We interrupt our regularly scheduled programming to ask you an admittedly strange question:

A bunch of brand new young enthusiastic preservice math teachers sit down in front of you. If you could pick 3-5 things about math education that you most want them to understand, what would they be?

Comment with whatever quick and dirty brainstorm you have. I know some of you have given this, or something like it, a stab in the past – in that case I’d appreciate a link. (I remember Dan did this semi-recently – others?)

1.
Know the culture;
Be the culture;
Change the culture.

2. You are here to teach children, not to teach math. You use mathematics as your vehicle…

3. Listen to the child. Listen in a way that allows you to see the world (see the problem, the question) through their mind. Listen to build a model for their ways of knowing and thinking.

4. Create scenarios from which the students pose the problems.

5. Recognize their are many ways of being mathematically smart; MANY ways. And let your students know this, repeatedly.

1. learn the math pedagogy
2. focus everything on clear communication of if the students are learning
3. get good mathematical conversation going – look at http://coxmath.blogspot.com/2010/08/exponent-rules.html for a nice video
4. do the math – do the problems every time
5. make sure every student has access and a way to help in the problem solving proccess – even if it’s just simple observations

btw, I asked the question because I am teaching a course for new math teachers in an MAT program and I’m thick into designing the course right now. I made my own list like this, I figured it should be the basis of the course, but felt sure I was leaving out important stuff. Y’all comments have already helped me think of some things I really care about that I’d neglected.

@blaw0013 – confused by #1 – can you make it more transparent? What culture? (Mathematical culture? American culture? Student culture?) And what does it mean to be it, and what needs to be changed? Sorry to be dense.

@hillby – thanks for the link to the video. Some clarification requests here too – when you say “learn the math pedagogy,” can you give me some specific facts or ideas or principles about math pedagogy that you’d most like them to understand? Also, #2 – do you mean, “focus your attention on whether the students are learning?” or “teach the students how to communicate whether they are learning?” or something else? And #4 – what do you mean do the problems every time? The teacher or the students? And what does “every time” mean?

Esoterically, any culture. It is a condition (requirement) of humanity to impact/affect the culture for which they are part.

Specifically: math education must change. So the culture I meant to point to was the math educational culture within the school they are a part of. The “change” demand is not necessarily one for “overthrow” (although in 80%+ of HSs I visit this is what is necessary), but “evolve”.

The message is basically, be a part of change, smartly, with intentionality. It is a moral obligation.

I can’t speak for Shawn, but what I think of when I see “Teach context before content” is to focus on purpose before product…on “when” and “why” before “what” and “how.”

Before teaching any math content, ask yourself, “What is it that I want students to understand about this content and what it means? What is this content’s ‘big purpose?'” When teaching about derivatives of composite functions, ask yourself, “What is a larger context to which this content applies?” Then help the kids care about the context first, which provides a need that can be filled by the content.

Just my two cents, though…like I said, I can’t speak for Shawn, but that’s that strikes me about it.

Try things (not in teaching) that are hard for you, or are confusing. Try a course in model theory? It is important to remember what it is like to feel confused, even lost.

Now, remember that when you are planning/teaching. Listen to the kids, let them explain back to you. They could be lost, and not saying anything. Remember, as hard as model theory was for me and whatever topic kills you/once killed was for you, the topic at hand may be for the kids.

What you teach is easy – for you. Don’t assume that silence = understanding.

That was preamble. My point:

teach modest amounts, well. and check for understanding as frequently as you can.

Some people say “less is more” and I hate the phrase, but love the point.

(there will come a time when racing through hard material is a reasonable thing to do. But it will not come soon.)

Each day, each lesson, come with more than you need, but think about what you need, and reduce that. The right amount to teach is usually significantly less than what a new teacher thinks is the right amount to teach.

And it takes a while (years in my case) to proceed from understanding this to actually implementing it well.

To quote Albert Einstein, “The mere formulation of a problem is far more often essential than its solution, which may be merely a matter of mathematical or experimental skill”

Teach your kids to construct problems and deconstruct problems, not to solve problems, especially poorly written ones in a textbook.

