We all have biases. We all process differently. Before you read on, I’m going to ask you to try to answer two questions. In the process, I hope you learn a little about one of your biases.
- Megan opens a bank account, depositing $50 that she earned taking photos. Each week, she takes $2 of her allowance and deposits it into her account. Write a model (equation) that shows the relationship between the balance, B, in her account and the number of weeks, t, that have passed. How much will be in her account at the end of 2 years?
- Write the equation of a line having slope 2 and y-intercept (0,50). Then use the line to predict the value of y when x = 104.
Which of the two problems is easier for you? Can you do them both? Can you do one but not the other? When you’re done reading, share your thoughts via a comment and I’ll reply to it!
Becky, from our last study learned a lot from her students once she started applying modern learning science to her teaching. One of the things that she learned was that they were very diverse in their mathematical thinking. Trained in algebra, she tended to approach problems in a very methodical way. But once she started asking open-ended questions, she started to see modes of thinking mathematically that had, frankly, never occurred to her. She started to realize how much she was biased towards her own training—both in how she taught students as well as in how she assessed them. It occurred to her that she often assessed methodologies rather than problem solving skill or conceptual understanding. That made her question the value of those methodologies a lot, which is something that she had never really done before. She also realized that she never really did much to TEACH problem solving skill or conceptual understanding. This made her rethink her teaching a lot. One of the things that happened once she started reflecting on her choices (rather than accepting them as prescribed absolutes handed down through the generations of educators) was that her instructional approaches and goals and her assessments diversified. And with that, she recognized a lot more talent and skill than she had been able to recognize with the narrower approaches, goals, and assessments that she had used previously.
For example, she was never especially interested or good at thinking about math in real-world contexts. That worked out fine for her, because she loved symbolic manipulation which was given really high priority traditionally, as opposed to contextualization. But once she started teaching contextually, she noticed that lots of her students were really good at solving problems in context. But those same students might be completely unable to answer an equivalent question, posed without a context. Consider those two questions that we started with. Mathematically, they are the same problem. Some people can do the first problem but not the second, some can do the second problem but not the first, some can do neither, and some can do both. Of those who can do both, some recognize them as the same basic problem and some do not. In a traditional class, any student who can do the second problem is likely to get recognized and rewarded equally for their abilities. Those who can do neither and those who can do only the first problem might be indistinguishable to the teacher, even though one has stronger mathematical skills than the other.
All of these experiences made Becky begin to question the use of algebra as a measure of whether a student was ready for challenging mathematical studies. Maybe the country was missing out on a lot of mathematical talent by requiring algebra proficiency up front as a prerequisite for all other math classes. Maybe those students who understand math contextually COULD learn algebra if it were taught in context, when it was needed. Maybe a just-in-time algebra approach could give a whole new sector of college students the opportunity to develop their mathematical skills. But that would require a department-wide shift in assumptions and expectations, which would require A LOT of work and a lot of cooperation and buy-in. Becky wasn’t sure how to make that happen. She wondered if it were even possible. Then she remembered! Those nationwide movements had been collecting data that suggested that students taught curriculum based on modern learning science could be better prepared for advanced study than those taught a traditional curriculum. Maybe that could be the starting point for the discussion that had to happen…