I've been traveling every weekend for the last two months, giving several talks and workshops on the pathways course we're teaching, Math Literacy (MLCS). It has been exciting to see the tremendous amount of momentum and progress being made in regards to courses like this. Many faculty are excited to see a new offering that is not algebra remixed and truly does something different for the developmental student. I've heard more than instructor say, "This is long overdue."
So what progress is being made?
First, we have begun to see very promising data. Carnegie has recently released their results, which are very encouraging. I'm a big supporter of all pathways courses, not just MLCS. Fundamentally, I believe in this type of course and the philosophy of it, even with slight differences amongst the various pathways options currently available.
Like Carnegie, the results of our 2 year pilot are exciting. Typically 55 - 70% of our students pass the Math Literacy course. It is not an easy course by any stretch, so there aren't lots of A's. But many students do pass. If they will work, they can pass.
Additionally, we've been tracking students in their outcome courses like intermediate algebra, statistics, and liberal arts math. And the data has so far has supported our hypothesis: there is no statistically significant difference in the performance in outcome courses when comparing MLCS students and students who take the traditional algebra courses. There is enough algebra for students to pass traditional intermediate algebra after MLCS. And there is enough rigor (and mathematics) for students to pass liberal arts math or statistics after MLCS.
Our sample sizes are not large yet, so we will continue to track students for likely 2 more years. As a veteran course redesigner, I know that redesign is not done when you roll out or even after a year or so. Time has to pass, more faculty have to teach the course, and bugs will need to be worked out. But I have seen that come to be over and over, so I have faith that will be the case with this course too.
Another point of progress are states allowing and considering allowing courses like MLCS to be piloted and accepted as an alternative to intermediate algebra. There are already several states who have changed their policies. Illinois has not yet, but we vote in less than 2 weeks. I will post on the outcome.
Definitely there is a shift going on in the U.S. Two years ago, the talk on redesign centered around emporium models. While that is a valid approach for certain students, many faculty are concerned about that model of redesign across the board. I believe the pendulum swung too far with using that approach at a very large scale, and am glad to see it swinging back to a balanced approach to redesign: models that support STEM students and skill remediation and models that support non-STEM students and their outcome courses. Faculty like seeing the emphasis on conceptual and applied understanding that exists in MLCS. And really, both redesigns can live happily in a department. We have modular (not emporium) algebra courses, an accelerated algebra course, and MLCS at our school. Different timeframes and options for the variety of students and instructors who work with these courses.
Another point of progress is materials. Our text, Math Lit, will be the first textbook for pathways courses published by a major educational publisher. It releases in July of this year. All the major publishers have projects in the works. Some educational foundations are also making progress on materials. While some of these projects have remarkable similarities on the surface, the functionality varies. Our goal when writing was not just to create a product with a new content approach in mind, but also to create a product that works in the classroom for all levels of educators. Pedagogy matters but more than that, instructor support is crucial. Our goal was to provide a product that works for faculty made by faculty who know what today's classroom is like because we are in it like they are. And we are in those outcome classes, so their specific needs are addressed too. We want students to enter a college level non-STEM class and be completely prepared for what will be expected of them.
We're already looking down the road at what needs to be created based on instructor requests. So we are planning our next projects, of which there will be several. I'll post when decisions are definite. But in the meantime, I can say this: we are committed to providing materials and support to schools with a variety of needs, be it a smaller version of our course or this approach with other content. Really, our work in this arena has just begun.
Our training and talks will continue. The online pathways course that Heather and I will teach for Canvas will be open for registration next month. I'll post info then. We have started building the course and are so excited to teach it. It will be engaging, informative, and also just fun. The platform, Canvas, is amazing. And we've really focused on working with students in that course the same as we do with our MLCS courses: engaging activities, lots of interaction between everyone in the class, and putting in practice the ideas so that you can successfully and confidently teach a pathways course.
It's certainly an exciting time to be in mathematics!
Tuesday, March 26, 2013
Friday, March 22, 2013
Handouts from ICTCM and a video link
Below are the handouts from the talk we gave at ICTCM today since we ran out. They include objectives from the course, a sample lesson, flowcharts to show possible implementation options, and a flyer for the book Math Lit with the unit structure and math topics listed.
The presentation today was brief and therefore it did not have as much detail as we often use. To compensate for that, I've included a youtube video below the handouts of a similar but more detailed version of the presentation we gave today. If you have questions, please email me.