Thanks everyone! This is awesome. Getting answers to a request like this is a benefit of having a blog I never would have predicted.

Sue, I think I know what Shawn means by “context before content,” though of course I invite him to clarify for himself too. I take him to be saying something like this:

Mathematical skills and ideas all exist for a reason, a context that makes them needed / useful / explanatory / elegant. E.g. trying to understand the nature of speed is a context for why the derivative is useful and elegant. Often more than one such context exists. For the derivative, I could also be interested in understanding something about geometry – slopes and angles, whatever. But if math content is introduced without any context – “Hello, welcome to calculus class, the derivative of a function f(x) at a point c is a number f'(c) with the property that for any given e>0, [f(x)-f(c)]/(x-c) can be guaranteed to be within e of that number by constraining x to be adequately close to c…” – then the content is both boring and unintelligible. What are you talking about and why does anyone care? So, make sure students understand the context of the content before they get hit with the content itself. This way it will be both interesting and make sense.

im designing a course for senior lecturers that are teachin engineering maths and i found that although most lecturers can cite a lot of context to attract the students towards the content, they have problems when it comes to …

1. how to make maths fun.
yeah, seein’ numbers all day long can be a bit dry. i mean partial differentiation, laplace etc aint really that colorful to some students. seems like when we are so obsessed with clarity, accuracy and what not, we lose sight of joy. we gotta train teachers on how to make it fun.

2. how to simplify maths
some maths can be quite daunting. if we quote examples directly from the textbook (what most lecturers are doing), learning can be quite hard, as compared to designing our own examples that can guide the students progressively from simple to complex (hull’s effect). we gotta imbue the flexibility of thought to teachers to make em more adaptive.

3. how to think about maths
metacognition is really crucial when it comes to maths. exam questions might not be like the ones discussed in class. if students are not prepared for that, panic is the usual response. so ..

how to handle ambiguity?
how to simplify a complex problem?
how to analyze and synthesize knowledge?
how to plan ur solution when one is not apparent?
how to exploit a formula to our advantage?
how to know which formula/concept is useful in a particular situation and why?

we can get so focused with content that sometimes, we forget that at the end of the day, the skills of how to think effectively with maths, will matter most.

Hey Ben, here’s a shot at this. Mostly it echoes what’s written above, but anything worth saying is worth saying multiple times and in multiple ways.

Teaching math means learning math.

You aren’t trying to transfer something—not trying to put a book in a kid’s head. You trying to cultivate curiosity and self-reliance.

Having an object to push around and tinker with means everything—a physical manipulative, a computer model, a sequence of calculations that are easy for a student to perform but point to something beyond his current understanding. Building this kind of context and allow intuition to grow are essential. Objects to push around give an in to every student, because there’s no excuse to shut down and give up—everyone can try to make the shape with the tangrams, everyone can drag the points around until a pattern is seen.

Problems come before solutions—don’t spoil a good problem by explaining how it isn’t really a problem at all, that “here’s how you do it.” If you train a student in a set of skills and the path is always smoothed in front of him, what wonder that he isn’t a patient problem solver when you finally put a real problem in front of him. He knows that if he waits, you’ll just show him what to do.

It is so fruitful for students to compose problems. It’s huge for ownership, they love to do each other’s problems, it’s a great way to see whether a student really understands a concept, it’s empowering, it provides an opportunity for creativity and surprises, and it’s a great source of practice problems that don’t feel like drudgery.

My beginning algebra students all failed (less than 85% is fail for my mastery tests) a test on pre-algebra topics. Their homework included redoing each problem, explaining how it’s done, and making up and solving another problem like it. You just got me thinking the next step is to have them do each other’s problems. I’ll do that on Tuesday, during out warmup time. Thanks.]

Take every opportunity to make math human. It makes you more human to your students, too.

By that I mean to highlight the interesting things that math does for you in your real life and try and involve students in it. I find bringing in the parts of my life where I’m a cook, fibercrafter and woodworker into my math teaching practice invaluable.

Last August I asked what math lessons my readers would offer elementary math teachers.

