## 1. Particular concepts

### 1.1. Joshua Greenbaum - what is the best way to both teach my kids the distributed property and use it in teaching other math concepts?

1.1.1. I experimented a bit with distribution as an idea. You can do it qualitatively through several actions, such as cutting or punching holes through multiple layers. On the other hand, coloring with a brush does not distribute through layers. Kids can look at a situation and predict (and then experiment) if the action will distribute or not. Can you distribute a mighty kick through all the bricks? :-)

### 1.2. Barbara Rowland - How early can the concept of the transitive property start making sense and in what ways do we learn it? I remember already thinking it made sense in 8th grade, so I must have learned it much younger!

1.2.1. If Jake loves Jill and Jill loves her hamster, does it mean that Jake loves the hamster, too? Young kids may think so, actually - but they usually realize many things in life don't transit well somewhere between one and five. This can be gently supported by experimentation, non-transitive games like "Rock, paper, scissors" and storytelling "If I sit on a chair, and the chair stands on the floor, does it mean I sit on the floor?" Then, when kids are making up their own relations, they can go ahead and analyze relations for transitivity.

### 1.3. Hope McMullan - I want to understand more about counting or not counting with young children. I haven't quite understood the concept and why it is detrimental

1.3.1. There are several mechanisms for building the big concept of NUMBER: - subitizing (instantly recognizing quantities) - counting/addition/subtraction (all three rely on the same mental mechanism) - unitizing/multiplication/division (grouping) - exponentiation (self-similar structures Delta loves very much, such as that fractal hand) None of these structures are dangerous or detrimental, not even counting. What is dangerous is the lack of balance. The four mechanisms have to live in harmony and form a nicely developed, balanced ecosystem together. If one takes over too much, the ecosystem becomes unstable and some concepts are hard to learn for kids.

### 1.4. Brian Vancil - How well do things like Glenn Doman's math program and dot cards work at building pre-numeracy in comparison to other activities or even compared to the opportunity cost (like developing language skills)?"

1.4.1. I have looked at the issue in some detail, and experimented with my kid a bit. I now prefer to develop and support subitizing (instant recognition) to ten only, or even to five (the range present at birth) and switch to unitizing (working with organized groups) past that. Subitizing by itself does not go anywhere mathematically, beyond very, very simple arithmetic, whereas unitizing leads to a lot of algebraic and geometric concepts. If you are into reading very early drafts of papers written by lots of people, here is a wiki page with some ideas: http://mathfuture.wikispaces.com/SubQuan+and+friends

### 1.5. Jacinda Bitely - in what order would you teach the multiplication tables to young children?

1.5.1. I have to confess I love multiplication, and collect multiplication activities as a hobby of sorts. This is a good question to ask, and there is actually some research and rules of thumb about it. We will discuss it more, but I can tell you my personal preferred order even now: Doubles, tens, fives, nines, squares, off-diagonals (e.g. 5*7 - they are squares minus one), fours, then 6*7 and 7*8. Also, you only memorize squares and 6*7 and 7*8 - you get the rest from patterns (and then possibly memorize, depending on preference). There are certain young kids who love number patterns and this is appropriate for them. I have to note they are a minority, and the rest like to use printed times tables for reference.

### 1.6. Linda Whipker - Are there benefits (if so, in what realm of math learning) to introducing an abacus early? Is it more tangible/concrete than some other methods of computing, and in what way?

1.6.1. yes, abacus is definitely an object to have around. One of its many values is in making all key concepts of the positional value system tangible - moreover, in making the mental actions involved into objects, if it makes sense. The biggest reason why the powerful algorithms of the positional system were developed relatively late in history is the short-term memory demands. They are actually beyond the capacity of many adults, hence many people are unable to learn the traditional long division algorithm at all, or struggle a lot. However, the abacus lifts these demands by visualizing all steps and concepts. The key concept abacus carries is unitizing. Young kids see equal groups of beads; older kids understand that positions mean different values of beads. I actually added abacus to the Wikipedia article on subitizing, which is a related idea: http://en.wikipedia.org/wiki/Subitizing#Mathematical_applications

### 1.7. Heidi Thoma - is with regard to using an abacus...perhaps too structured for your forum? But I have been searching for abacus activities for the young or old....and would love insights on this beautiful and historically and culturally fascinating tool.

