Mathematics education is often described as “an inch deep and a mile wide”. Most teachers would agree that mathematics courses often lack depth. As teachers, we are expected to cover a prescribed amount of territory throughout a course. However, covering such a large amount of material rarely allows students to learn and understand the concepts at a deeper level. Feeling the pressure to try to teach all standards assessed on state exams, teachers often settle for basic student understanding rather than deeper student understanding.
This has several negative consequences. Since few students really had a good initial understanding of the material, too much time is being wasted at the beginning of the year to review what should have been “previously learned”. Also, teachers often focus on transferring information (and hoping it’s received!) rather than having students problem solve and use higher order thinking to make sense of it. This can lead to a much lower level of understanding.
I find there are several teacher characteristics that contribute to this problem. One is that teachers (myself included!) too often teach procedures. When teachers lay out a step by step guide to solving problems, all higher level thinking is thrown out the window! We need to have students think about strategies to solving these problems and then determine whether their strategy works.
Clearly, this is a quite a problem in the world of mathematics education. If we expect to see change from student performance, I believe teachers must first be willing to make the change. This leads to a couple big questions:
- How do you balance eliciting deeper student understanding while still covering all of the content?
- What additional common teaching characteristics can prevent students from using their higher order thinking skills? How can these characteristics be changed to elicit deeper understanding?
-Katelyn
I think the issue of teaching for higher thinking while simultaneously covering curriculum (which I have also struggled with) is a mirage. You can’t do both. One must do one or the other. Period.
For example, a good cup of tea or coffee requires time for the hot water to penetrate and permeate the leaves/grounds in order to make a quality infusion of good drink. However, some people are far too busy to wait for this phenomenon and are satisfied with the instant version of these beverages. We both know that the instant version is of much lesser quality.
My point is that we can’t have it both ways. Good tea or coffee takes time. Period. So does acquiring deep understanding.
Now if we lived in Japan, our students would attend after school ‘cram schools’ in order to make up for their deficiencies. The responsibility is the students’ not the teachers’. In America, however, we have created this mirage (something that appears real or possible but is not) that our students can keep up with the rest of the world in our existing system if our teachers were up to snuff. No extended school year, no additional tutoring, no homework, etc.
My advice. Do one or the other. You’ll be happier and the students won’t care.
I think you can elicit deeper understanding of your students and still cover the necessary curriculum by giving a short 10-15 minute open-ended problem-solving situation that goes along with the concept covered. You will have to give less in-class time for homework, but it is called homework for a reason! That will help you stay on schedule, curriculum-wise. You can present this problem as a motivating problem at the beginning of the lesson or as a summation problem to the lesson. I suggest you present the problem-solving situation as a motivating start to the unit or as a summing up of all the concepts contained in that chapter. I do think it is possible to achieve higher level thinking, but it requires work and creativity on the part of the teacher.
I must admit that I have been guilty of just teaching procedures in the past. This year I want to make sure I use an opening problem-solving situation as motivation for studying the math of the unit and making sure there are some good, open-ended problem-solving situations on the homework.
Giving answers, steps to finding the answer, or obvious hints to students hinders their ability to strategize and problem-solve.
The students would rather do computation problems than open-ended problem-solving. I will have to learn not to give in to their asking for the answer or the steps to find the answer. Providing some problem-solving strategies should help them become confident problem solvers!
Thanks to both of you for your advice and ideas!
I agree Suzanne that creating problem solving lessons takes a lot more time than the more traditional lessons. Hopefully I’ll be able to set some time aside and work on incorporating a couple of these lessons into each unit!
Thanks again!
-Katelyn
I agree with both these veteran teachers. Josh is right and the answer is you don’t have to cover every topic in the textbook. Teach the big ideas and teach them well. Hopefully teaching the big ideas well will include making your students think by having them problem solve. Most teachers will not be able to do this with out support. Hopefully we (other teachers you have met this summer) will have good ideas you can use or point you to good resources on the web. It would be great if you could use your planning time to create at least one deeper problems solving type activity a week. I don’t know how many preps you have but you need to set goals. Stay in touch!