1.

Know the culture;

Be the culture;

Change the culture.

2. You are here to teach children, not to teach math. You use mathematics as your vehicle…

3. Listen to the child. Listen in a way that allows you to see the world (see the problem, the question) through their mind. Listen to build a model for their ways of knowing and thinking.

4. Create scenarios from which the students pose the problems.

5. Recognize their are many ways of being mathematically smart; MANY ways. And let your students know this, repeatedly.

1. learn the math pedagogy

2. focus everything on clear communication of if the students are learning

3. get good mathematical conversation going – look at http://coxmath.blogspot.com/2010/08/exponent-rules.html for a nice video

4. do the math – do the problems every time

5. make sure every student has access and a way to help in the problem solving proccess – even if it’s just simple observations

Thanks, folks!

btw, I asked the question because I am teaching a course for new math teachers in an MAT program and I’m thick into designing the course right now. I made my own list like this, I figured it should be the basis of the course, but felt sure I was leaving out important stuff. Y’all comments have already helped me think of some things I really care about that I’d neglected.

@blaw0013 – confused by #1 – can you make it more transparent? What culture? (Mathematical culture? American culture? Student culture?) And what does it mean to be it, and what needs to be changed? Sorry to be dense.

@hillby – thanks for the link to the video. Some clarification requests here too – when you say “learn the math pedagogy,” can you give me some specific facts or ideas or principles about math pedagogy that you’d most like them to understand? Also, #2 – do you mean, “focus your attention on whether the students are learning?” or “teach the students how to communicate whether they are learning?” or something else? And #4 – what do you mean do the problems every time? The teacher or the students? And what does “every time” mean?

Esoterically, any culture. It is a condition (requirement) of humanity to impact/affect the culture for which they are part.

Specifically: math education must change. So the culture I meant to point to was the math educational culture within the school they are a part of. The “change” demand is not necessarily one for “overthrow” (although in 80%+ of HSs I visit this is what is necessary), but “evolve”.

The message is basically, be a part of change, smartly, with intentionality. It is a moral obligation.

1. “Homework” is ungraded, feedback-oriented, practice, not some magic way to teach responsibility (which does not exist)

2. Do not be boring. If you are bored, they want to die.

3. Teach context before content.

Shawn, I want to know more about what you mean by #3.

I can’t speak for Shawn, but what I think of when I see “Teach context before content” is to focus on purpose before product…on “when” and “why” before “what” and “how.”

Before teaching any math content, ask yourself, “What is it that I want students to understand about this content and what it means? What is this content’s ‘big purpose?'” When teaching about derivatives of composite functions, ask yourself, “What is a larger context to which this content applies?” Then help the kids care about the context first, which provides a need that can be filled by the content.

Just my two cents, though…like I said, I can’t speak for Shawn, but that’s that strikes me about it.

Try things (not in teaching) that are hard for you, or are confusing. Try a course in model theory? It is important to remember what it is like to feel confused, even lost.

Now, remember that when you are planning/teaching. Listen to the kids, let them explain back to you. They could be lost, and not saying anything. Remember, as hard as model theory was for me and whatever topic kills you/once killed was for you, the topic at hand may be for the kids.

What you teach is easy – for you. Don’t assume that silence = understanding.

That was preamble. My point:

teach modest amounts, well. and check for understanding as frequently as you can.

Some people say “less is more” and I hate the phrase, but love the point.

(there will come a time when racing through hard material is a reasonable thing to do. But it will not come soon.)

Each day, each lesson, come with more than you need, but think about what you need, and reduce that. The right amount to teach is usually significantly less than what a new teacher thinks is the right amount to teach.

And it takes a while (years in my case) to proceed from understanding this to actually implementing it well.

Jonathan

To quote Albert Einstein, “The mere formulation of a problem is far more often essential than its solution, which may be merely a matter of mathematical or experimental skill”

Teach your kids to construct problems and deconstruct problems, not to solve problems, especially poorly written ones in a textbook.

Thanks to @ddmeyer for this idea.