1.7.1. I love abacus, for many reasons, including cultural and pedagogical. Actually, another person, Linda W. asked about abacus as well. We are aggregating questions and people will be able to see them soon. Meanwhile, here is a quick activity idea: design an abacus of your own. There are many different versions in different countries, and modern ones sometimes work in binary or other bases.

### 1.8. Melissa Collins At what age would you introduce the concept of negative numbers? I am considering doing some basic number line stuff with my 6 year old and his not-yet-4 brother.

1.8.1. I would start building the foundation for negative numbers in the first year. The idea is based on the metaphor of opposites, and is the source of several excellent games. For example, you can roleplay with "underwater elevator" in the bathtub, marking up levels over and under water, their dwellers and so on. You can also play with "objects and anti-objects" that explode, eat, or otherwise cancel one another. For example, three hungry caterpillars can cancel 2 leaves and one will still be hungry (in adult talk, -3+2=-1). You can also play with multiplication of negative numbers (the metaphor of a carousel) pretty early.

## 2. Ages and stages

### 2.1. Brooke McCallum-Bont - How can I teach math to a variety of ages as a group? My kids are ages 1- 12 and I'd love to be able to teach them math in a group setting as opposed to only one on one. How can I adapt the same info to different age levels?

2.1.1. The general idea is to keep tasks open, so that different people can approach them at different levels. Obviously, the task "what is 2+2?" is hard to make open, but something like, "How can you build a cylinder?" or "What types of symmetry are there in the room and why do they appear?" can invite participation from babies to adults. We will talk about opening tasks up throughout the course.

### 2.2. Debbie Harne - What are three of the first concepts that children learn in the area of mathematics and what age do they typically learn them?

2.2.1. I think learning is both cultural and biological. The first few things are probably biological and shared in some form with animals - my pick would be subitizing (instant recognition of small quantities), recognition of symmetry and categorization. All three are present at birth, and also present in some animals.

### 2.3. Maria Muscarella - my hardest struggles these days is being able to help my daughter focus on more complicated math issues while my 2 year old son is being his crazy self. I'd love some tips on having math time with kids who are in very different places learning-wise.

2.3.1. What comes to my mind is the realtor's motto "Location, location, location!" - holding activities by a bathtub where the younger one plays, by a playground, on a hike and so on really makes a difference. Just being amidst some grass and trees calms people down, and things like loud voices or running around aren't annoying outdoors.

### 2.4. Diana - How old the child should be to realize the sense of counting

2.4.1. Children can recognize quantities up to 4 or 5 at birth - the ability called subitizing. It is separate from counting, though. You can do a lot of fun activities with quantities in the first year or two based on subitizing. Since counting involves speech and keeping track of objects one by one, it develops together with words and with general spatial abilities, usually some time in the first five years of the child’s life. Most young children mix up names of counting numbers and can’t reliably keep track of which objects they already counted and which they did not. Do not delay other math fun because of these natural properties of a young child’s mind!

### 2.5. Carolyn Kelley - How do you involve a 2 year old?

2.5.1. A very generic and quick answer, for now, is that you invite a 2 year old in ways similar to any age: following their interests so math is useful, appealing to the sense of meaning and significance, and engaging their feelings of beauty and fun.

### 2.6. Martha Myers - How do you include all ages in the fun learning of math?

2.6.1. one of my main "tricks" for multi-age activities are open problems that invite kids to make their own math. For example, a couple of weeks ago we spent an hour of the math club, literally, around these tasks: - Make a circle - Make a circle that is more perfect than this This led more philosophical kids into a long mathematical discussion of perfection, as well as working with compasses and symmetry, while kids who were more into hands-on work (and younger) honed their scissor and drawing skills and made puppet dolls out of circles and ovals.

### 2.7. Brenda Weiss - What mathematics concept that very young children naturally understand has been the most surprising to you? (or What is the first math concept very young children understand?)

2.7.1. I think for me the most surprising thing was exponentiation. It's easier than addition, subtraction, division and multiplication - through self-similar structures.

