Emotions are powerful and immediate, so if you can harness an emotion -- any emotion -- and link it to mathematical insight, you create a visceral experience for the student: the initial insight is immediate ("ughh that looks horrible!"; "pffft, that one would be so easy!") and they can marshal reasons for justifying that insight *after the fact*; thus creating the sense of having intuition or natural insight before "learning" anything.
Now every student has some expertise. Now the stage is set: the student's feelings towards the easiest and hardest problems are *different* (e.g. confident about A but intimidated by X) but clearly of the *same* type. Suddenly there is a natural motivation to discover how the two are related ("can we transform X so that it would be more like A?"). I'll often follow this up by asking students if they could think of ways we could make A appear more like X, without affecting the solution; now they're thinking like a teacher.
Implicitly, it also reinforces an important lesson in learning: feelings *aren't* facts. You might be intimidated by X, it might look impossible but now you see it's on the same spectrum as A, which is not intimidating, so you know not be afraid of a new problem simply because it looks hard.
This is one of my favourite approaches to use. It is especially effective with intimidating content *because* of the strong emotional response. In some variations, I also emphasize to my students that this really is how mathematicians work: we look at a problem and notice what we don't like about it ("hmm, this would be a lot easier if we didn't have this part") and this emotional response can often be trusted to guide us towards an appropriate approach. Students learn that mathematical intuition and common intuition are not fundamentally different -- and who doesn't feel great about being "a natural"?! It feels wonderful knowing you can trust yourself.
Another variation on this approach is to ask "if you could change one thing about this problem, what would change?"; however, it's more appropriate at a later stage, when students have already developed a sense of the diversity of possible problems.
Every word of this, especially the ones connecting the visceral common intuition and the abstract mathematical intuition. What you're doing here is helping students realize "I know math" much earlier than is typical.
Giving students the opportunity to choose the "least difficult" of problems, offering student agency - giving them a choice and an entry into their learning. However, it also provides those who are confident to be looking for the most challenging to solve if they want. I recall advanced students who would jump to looking for the MOST difficult to answer, looking for that challenge, too. Ms. Esmende also includes the discussion between students giving voice, and then, I'm sure, she would use routines for students to explain the reasoning and continue with the lesson from there.
Love that move. The ranking offers us an angle into a conversation about the features of the systems but it also helps students locate themselves in their zone of proximal development.
Talk about a low floor! The question is an invitation that pretty much anyone can engage with. It is also a question that does not have one right answer. At the same time, it is an invitation to look at all the problems and evaluate how you think about them -- to consider them critically -- so it is not an "easy" ask either. In addition, by saying, "don't solve them," Gen is taking the need to get an answer off the table -- for this time, everyone is just considering the problems from their own points of view without the pressure to solve.
Decidedly NOT how systems of equations are treated in most curricula, including the ones I learned from as a kid and every one I taught from as a teacher. "Figure out if it's substitution or elimination. Execute. Don't mess up."
As a forever student, I enjoyed thinking about systems this way. I suppose that even when getting my undergrad in math and solving matrices, I would look for the easiest to solve first. I've never thought about my thinking in that regard. Thanks, Ms. Esmende!
Engaging way to start the lesson that encourages analysis and comparison. Without realizing it the students are comparing several systems to each other to determine which ones are easier or more challenging. It helps reluctant students to become part of the conversation. To take it one step further, have students work on vertical white boards and up and moving, allowing them to walk around and see each groups' choices and reasoning.
I only learnt the "formulaic" way to solve these. I had a good intuition for math and loved patterns and abstractions as a child so I thrived on that. what this question does is creates a situation for which I will always have an answer - because its asking my opinion not absolute right or wrong
Recently, in the playstore review of a popular educational app a user had commented that he/she used the answer from the AI in the app in a class assignment and the answer was incorrect. Only when teacher evaluated it, she got to know that it was wrong. This could be happening to many learners.
