they can hold, and I feel you will get a better response. If, instead, we started Calc 1 looking at the simple stone dropped from a cliff example, they have a basis from which to connect ideas.
This is great. I agree with you 100%. Motivation, particularly for primary and secondary students needs to be palpable. Dan Meyer does a really great job doing just this sort of thing.
>>Now, I want to distinguish that I am not suggesting a truly experimental approach; more that we actually learn from the College of Education, and try to build on our students’ intuitive knowledge rather than give them several ideas which appear disjoint to them, and expect regurgitation without connection. I think, in short, most have forgotten what it was to be a beginner, and to sit on the opposite side of the desk.
Preach it! Thanks for your comments!
When I was nineteen years old, many of my classmates still had no concept of natural numbers. That *** is always *** regardless of calling it “three” or “four” was a source of confusion. Teachers never addressed it.
I have been a high school math teacher for close to 10 years, the problem is much deeper than common core can fix at least quickly. It may be a solution, but it will have to work its way through the system. That being said, here are the problems CCSS will not be able to fix.
1. Too many kids in a classroom – 35+ has become the norm. No matter how great a teacher you are, no matter how engaging you are, you cannot get to a class that size, classroom management becomes your biggest goal.
2. Attendance – when 25% of your class misses 50% of the classes, they cannot learn. Calls home, personal talks with parents, students, counselors, admin, this does not resolve the problem of attendance.
3. Insufficient prior knowledge – Students are taking geometry courses without having passed prerequisite courses. You need Algebra 1 skills to do geometry!
4. Increase in English Language Learners – This is often combined with students who have interrupted education. To differentiate enough, be dynamic enough, to be engaging enough is difficult in a small ELL class, but put them in a class of 35, you are asking the impossible.
I could go on, but I think the top 4 here state the facts. No matter how good common core is, if we don’t address class size issues, social passing, and class size issues, it will not matter one iota and nothing will change.
Christ, you’re an asshole. Go and crawl back into whatever hole you came out of.
jenny, after the shit you pulled over at Pharyngula on the ‘here’s the situation’ thread, including rape victim blaming, you have absolutely zero credibility on any subject. fuck off.
We’ve been tutoring a number of high school kids in geometry and algebra lately, so we’ve gotten some insights.
1) Lack of mastery is a big problem. Every child had trouble with fractions, and they had trouble with fractions because they didn’t understand grouping, that is, the order of operations and when you can and cannot apply various operations. (They had the same problem with radicals. Math is a language and should be taught as one.) We tutored a 5th grader who hadn’t quite grasped base ten notation yet, and while she could mechanically divide 50 by 10, she couldn’t just “drop the zero”. Kids learn at different paces, so math should be decoupled from age, but certain things need to be learned before moving on.
2) A lack of “plot focus” makes it harder for the kids. Math is cumulative, so there are certain things you need to know to move on and other things that would be nice to know, but you can move on nicely without them. Learning about exponents and logarithms is much more important than learning about holes and asymptotes of rational polynomials, but much less time was spent on them. You can go for miles without the details of rational polynomials, though at a certain point you should be able to derive them from more general principles, but if you don’t understand e-to-the-x and log-x and the related operational mechanics, you aren’t going to get far.
3) An over-emphasis on applications is sometimes silly. Yes, there are lots of applications for geometry, but the whole point of learning geometry is to understand the power of proofs and how mathematics is structured. You start with a handful of rules and concepts and build a huge, powerful structure from it. If you spend more time doing the applications of the rhombus than proving thing using the fifth postulate, the students are going to miss the entire point. It’s like stopping the Star Wars tape to explain some details of X Wing fighter maintenance. Math is sometimes neat, because it is neat, and most kids get this.
Here are Cliff Mass’ objections to the adoption of the Common Core Curriclulum in Washington. He says Washington’s 2008 math curriculum is already better.
I do like the Common Core because it focuses more on concepts, but it’s going to take more than stating that we’ll spend more time on concepts to make it happen, especially if things like the Khan Academy are talked about as if they support that… when it’s a mess of sloppily done videos all about procedures, and full of mistakes (“two plus itself times one” is 2 x 1 was the one that made me just have to stop watching them).
Insufficient prior knowledge is a huge problem… especially when “passing courses” is considered learning. http://pathways.carnegiefoundation.org/wp-content/uploads/2011/05/stigler_dev-math.pdf is just one exploration of where students’ “understanding” of math is.
Basically reviewing the post plenty will like the above because it’s reality so it is nice to read from a person thats writing this for all to see to think about
Though there are many key elements of the Common Core that are worthy, and I should point out well represented by this blog post, I am strongly opposed to the Common Core standard. My objections are primarily concerned with the implementation portion of the Common Core standard package. My concerns are also narrowly focused on the implementation in Utah, and some of the missteps might have been avoided in other parts of the country whose implementation particulars I am ignorant of.
In Utah the common Core has been implemented to replace all math classes. It has been implemented as a three-track system where there is a minimal, standard, and honors track.
The honors track is the standard track with unspecified “enhancements”. It has replaced a track that was previously called algebra-geometry-algebra2-precalculus-calculus with CCgrade7-CCgrade8-CCgrade9-CCgrade10-CCgrade11-Calculus. Simply counting the number of classes shows that an extra class has been added. What does the extra class do? I have asked many people (including the members of the national implementation committee) and the extra class apparently performs hand waving and palliative jargon.
Although the national implementation plan does state that there should be allowances for highly-motivated bright students to progress at an accelerated rate the boot-on-the-ground implementation has been blind to that possibility. The grade-level common-core placement has been given as an invariant structure critical to the success of the common core. So in addition to adding an additional class the idea is to cut off any access to accelerated learning potential present in the current system.
Both of my daughters have worked hard to accelerate their movement through mathematics in the Utah school system. My youngest was caught in the CC implementation when the Common core turned her geometry classes into CCgrade8 honors. Instead of learning geometry she was treated to a review of algebra and pre-algebra using the same textbooks and homework assignments she had completed in 6th and 7th grade.
One of the reasons stated for eliminating advanced math tracks is that many kids don’t learn the material. My eldest just completed her sophomore year in high school (grade 10) and was able to take both AP calculus and AP statistics (CC standards allow only for taking either statistics or calculus and then only in the senior year); she just got her scores from the AP tests in the mail, and her top scores (a 5 in both Calculus AB and Statistics) should be indication enough that comprehensive learning is taking place in the current advanced math track.
Only by switching my daughters to a different school district have we been able to keep them on track in their math education. The district they are in now is introducing CC in a phased plan. The plan is to do away with advanced math tracking for every K-12 Utah student within the next couple of years.
How is eliminating the ability to excel worthwhile? Explain how eliminating demonstrably excellent mathematics programs will benefit anyone.
My paranoid, and unsubstantiated, theory is that the elimination of advanced mathematics is calculated to increase relative test scores. By taking the 10% of students that would have been categorized in classes other than the minimal “grade-X mathematics”, and placing them in something called CCgradeX mathematics one creates a group that should perform better on tests than the former “grade-X mathematics” group.
Think of the great emotional high I will get when I am shown comparisons between grade8 mathematics student scores with CCgrad87 mathematics scores. I will both feel correct about my paranoid conspiracy theory, and know that my youngest daughter contributed to the surge of excellence that is CC implementation.