### 2.8. Dawn La Du - At what age do you transition to writing math on paper?

2.8.1. Paper is an excellent cheap medium, and you can use it from birth. However, it's a good idea to use pictures and cutouts, as well as symbols, at all ages. So, my answer is "always use paper, but not only for symbols!"

## 3. General

### 3.1. Jennifer Ramsey - How best can I provide a 'maths rich' environment for my child to learn and develop?

3.1.1. I think your question is something the whole experience will answer. Meanwhile, here is a quick thought from me: the main feature of a young child's environment is the primary caregiver - usually mom. So, to make the environment richer in math, we have to make our thoughts, activities and words richer in math.

### 3.2. Sue Stern - Should I do more math with my kids? How? ...Should I use a curriculum?

3.2.1. a curriculum does not typically answer the need to make your environment rich in mathematics, and can cause the opposite! And if we talk about very young kids, most curricula are especially dreary and depleted of real math. As Richard Rusczik (the creator of Art of Problem Solving) said about a large curriculum-making group, "I don't think they like mathematics!" My hope with this course is to aggregate resources and people who want to change this.

### 3.3. Jeanette Wilson - What are the best materials to have on hand for kids to play with math?

3.3.1. For my part, I won't do math without graph paper and color pencils or markers.

### 3.4. Maria Levin - My youngest daughter is 7 months. I want to start playing early development games with her. But which ones? It's not about particular skills or knowledge, but about instilling the general interest in math, developing her imagination and desire to create.

3.4.1. here is a very simple activity we do with babies to adults, for about 5 minutes at a time. Go to Google Image Search and think of something your daughter is interested in right now. Let's say she's playing with a toy cat. Try these searches: symmetry cat tessellation cat math cat Just let her poke the screen to point at things she likes, and click on those to enlarge them. Talk about what YOU see there (symmetry, transformations, whatever math) even though the kid may not understand. Also, enlarge some of the images you like - pick those that seem more mathematical. Have fun!

### 3.5. Denise Gaskins - Which math concepts come easily to young children, and which ones do they struggle with? Or do these categories vary from one child to another?

3.5.1. It looks like there are no hard concepts, but there are hard representations, if it makes sense. Here are a few features that make concepts inaccessible to young ones: - Chains of symbolization (this stands for that which stands for something else etc) - kids really benefit from WYSIWYG! It does not have to be "hands-on" but it has to be direct. - Necessity for more than 2-3 registers of short-term memory at once (and most curriculum developers have no idea how to "count" memory requirements). Nobody would send a three-year-old to bring a plate from the second room on the right down the left corridor, where it's in the bottom drawer of the third cupboard on the right. Yet people routinely expect feats like this in math. Some people never develop more than 3-4 registers, and 5 is the average - yet most lessons for the long division algorithm or fraction division require seven... - Task and time management requirements. Montessori had a word or two to say about this. Toddlers aren't strong on organizing their space, their sequence of actions, or their attention patterns (rest/concentration), and need much help and scaffolding there.

### 3.6. Maria Droujkova - How can I help people to reflect on math activities and become co-creators?

## 4. Methods

### 4.1. Irina Abramson - I'd like to get out of the habit of evaluating my kid's math solutions as "right" or "wrong", and to encourage him to try to find a way to check whether his solution is right on his own. How should I start?

4.1.1. Any object (answer, number, shape) is an example of something. It may not be the example you were originally looking for, but that's OK. Maybe you can take the answer and love it and find a problem that would fit it? Another trick is to stop, at least for a while (say, a week or two), asking questions for which you know answers. Just think before asking questions, and if you know the answer, don't ask.

### 4.2. DalaleN - What is the most effective way to engage my 3 year old when introducing and teaching math

4.2.1. The most effective way to engage your kid is to be engaged yourself, play for your own personal enjoyment, in a space where the kid is present and welcome, with materials accessible and interesting to the kid.

