I’m keen on looking for math in our FIAR books, but this time I’m thinking about doing some math in a new way. There’s math in poetry – tons of patterns with the iambic pentameter and such. But this picture book took a short poem (it’s only 16 lines) and added illustrations to give it that extra touch.
Illustrations… hmm…
There’s a classroom teacher who created a math assignment around photography. What a cool project for homeschool families; you can make it as strict or as loose as you want.
He blogged all about the process, including the challenges and results,which makes modifying it for homeschool super easy and fun!
Snowy Evening Math Photos
If you have snow around, use the beautiful images in Stopping by Woods on a Snowy Evening to inspire a photo field trip in the neighborhood. Use the assignment and rubric from the above links or just go with the flow. Encourage your children to look at the world with math eyes!
I live in Houston, where it snows once every nine years. So I had to depend on Flickr for these examples.
Abstract, curious, math-y!
There’s some fun math in the angle of the snow, and also in the shadow in this pictures. It makes me wonder if the camera is tilted or if the snow is on a hill. Click on the image to get some math thoughts from the photographer.
by windiepink | Flickr.com | CC BY
Mr. Rene’ Descartes would love this one!
Lots of snow, but also some coordinate plane stuff here:
by Jan Tik | Flickr.com | CC BY
Blocks and smiley faces!
These look like snow covered blocks – big ones! Fancy math term: rectangular parallelepiped. (Get your two year old to say that – it’s SO cute!)
by plizzba | Flickr.com | CC BY
Graphing animal tracks is fun!
You know how you put a bunch of dots on a grid and then connect them? That’s graphing in math. That’s what these animal tracks look like! Incidentally, this photo is from my favorite Flickr person who has a great collection of math photos!
by woodleywonderworks | Flickr.com | CC BY
What do your photos look like?
What kinds of math do your kids see in the snow or other weather you might have? Share the links and their thoughts in the comments!
I previously wrote about preschool math readiness after I found a list for preschool studies. There are many more preschool math concepts inside all the other subjects, too.
Here’s the list for reading readiness and how math is really a part of these!
Reading Readiness
Remembers objects from a given picture.
If a student can remember things that happened in previous problems in their math homework, they will make the connections faster. This remembering thing is huge in math!
Knows what a letter is.
This helps them to understand the difference between symbols with value (numbers) and sybols with no value (letters). Later they’ll be able to see that both letters and numbers can “signpost” things – like in an outline or the answer options on a test.
Has been read to frequently.
And count as many things in those books that you can!
Looks at books and magazines. Make sure to include magazines like Time that have graphs and statistics in them.
Recognizes some nursery rhymes.
Great silly math one: “A dillar a dollar a 10 o’clock scholar – why do you come so soon? You used to come at 10 o’clock and now you come at noon!”
Identifies parts of the body.
The body is full of the symmetry and pairings. Not to mention patterns – like 5 fingers, 5 fingers, 5 toes, 5 toes.
Identifies objects that have a functional use.
Scissors cut things in half, glue allows you to add things.
Knows common farm and zoo animals.
Notice the pattern in the number of feet animals have. Also, if you multiply the number of animals by 4, you get the number of feet!
Pronounces own first name.
Identifying self helps to “name” things – in particular name a numeral with a word (like 2 is “two).
Pronounces own last name.
If a child has the same last name as one or both parents, you can point out that as a pattern.
Expresses self verbally.
Toddlers have to be ready for everything!
This includes desires – so it’s a great opportunity to teach them to desire “three more,” and then count down. When they say, “one more,” they think it means “another” – unless they see they can go from three to two to one, etc.
Identifies other children by name.
Again – this is a forerunner to identifying the names of numerals.
Tells the meaning of simple words.
Number words are words too. Knowing the meaning of number words is important.
Repeats a sentence of 6-8 words.
When kids can repeat a sentence, they are ready to start repeating math facts. The sentence, “Four times eight is thirty-two” is pretty much a sentence of six words.
