Sunday, February 14, 2016

Teaching Elementary Math Without a Formal Curriculum

This post was updated in September 2017 to include more math games and books.

We've Come a Long Way

We started homeschool kindergarten nearly 5 years ago using the typical math education methods such as textbooks, workbooks, and flash cards. By halfway through 1st grade, my daughter was dreading math. When I had my epiphany a few years ago about how my schooling methods were actually detrimental to teaching my daughter to love learning, math was one of the subjects that was forefront in my mind. How could my only-6-year-old daughter be starting to hate math?

Over the last few years I have been implementing Leadership Education principles into our homeschool, and our math work has been transformed. Instead of dreading and hating math, my now nearly-9-year-old daughter thinks math is fun and interesting. And her little brother, who just turned 6, is coming right along with us, enjoying it as much as his sister.

How I Teach Math Without a Formal Curriculum

I focus on three specific approaches for teaching math: games, read-alouds, and everyday math. These three approaches form the cornerstone of our homeschool math curriculum. My children love them all, and that means that they love their math studies.

I don't push my children to engage in any of these resources. Instead, they are always free to decide whether or not they want to participate. But the thing is, our math work has become so fun and un-pressured that they almost always want to participate. I don't do math read-alouds and games with my children every day; that would take some of the fun and excitement out of it. Rather, I aim for about 3 times a week (and of course, everyday math does happen pretty much every day). My children are getting to explore math rather than getting bogged down in repetitive drills, and this exploration fosters a high level of interest in math. 

Math Read-Alouds

Math read-alouds provide a great opportunity to introduce new mathematical concepts to my children. Often I will read these books alongside a lap-size dry erase board where I can illustrate things further, or where we can write our answers to questions posed in the books. Instead of forcing my children to answer the questions in the books, I give them the opportunity to do-so; if they don't feel like it, then I will keep it lighthearted and work through the problems myself. We read math read-alouds usually once or twice a week.

  • Bedtime Math - Each page includes some engaging facts and then three math problems (ranging from easy to difficult) that give my children a chance to put math concepts into practice. My children absolutely LOVE Bedtime Math, and they are always begging for one more page.
  • Life of Fred - Life of Fred is a series of books that tells the story of Fred Gauss, a 5-year-old math genius who teaches at a university.  Life of Fred books range from elementary math all the way up through Calculus. Besides mathematical concepts, Life of Fred books also weave other topics into the story such as constellations, carnivores, and languages. At the end of every chapter, there are a few math problems to answer, but they are much more interesting than the problems in most math books. Fred and his doll Kingie are an unlikely duo that my children just adore. I wrote a post on how to use Life of Fred with kids of multiple ages here.
  • Sir Cumference books - These are engaging picture books that cleverly wind mathematical concepts into the stories. For instance, in Sir Cumference and the First Round Table, my children learned about diameter, radius, and circumference in a fun, easy-to-remember way. Sir Cumference and All the King's Tens introduced place value in a way that my daughter, especially, loved.   
  • Anno's math books - Anno's books are beautifully illustrated and they show math concepts such as multiplication very clearly. My children especially love Anno's Magic Seeds, and Anno's Mysterious Multiplying Jar.    
  • Math and Magic in Wonderland - This delightful book tells the story of two sisters and their magical adventures with math. I love that the language in this book is beautiful, that the storyline is interesting, and that the math problems are designed to elicit curiosity instead of boredom.  The math problems in Math and Magic in Wonderland were too advanced for my 6-year-old son when we started reading this, but at age 7&1/2 he is ready to participate more. My daughter was 9-years-old when we started this book, and she enjoyed the math problems right away. 
image from

Everyday Math 

With Everyday Math, I look for opportunities to teach math in the context of real life. This allows my children to see that math is relevant to their lives. I don't get preachy about math in our everyday lives; rather, I just use math in meaningful ways as I go about my days, and I encourage my children to do the same. 

Here are some examples of how Everyday Math can be used to teach math.