### 4.3. ddomar - what is the best way to teach how to write numerals?

4.3.1. To learn any motor skill, the brain has to guide the body. One of the easiest ways to make it happen is to first let the body guide the brain! In the case of writing or drawing skills: - Make a large textured numeral (a foot in size at least). You can use felt, sandpaper, wool, leather or anything you have handy - Help the child trace it with his or her hand. You can gently guide the hand by your own. Make it a relaxing experience - you can combine it with hugs, shoulder massage or other pleasant body-oriented actions - whatever the kid likes! - After you traced several times, attempt to make a copy of the numeral next to it. Whiteboards or chalkboards are a must - the copy must be large at first. You can start by drawing the numeral with your kid's hand in your own. - Draw other simple shapes the kid likes - cats, faces, rockets - in the same manner. I use the same techniques with older kids and adults who learn to draw. Just trace contours with your finger, then with your eyes, then draw it.

### 4.4. Casey Price - how I can help my son remember his numbers (above 10) without just forcing him to count over and over.

4.4.1. "the number song" (words) does very little for mathematical understanding... Mathematically interesting things happen when you work with quantities and lengths. We will have an activity where we search for beloved objects that represent quantities. For example, 18-wheeler truck! But the biggest thing that will help to understand past ten is actually the notion of the UNIT. We will work a lot on it through symmetry and other visual, hands-on ideas.

### 4.5. Marina Kopylova - Many people are getting tired when kids are working on a particular skill, when they repeat the same game again and again, asking the same question, counting to 10 twenty times in 5 minutes and so on. How could parent stay focused and enthusiastic?

4.5.1. Kids get into "vicious cycles" of all sorts where they keep repeating themselves again and again with diminishing returns on learning and sometimes enjoyment. At other times, kids seemingly have fun, but later suddenly burst into tears or otherwise break the cycle dramatically. I think it is up to older helpers to turn cycles into spirals leading to new heights. For questions, my rule of thumb is: every next answer has to be twice as long as the previous one, with more details and extra info. Also, it has to be followed by my question to the child. This also applies to thoughtless "Why, why, why?" and other "form-driven" questions some kids repeat without concentration. The goal is for kid to pause and smell the roses, so to speak: to concentrate, to think, and to engage me meaningfully. Maybe that's what kids want in the first place when they repeat themselves, eh?

### 4.6. Amy Pott - What are some good ideas for combining math with music for the very young?

4.6.1. Music is such a rich area for baby and toddler mathematics! There are rhythms, gradients (in tone and in volume), all sorts of patterns! You can even play some sound "games" with babies in the womb, in the last couple of months of pregnancy.

### 4.7. KZandber - how do you collect and make hands on materials without feeling overwhelmed by "stuff"? I love to recycle objects and use them as manipulatives. And I love to look for new things to make for the kids to use during math time. But I feel like I have so many containers of things to count and things to sort etc. That it is cluttered. And I also look at these lovely things to make but then think, I will go through all this time etc. to make the object for the lesson/activity, and then it will never be used again. So I don't usually make it. I do make the really good ones, but I wish I would make more.

4.7.1. As someone who loves crafts and manipulatives both, I can definitely relate! We will discuss it in more detail when we start activities. Meanwhile, one trick I learned over the time: I make all manipulatives together with kids. This makes for excellent learning tasks (better than using the manipulative, in many cases), it's a lot of fun, and I don't feel guilty over spending too much time on preparation: kids area learning throughout the preparation.

### 4.8. gail elliott - What help can I give my child when using problem solving for math?

4.8.1. Problem solving is a cool topic. There are many things you can do - let's discuss it in more detail during the course. Meanwhile, one of the more powerful thing you can do for problem solving, especially with young kids, is modeling. Represent the problem with objects and toys, or roleplay it with people. Consider using drawings and paper models, too. Colored markers are often very useful to model different mathematical features of the problem. Tell the story of the problem in words, if your kid likes stories. Some problems work very well with diagrams (of different kinds), such as grids.

### 4.9. Heather Brandt - What should we be doing to promote problem solving skills in our 4 year old or to help prepare him for the problem solving aspect of mathematics?

4.9.1. here are three activities I invite young kids to do that really make a difference for problem solving: - Make a model - Tell a story - Ask questions For example, the classic river-crossing problems invite all three. You can invite young kids to roleplay the problem with toys or with themselves as characters; report on what happens as they keep trying (tell stories, reflect); and ask questions that may help them. The most important math question is "Why?"