Completes incomplete sentence with proper word.
Can you say more math facts? Sweet!
Has own books.
And if some of these are counting books, how wonderful!
Understands that print carries a message.
This is the basis of equations too. An equation is a “printed” thing that means something, that has a message.
Pretends to read.
If your books also have basic math concepts, they can pretend to read those too.
Uses left-to-right progression.
Math is also read left to right.
Answers questions about a short story.
Include fun math stories and they can practice answering questions about those too.
Tells the meaning of words heard in story.
Use stories that have more than one of the same thing like two children – Hansel & Gretel – and Three Little Pigs. Discuss what it would change in the story if there were three children, or four little pigs.
Looks at pictures and tells a story.
If the pictures have multiples, they can use some counting and subitzing to explain what’s happening with those many objects.
Identifies own first name in manuscript.
Identifying manuscript letters is like identifying written numbers. It takes some time to see the differences between a 4 in print and a 4 in writing.
Prints own first name.
When they can do this, they also can print their phone number, zip code and street number.
What you can do
Notice that there’s math in so much – even toddler reading readiness. When you get your children ready for school – whether it’s homeschool or classroom school – you remind them that they are also doing math.
The more they hear that they’re doing math, the better they’ll be when the get into an “official” math class.
This article is based on the talk “Improving Creativity with Technology” given at the University of Houston teachHouston T3 Regional Conference.
Traditionally we give a concept or algorithm to the students and ask them to memorize, understand and use it. And by “give” I mean we serve it to them well done, fully baked, nothing left to do but eat it.
Piaget’s research (and subsequently others) suggest that allowing students to discover or create the methods is more effective than handing them the method and asking them to take it, eat it, no questions asked.
But how do you let them discover it?
Since the math we teach in middle school and high school is based on real numbers, every concept can be demonstrated with “plain” numbers. Which means it can be discovered by playing with numbers.
Calculators make this playing or experimentation fast – giving a student the ability to quickly see patterns and construct concepts.
Introduce the topic with numbers.
When you introduce a topic, give 10-20 “examples” of it with real numbers. Ask the students to play with them and notice any patterns they find.
Notice you’re creating an inquiry-based learning environment, an Inquiry Zone for Math Learning. Remember to maintain positive body language, ignore negativity, and super-validate what any student says.
Give the students the power!
Until the student decides differently, everyone is wrong. Even the teacher and textbook. They get to validate it for themselves – and they can do this with real numbers.
Ultimately, if they grow to be mathematicians, they’ll learn that verifying with lots of real numbers doesn’t mean “proving” it – but for the time being, this works fine.
Giving them this power lets them experiment as much as they need, and only as much as they need, to verify a concept for themselves.
Use the Play & Say method.
You’ve heard of the “Plug & Chug” method? You take a formula, plug in the numbers and chug through the arithmetic. Plug & Chug is a non-discovery based practice tool. The practice is good, mind you, but the formula is given, not discovered.
Remember, something discovered is more likely to be remembered than something given.
So use the “Play & Say” method. Each student plays with the numbers until he or she can say what the formula or concept is.
Caveats
If you’re trying to teach a concept with this and one student discovers a different formula or concept. Run with it – as long as it’s mathematically sound. Don’t discourage the discovery of anything, even something not on the current curriculum.
If a student gets frustrated, don’t force them to discover it themselves. Give them the concept or formula and encourage them to experiment later with it.
Suggestions
If you find there is a big difference in how much time each student takes, send the experiments home with them. Give them five minutes at the beginning of class to play – the students who realize they need more time will have done more the night before.
How will you do it? Share your thoughts and experiences in the comments.
In a previous article, I suggested two ways to introduce inquiry-based math instruction into your teaching structure. One of them was to create a safe discovery environment – an Inquiry Zone. This is where students could ask anything and not be held responsible for performance.
It can be a location or a time – or both. It can be announced or understood.
As the facilitator, here are some things to do to create the safe Inquiry Zone.
Maintain positive body language.
