Using Living Math Materials and Plans with Structured, Incremental or Various Other Math Teaching Approaches

Note: This article was written for families using the Living Math Lesson plans in response to questions by families wanting to integrate the material with their math approach. Most of the comments, however, are applicable to families using various living math resources, and are not limited to lesson plans users.

The Living Math lesson plans were written in a format to facilitate teaching mathematics to all ages within a framework of its historical development. Materials facilitating this have been available for advanced high school levels on up, but I have yet to run across materials beyond the Luetta Reimer "Mathematicians Are People, Too" support materials, Historical Connections in Mathematics available from AIMS Education, that attempt to provide more than tidbits to students who have not mastered high school level algebra and geometry.

Because the lesson plans follow math development through history, they refer to more and more complex ideas. This structure / organization does not directly facilitate the sequence of elementary math learning topics that a traditional math curriculum does. It becomes more naturally sequential by the advanced / high school levels, because the math skills and cognitive development necessary to understand advanced ideas have more likely been attained. As such, the materials aren't written to be used back to back in levels.

The Primary Level plans suggest readings and activities appropriate for early elementary students, but it would be entirely appropriate to use the materials over a three to four year period,often repetitively as will be explained below, rather than used once through in order and then assumed to be ready to move on to the Intermediate Level. The same goes for Intermediate and even to a degree to Advanced. High School is different, in that the skills needed to complete this level of work may be attained to allow for a sequential, college-model study of mathematics in the full context of history. Therefore the comments below contain more for the elementary years than beyond.

When I taught weekly co-op classes using the plans and activities, parents approached the material in several ways. Unschoolers, or advocates of delayed formal academics, tend to find it easiest to adapt the plan materials, because the philosophy of this learning style is not generally incremental learning based. For a child who enjoys reading aloud with a parent and doing hands on activities, it can work quite well to provide math exposure in a wide range of real life situations. It can also provide the parent with many experiences to enrich their ability to stimulate interest in mathematics, especially if they enjoy history.

Relaxed or "eclectic" schoolers often took the opportunity of the classes or lesson plans to take a completely different approach to math for a given period of time. For many families, it was a break from a structured approach to help shift attitudes in a more positive direction toward math learning. Some families continued with math curriculum on certain days of the week, and did "living math" on other days. Others immersed themselves in the math history studies and left curriculum aside for months, if not a year. If their children enjoyed the materials, this was a beneficial process. Many went back to curriculum work after a time, reporting their children enjoyed it more, and together they had "hooks" to place ideas they were encountering in the curriculum on that were simply abstract before.

Some families continued a fully structured approach to math, supplementing with Living Math readings and activities for enrichment. They still used a daily curriculum such as Saxon, but reported that they could reduce the number of problems they assigned their children because they were "getting" the math more quickly after the classes or doing activities and readings at home. Many of these families chose to use the Living Math plans as their history curriculum, not worrying about the order of math concepts reported, as they continued to use the curriculum for sequential learning.

The most challenging model to teach and/or describe would be one wherein the sequential teaching is done through the Living Math plan materials, without using a formal math curriculum as the base. This is, however, in essence how I myself use the materials with my own children now, although we still use texts and workbooks that appeal to my children. Living math has been a feature of our home for so many years, my children have had read to them and/or have read to themselves the math readers that are scheduled in the lesson plans, often repeatedly. We do the activities – the best ones are often repeated at least once, if not more times over the course of several years. We as a family read the historical and thematic readers and refer to them as we experience things that relate to what we've read.

The reason we can repeat an activity in as short a period of a year is the fact we are not trying to limit the learning objective. The emphasis of a repeated activity tends to follow whatever concepts I know my children are working on learning at a given point in their math development. This natural approach took a few years to develop, and required me to be familiar with the tools that were out there to use. I will give examples below.

Dialoguing with families about the various ways they used the materials, and the experiences I've had with my own children, have given me confidence that the lesson plan materials can be used with a wide range of homeschooling and teaching approaches. I hope to give parents ideas of ways the materials can be adapted in various situations.

Primary/ Elementary Levels (approximately ages 6 to 8)


In order to use the Living Math materials as the primary basis to teach incrementally, it requires the parent to be familiar with the math activities available, and be able to identify when activities can be used to facilitate learning of the concepts the parent wants to emphasize in a given period. In other words, get to know your toolkit.

When I first began using these materials myself, I had no guides or manuals as to how to teach math through history at the pre-high school levels. The first step I took years ago was that of keeping us on a curriculum through the week, but having math history days wherein we would not work on curriculum at all, but rather read the materials and follow any bunny paths they led us on. This allowed me to keep the structure I felt I needed, but blend in the other materials for interest and relevancy.