1. Every educational product that is using AI gives a caution message that you should check the relevancy of the response. They are not considering about the 'Critical Thinking' skill required for a learner(or anyone) who uses AI for guidance. Not thinking how a learner is supposed to have this ability to determine whether the generated answer is correct or not. Even companies which are training their own models to handle STEM subject questions and answers are hiring only highly qualified (PhD) candidates to validate and improve the responses. Not even subject experts. Which indicates the level of knowledge required to analyze the AI response.
2. The quality of the answer depends on the quality of the question. Do learners ask the right questions always? Sometimes they don't even realize there is some mistake in the question that they are asking.
There is definitely scope for innovation and improvements, but current trend of pushing young learners towards AI products doesn't seem to be ideal. The approach should be different.
Kids need to feel seen and heard, so asking them to articulate their opinions about the difficulty of equations to open rather than showing them procedures invites all learners to the lesson by meeting an essential need. Most (~95%) kids do not feel they have an essential need to follow procedures to solve systems of equations, so we have to welcome them into the learning by honoring what they do need: social connection, belonging to a community, being seen and heard by community members,...
Grabbed this piece for next week's featured comments, thanks!
> Kids need to feel seen and heard, so asking them to articulate their opinions about the difficulty of equations to open rather than showing them procedures invites all learners to the lesson by meeting an essential need.
OK there's all the emotional stuff... but I gravitate towards the cognitive.
.... but the thing that jumps out to me is how often numbers are given meaning. The gestures and comments about how much cake that would be (and the diagnostic that saying "two wedding cakes!!" suggests that thinking of amounts and making connections is a habit that's been developed) mean that this is so much less likely to turn into "KEEP CHANGE FLIP!"
Now, I *do* fervently hope attention is made to translating the stories into reasonably fast processing of the symbolic math language, but so many of my adult learners (I had several today) just don't make meaning of numbers when it's in math class (and some, never).
I hope their experiences include seeing and doing the stuff of figuring out what 24 cups of flour using 1/3 measuring cups is like, too!
Emotions are powerful and immediate, so if you can harness an emotion -- any emotion -- and link it to mathematical insight, you create a visceral experience for the student: the initial insight is immediate ("ughh that looks horrible!"; "pffft, that one would be so easy!") and they can marshal reasons for justifying that insight *after the fact*; thus creating the sense of having intuition or natural insight before "learning" anything.
Now every student has some expertise. Now the stage is set: the student's feelings towards the easiest and hardest problems are *different* (e.g. confident about A but intimidated by X) but clearly of the *same* type. Suddenly there is a natural motivation to discover how the two are related ("can we transform X so that it would be more like A?"). I'll often follow this up by asking students if they could think of ways we could make A appear more like X, without affecting the solution; now they're thinking like a teacher.
Implicitly, it also reinforces an important lesson in learning: feelings *aren't* facts. You might be intimidated by X, it might look impossible but now you see it's on the same spectrum as A, which is not intimidating, so you know not be afraid of a new problem simply because it looks hard.
This is one of my favourite approaches to use. It is especially effective with intimidating content *because* of the strong emotional response. In some variations, I also emphasize to my students that this really is how mathematicians work: we look at a problem and notice what we don't like about it ("hmm, this would be a lot easier if we didn't have this part") and this emotional response can often be trusted to guide us towards an appropriate approach. Students learn that mathematical intuition and common intuition are not fundamentally different -- and who doesn't feel great about being "a natural"?! It feels wonderful knowing you can trust yourself.
Another variation on this approach is to ask "if you could change one thing about this problem, what would change?"; however, it's more appropriate at a later stage, when students have already developed a sense of the diversity of possible problems.
Every word of this, especially the ones connecting the visceral common intuition and the abstract mathematical intuition. What you're doing here is helping students realize "I know math" much earlier than is typical.
Giving students the opportunity to choose the "least difficult" of problems, offering student agency - giving them a choice and an entry into their learning. However, it also provides those who are confident to be looking for the most challenging to solve if they want. I recall advanced students who would jump to looking for the MOST difficult to answer, looking for that challenge, too. Ms. Esmende also includes the discussion between students giving voice, and then, I'm sure, she would use routines for students to explain the reasoning and continue with the lesson from there.