When a student asks a question, make sure your body language and facial expressions say, “Wow, that’s a very smart thing to ask.” Do this regardless of how you feel about the question personally.
Ignore negativity.
If other students laugh at a question, ignore them. Instead validate the asker and the question through words and body language. Chastising the negativity only validates it.
Super-validate every question.
Follow up every question with a similar or extended question. If you can’t think of one, say, “That’s really interesting. Let’s write it down so we can look it up on Google later.”
Post the rules.
Set up rules so that all the students know what inquiry really means in math. The rules I use are:
Until you decide differently, everyone is wrong. Even the teacher and textbook.
Every question is a great question.
These two rules build confidence. Not in getting the right answer, but in deciding what it means to be a right answer. If a kid gets to decide when the answer is right, they will ask more questions. Inquiry!
Be wrong – often.
Modeling this “wrongness” makes kids comfortable with it. The more comfortable they are with being wrong, the more likely they are to engage in inquiry.
You can be wrong by not knowing something or actually doing a problem wrong (doing your arithmetic wrong, for instance).
If you are normally perfect, fake it. Mess up. Do it for the kids.
Destroy the back of the book.
I’m not sure you can get away with this without being fired.
If you don’t mind getting fired from your teaching job, or if you homeschool, get the kids to rip out the back of the book – you know, the part with all the answers in it. And don’t let them talk you into letting them “check their answers” on the calculator. That’s just as damaging.
The more they get to decide if their own answer is right, the more they will ask. The more inquiry will happen!
The math education revolution has been growing. Last year Salman Khan and the Gates Foundation brought Khan Academy to everyone in the world with a computer.
Dan Meyer jumped into the ring with his math class makeover. With his Any Questions? and Three Acts, he’s fighting the good fight to get kids to learn math.
Social learning math games like Sokikom, iPhone apps like Motion Math and face to face programs like Mathnasium have joined the math education revolution, too. Thousands of tutors are taking part.
Experts all over are helping kids understand that math is important, necessary and valuable to learn.
Kids still resist math homework and avoid participating in math class. Why?
Because everyone knows that math is boring, hard and has nothing to do with real life. Nobody really likes it – unless they’re an engineer or accountant.
Math teachers are mean and professors write math books just to mess people up.
Everyone knows this.
Who is this “Everyone”?
Grownups.
Yup – you and me.
When you hear people talk about math or math education, what do they say? Anything positive?
The best I’ve ever heard was, “Actually, I kinda like math.”
Which means, “I know it’s not cool, and I’m sorry for saying it, but I like math.”
Would you do something nobody liked?
No! Of course you wouldn’t. At least not on a regular basis.
If everyone you knew and respected avoided something – you would too. If all your friends jumped off a cliff – so would you.
We’re human. That’s what we do. We stick together.
So we’re losing the math education revolution.
By the very design of our society, Khan Academy, Sokikom, teachers, tutors and everything designed to help kids learn math are failures. The math eduction revolution is bust.
We want kids to learn math because it’s important. Math is necessary and valuable to learn.
But since nobody really likes math, or even does math, kids aren’t buying it.
And I can’t blame them.
Is it fixable?
At this point it’s easy to throw in the towel. Give up. Quit. Decide that the world is going to end up like that movie Idiocracy.
But we’re so close to the solution.
The solution involves something that’s very cheap – and research based!
Ready for it?
The missing piece of the math education revolution is that we need to teach parents positive influence skills to encourage math.
What? Will that work?
We have seen this happen with reading – remember the Reading is Fundamental campaign of the 80s? It’s still going strong along with other programs like the “Read 3” program from HEB. Parents are encouraged, even pushed, by teachers to read to their children every day.
These efforts have changed the culture in our world so that reading is viewed as something “everybody does.” Parents now have positive influence skills in encouraging reading.
And those skills have extended across our entire culture!
That’s the missing piece!
Parents can develop the same skills for encouraging math. And when we do, everything will change.
Just like it did with reading.