  • We have a thermometer outside our kitchen window that allows us to see what the temperature is outside. We use this daily to see if it is more appropriate to wear long-sleeves versus short-sleeves, sandals versus close-toed shoes, etc. Using our thermometer can also be tied into Nature Study since the children can observe that there is frost on the ground when the temperature is below freezing, that the humidity increases before it rains, etc.
  • We bake or cook together. I allow the kids to measure out ingredients using measuring cups and spoons, which teaches fractions as well as awareness of what different amounts look like.
  • When we are reading about something that mentions a size (such as the length of a snake or a distance that has been traveled), I take the time to put the measurement into context by comparing it to some known amount. The tiles in our house are 1-foot-across, so they can easily be used to see how long specific measurements are (and my daughter especially loves walking the tiles to see how large different creatures are). We relate distances to places we regularly visit [such as from the front of our property to the back, or to Grandma's house in Albuquerque (220 miles away), or to El Paso (40 miles away)].
  • The children help in grocery shopping by price-checking different items, counting and weighing produce, comparing prices, etc. I also send them on errands to get items from our grocery list. They thoroughly enjoy grocery shopping because they have real, meaningful work and purpose while we are at the store.
  • Each winter, we participate in Project Feeder Watch, where we observe the species and numbers of birds in our backyard about once or twice a month. Feeder Watch is a great way to integrate math and science into our lives. In addition to reporting the number of birds we see, we are also required to report the weather conditions (low/high temperatures, precipitation, etc). Both kids love participating in this program.
  • We regularly reference a calendar to see what the date is, or when specific dates are coming up. Birthdays and holidays are referenced often on the calendar, and the kids like to count down the days until they arrive.
  • My children can earn money any time by pulling weeds. I generally pay 1-cent per weed. Sometimes, when the weeds are getting especially out-of-control, I will run a special where I pay double or even triple for certain weeds. The children are required to count up their weeds so I know how much to pay them. For instance, this week my daughter pulled 342 weeds (!) and my son pulled 26 weeds.  (I didn't even know they were doing it until they were ready to be paid.) My husband also pays the children for killing flies or ants in the house.
  • We have a number line that wraps around the living room which goes past 400. The kids like to use this to practice counting (on their own initiative) or they can use it to say where something is ("over by 320"). Children who visit our house also seem to love this visual way of seeing how large numbers are.  

  • Each child has their own wallet as well as an envelope for long-term savings and one for charity.  They are required to put at least 10% of their earnings into their long-term savings (otherwise known as the "car-fund" or the "horse-fund"). My daughter also has her earnings from her chicken business to manage, and she sometimes pays her brother to help with her chicken chores. She has become very proficient at making change and using coins.
  • My children save for and buy items from the store. Earning, saving, and spending real money is invaluable in teaching them math concepts such as addition, subtraction, multiplication, and money management.

Math Games

Math games provide a fun way for my children to practice their math facts without even knowing it. I don't tell my children, "let's play a game so you can practice your math." Instead, I just let them choose a game to play and the math happens during the game without anyone making a big deal about it. If a child gets stuck on the math, then I help them, without making them try to figure it out until they are frustrated. I try to give them the freedom to guess and make mistakes, and keep it light-hearted and fun, instead of making them feel like they are "wrong" or putting them on the spot.

With all of these games, I often make special rules or modifications (described below) so that both children can play and enjoy the game, irregardless of their age difference. In some games, it works best for my youngest child to be on my "team" when the rules or math-involved are too complicated for his current understanding. We usually play math games once or twice a week. (More games specifically for preschool-age children are described in a comment below the post.)