### 4.10. Kim Barnett - how do I present math to a 3-year old in a more conceptual way? Or is it all conceptual for him

4.10.1. It's true that young kids may find some very mundane experiences meaningful because they are still exploring so much of the world. However, no given kid finds every experience meaningful! Actually, even newborns have their likes and dislikes in music, toys, styles. By now Alan has developed complex relationships with many parts of the world, and I am sure you can tell he finds some ideas, activities, interactions more meaningful than others

### 4.11. kim meltzer - When I ask my daughter word questions she always gets it, but if I ask her what 2+3 she doesn't know.

4.11.1. there is the so-called Rule of Three that we can break, but at great risk of confusion. Any idea needs at least three examples. If you see confusion, most likely there aren't enough examples. In this case, you offer two representations: stories and numbers. Offer more representations: jumps (two jumps and three jumps), lengths (two units and three units), claps, counters (two raisins and three raisins) and so on.

### 4.12. Dawn La Du - my 2 youngest child,an 11 yr old has dyslexia. She needs constant review and reiforcement but yet she likes to move forward. How can I make math fun and visual for her and progress learning new concepts?

4.12.1. For your dyslexic kid, use visual organizers such as tables and grids extensively. For example, she might like partial quotient division instead of traditional long division, because it's more visual. Also, use spreadsheet software and tools like Wolfram|Alpha and grapher software when she works on math. I would not use calculators because their screens are small and confusing for many, but instead full computer screens. A spreadsheet keeps track of patterns so nicely!!!

## 5. Seeking content

### 5.1. Kelli Brandon - sometimes I let my older son play "computer games" (Starfall.com, SesameStreet) for short periods of time. He is 4 years old - do you have any educational/fun websites you might recommend for his age group?

5.1.1. Most math games out there are total trash - they aren't about mathematics (but arithmetic) and they aren't games, but sugar-coated drills. I would recommend Crayon Physics and Lego games. Also, you can start him on making his own games using excellent Scratch from MIT. I've done it with kid and parent pairs as young as 2.5 year olds, though pre-literate children need to be in mommy's lap for this, obviously. It's very intuitive and fun.

### 5.2. Sharon Bliss - What are some fun games for a 5 and 3 year old that helps them learn some basic math concepts?

5.2.1. Any idea can be made accessible to anyone in some basic form. There are some pretty neat ideas in math that can be made available for young ones, such as symmetry, grids, transformations, functions, equivalence - but at the same time, the ideas can grow deep (non-basic) roots with time.

### 5.3. Mandy Lucero - Where can I find a list of Living math books for the preschool age

5.3.1. There are some nice once online, too. Meanwhile, a couple of my favorites: "The dot and the line" - here's the cartoon made from it http://www.youtube.com/watch?v=OmSbdvzbOzY Molly's books http://mathplace.net/books.html We also started to scan some of the old out-of-print favorites here: http://naturalmath.wikispaces.com/Young+Math+book+series Out of these, my favorite is "3D, 2D, 1D"

### 5.4. Dawn La Du - Many members have recommended board games. Do you have a master list of board games and card games and what mathematical concepts the games cover?

5.4.1. There is a good master list of board games at http://livingmath.net/GamesMisc/BooksDiceCardsBoardgames/tabid/1094/language/en-US/Default.aspx

## 6. Everyday math

### 6.1. Louise - how to introduce mathematical concepts in daily living. We often work with fractions, addition, and measurements in the kitchen, but other concepts would be good to talk about -- how to play with fractals in day to day experiences for example?

6.1.1. In some sense, the whole course will be an answer to this! Meanwhile, a quick note about fractals. They are very frequent in nature, so you can easily find them in a park or a forest - in ferns or branching trees. Most of the time, you only see 2-4 clear fractal levels in nature. For example, a feather under the microscope has three levels, but human hair only has two. Kids like to draw fractals, natural or fancy and silly. You can find fractals in many everyday designs. For example, the symbol of Windows operating system is a substitution fractal (or used to be)

### 6.2. Erin Sloan - What are some common tasks that involve math that we do every day and I don't even realize it.

6.2.1. Most people don't realize the complicated logic of their decision-making. And most people underestimate the complexity of spatial reasoning they routinely do!