When parents start talking about math in a positive way, all of society will.
The math education revolution will succeed!
It’s your turn…
If you’re a parent, learn how to use some positive math talk. Join a program like That’s Math, read articles on Math for Grownups or any other math blog that strikes your fancy.
If you’re a math teacher, blogger, tutor or developer of math products – make something teaching parents how to talk positively about math. We’re developing That’s Math, but there needs to be more of these.
Inquiry-based math instruction is the opposite of performance-based math instruction. And the research shows that inquiry-based math instruction is more effective than performance-based.
But performance is required if you’re teaching anywhere other than a homeschool. So what do you do?
You have to balance the requirement of performance with the need for discovery and inquiry-based learning. Which means you have to integrate discovery and inquiry-based elements into your math teaching.
You can do this formally or informally. You can say it out loud, or just do it. And which method you choose will depend on how much you are supported.
Set aside time for inquiry-based instruction.
Bring inquiry-based math instruction into your lessons by breaking up class time into “performance-for-tests learning time” and “discovery/inquiry-based learning time.”
You can have specific topics for the discovery learning time or even a “free math time” where there is no specific topic.
Make sure you keep the discovery topics independent from test based/performance-based topics. As much as you’re tempted, don’t connect the two. If the students connect them, acknowledge it briefly and keep moving.
Keeping them separate keeps the “must get the right answer” attitude out of the discovery time. This is very important.
For example, if you’re teaching multiplication this week in class, set aside the discovery time as patterns or geometry. This can be 5-10 minutes at the beginning of each class period or one day of the week devoted to it.
Create a safe discovery/inquiry zone.
If you want to include discovery in the regular curriculum, you have a little bit more of a challenge. Children learn early that performance is required in math classes. So they avoid being creative and asking questions. (See the research paper on that here.)
Which means you have to undo years of creativity-destruction to get them to participate in inquiry based activities. Depending on the students’ ages, you might have to start with introducing really goofy stuff to break them out of their comfort zone.
Let them wear fuzzy red hats and crazy glasses and tell tall math tales for the first few times. They’ll get warmed up to dig into some creative math inquiry. Anything’s easy when you’re wearing fuzzy red hats!
After a while they’ll be used to the safe environment and the inquiry-based math learning will start to flow from them.
Do both.
Ideally, you can have a “fun” time – where the math you have to do is set aside and you let the students dream about crazy math stuff. In addition, all your lessons involve the safe zone.
If you can make it work – do it. You’ll grow confident, adventurous, smart kids – the research supports it!
* Inquiry based math instruction is really the wrong term. Instruction means giving of something – in this case knowledge. It really should be inquiry based math learning. Teachers and parents are inquiry-based facilitators.
Feature image by philcampbell | Flickr.com | CC BY
I was reading a research paper about math teacher shortages by D.R. Sterling. Turns out that less than 12% of math teachers leave the profession through retirement.
So 88% of the time math teachers quit, get fired or die!
Have you avoided the profession because it doesn’t pay anything near what you make now?
Could you use a 9 month vacation to do something totally different, worthwhile and exciting?
Can you afford a 70% cut in pay for 9 months to really make a difference?
If you want to change things in math education, you can!
You can give 9 months to make a difference in the lives of 100 kids!
Is that do-able?
In Texas, and I’m guessing lots of other places, there is an emergency teaching certification some schools are allowed to offer when they can’t find any certified teachers. You go through an interview process and if the school decides you’re qualified – then you get to teach.
It expires after three years, but hey – you’ll probably be gone by then anyway!
Find one of these schools – it shouldn’t be hard. Go to work making a difference for a lot of kids – and making a difference in the system!
Why would you do this?
I just finished Seth Godin’s book Tribes. The premise is that a tribe (a group of people with common thinking) needs a leader – and you are that leader. So get up off your tooshie and lead, already! (That’s not a quote – it’s just what I got from the book. But it should be a quote.)