The math games we use are:
  • Cribbage is a card-and-board game where the players try to be the first to score 121 through making card combinations such as cards that add to 15, 31, etc. 
    • Teaches addition, multiplication,  and knowledge of numbers up to 121. 
    • Modifications for younger players: a number line is helpful for players who can't easily add to 31 in their head; my son would play on my team by helping to look for card combinations and moving our counter around the board. 
  • Yahtzee is a dice-rolling game where the players see who can achieve the highest score as they fill in the scores for various dice combinations (such as 3-of-a-kind and Full House).
    • Teaches addition and eventually multiplication, strategy through determining the best ways to use the high rolls versus the low rolls, and writing.
    • Modifications for younger players: my son usually plays on my team by rolling the dice, helping me decide which dice option we are aiming for on the scorecard, and helping me add up the dice.
  • Mille Bornes is a card game where the players are in a car race; the first to reach 1,000 miles wins.
    • Teaches addition, knowledge of numbers up to 1,000, and knowledge of which numbers are greater.
    • We use lap-size dry-erase boards during this game to keep track of how far our cars have traveled.
    • Modifications for younger players: I keep track of my son's score for him.
    • Although this game has cards that can be used to sabotage other players (such as giving them a flat tire or Stop sign), we typically play this as a "sweet" game, where we don't sabotage each other. I don't force the kids to play it this way, but I did demonstrate non-aggressive playing through my own behavior, and my children decided to follow suit. This makes the game a chance practice choosing kindness, as well.
  • Uno is a card game where players try to match colors or numbers to be the first to use up all of their cards.
    • Teaches numbers and colors. Can also teach addition and subtraction using the variant described below.
    • My daughter and I will often play Add and Subtract Uno, where we can combine two cards through addition or subtraction to match a number being shown. For instance, if there is a 4 showing, we could use a 5 and 1 (to make 5-1 = 4). When we play this way, little brother still plays the usual way and is dealt a few less cards to make it fair. 
  • Monopoly is the classic game of buying and selling property.
    • Teaches addition, subtraction, how to make change, knowledge of large numbers, and concepts such as mortgage, bankruptcy, etc.
    • We usually limit the length of this game to 1-hour, and we start the game with 3 properties per player (chosen randomly from the deck and paid for from our individual banks).
    • Modifications for younger players: my son will often play on my team by rolling the dice, moving our player around the board, managing our small bills ($1's, $5's, and $10's) and helping me decide whether or not to buy/sell properties.
  • Poker is a classic card game where players compete to see who has the best 5-card hand.
    • Teaches strategy, analytical thinking, money management (if played with poker chips), and weighing of risk versus reward.
    • Modifications for younger players: my son often plays on my daughter's team. They love being in on the secret of what cards she holds, and giggle delightedly when she has any "wild" cards.
  • Sum Swamp is an Addition and Subtraction board game using multiple dice which are combined to make math problems (such as 1 + 3 or 5 - 1).
    • Teaches addition, subtraction, understanding of math problems and symbols, odd and even.
    • Modifications for younger players: when my son was younger, he played Sum Swamp with just one die instead of using all of the dice to make math problems. 
  • Pretend Store is a game where my children set up stores, usually with stuffed animals being the shopkeepers. Then we go shopping at the Pretend Stores.
    • My children like to create price tags and signs for their stores. 
    • We shop at the stores using paper play money that I printed years ago, or using real coins.
  • Sorry is a card-and-board game where players race to be the first to get all of their pieces to "Home".
    • Teaches numbers, memory for special rules, and sportsmanship.
    • Modifications for younger players: since my son is not yet reading proficiently, his sister and I help him read the cards when necessary.
  • Carcassone is a tile-laying game where players create settlements, farms, abbeys, and roads.
    • Teaches addition, skip-counting, puzzle skills, and strategy. 
    • Modifications for younger players: my daughter and I help my son in adding up his points for completed settlements, roads, etc.
  • Cuisenaire Rod Games are played with math manipulatives.
    • Teaches addition and awareness of what numbers actually mean through hands-on math experience.
    • There is a free printable Cuisenaire Rod game book here. My kids especially love playing Snakey Rods and Chutes and Ladders using the Cuisenaire Rods.
  • Settlers of Catan is a "strategy game where players collect resources and use them to build roads, settlements and cities on their way to victory".
    • Teaches addition, strategy, and risk management.
    • Modifications for younger players: my son would play on my team, rolling the dice, managing our resource cards, and helping me decide what to build next.
  • Settlers of Catan Dice Game is like a cross between Yahtzee and Settlers of Catan. Players roll the dice as they try to build roads, cities, and settlements for points.
    • Teaches addition and strategy.
    • Modifications for younger players: my son would play on my team, rolling the dice for me and helping me decide what to build/buy next.

Love of Math

Our math curriculum, based around games, everyday math, and read-alouds, is creating a love of math in my children.  It took me awhile to get used to the idea of teaching math without worksheets and repetitive drills, but yet these new methods are much more effective. Instead of math being something my children have to suffer through, they are eager to use these methods. Because these methods emphasize hands-on math, my children understand math concepts much better than they would if they had only been working math problems on paper. Want to read more? Check out my post about Why I Don't Try to Meet Math Grade Standards.