The presentation today was brief and therefore it did not have as much detail as we often use. To compensate for that, I've included a youtube video below the handouts of a similar but more detailed version of the presentation we gave today. If you have questions, please email me.
This video was recorded in September 2012 so several things have changed since then. The book is almost finished and is coming out in July of this year. It will have a full MyMathLab course using Pearson's new design, releasing this summer. A sample MyMathLab course already exists. Please search for the book title, Math Lit, in MyMathLab to create a course. Several schools have been class testing our materials and interim MML course over the past academic year. We have conducted several workshops and plan to do more with training. We will be offering an online training course in June using the LMS Canvas. Registration information will be posted on this blog next month.
Friday, March 1, 2013
The Role of Algebra in Math Literacy
We don't approach algebra in the traditional order or using traditional methods in the MLCS course. One reason for that is the audience. The student taking this course has taken somewhere between 1 and 3 years of algebra, and yet they have placed into an algebra class at the high school level. Taking the same approach, when it was not effective the first or second or third time, will likely not yield a new result.
So we approach problems and ways to solve them, not algebra and then problems using it. It seems like a subtle change but the emphasis is incredibly different. In MLCS, algebra is a means to an end, not the end itself. We value algebra tremendously. But there is an overemphasis on this wonderful subject at the developmental level. I've often thought calling this area "developmental math" is a misnomer. It should be called developmental algebra, because that's what we spend 80% of the time working on. The remaining 20% is spent on numbers, geometry, and applications.
In reality, students will face problems with numbers in their lives and future classes, not necessarily algebra problems. Knowing when and how to use algebra matters more if we really do believe this subject is important. But that's not the traditional focus. It's just on the mechanics, as though linear equations and polynomials are falling from the sky waiting to be solved and factored.
Another reality is that in real life, the numbers aren't nice. Polynomials exist but they often don't factor. And graphing by m and b is a lovely idea but real data doesn't make it nice to do. And it's often possible to come up with an equation that is not easily solved without technology or numeric methods. We do students a disservice by shielding them from these truths. And in teaching them how to deal with them, the problems are far more interesting and students are more engaged.
Our approach is to always begin concrete and with numbers. Bring in the algebra when it will make the problem easier in terms of solving and/or organizing the details. Students will gravitate toward it often if they are not forced to use it because it is so powerful and often helpful. And sometimes algebra completely obscures the issue at hand and is overkill. Again, knowing when to use it is just as important as being able to use it.
This can be a difficult shift at first, especially if an instructor has taught traditional algebra for many years. Nearly every book has the same ordering of topics and the same emphases. So it's unusual at first to see commutative property after integer operations have been addressed. But students can still be successful with integer operations because they've been using the commutative property since first grade as a natural behavior for numbers and certain operations. It's not the name that matters; it's the understanding of the concept so that it can be used. Intuitively, students can solve many problems without the formality we often put on them.
Take Pythagorean theorem for example, a topic I taught this week. We came at the theorem and its uses from every angle. Using it requires algebra but really an understanding of numbers, something we've been working on for weeks, will suffice when solving the equations involved. When we get to equations like this, this is the approach we used since we haven't gotten to formal equation solving:
152 + leg2 = 222
Simplify the exponents.
225 + leg2 = 484
We need to find what adds with 225 to achieve 484. That can be found by subtracting 225 from 484. This approach uses the concept of addition and subtraction being inverse operations. The result is 259.
leg2 = 259
The length of the leg will be a positive number and it should square to be 259. That is the definition of the principal square root of 259. We write that next and then use the calculator to get a decimal approximation.
leg = √259 ≈ 16.1
It was also interesting when using the calculator for this problem. Students used the square button and then the square root button. I asked them if they notice anything about the placement of those functions on the calculator we typically use, the TI-30. They are paired on one button with one function being the direct operation and one requiring the 2nd function to get to it. I told students this is not a coincidence and asked if they could figure out why. A student's eyes lit up and said, "they're inverses of each other!" Exactly! That's a valuable concept to know.
Can students solve problems like this without formal equation solving techniques? Absolutely! And it makes sense because we are working at the numeric level. When we get to the formality of algebraic manipulation, it is always easier because a strong foundation in numbers has been established.
Some students remember equation solving from a previous course and prefer to use those steps. We encourage them daily in this course to use methods that make sense. Every concept that is taught uses multiple approaches to understand it, be it visual, numeric, algebraic, or verbal. After we learn those methods, we ask students to reflect on the method they prefer going forward. Learning how they learn provides a student tremendous insight on their understanding and is very helpful throughout college.