If you can afford it, and you’re passionate about helping kids learn math, then you have an obligation to help. Teachers in the system won’t risk getting fired to make change. You can!
But really, WHY would you do this?
Tons of people – teachers and parents – know beautiful and encouraging ways to teach math.
But there’s a group of people who have put together these nutty rules of what performance in math should look like each step of the way (the Common Core Standards and NCLB). Those rules prevent teachers from doing what they do best – teaching. Instead they fear for their jobs and teach the tests. All the while struggling to fit in a little learning here and there.
If you jumped in there, with no intent on following the rules, think of what would happen:
You’d teach. Not some crazy set of rules, but you’d teach math the way only you know how. And that’s probably a beautiful, experiential, exciting way.
You’d change the lives of the kids. They would see what real math might look like. They would see someone not paralyzed by fear of having your students fail the latest standardized test.
You’d inspire the teachers. Sure, some of them would call you a heretic. And so what – you would be! The ones who really want kids to learn math would follow your lead.
You’d be giving of yourself in the most valuable way there is. Talk about donation – you’re likely to need an extra $40K to make your bills. Is it tax deductible? If you consider that it puts you in a seriously lower tax bracket – yes!
You’d get fired. Yup – and isn’t that the point? Then you could go back to your high paying job, hire an attorney to sue the state and make big news. Show the public what it means to teach math!
Are you in?
I’ve put in notice at my day-job. I’m fixing to hit the pavement as full time Math Education Advocate at MathFour.com. Last year I made $16 in Amazon.com affiliate sales from this website, so I’m not falling back on any income here. I’ve also picked up teaching two community college developmental math classes. Which means I’m going to be gaining 32 hours a week to work on math ed advocacy – and taking a 90% pay cut.
I’m scared out of my mind.
But Daughter is 2 years old. I can’t afford to not go at this full time.
Will you join me?
Feature and post images by ewen and donabel | Flickr.com | CC BY
This is a feature article by William Devine, MA, LPC, CART. Wil is the Research Guy at MathFour.com.
The “old guard” of academics focused on rote memorization, correct answers, and academic performance – performance based, measurable objectives. To a large degree some of these focuses continue to prevail.
But a different way of teaching is emerging. Based on the research, this new way is more helpful, sustainable, and effective.
They were studied in an attempt to understand the affects of teaching styles on the students. And they were observed during regular, non-testing times.
It was noticed that their teaching styles tended to fall into roughly two categories: performance based and understanding based.
Performance-based teaching was found lacking.
When the teachers focused only on math performance, students were more likely to disconnect and feel intimidated.
In these cases, if a student gave an incorrect answer, they were told as much. Then another student was called on and so on until the correct answer was given. The problem with this search for the right answer was the lack of instruction that followed. It became all about doing it right instead of teaching the kid how to do it.
The instructor would then approach the next math problem with little or no discussion on how the correct answer was determined. Where was the teaching?
As we all know, if we feel uncomfortable with something, have anxiety, we tend to disengage or avoid the situation. This was observed to be the case in the classrooms of performance-focused teachers .
Students disengaged and were concerned about not only doing something the wrong way but feeling unable.
Understanding-based teaching was effective, helpful and encouraging.
Other teachers focused on helping students understand where he or she may have come up with the incorrect answer. The observed results were very different.
Teachers would help students arrive at the correct method. They would work to help them understand how to do the problem. Ultimately, students would arrive at the correct answer.
Furthermore, they would engage on each of the next questions – and get those right too!
So what can we do right now to move toward this idea of supportive academic encouragement?
Be careful with the words “wrong”, “incorrect”, and “bad”.
Kids are sensitive to these terms because they imply doing something they aren’t supposed to do and that lead to things they don’t want. And then we wonder why they don’t want to try!
We want them to continue engaging and putting forth effort because this is how learning happens. A “wrong” answer is so much better than no answer at all.
We’re not proposing that you stop using these terms altogether. Just do so sparingly.
Encourage improvement, not performance.
