Below is a letter to parents, summarizing our school’s philosophy around summer homework and math support. It ends with a set of links to dozens of resources parents can use to help their child continue to build number sense, problem-solving skills and a love of learning.
Every parent wants the best for their child’s future, and of course, thriving in math class is an important component of that goal. How can parents support their child’s math learning, though, if what we understand about learning math has changed so much?
1. What is Our Conceptual Approach to Math, and Why Do We Use It?
The column at the far right summarizes our beliefs about how children best learn math. Most of us adults, however, were educated in the traditional system on the left. Some of my classmates excelled, others of us did not. Traditional instruction was successful at preparing a small group of students for professional careers.
A large majority of students, though, were simply adequately prepared for the skills they needed for factory jobs. This has all changed. The recognized educational priorities for the 21st century include critical thinking, communication and collaboration, creativity, and technology skills.
If we expect the citizen of the future to be foremost a creative, flexible problem solver with good communication skills, then it is our duty to align education with that goal.
2. What does this look like in the classroom?
- “Ask, Don’t Tell” – We keep our own telling to a minimum, and instead ask questions. “What if?” questions with overtones of curiosity and wondering. “What if we moved this block here? What if we had twice as many blocks? What if a mouse stole some of the blocks at nighttime? What if you and I shared them equally? What if we built a bunch of houses with 6 blocks each?” Open-ended wonderings and stories that feed a child’s natural interest in how the world works.
- “Let’s Ask the Blocks” Students don’t need to be told how to understand math if they trust their own ability to deduce math truths by exploration.
A first grader might remark that these 4 sixes are easier to see as 4 fives plus 4 ones (therefore 20 + 4) . She is benefitting from having used ten-frames so often that she sees TEN blocks on the ten frame as “The Quantity that Does Not Need to Be Counted”, and FIVE as “No Way Do I Need to Count That Row”. She is now well on her way to a conceptually-based understanding of multiplication. The blocks give way to drawings, then to visualization, and finally to mental math with confidence.
- “Can You Prove That?” – We use this sentence or its twin “Can you show me why that works?” to encourage students to think critically about facts they think they know. I feign surprise at their new insight, and ask them to explain it to me. If they can use blocks to show me why their answer works, they have solid, conceptual understanding.
- “Speed Is Not Intelligence” – We avoid games based on speed and competition. We don’t use timed tests or flash cards. Here’s the thing: Thinking is hard. And good thinking is often slow. Actually, good thinking probably alternates between a fumbling slowness and sudden flashes of insight. But memorization is a shortcut that builds neither number sense nor thinking skills.
Yes, fluency is necessary for later math subjects, but there are many roads to fluency, and memorization is the dullest.
Note – Like Stanford professor Jo Boaler? Read her take on speed in math learning here.
3. Doesn’t All This Take a Long Time ?
Why yes, it does. Especially at the beginning, in the first 3 or 4 years of school. But we’re looking at the long game. Young children need TIME to build a huge foundation of number sense, mental math, conceptual understanding and problem-solving experience. For example:
- 10 ones make a ten, and then we bundle those ones and start counting over again, (“10+1, 10+2… I mean eleven, twelve…) And one ten can be unbundled again, into 10 ones, if you want to subtract something and need some ones.
- 10 tens bundle to 100, and unbundle again, too. “It’s always ten of this equals one of that, unless we’re learning fractions, and then five fifths bundle to one… and one and 2 fifths unbundles to 7.”
- 5 – 0 is five, unless we’re counting all the trees placed at five intervals along a street, and then it’s six.
- If you’re multiplying 2 numbers, you can double one of them and halve the other, and get the same answer, if it’s easier. But not if you’re dividing. Then you have to double or halve both of them.
This sounds confusing, but with BLOCKS, it’s not. It makes perfect sense. (Email me and I’ll send you a short video.)
Important though, is the outcome of all this conceptual work in the long run.
What takes so much time in grades Kg – 3rd leads to rapid gains in grades 4 and beyond. The investment in a strong conceptual foundation of number sense pays off!
I have spent 26 years teaching math at the same school. We adopted Singapore Math 16 years ago. We slowed down the curriculum, used more manipulatives and problem-solving strategies, and spent more time listening to children. Since then, I have seen a steady increase in our middle schoolers’ math achievement:
- A far greater number of our students finish and test out of a complete high school algebra course in 8th They comment that “Algebra is just arithmetic with letters instead of numbers.”
- All through MS, our students show greater number sense, solve word problems with more ease, engage in math discourse freely, and show dogged resilience in problem solving.
4. What Can I Do As a Parent to Support My Child ?
- Play games! Games that are not speed based. Games that involve both strategy and luck, to even the playing field.
- Ask questions, and pretend not to know the answer.
- Listen to your child solve a problem. (Be patient – they’re not yet good at details like brevity and clarity)
- Play shopping with real coins. Cook and bake following a recipe. Talk about daily math as you do it – mileage, discounts, growth.
- Estimate quantities (maybe a handful of jelly beans!) , then count them in different groupings.
- Use puzzles, logic problems and conundrums if your child enjoys math and wants more depth.
- Load them down with worksheets and drill, memorized algorithms and facts. This seems to work in the short run, because children want to please us. But in the long run, they often burn out. Here is Jo Boaler’s film about being a gifted math student. (Yes, it’s emotional, but there’s truth in it.)
- Don’t feel you have to know all the answers. Resist the desire to explain everything to your child. The children will learn more if THEY do the thinking.
5. What If My Child Desires More Challenging Math ?
This is wonderful! We love watching students delve into challenging problems with gusto and resilience. (It makes my math teacher’s heart glow!) But enrichment does NOT mean bigger numbers and more algorithms to learn. It means more depth. It means hard problems – problems we get wrong and have to start over again. It means discussion with like-minded peers – math clubs and math circles. It means learning to love the “thrill of the chase.” In the 3rd tab of each link below, you’ll find resources that build thinking skills in our young math lovers’ minds. Let me know if you know of others!
6. Parent Resources for Summer and Beyond
We teachers always hope that students can use their summer vacations to play, explore, spend time outdoors, and take on responsibilities at home. Part of school readiness is a maturity level that doesn’t necessarily come from the classroom!
However, a moderate amount of basic subject review can be helpful. Try to build this into a comfortable weekly schedule, and try to keep it fun.
Below is a chart with links to lists of resources and activities, if you choose to use them. Choose activities that best match your own and your child’s schedule.
- 20 – 30 minutes sessions, two or three times a week is a good guideline. (Of course, spend more or less time depending on your circumstances)
- Games or hands-on investigations are most productive and enjoyable, but we’ve included some work children can do independently, too.
- Giving students choice in what activities to do also increases learning effectiveness.