I really enjoyed the Jo Boaler article this week. I think that if we adopted her techniques for teaching mathematics, and took the evidence from research done on brain development and math more seriously, a lot more students would like math and succeed in it. I always used to be quick with math, but not anymore. I struggle to get all the questions answered in pre-tests for the Praxis exam, and it really bothers me that it is timed. It makes me stress, and my own experience tells me this is not the way to go about doing math. I am taking an applied math concepts course at CCV (for my teaching license) and I find it to be unbelievably old school. It is in opposition to everything we are learning in this class! I have started a running list of quotes from the teacher. (I feel badly for my classmates who really don't understand the material). Tonight, she actually said: "Math is 99% memorization." She is only teaching it so we will pass the QT (?) test at the end of the semester, not to help us develop our understanding. I like how Boaler insists on teaching number sense, and I have realized how valuable this is for students. One more thing: Boaler writes that successful math students use "different brain pathways-one that is numerical and symbolic and the other that involves more intuitive and spatial reasoning." This reinforces to me the ideas behind the chart in our PTA text (p. 25) linking visual, symbolic, verbal, contextual, and physical representations of math. The more connections we can make about math, and the more areas of our brain we can get involved in learning math, the stronger our understanding will be and the more likely we will develop true fluency.
It does seem like a tremendous cultural shift will have to happen in the math world (similar to that which is happening in the science world) - a shift away from old-school memorization to one which develops number sense through working with numbers, reasoning, and building those pathways you mention - visual, symbolic, verbal, contextual, and physical. And this shift starts with new teachers embracing the new research and the techniques J. Boaler and others are promoting. One challenge is that as a teacher is sure seems easier to fall back to the familiar ways we learned...right or wrong.
Meg I think you nailed the review. For the students struggling it is often the linking of visual, symbolic, verbal, contextual and physical representations of math that help break through the wall that says "I can't". But the defense that the skills are important and if students are not memorizing the facts that means they must get the knowledge in another method which is just as worthy and valuable.
It was interesting to learn that memorization of math facts (the old fashion way) is an innate ability - some students are better at it than others. What I found most interesting about this is how past practices encouraging memorization would reinforce the idea that some students are good at math and some are not. The new research shows that in fact everyone can be good at math - everyone can master math facts - if given the opportunity to reason and work with numbers in different ways. Those that memorized in the past were not necessarily mastering the content.
So then I had difficulty distinguishing between memorization vs. mastery...difficulty, that is, until the word "automaticity" was used on p. 4. So now, in my mind, there's memorization (achieved through old school methods) and automaticity (mastery achieved through reasoning, working with numbers, and number sense).
I like the How Close to 100 activity the best. I like how it's interactive and visual, it reinforces the fact that multiplication can be represented by a rectangle (length x width = area), and it can be a strategic challenge for students to fill up as much of a 100 grid as possible.
Finally, I thought a variation of the Math Cards game (more advanced) could be having all the cards face down and playing the 'memory' card game - students flip over two cards at a time looking for matches, a match is the same answer, not the same representation. If a match isn't found, the two cards are turned face down again, and the next player gets a turn.
Sorry, one more thought. Our culture values speed...often to the expense of other, useful/valuable characteristics. Because of this, it will be difficult to de-emphasize this as a teacher. For example, students LOVE competitive games of speed - who can answer the question the quickest. Some how competition needs to be created differently in the classroom.
To me, this article was very relatable. I have so much math anxiety and I STILL distinctly remember being in fifth grade doing tons of math drills. It was terrible!!
When I was reading this article I kept thinking of the Facebook posts I saw when the common core standards first went into place. People were really freaked out by how different the math problems looked. I admit that I too was a bit confused initially. Now I realize that these problems are to help students use numbers flexibly and to develop a strong number sense.
I agree with Boaler when she says that 'fluency' is still commonly incorrectly interpreted in mathematics. The key words in the 'Engage New York' math curriculum are speed, accuracy, and memorize. A students ability to quickly solve math problems is not an accurate representation of a high versus low achiever. Still to this day it is ingrained in my head that if you are able to quickly solve math problems you are better at math.
