Another good source for information about cognitive science and math (as well as other subjects) is Holly Korbey's Substack, The Bell Ringer: https://hollykorbey.substack.com/.
I love this! Sometimes we have to memorize things! I have my World students memorize 10 countries on each continent and am TRYING to help (force?) my own math-averse child to memorize her multiplication tables.
Thanks for your comment, Jenna. I'm still thinking about the relationship between memorizing and learning/understanding. Memorizing without context and understanding is probably not so helpful. But sometimes we need to memorize in order to learn more.
To me, one key part of incorporating memorization is doing it efficiently. Asking students to memorize something but not helping them understand how to do so can lead to stress, wasted time, or students giving up. In my experience, the best way to help students memorize things is to break them down into small chunks, use spacing and retrieval practice, and increase the number of things to be memorized gradually. It won't take so much time it crowds out everything else, and students are more likely to feel successful and get both the benefits of more knowledge and the confidence that comes with the feeling of success.
Good point. That's why, when I asked my students to memorize the 50 states, we did it in chunks, starting with the 13 colonies, then to the Mississippi River, and so on. In a comment on Michael Pershan's post, Adrian Neibauer said he allowed students (I think grade school) to refer to a copy of the times tables regularly. Thanks for your comment, which also points to the problem of memorizing things and quickly forgetting them because you don't use them. That would likely be more of an issue with memorizing things like geographic places and obscure vocabulary words, perhaps, than in math class or memorizing verb conjugations in a foreign language. Again, the larger point is that one shouldn't memorize anything unless it is useful for building future knowledge.
I appreciate this, because learning is integrated. What makes math mean something are the contexts and examples you showed above. Collaboration around scenarios that make math relevant to students lives is critical-- when teachers share and plan with the intent of helping students problem solve in and out of math is when we all win.
To me, the questions posed (particularly the raw number vs. percentage of bison) above are not the lack of "basic math", but instead reveal the lack of basic problem solving. With most of our students, starting from a young age we teach them what to do instead of how to make sense of problems and how to decide what to do. If we want our students to be able to transfer skills from math to other classrooms & life contexts then we must consider the behaviors or a mathematician-- problem solving, making sense of a problem, persevering in solving when the first attempt doesn't work, explaining our thinking, understanding the perspectives of others. And if the student can do that, they surely will be able to memorize their multiplication facts.
To me the skills listed above are those of a mathematician, a historian, a citizen, a reader, a friend, etc. They are the skills needed to solve the problems our world needs solved.
I didn't think of that--that the bison example shows a lack of problem solving and not basic math facts. I think you're right. It reminds me of how students struggle to transfer writing skills, vocabulary and grammar to their writing in history class. The solution can only be, as you suggest, greater collaboration among teachers.
Again, I'm not a math teacher, so maybe this isn't practical, but I wonder if it would be useful or worthwhile for math teachers to occasionally introduce random problems from "the real world" or other classes (like my 2 examples) so that students have the opportunity to problem solve. Like when I'm teaching that lesson on the bison and the wars with Native Americans, we go back to General William T. Sherman and his "total warfare" policy of the Civil War because similar tactics were then used in the West to fight Native Americans.
Regarding "the real world," which I put in quotation marks because of my discomfort with the term. It suggests a sharper line between academic work done in schools and the skills and knowledge used elsewhere that I think actually exists. It also suggests a narrow view of the purpose of education, one that is job-focused. I am thinking of a quote from Greg Ashman's post that I linked to in footnote #4 above. It is this: "To someone who loves mathematics, it is not a subject that only becomes valuable when it can be used to make a leaden social justice argument about the minimum wage or about including people with disabilities, it has intrinsic value and, dare I suggest it, beauty." I think finding beauty in our subjects as a primary goal; the goal of connecting it to real world experience is secondary. At least I think I think that. I wonder what your perspective is on that, relating to math education?
Another good source for information about cognitive science and math (as well as other subjects) is Holly Korbey's Substack, The Bell Ringer: https://hollykorbey.substack.com/.
Yes, I'm a subscriber. Thanks, Natalie.
