“As we go through each lesson, it seems like my daughter has a good handle on the concepts, but when we get to the test she forgets everything. When I ask her about it, she shrugs and says, ‘I don’t know.’ What do you do when your child completely loses what she has learned?”
Forgetting is the human brain’s natural defense mechanism. It keeps us from being overwhelmed by the abundance of sensory data that bombards us each moment of every day.
Our children’s minds will never work like a computer that can store a program and recall it flawlessly months later.
Sometimes, for my children, a gentle reminder is enough to drag the forgotten concept back out of the dust-bunnies of memory.
Other times, I find that they answer “I don’t know” out of habit, because it’s easier than thinking about the question. And because they’d prefer to be doing something else.
My hundred chart list began many years ago as seven ideas for playing with numbers. Over the years, it grew to its current 30+ activities.
Now, in preparing the new book, my list has become a monster. I’ve collected almost 70 ways to play with numbers, shapes, and logic from preschool to middle school. Just yesterday I added activities for fraction and decimal multiplication, and also tips for naming complex fractions. Wow!
Gonna have to edit that cover file…
In the “Advanced Patterns” chapter, I have a section on math debates. The point of a math debate isn’t that one answer is “right” while the other is “wrong.” You can choose either side of the question — the important thing is how well you support your argument.
Here’s activity #69 in the current book draft.
Have a Math Debate: Adding Fractions
When you add fractions, you face a problem that most people never consider. Namely, you have to decide exactly what you are talking about.
For instance, what is one-tenth plus one-tenth?
1/10 of 100
Well, you might say that:
of one hundred chart
+ of the same chart
= of that hundred chart
But, you might also say that:
of one chart
+ of another chart
= of the pair of charts
That is, you started off counting on two independent charts. But when you put them together, you ended up with a double chart. Two hundred squares in all. Which made each row in the final set worth of the whole pair of charts.
So what happens if you see this question on a math test:
+ = ?
If you write the answer “”, you know the teacher will mark it wrong.
Today we have a guest post — an exclusive tale by Sasha Fradkin and Allison Bishop, authors of the new math storybook Funville Adventures. Enjoy!
Funville Adventures is a math-inspired fantasy that introduces children to the concept of functions, which are personified as magical beings with powers.
Each power corresponds to a transformation such as doubling in size, rotating, copying, or changing color. Some Funvillians have siblings with opposite powers that can reverse the effects and return an object to its original state, but other powers cannot be reversed.
In this way, kids are introduced to the mathematical concepts of invertible and non-invertible functions, domains, ranges, and even functionals, all without mathematical terminology.
We know about Funville because two siblings, Emmy and Leo, were magically transported there after they went down an abandoned slide.
When they came back, Emmy and Leo shared their adventures with their friends and also brought back the following manuscript written by their new friend Blake.
But my favorite way to celebrate any new year is by playing the Year Game. It’s a prime opportunity for players of all ages to fulfill the two most popular New Year’s Resolutions: spending more time with family and friends, and getting more exercise.
So grab a partner, slip into your workout clothes, and pump up those mental muscles!
For many years mathematicians, scientists, engineers and others interested in mathematics have played “year games” via e-mail and in newsgroups. We don’t always know whether it is possible to write expressions for all the numbers from 1 to 100 using only the digits in the current year, but it is fun to try to see how many you can find. This year may prove to be a challenge.
Use the digits in the year 2018 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.
You must use all four digits. You may not use any other numbers.
Solutions that keep the year digits in 2-0-1-8 order are preferred, but not required.
You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.
My Special Variations on the Rules
You MAY use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal. But students and teachers beware: you can’t submit answers with repeating decimals to Math Forum.
You MAY use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. I’m including these because Math Forum allows them, but I personally try to avoid the beasts. I feel much more creative when I can wrangle a solution without invoking them.
This lovely puzzle (for upper-elementary and beyond) is from Nikolay Bogdanov-Belsky’s 1895 painting “Mental Calculation. In Public School of S. A. Rachinsky.” Pat Ballew posted it on his blog On This Day in Math, in honor of the 365th day of the year.
I love the expressions on the boys’ faces. So many different ways to manifest hard thinking!