Oh, if only I could have grown up going to school with Jo Boaler. Reading Jo's article shined a light on the way I learned math. Math was always a memorization game. It was flash cards and timed test. It was anxiety and feeling rushed. And get this.. My dad had been a math teacher. The idea of understanding deeply really resonated with me. I remember the feeling of not quite understanding the math concepts when my teacher taught them in class, but feeling like because everyone else "got it" enough to move on that I didn't have the courage to ask or get to a deeper understanding. The idea of number sense = number fluency turns the way my generation learned math upside down. They way Jo Boaler describes how best to learn math takes the pressure off and makes math more like doing puzzles to me. Puzzles are fun and can be done many different ways. It's so freeing and fun to think of numbers as Boaler does as being "flexible". I know I was one of the Low achievers, for all the reasons she states. However, reading this article makes me feel empowered and excited about sharing mathematics to the next generation.
Hi Rachel - I love the connection to puzzles - who doesn't like puzzles? I, too, was an anxious math student and felt rushed much of the time like you did. I think one of the things I like the most about what I have read and seen about Jo Boaler's work is her pacing. She is calm and slow when she talks and it feels like she is giving space for students brains to relax and do their work. This is a shift from how I remember math as a kid - and a good reminder in general for me in the classroom.
For some reason, when I initially began reading this article I was surprised that timed testing caused so much anxiety and discouraged students from wanting to practice math. I guess I had never really made the connection before but it makes perfect sense. Her statements in the article are also supported in Chapter 10 as ineffective and inefficient. I am currently working in a 4th grade classroom that does timed tests on math facts every week. Once students pass addition they move onto subtraction, then multiplication, and finally division. As the primary person correcting these tests, I see how students struggle to memorize the facts and are sometimes wrong. I also see the students who struggle with math in general and how unsuccessful they are at these assessments. It must be discouraging to do the same test every week because they can't pass it. In addition to the timed tests, students spend time every week and even at home playing a computer game that is timed memorization of math facts. The article states "Number sense is the foundation for all higher-level mathematics" yet these timed tests do not encourage deep careful thinking about number relationships. It seems to me that the time spent every week working on memorizing these math facts could be better spent by teaching a variety of math strategies and providing meaningful math experiences.
I really enjoyed the Jo Boaler article this week. I think that if we adopted her techniques for teaching mathematics, and took the evidence from research done on brain development and math more seriously, a lot more students would like math and succeed in it.
ReplyDeleteI always used to be quick with math, but not anymore. I struggle to get all the questions answered in pre-tests for the Praxis exam, and it really bothers me that it is timed. It makes me stress, and my own experience tells me this is not the way to go about doing math.
I am taking an applied math concepts course at CCV (for my teaching license) and I find it to be unbelievably old school. It is in opposition to everything we are learning in this class! I have started a running list of quotes from the teacher. (I feel badly for my classmates who really don't understand the material). Tonight, she actually said: "Math is 99% memorization." She is only teaching it so we will pass the QT (?) test at the end of the semester, not to help us develop our understanding. I like how Boaler insists on teaching number sense, and I have realized how valuable this is for students.
One more thing: Boaler writes that successful math students use "different brain pathways-one that is numerical and symbolic and the other that involves more intuitive and spatial reasoning." This reinforces to me the ideas behind the chart in our PTA text (p. 25) linking visual, symbolic, verbal, contextual, and physical representations of math. The more connections we can make about math, and the more areas of our brain we can get involved in learning math, the stronger our understanding will be and the more likely we will develop true fluency.
It does seem like a tremendous cultural shift will have to happen in the math world (similar to that which is happening in the science world) - a shift away from old-school memorization to one which develops number sense through working with numbers, reasoning, and building those pathways you mention - visual, symbolic, verbal, contextual, and physical. And this shift starts with new teachers embracing the new research and the techniques J. Boaler and others are promoting. One challenge is that as a teacher is sure seems easier to fall back to the familiar ways we learned...right or wrong.
DeleteMeg I think you nailed the review. For the students struggling it is often the linking of visual, symbolic, verbal, contextual and physical representations of math that help break through the wall that says "I can't". But the defense that the skills are important and if students are not memorizing the facts that means they must get the knowledge in another method which is just as worthy and valuable.