Thanks everyone! This is awesome. Getting answers to a request like this is a benefit of having a blog I never would have predicted.

Sue, I think I know what Shawn means by “context before content,” though of course I invite him to clarify for himself too. I take him to be saying something like this:

Mathematical skills and ideas all exist for a reason, a context that makes them needed / useful / explanatory / elegant. E.g. trying to understand the nature of speed is a context for why the derivative is useful and elegant. Often more than one such context exists. For the derivative, I could also be interested in understanding something about geometry – slopes and angles, whatever. But if math content is introduced without any context – “Hello, welcome to calculus class, the derivative of a function f(x) at a point c is a number f'(c) with the property that for any given e>0, [f(x)-f(c)]/(x-c) can be guaranteed to be within e of that number by constraining x to be adequately close to c…” – then the content is both boring and unintelligible. What are you talking about and why does anyone care? So, make sure students understand the context of the content before they get hit with the content itself. This way it will be both interesting and make sense.

im designing a course for senior lecturers that are teachin engineering maths and i found that although most lecturers can cite a lot of context to attract the students towards the content, they have problems when it comes to …

1. how to make maths fun.

yeah, seein’ numbers all day long can be a bit dry. i mean partial differentiation, laplace etc aint really that colorful to some students. seems like when we are so obsessed with clarity, accuracy and what not, we lose sight of joy. we gotta train teachers on how to make it fun.

2. how to simplify maths

some maths can be quite daunting. if we quote examples directly from the textbook (what most lecturers are doing), learning can be quite hard, as compared to designing our own examples that can guide the students progressively from simple to complex (hull’s effect). we gotta imbue the flexibility of thought to teachers to make em more adaptive.

3. how to think about maths

metacognition is really crucial when it comes to maths. exam questions might not be like the ones discussed in class. if students are not prepared for that, panic is the usual response. so ..

how to handle ambiguity?

how to simplify a complex problem?

how to analyze and synthesize knowledge?

how to plan ur solution when one is not apparent?

how to exploit a formula to our advantage?

how to know which formula/concept is useful in a particular situation and why?

we can get so focused with content that sometimes, we forget that at the end of the day, the skills of how to think effectively with maths, will matter most.

Hey Ben, here’s a shot at this. Mostly it echoes what’s written above, but anything worth saying is worth saying multiple times and in multiple ways.

Teaching math means learning math.

You aren’t trying to transfer something—not trying to put a book in a kid’s head. You trying to cultivate curiosity and self-reliance.

Having an object to push around and tinker with means everything—a physical manipulative, a computer model, a sequence of calculations that are easy for a student to perform but point to something beyond his current understanding. Building this kind of context and allow intuition to grow are essential. Objects to push around give an in to every student, because there’s no excuse to shut down and give up—everyone can try to make the shape with the tangrams, everyone can drag the points around until a pattern is seen.

Problems come before solutions—don’t spoil a good problem by explaining how it isn’t really a problem at all, that “here’s how you do it.” If you train a student in a set of skills and the path is always smoothed in front of him, what wonder that he isn’t a patient problem solver when you finally put a real problem in front of him. He knows that if he waits, you’ll just show him what to do.

It is so fruitful for students to compose problems. It’s huge for ownership, they love to do each other’s problems, it’s a great way to see whether a student really understands a concept, it’s empowering, it provides an opportunity for creativity and surprises, and it’s a great source of practice problems that don’t feel like drudgery.

Justin, I love the way you put it.

[And, you just gave me an idea for class.

>they love to do each other’s problems

My beginning algebra students all failed (less than 85% is fail for my mastery tests) a test on pre-algebra topics. Their homework included redoing each problem, explaining how it’s done, and making up and solving another problem like it. You just got me thinking the next step is to have them do each other’s problems. I’ll do that on Tuesday, during out warmup time. Thanks.]

Take every opportunity to make math human. It makes you more human to your students, too.

By that I mean to highlight the interesting things that math does for you in your real life and try and involve students in it. I find bringing in the parts of my life where I’m a cook, fibercrafter and woodworker into my math teaching practice invaluable.