Love that move. The ranking offers us an angle into a conversation about the features of the systems but it also helps students locate themselves in their zone of proximal development.
Talk about a low floor! The question is an invitation that pretty much anyone can engage with. It is also a question that does not have one right answer. At the same time, it is an invitation to look at all the problems and evaluate how you think about them -- to consider them critically -- so it is not an "easy" ask either. In addition, by saying, "don't solve them," Gen is taking the need to get an answer off the table -- for this time, everyone is just considering the problems from their own points of view without the pressure to solve.
Decidedly NOT how systems of equations are treated in most curricula, including the ones I learned from as a kid and every one I taught from as a teacher. "Figure out if it's substitution or elimination. Execute. Don't mess up."
As a forever student, I enjoyed thinking about systems this way. I suppose that even when getting my undergrad in math and solving matrices, I would look for the easiest to solve first. I've never thought about my thinking in that regard. Thanks, Ms. Esmende!
Engaging way to start the lesson that encourages analysis and comparison. Without realizing it the students are comparing several systems to each other to determine which ones are easier or more challenging. It helps reluctant students to become part of the conversation. To take it one step further, have students work on vertical white boards and up and moving, allowing them to walk around and see each groups' choices and reasoning.
I love this. A conversation on what the problems look like ( difficulty) amongst themselves makes it real.
I only learnt the "formulaic" way to solve these. I had a good intuition for math and loved patterns and abstractions as a child so I thrived on that. what this question does is creates a situation for which I will always have an answer - because its asking my opinion not absolute right or wrong
This makes me wonder where else in mathematics might opinions and intuition apply.
Recently, in the playstore review of a popular educational app a user had commented that he/she used the answer from the AI in the app in a class assignment and the answer was incorrect. Only when teacher evaluated it, she got to know that it was wrong. This could be happening to many learners.
1. Every educational product that is using AI gives a caution message that you should check the relevancy of the response. They are not considering about the 'Critical Thinking' skill required for a learner(or anyone) who uses AI for guidance. Not thinking how a learner is supposed to have this ability to determine whether the generated answer is correct or not. Even companies which are training their own models to handle STEM subject questions and answers are hiring only highly qualified (PhD) candidates to validate and improve the responses. Not even subject experts. Which indicates the level of knowledge required to analyze the AI response.
2. The quality of the answer depends on the quality of the question. Do learners ask the right questions always? Sometimes they don't even realize there is some mistake in the question that they are asking.
There is definitely scope for innovation and improvements, but current trend of pushing young learners towards AI products doesn't seem to be ideal. The approach should be different.
Kids need to feel seen and heard, so asking them to articulate their opinions about the difficulty of equations to open rather than showing them procedures invites all learners to the lesson by meeting an essential need. Most (~95%) kids do not feel they have an essential need to follow procedures to solve systems of equations, so we have to welcome them into the learning by honoring what they do need: social connection, belonging to a community, being seen and heard by community members,...
Grabbed this piece for next week's featured comments, thanks!
> Kids need to feel seen and heard, so asking them to articulate their opinions about the difficulty of equations to open rather than showing them procedures invites all learners to the lesson by meeting an essential need.
OK there's all the emotional stuff... but I gravitate towards the cognitive.
.... but the thing that jumps out to me is how often numbers are given meaning. The gestures and comments about how much cake that would be (and the diagnostic that saying "two wedding cakes!!" suggests that thinking of amounts and making connections is a habit that's been developed) mean that this is so much less likely to turn into "KEEP CHANGE FLIP!"
Now, I *do* fervently hope attention is made to translating the stories into reasonably fast processing of the symbolic math language, but so many of my adult learners (I had several today) just don't make meaning of numbers when it's in math class (and some, never).
I hope their experiences include seeing and doing the stuff of figuring out what 24 cups of flour using 1/3 measuring cups is like, too!