I love this! Sometimes we have to memorize things! I have my World students memorize 10 countries on each continent and am TRYING to help (force?) my own math-averse child to memorize her multiplication tables.
Thanks for your comment, Jenna. I'm still thinking about the relationship between memorizing and learning/understanding. Memorizing without context and understanding is probably not so helpful. But sometimes we need to memorize in order to learn more.
To me, one key part of incorporating memorization is doing it efficiently. Asking students to memorize something but not helping them understand how to do so can lead to stress, wasted time, or students giving up. In my experience, the best way to help students memorize things is to break them down into small chunks, use spacing and retrieval practice, and increase the number of things to be memorized gradually. It won't take so much time it crowds out everything else, and students are more likely to feel successful and get both the benefits of more knowledge and the confidence that comes with the feeling of success.
Good point. That's why, when I asked my students to memorize the 50 states, we did it in chunks, starting with the 13 colonies, then to the Mississippi River, and so on. In a comment on Michael Pershan's post, Adrian Neibauer said he allowed students (I think grade school) to refer to a copy of the times tables regularly. Thanks for your comment, which also points to the problem of memorizing things and quickly forgetting them because you don't use them. That would likely be more of an issue with memorizing things like geographic places and obscure vocabulary words, perhaps, than in math class or memorizing verb conjugations in a foreign language. Again, the larger point is that one shouldn't memorize anything unless it is useful for building future knowledge.
I appreciate this, because learning is integrated. What makes math mean something are the contexts and examples you showed above. Collaboration around scenarios that make math relevant to students lives is critical-- when teachers share and plan with the intent of helping students problem solve in and out of math is when we all win.
To me, the questions posed (particularly the raw number vs. percentage of bison) above are not the lack of "basic math", but instead reveal the lack of basic problem solving. With most of our students, starting from a young age we teach them what to do instead of how to make sense of problems and how to decide what to do. If we want our students to be able to transfer skills from math to other classrooms & life contexts then we must consider the behaviors or a mathematician-- problem solving, making sense of a problem, persevering in solving when the first attempt doesn't work, explaining our thinking, understanding the perspectives of others. And if the student can do that, they surely will be able to memorize their multiplication facts.
To me the skills listed above are those of a mathematician, a historian, a citizen, a reader, a friend, etc. They are the skills needed to solve the problems our world needs solved.
I didn't think of that--that the bison example shows a lack of problem solving and not basic math facts. I think you're right. It reminds me of how students struggle to transfer writing skills, vocabulary and grammar to their writing in history class. The solution can only be, as you suggest, greater collaboration among teachers.
Again, I'm not a math teacher, so maybe this isn't practical, but I wonder if it would be useful or worthwhile for math teachers to occasionally introduce random problems from "the real world" or other classes (like my 2 examples) so that students have the opportunity to problem solve. Like when I'm teaching that lesson on the bison and the wars with Native Americans, we go back to General William T. Sherman and his "total warfare" policy of the Civil War because similar tactics were then used in the West to fight Native Americans.
Regarding "the real world," which I put in quotation marks because of my discomfort with the term. It suggests a sharper line between academic work done in schools and the skills and knowledge used elsewhere that I think actually exists. It also suggests a narrow view of the purpose of education, one that is job-focused. I am thinking of a quote from Greg Ashman's post that I linked to in footnote #4 above. It is this: "To someone who loves mathematics, it is not a subject that only becomes valuable when it can be used to make a leaden social justice argument about the minimum wage or about including people with disabilities, it has intrinsic value and, dare I suggest it, beauty." I think finding beauty in our subjects as a primary goal; the goal of connecting it to real world experience is secondary. At least I think I think that. I wonder what your perspective is on that, relating to math education?
Lastly, a comment about your last 2 sentences on skills. I am thinking of work by Daniel Willingham and others about the ability to actually teach students to think like a mathematician, historian or what have you without teaching content first. Here's a long read on that by Willingham: https://thenext30years.substack.com/p/curriculum-for-deep-thinking (but you can just read the 3 takeaways in the margin) and a recent post by Robert Pondiscio: https://thenext30years.substack.com/p/curriculum-for-deep-thinking.
Thank you for commenting!