Practicing Like an Expert (instead of a math student)
(here are some excerpts from a paper I wrote for grad school about structuring math homework for better learning – the full paper is below)
The traditional structure of math homework (e.g. 1-67 odd) forces students to work hard, but not effectively, as all students blindly do the same assignment consisting of a similar number of each type of problem regardless of each student’s personal weaknesses. In “Practice Perfect,” Doug Lemov’s book on how to practice more effectively, the authors compare this type of practice to shampooing your hair, something we repeat daily but probably never improve on.
Math students ought to practice math the way that experts in other fields practice. When a musician learns a piece of music, they do not just play whole piece over and over. Instead, they workshop specific parts that they need to work on. When I was learning to play piano, my teacher would have me play difficult parts repeatedly – first each hand separately, then both hands together slowly, and finally at full speed. Students do the exact opposite on math homework – they do the problems with which they are comfortable and then leave blank those they do not know how to do. Thus, they are only practicing the very material that they do not need to.
If teachers would like them to engage with it differently, they need to make intentional changes in the structure. To make math homework more like expert practice, teachers should:
Force students to differentiate their homework experience.
Though the other suggestions below would be a helpful addition to traditional homework assignments, this first one would require a more radical shift. Instead of a linear assignment that encourages students to spend an equal amount of time on each part of the course, a math assignment should consist of minimal core problems for each learning objective that each student must complete, and a bank of other problems that the student could use to remedy any misconceptions. The set of core problems should be small enough that students can complete all and still have time to tackle their weak areas. Students should be instructed that a wrong answer should be a sign to reflect for a moment about what went wrong, perhaps even formally, and then immediately try more problems until they understand. To give space for this thoughtful type of work, a teacher might have to assign less work, but the quality of the work that is completed has the potential to be much higher.
(some thoughts on grading and accountability in the full paper)
Make homework objectives transparent.
With a differentiated homework assignment that required metacognition about weaknesses, students would need the tools to pick out their own weaknesses. When picking out problems from a math textbook to assign, I have an objective in mind for each group of problems. In retrospect, it seems obvious that I should simply share these objectives with students, paralleling Standards Based Grading for the wider structure of the class. Simply grouping math problems into learning objectives would help students focus their effort more effectively by allowing them to isolate skills and measure their success.
Ensure students can get immediate and actionable feedback
But even with clear learning objectives, students can’t make progress without feedback. Too often, students will power through their entire homework doing something wrong the entire time, encoding something in their brains the wrong way. Worried that students will copy answers out of the back of the book, teachers will assign problems that do not have attached solutions. We have to get over this fear – if students cannot check their work, or are not in the habit of doing so even when they are confident they got a problem correct, they risk not knowing that they are doing something wrong. Doing one problem wrong, fixing a misunderstanding and then doing a few more correctly will lead to far better results than doing five wrong and having to unlearn something incorrect a week later. For complex problems that only have a simple answer in the back of the book, teachers could post a solution guide that details not only the answer but the process that it takes.
No idea if this will work! I’m interested to try it out when I get back to the classroom. Here is the full paper:
Posted on June 5, 2014, in Homework, Teaching. Bookmark the permalink. 12 Comments.
Thanks for sharing these thoughts! I find myself wondering about this in writing instruction as well: teachers of writing can spend so much time writing comments, but how do we know if students are learning anything from all the time we spend on them? (Put more starkly, how do we know if our time spent grading their writing translates into meaningful learning for them?)
The piano lesson analogy is a compelling one. It’s much easier to do what we’re good at, but it won’t actually help us improve. How can we help students learn to identify areas where they need to work…and then give them the opportunity to do so?
Making this whole process transparent and explicit seems like the way to go as well. Thanks for such a rich post!
Thanks Andy! Yeah, comments are tricky! I know from being a grad student this year how useless some are.
To me this feels like it would be very tricky to get right. If they have any choice over which questions to answer, the less confident students would find it very hard to resist the temptation to stick to questions they already know. If they have access to solutions then they would find it hard to resist the temptation to go straight to them before trying on their own.
I haven’t read the report yet, so maybe my concerns are discussed in there, but how will you try to avoid these problems?