There are many people who feel this way about algebra, that our focus could use some updating. But often they are the ones using the mathematics instead of teaching it. I think it's worth listening to the people who use math for a living. Of course not every concept has to be immediately useful to be worth learning. But with the current traditional approach, there is too little that is useful and too many topics developed in a way that is unrealistic in terms of the real world uses. For more on this idea, please watch this short video by Richard Feynman, Nobel prize winning physicist.
So we approach problems and ways to solve them, not algebra and then problems using it. It seems like a subtle change but the emphasis is incredibly different. In MLCS, algebra is a means to an end, not the end itself. We value algebra tremendously. But there is an overemphasis on this wonderful subject at the developmental level. I've often thought calling this area "developmental math" is a misnomer. It should be called developmental algebra, because that's what we spend 80% of the time working on. The remaining 20% is spent on numbers, geometry, and applications.
In reality, students will face problems with numbers in their lives and future classes, not necessarily algebra problems. Knowing when and how to use algebra matters more if we really do believe this subject is important. But that's not the traditional focus. It's just on the mechanics, as though linear equations and polynomials are falling from the sky waiting to be solved and factored.
Another reality is that in real life, the numbers aren't nice. Polynomials exist but they often don't factor. And graphing by m and b is a lovely idea but real data doesn't make it nice to do. And it's often possible to come up with an equation that is not easily solved without technology or numeric methods. We do students a disservice by shielding them from these truths. And in teaching them how to deal with them, the problems are far more interesting and students are more engaged.
Our approach is to always begin concrete and with numbers. Bring in the algebra when it will make the problem easier in terms of solving and/or organizing the details. Students will gravitate toward it often if they are not forced to use it because it is so powerful and often helpful. And sometimes algebra completely obscures the issue at hand and is overkill. Again, knowing when to use it is just as important as being able to use it.
This can be a difficult shift at first, especially if an instructor has taught traditional algebra for many years. Nearly every book has the same ordering of topics and the same emphases. So it's unusual at first to see commutative property after integer operations have been addressed. But students can still be successful with integer operations because they've been using the commutative property since first grade as a natural behavior for numbers and certain operations. It's not the name that matters; it's the understanding of the concept so that it can be used. Intuitively, students can solve many problems without the formality we often put on them.
Take Pythagorean theorem for example, a topic I taught this week. We came at the theorem and its uses from every angle. Using it requires algebra but really an understanding of numbers, something we've been working on for weeks, will suffice when solving the equations involved. When we get to equations like this, this is the approach we used since we haven't gotten to formal equation solving:
152 + leg2 = 222
Simplify the exponents.
225 + leg2 = 484
We need to find what adds with 225 to achieve 484. That can be found by subtracting 225 from 484. This approach uses the concept of addition and subtraction being inverse operations. The result is 259.
leg2 = 259
The length of the leg will be a positive number and it should square to be 259. That is the definition of the principal square root of 259. We write that next and then use the calculator to get a decimal approximation.
leg = √259 ≈ 16.1
It was also interesting when using the calculator for this problem. Students used the square button and then the square root button. I asked them if they notice anything about the placement of those functions on the calculator we typically use, the TI-30. They are paired on one button with one function being the direct operation and one requiring the 2nd function to get to it. I told students this is not a coincidence and asked if they could figure out why. A student's eyes lit up and said, "they're inverses of each other!" Exactly! That's a valuable concept to know.
Can students solve problems like this without formal equation solving techniques? Absolutely! And it makes sense because we are working at the numeric level. When we get to the formality of algebraic manipulation, it is always easier because a strong foundation in numbers has been established.
Some students remember equation solving from a previous course and prefer to use those steps. We encourage them daily in this course to use methods that make sense. Every concept that is taught uses multiple approaches to understand it, be it visual, numeric, algebraic, or verbal. After we learn those methods, we ask students to reflect on the method they prefer going forward. Learning how they learn provides a student tremendous insight on their understanding and is very helpful throughout college.
There are many people who feel this way about algebra, that our focus could use some updating. But often they are the ones using the mathematics instead of teaching it. I think it's worth listening to the people who use math for a living. Of course not every concept has to be immediately useful to be worth learning. But with the current traditional approach, there is too little that is useful and too many topics developed in a way that is unrealistic in terms of the real world uses. For more on this idea, please watch this short video by Richard Feynman, Nobel prize winning physicist.
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