What are your favorite math games and math read-alouds?


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Elizabeth said...

We recently discovered a dice game called Quixx. Super fun. Great for strategy & addition. We all love it.

-Danica- said...

This is so helpful!! Thank you for putting this together. I want to use all of these ideas!

Beth said...

I LOVED all these ideas! My daughter just turned 4 and I am really considering home schooling her, but would like to start dappling in it a bit this year to see if I can actually pull it off (haha). Articles like this really make me feel like I"ve got this!

My daughter is great at counting, and just figured out how to count past 20. She is eager to keep learning. What kind of math subjects did you touch upon around this age - or do you know a good resource I could turn to for help?

THanks again, really enjoyed these ideas!

Sarah Smith said...

Hi Beth,
At your daughter's young age, a lot of the same resources could be used, but with a few modifications:

-A small number line on the table or wall, going up to perhaps 20 or 30, where you can demonstrate addition or subtraction of small numbers. A small animal figurine could be jumped forward or backward on the number line to show addition and subtraction (but only in a fun way, and not as a quiz or frustrating problem for your daughter)

-Life of Fred and the Anno books would likely be interesting to her. (With the Life of Fred, you could work the problems at the end of each chapter out loud, or even skip them entirely at this age.)

-She could get involved in baking with you, not necessarily to start learning the fractions or amounts yet, but more to illustrate more vs less, spoons vs cups, etc.

-Use a ruler or tape measure to demonstrate the size of different things, such as animals you read about, etc

-Math manipulatives of some kind, that she can play with and in doing so gain an understanding of what math numbers actually "look" like; our favorite manipulatives are cuisenaire rods

-Math games that could work well at her age are the following, so long as you are open to helping her out whenever she gets stuck:

Hi-Ho Cherry-O game board game (both of my kids LOVED that game when they were little)

Uno card game

Cuisenaire Rod Games (which are described in the post; the Cuisenaire rod games download I mentioned was designed specifically for preschool age kids)

21 (the card game also known as BlackJack; you could remove the jacks, queens, and kings, and use only the number cards; let her say "hit me" to get more cards and play with them all face-up so you can see and help her; this can be tied in to the number line, where you use a token or figurine to illustrate where she is on the number line and jump ahead when she gets more cards, trying to reach 21 without going over)

Yahtzee (if you do the writing and addition; she will start being able to recognize the numbers on the dice and probably have a lot of fun rolling the dice)

Sorry (if you read the cards for her; this will help in teaching one-to-one correspondence as she learns to move her pieces around the board the number shown on the cards)

I hope this helps!

Laurie Paul said...

Thank you so much! I just checked the library for the books and they had most of them. I always like to check books out before I decide to buy them. Now I will look for the games. For my grandchildren. It is so wonderful to second generation homeschool.

Tara said...

Hi Sarah, where did you get your super-long number line for your walls? I especially love the part about indicating where in the room something is (it's "over by 320"). Definitely falls into the category of "homeschooling is a lifestyle"! :)

Sarah Smith said...

Hi Tara,
I made the number line using Artist Tape and a Sharpie. Artist Tape is way better than masking tape or painter's tape, because it doesn't remove the paint even when left in place for awhile (whereas with painter's tape, the paint would eventually get stuck to the tape). I haven't checked underneath the number line, but did have some other papers taped to the wall (kid art) for months and had no issues when they were taken down.

Unknown said...

This is the best article I have read on this subject! Thank you for putting it together so well. I have homeschooled for 13 years and taking a different approach to math didn't "occur" to me until last year (I'm a little embarrassed to say...but three of my five children are boys who seems to be born with math youngest, a girl, is struggling with math).
Thank you, thank you, thank you!

Another math game we are loving at the moment is Rummikub. My 8 year old daughter and 13 year old son love it (and the 8yo is winning more than she is losing!)

Sarah Smith said...

Dixie, I'm glad you found this article to be helpful, and thanks for sharing the recommendation for Rummikub. I'll add that one to my wish list!