Phil, you’re spot on. This is definitely a brainstorm of an idea where the concept is far easier than the execution. I deal with some of those issues in the full paper, but again only from a theoretical perspective. Not usually how I liken to blog, but theoretical is the grad school life! I hope to try it out in the fall.
And another quick thing: I think this would take work not only teaching them the math concepts but also HOW to do homework. Not going to work automatically for the reasons you mentioned.
I think it can work. Have you read Daniel Pink’s Drive? His big assertion (which is pretty ambitious but I think is true) is that 85% of people in this world are looking to do their best on a given task, without the need to be supervised. In fact, Pink suggests that we’re far more likely to be motivated intrinsically if we are trusted to do so. If we go by that assumption and you provide clear tools for your students to help them diagnose/select their areas of weakness to continue working, it can work.
Yeah, that makes sense. It sounds counter-intuitive in a school environment, no? But it’s something to keep in mind.
I started teaching 6 years ago after 10 years as a CPA and this was the philosophy that I have always taught with. Many veteran teachers looked at my like I was nuts. I explain that I choose the homework that I think the average student needs to complete in order to master a topic. If a student does 70% of what I ask in general, then they can retake a test that they fail and there is a workplan depending on what they missed but there is a large amount of reflection on their learning first. It is my goal to not only to help them learn math, but to help them be successful in college, where most professors won’t have the relationship to do the same.
I don’t collect homework until we have a some sort of unit test as I explain to them that they are becoming young adults and they have a lot of competing demands for their time and it is their job to learn how to balance these demands. They are required to attempt some of their homework each night in order to make the discussion the next day meaningful. By the time we have an assessment, they should be prepared. Usually by January, they slack off a bit so we have a reprioritizing activity.
I explain my philosophy like basketball. I am 5’6″ and 47. If I want to play basketball, I have a lot of practice and some genetics to overcome. I probably have to practice more than another 47 year old who is 6’2 and played basketball for years. After some conversation they come to understand that watching me or others do math isn’t the same as doing it themselves. They all agree that watching a basketball game doesn’t mean that I can play basketball in the same way that watching me introduce a math topic doesn’t mean that they can do math.
It has been a very successful strategy. I’m glad that you have voiced a similar opinion. Now I am not the only nut.
That’s just why situated learning plus rote technique mastery makes a powerful combination! There’s never a concern of “Jeez, what it I can’t do the problem once I’ve set it up (common among middle school students)?” I played around with the idea of STOPPING the problem on the assessment when the problem is set up and letting the students practice the arithmetic in the computer lab.
Now I blog for a company (www.mathnook,com) that writes the games I’d set my classes loose on in the computer lab, so I am looking forward to seeing what you come up with. Let the kids make sport of the lower-level stuff on lab time, and spend all the time in math class learning to think mathematically.
I like this idea.
I’ve been experimenting with setting more but better homework this year, it’s not quite what you’ve suggested (mine runs into the starter of the next lesson) but thought i’d share: kids get a “guided” homework each week where each key topic has a worked example with it – they can work through that example to check understanding then try 1 similar question.
For the starter next lesson, they choose the one they find most difficult and try two more problems in that area.
You’re right in what you say, the quality of work (and retention) is way improved over doing 30 trig questions for HWK.
Cheers for sharing.
We try to do something similar to this idea when it comes to test review for our grade 9 students. We have students write a quick pre-test and then based on their answers they are divided into groups the next day to review and practice the skills they did not do well on. Most students look forward to “Gap Day” as we call it where they fill in the gaps before they write the test.
I am still trying to find the right balance with homework myself. One thing I find crucial is to really emphasize that homework is formative and to not count it as a significant part of their grade. If students buy in to the idea that homework is formative and see the link between the homework assigned and their assessments then you can have more success with providing full solutions and asking students to reflect and differentiate their practice as needed. I’ve done a few successful experiments in my classroom along these lines. In one class I gave my students the Standards Based rubric and asked them to find questions through their homework and quizzes to create a test review. I then posted the questions they came up with and full solutions to the class website for studying. Tying the learning outcomes and criteria for success directly to the test had a huge impact. I wonder if something similar could be done with regular homework? Provide a rubric for the learning outcome the homework is designed to practice and get students in the habit of matching the homework to the rubric to draw that connection and validation for the purpose of the assigned problems?