### 6.3. msutaria - I am looking for new fun and exciting ways to teach my 2 year old the wonders of mathematics. I guess a good way to teach them how to add is with fruit? show diff baskets with different # of fruts in them to get them to count them.

6.3.1. Two year olds can explore a whole lot of topics - addition isn't actually particularly easy to them, though they can usually manage, especially with something tasty like fruit! Try also looking at symmetry, interesting shapes such as ferns (they are fractals), transformation in space such as rotation and reflection (toddlers love mirrors) and building or dress-up games for combinations and permutations.

### 6.4. Yelena McManaman - How to build a "creative routine" of doing interesting math activities with a child every day (or almost every day)?

6.4.1. My main tools for building routines are: Visual reminders - pictures and objects on the walls and in highly visible places, such as hanging from the ceiling Reflection on what I am trying to do, several times a day (based on a list) Tying activities to easy milestones within the day, for example, doing something right after breakfast, or during a car ride, or right before bedtime.

## 7. Feelings

### 7.1. Kerry Rochford - how do you inspire kids to see 'real math' all around them if the parent/teacher is not skilled at seeing those natural teaching moments?

7.1.1. One of the best thing to do to inspire kids is to get inspired yourself. Even if you aren't very skilled, make it a rule to find something fun, neat, beautiful mathematically every day. Use YouTube and blogs for the purpose. Just find something that speaks to YOUR heart, and kids will follow. There is math in everything, from quilting to robotics.

### 7.2. Cocoj - How can I make math entertaining for my child

### 7.3. Narelle Smith (hakea) - How do I make maths apealing and accessible to parents so they have the confidence to talk and play with maths with their children?

7.3.1. I offer parents tasks that help with math club. For example, we play parent bingo sometimes: http://naturalmath.wikispaces.com/bingo Also, parents take turns photographing and writing stories about club events, to send to tthe Natural Math email group, and they tell me both activities help them notice children's behavior much more, and be involved. Also, we have show-and-tell before each club, where kids bring their favorite objects and we search for math in them. I can tell parents discuss show and tell with kids ahead of time, and I try to invite them to contribute to the story, too.

### 7.4. Michelle zakolski - how to make math fun and not just about putting it on paper

7.4.1. There is some math fun to be had with paper too - you can draw, you can make paper sculptures, or do cool graphs and charts. However, there are many more representations in math: you can model math with toys or roleplay with people, you can use manipulatives (or virtual manipulatives), or you can tell math stories, create math music and even math dances. Thank you for the excellent question - this topic is dear to me. Last year I invited my graduate class to create this Wikipedia article which relates to it: http://en.wikipedia.org/wiki/Multiple_representations_%28mathematics_education%29

### 7.5. Kerry Rochford - How to make math fun amidst the tasks of learning the processes (i.e. learning multiplication facts)?

7.5.1. To make work fun, it has to be meaningful, productive and tracked well. The first task is to have a very good, deep reason kids believe in for any math task. It has to be meaningful and relevant for them before they start. Kids find meaning in play, beauty, community or utility, just like adults do. Second, visualize and track progress. Use time and task management tools, memory management techniques, and in general consider yourself a manager of a complex project. How can you help kids see their progress with every task? Observe how computer games track progress with levels, achievements, badges. Don't do rewards, but celebrate milestones - and track them visibly. What is a good to-do list for times tables? And third, create work flow so everybody feels productive. This may involve eating before or during math time (math is hungry work), exercising vigorously for 5-10 minutes before doing any memory task, finding best time of the day for math (kids know, just ask them), using music or quiet room, and so on.

### 7.6. Kerry Rochford - ow to not pass on a generalized apprehension about math

7.6.1. To avoid passing math fear, find little things in math you love, personally. Focus on those. Love conquers fear.

### 7.7. Sue VanHattum - He says he doesn't like math. (Shocking, I know.) I think it's because he feels he's not good at it. I debate whether I should continue to let this pass, or do something to pull him in to this joyous part of our minds

7.7.1. If you do not let this pass, what are options? First of all, what does he accomplish or hopes to accomplish by saying this? Does he dislike everything in math, and if not, which parts?