ReplyDeleteIt was interesting to learn that memorization of math facts (the old fashion way) is an innate ability - some students are better at it than others. What I found most interesting about this is how past practices encouraging memorization would reinforce the idea that some students are good at math and some are not. The new research shows that in fact everyone can be good at math - everyone can master math facts - if given the opportunity to reason and work with numbers in different ways. Those that memorized in the past were not necessarily mastering the content.
ReplyDeleteSo then I had difficulty distinguishing between memorization vs. mastery...difficulty, that is, until the word "automaticity" was used on p. 4. So now, in my mind, there's memorization (achieved through old school methods) and automaticity (mastery achieved through reasoning, working with numbers, and number sense).
I like the How Close to 100 activity the best. I like how it's interactive and visual, it reinforces the fact that multiplication can be represented by a rectangle (length x width = area), and it can be a strategic challenge for students to fill up as much of a 100 grid as possible.
Finally, I thought a variation of the Math Cards game (more advanced) could be having all the cards face down and playing the 'memory' card game - students flip over two cards at a time looking for matches, a match is the same answer, not the same representation. If a match isn't found, the two cards are turned face down again, and the next player gets a turn.
Sorry, one more thought. Our culture values speed...often to the expense of other, useful/valuable characteristics. Because of this, it will be difficult to de-emphasize this as a teacher. For example, students LOVE competitive games of speed - who can answer the question the quickest. Some how competition needs to be created differently in the classroom.
To me, this article was very relatable. I have so much math anxiety and I STILL distinctly remember being in fifth grade doing tons of math drills. It was terrible!!
ReplyDeleteWhen I was reading this article I kept thinking of the Facebook posts I saw when the common core standards first went into place. People were really freaked out by how different the math problems looked. I admit that I too was a bit confused initially. Now I realize that these problems are to help students use numbers flexibly and to develop a strong number sense.
I agree with Boaler when she says that 'fluency' is still commonly incorrectly interpreted in mathematics. The key words in the 'Engage New York' math curriculum are speed, accuracy, and memorize. A students ability to quickly solve math problems is not an accurate representation of a high versus low achiever. Still to this day it is ingrained in my head that if you are able to quickly solve math problems you are better at math.
Oh, if only I could have grown up going to school with Jo Boaler. Reading Jo's article shined a light on the way I learned math. Math was always a memorization game. It was flash cards and timed test. It was anxiety and feeling rushed. And get this.. My dad had been a math teacher. The idea of understanding deeply really resonated with me. I remember the feeling of not quite understanding the math concepts when my teacher taught them in class, but feeling like because everyone else "got it" enough to move on that I didn't have the courage to ask or get to a deeper understanding. The idea of number sense = number fluency turns the way my generation learned math upside down. They way Jo Boaler describes how best to learn math takes the pressure off and makes math more like doing puzzles to me. Puzzles are fun and can be done many different ways. It's so freeing and fun to think of numbers as Boaler does as being "flexible". I know I was one of the Low achievers, for all the reasons she states. However, reading this article makes me feel empowered and excited about sharing mathematics to the next generation.
ReplyDeleteHi Rachel - I love the connection to puzzles - who doesn't like puzzles? I, too, was an anxious math student and felt rushed much of the time like you did. I think one of the things I like the most about what I have read and seen about Jo Boaler's work is her pacing. She is calm and slow when she talks and it feels like she is giving space for students brains to relax and do their work. This is a shift from how I remember math as a kid - and a good reminder in general for me in the classroom.
DeleteFor some reason, when I initially began reading this article I was surprised that timed testing caused so much anxiety and discouraged students from wanting to practice math. I guess I had never really made the connection before but it makes perfect sense. Her statements in the article are also supported in Chapter 10 as ineffective and inefficient. I am currently working in a 4th grade classroom that does timed tests on math facts every week. Once students pass addition they move onto subtraction, then multiplication, and finally division. As the primary person correcting these tests, I see how students struggle to memorize the facts and are sometimes wrong. I also see the students who struggle with math in general and how unsuccessful they are at these assessments. It must be discouraging to do the same test every week because they can't pass it. In addition to the timed tests, students spend time every week and even at home playing a computer game that is timed memorization of math facts. The article states "Number sense is the foundation for all higher-level mathematics" yet these timed tests do not encourage deep careful thinking about number relationships. It seems to me that the time spent every week working on memorizing these math facts could be better spent by teaching a variety of math strategies and providing meaningful math experiences.
ReplyDelete