Focus: How much time do we focus on the incorrect answers? Instead, acknowledge what was correct and build on this. Recognition and praise for what they are doing well will encourage them to continue to stay engaged in the exercise.
Before: “I graded your paper and you missed 4. Let’s try those again until you get them right.”
After: “Good job! You got 6 right. Let’s try a few more.”
Help them discover how they got to where they got.
How: Understanding how they arrived at a particular answer can help determine what needs to be corrected in how they are doing something. It becomes a truly instructing experience rather than a performance (you got this one wrong and this one right). We want them to learn how to do something. If they feel pressure to get the right answer the first time, they will hesitate to offer any answer at all. This stifles trial and error learning and instills an aversive experience (fear) into the learning process.
Before: “5 is the wrong answer. Try this next one and really concentrate.”
After: “I see that your answer is 5 here. Walk me through how you got there.”
A note of concern: I have found in the research literature an indication that some teachers think if a child is having fun while learning, they aren’t really learning. Wow.
William Devine is a Licensed Professional Counselor in private practice. He has joined the MathFour.com team as the “Research Guy”. Connect with him in the comments, on the contact page or via twitter @MathPsych.
Feature and post images by ukanda | Flickr.com | CC BY
Through the Living Math Yahoo Group, I got a list for the typical courses of study for various grade levels. In each grade level link, there’s a list of math concepts.
Since daughter is two, I immediately clicked on the preschool link. I was a little offended to find such a long reading list and no list titled, “Math.”
And then I read the list. In almost every line I found math concepts!
If you teach your little one these things on this list, check out the advanced math they will also be getting!
(Note: my comments are italicized.)
Size
Understands big and little.This preschool math concept sets them up for volume comparisons.
Understands long and short. This one gets them ready for length comparisons.
Matches shapes or objects based on size. Forerunner to inscribing shapes in other shapes, dissecting shapes to understand area formulas, etc. Also getting them ready for pattern recognition.
Colors and Shapes
Recognizes and names primary colors. This preschool math concept is the beginning of pattern recognition.
Recognizes circles. Getting ready for the idea of a shape with no sides and a shape with infinite sides.(whoa!)
Recognizes rectangles. Setting up for all sorts of area formulas – since just about every regular shape can be made into a rectangle with scissors and glue.
Matches shapes or objects based on shape. The math concepts here get them ready for the geometric concept of similar and even more pattern recognition.
Copies shapes. Kinesthetically understanding concepts of geometry. This one also helps them get ready for pattern recognition – or making their own patterns!
Numbers
Counts orally through 10. This is less of a preschool math concept and more of a “you just really gotta know this” concept – kinda like the alphabet.
Counts objects in one-to-one correspondence. One-to-one correspondence is a huge and important concept in algebra.
Understands empty and full. Volume comparisons as well as the grouping of objects to fill a volume.
Understands more and less. The foundation of ordering numbers and understanding relations.
Position and Direction
Understands up and down. Forerunner of the number line.
Understands in and out. This preschool math concept gets them into early set theory.
Understands front and back. More number line stuff.
Understands over (on) and under. The math concepts here get them ready for setting up fractions.
Understands top, bottom, middle. Standard preschool readiness and also the basics of ordering.
Understands beside and next to. Gets them ready to multiply a number by a variable by just standing the number next to the variable: 3x means 3 times x.
Understands hot and cold. Getting ready to measure temperature.
Understands fast and slow. Getting ready to measure speed as well as rates.
Time
Understands day and night. This preschool math concept gets them ready to expand into bigger and bigger time units – which will allow them to eventually understand the ideas of infinity and infinitely small. Curiously, this is also pattern recognition – every day you get day and every day you get night!
Knows age and birthday. This seems to be another “good to know” thing. However, it is another forerunner to big time units.
More…
There are many more preschool math concepts that can be spun off from the other lists, like Reading and Social-Emotional Development. I’ll be writing up how to integrate math in those in a few days!
Am I missing anything? Let me know in the comments.