Here’s the question:
No calculator allowed. But you can talk it over with a friend, as the boys on the right are doing.
You can even use scratch paper, if you like.
Thinking About Square Numbers
And if you’d like a hint, you can figure out square numbers using this trick. Think of a square number made from rows of pennies.
Can you see how to make the next-bigger square?
Any square number, plus one more row and one more column, plus a penny for the corner, makes the next-bigger square.
So if you know that ten squared is one hundred, then:
… and so onward to your answer. If the Russian schoolboys could figure it out, then you can, too!
Update
Simon Gregg (@Simon_Gregg) added this wonderful related puzzle for the new year:
You want your child to succeed in math because it opens so many doors in the future.
But kids have a short-term perspective. They don’t really care about the future. They care about getting through tonight’s homework and moving on to something more interesting.
So how can you help your child learn math?
When kids face a difficult math problem, their attitude can make all the difference. Not so much their “I hate homework!” attitude, but their mathematical worldview.
Does your child see math as answer-getting? Or as problem-solving?
Answer-getting asks “What is the answer?”, decides whether it is right, and then goes on to the next question.
Problem-solving asks “Why do you say that?” and listens for the explanation.
Problem-solving is not really interested in “right” or “wrong”—it cares more about “makes sense” or “needs justification.”
Homeschool Memories
In our quarter-century-plus of homeschooling, my children and I worked our way through a lot of math problems. But often, we didn’t bother to take the calculation all the way to the end.
Why didn’t I care whether my kids found the answer?
Because the thing that intrigued me about math was the web of interrelated ideas we discovered along the way:
How can we recognize this type of problem?
What other problems are related to it, and how can they help us understand this one? Or can this problem help us figure out those others?
What could we do if we had never seen a problem like this one before? How would we reason it out?
Why does the formula work? Where did it come from, and how is it related to basic principles?
What is the easiest or most efficient way to manipulative the numbers? Does this help us see more of the patterns and connections within our number system?
Is there another way to approach the problem? How many different ways can we think of? Which way do we like best, and why?
What Do You think?
How did you learn math? Did your school experience focus on answer-getting or problem-solving?
How can we help our children learn to think their way through math problems?
I’d love to hear from you! Please share your opinions in the Comments section below.
CREDITS: “Math Phobia” photo by Jimmie (blog post title added) via Flickr (CC BY 2.0). Phil Daro video by SERP Media (the Strategic Education Research Partnership) via Vimeo.
Playing with a new image editor, I came across this Winston Churchill quote. What a great description of how it feels to learn math!
If you have a student who struggles with math or is suffering from a loss of enthusiasm, check out Jo Boaler’s free online course on developing a mathematical mindset:
It’s a short book with plenty of great stories, advice, and conversation-starters. While Danielson writes directly to parents, the book will also interest grandparents, aunts & uncles, teachers, and anyone else who wants to help children notice and think about math in daily life.
“You don’t need special skills to do this. If you can read with your kids, then you can talk math with them. You can support and encourage their developing mathematical minds.
“You don’t need to love math. You don’t need to have been particularly successful in school mathematics. You just need to notice when your children are being curious about math, and you need some ideas for turning that curiosity into a conversation.
“In nearly all circumstances, our conversations grow organically out of our everyday activity. We have not scheduled “talking math time” in our household. Instead, we talk about these things when it seems natural to do so, when the things we are doing (reading books, making lunch, riding in the car, etc) bump up against important mathematical ideas.
“The dialogues in this book are intended to open your eyes to these opportunities in your own family’s life.”
If you’d like to help fund the blog on an on-going basis, then please join me on Patreon for mathy inspiration, tips, and an ever-growing archive of printable activities.
If you liked this post, and want to show your one-time appreciation, the place to do that is PayPal: paypal.me/DeniseGaskinsMath. If you go that route, please include your email address in the notes section, so I can say thank you.
My youngest daughter wanted to do Singapore math. Miquon Red was her main math text at the time, but we added a bit of Singapore Primary Math 1B whenever she was in the mood.
We turned to the lesson on subtracting with numbers in the 30-somethings.
of one hundred chart
of that hundred chart
of the pair of charts
of the whole pair of charts.
Denise Gaskins' Let's Play Math 