What They Do Know Can Hurt Them: Identifying and Addressing Math Misconceptions
Presented by: Phil Daro and Cindy Fielder- America’s Choice
Early in the presentation, Daro shared some compelling results of International Comparisons that have been done in an extensive N.A.E.P. study (1991-2007). He shared that in Reading, we are at the head of the pack. In Science, we are in the middle of the pack. In Math- we are at the back of the pack. The point was made that these results are based on the same kids with the same teachers- same parenting, same culture, etc. This suggests the idea that there is something wrong with the way math is being taught in our country. While reviewing the data collected in this study, the comment was made that the data shows that we are making progress in our country, which is very encouraging…but the data also shows improvements on an international level. The charge was made to move more aggressively toward closing this gap.
On another note: Daro also commented that when doing data comparisons, it is best to compare data against other states to assess your own state’s progress.
Three Areas of Needed Focus
Suggestions were made on where we need to focus in order to facilitate this needed change. 1) Time- Kids are running out of time to seize the opportunity to go to college, etc. Problems must be faced before change can occur. Students with a history of learning more slowly are not going to catch up without spending more time on their learning. The strategy must be to accelerate their learning with their time…shift more of their time to learning mathematics (which means giving up something else). 2) Attention: “Who Teaches Whom” – One problem that we face is that first year teachers in inner city schools spend much of their time on behavior management, some leave after Christmas…etc. The professional development they receive during that first year of teaching will begin to help the following year, but many of these students will have moved on to another grade level or teacher (who may likely be another inexperienced teacher). Middle school math departments must then take ownership of this problem.
3) Algebra in 8th grade? If we want kids to enter college without remediation, they have to complete Algebra 2 by 12th grade. This leads many districts to accelerate their course work, and many times, students struggle to pass other courses to get to that point. There is time if we spend it wisely. If kids are not ready for Algebra, Daro suggests giving them algebra readiness courses instead – this can happen even at 9th grade. There will still be time to complete Algebra 2 by the end of 12th grade.
Offering Students Feedback on their Work
Daro emphasized that you must ask children questions about their work. Simply saying “good job” does not offer enough. Questions drive learning and help to clarify misconceptions.
He emphasized that in a classroom of 30 kids and 1 teacher, each kid gets 1/30 of the teacher’s time. That is where partner work can benefit in offering feedback to students, when half the class talking and half the class listening (as long as the conversations are “on topic” conversations).
Offering homework help is also a vital part of offering feedback. Sometimes, if you help a student with just one of the problems on a homework assignment, they are able to complete the rest of the assignment without incurring problems…this can help them not become one year behind. (closing the gap).
Reasons Students Struggle in Mathematics
1-Misconceptions- Students need to develop their meta-cognitive development of analyzing their own thinking and work.
2-Bugs in procedural knowledge
3-Mathematics language learning – There is a lot of math vocabulary that kids need to understand, which needs to be developed meaningfully and in context.
5-Lack of knowledge (gaps) – You should not look at data and look at not what they are getting wrong, but why they are getting it wrong; Extra feedback for kids is the key- they need to receive feedback early in their schooling and often.
6-Disposition, belief, and motivation- If students believe if they can change for the better, their behavior can change. Research shows that teachers can build student confidence with practice, feedback, and encouragement- and make a difference. He also stated that sitting a kid alone in front of a computer is poison…. They need to have conversations with someone asking them the right questions.
Ïmportance of Number Lines
If a number line is not in your classroom, there is a problem.
Number lines are the most important key in moving from the concrete to abstract.
Rulers are also important number lines in showing fractional parts of wholes.
~As teachers we need to think~
“What is the math that my kids need to learn by doing these problems?”……not……
“How can I show the kids how to get the right answer?” (using math they already know- this means I am not teaching you the math you need to know…I am just helping you get the point to do well on the next test)……this is where misconceptions come from. It was emphasized that answers are important, they are part of the process, but they are not the product. Even incorrect answers are important. (Why didn’t that strategy work?)
One of the most compelling statements Daro made was the following;
**If kids forget how to do something, it means they did not learn it in the first place. The brain forgets very little of what is learned.