The bunny paths we took often involved math concepts the kids hadn't yet fully learned, yet they wanted to keep going. We found ourselves spending hours and hours on math ideas and activities, whereas had we set that time aside for working math curriculum we would have spent far less time. Interest and relevancy provided energy to spend many more hours on math learning, and in context. Most of the plan activities in the early lesson plans came from these bunny path explorations. I became more and more educated myself about these ideas to be able to naturally insert them when the opportunity came up with a younger child, or if a different concept that was linked to it came up. As we spent more and more time on bunny paths, we cut back on the curriculum use, as it was simply becoming unnecessary and a distraction from our our highly productive and enjoyable studies.

I also realized many of these activities did not require as advanced math skills as I assumed. Many times doing an activity with my 9 year old and 6 year old, I could see that the difference was only in how much of the work I myself did to complete the activity, or whether we completed the activity at all. It was okay to stop when substantial learning had occurred and interest began to wane with a younger child, just like leaving some food on your plate when you're full.

Similarly, I might be able to do an activity written for an older child with a younger child if I rounded every number to numbers the younger child could comprehend, or if the fractions we encountered were rounded to whole numbers, or to easy fractions they could easily work with.

To give an example, in the Pythagoras lessons, ideas of number theory are introduced. Number theory is usually considered a high-level, complex mathematical area. But the simple idea of even and odd numbers is number theory that originated with Pythagoras. And while simple, it is profound; math theorists often rely on even and odd properties when constructing complex proofs. While I might spend time with my youngest working on even and odd numbers, and introduce figurate numbers to them in concrete / pictorial ways, I would investigate further with my older child relationships between figurate numbers to the level they could understand. We would simply stop when it was clear the child could not comprehend anything more.

In the Ancient Mathematics units different number bases are introduced. Again, in the traditional curriculum, number bases are usually considered middle school level math. But in co-op classes with children as young as 4 or 5 years old, I could demonstrate the Mayan base 20 system quite effectively, if they understood ones, fives and tens, as the numbers are written pictorially. Binary would be more difficult as it is more abstract, but it could be presented concretely with objects, games and activities. More advanced number bases would be an idea for older children.

The relationships between numbers can be analyzed by very young children in terms of their additive properties, or their multiplicative properties – big words, but concepts that are shown in math picture books quite easily. The doubling sequence is so prevalent in the history of mathematics it comes up in many activities and stories. The ubiquity of the idea itself communicates to a child to the notion that it is a very important idea. I've had young children who could recall and chant the binary sequence as well or better than they could skip counting by 2s. A young child might go as far as 1, 2, 4, 8 . . . . my older child may go up to 32 . . . and the oldest as far as they like. And of course, a middle school child on up can understand that this is an exponential pattern, and relate it to other bases. I would not attempt to teach exponents to a younger child, relying on concrete examples, but it they may in fact pick it up themselves, especially if it is compared to our base 10 system.

Having taught my oldest children math using more standard math programs such as Math U See and Singapore Math, I found that the activities over time were working the same principles that were presented in the curriculum. But it was up to me as the math mentor to bring the concept teaching in when it was appropriate for their level. As I worked with the materials and ideas more and more, I became better at identifying when a specific activity might work well with my child to work on a concept they were learning. I became an expert at adapting an activity to a child's level, because by exposing myself to these repetitively, I began to intuitively see the basic math structure underneath them. When you yourself actually do an activity with your child, you can observe and participate in the process required to do it. When you realize that multiplication is simply fast addition, any activity involving multiplication can be converted to an activity in addition, by bringing the numbers down to simple terms the child can add, or ending the activity when the terms become too large. Any simple multiplication activity can be used if a child can skip count,reinforcing the upcoming link to multiplication.

Once I'd read through the historical materials sequentially with my children, I did not need to stick to the plan order anymore for activities. We can read Hans Magnus Enzensberger's The Number Devil: A Mathematical Adventure and decide to go back to Pythagoras ideas we visited a few months or even a year earlier. If you read the book and did the activity with your children, you can make the connections with them. We can read Theoni Pappas's Penrose the Mathematical Cat books, and revisit the numerous activities and ideas we covered in other units. Repetition in these activities is usually well tolerated, and even welcomed, if it isn't immediately after the first exposure, because in the repetition they see and understand things they didn't understand the first time. Because the activities take more time than a typical math worksheet, they are often remembered for a long time, but even more so if they are repeated at a later date. The "ah-ha" moments are very empowering, showing them how much more they are able to understand than when they saw an idea presented before. This happens to my children very often when a younger child reads an older reader repeatedly over years.

Here are some examples of blending living math materials with incremental teaching at younger levels, and how I've identified opportunities quite often by teaching to older children.

My 7th grader was going through the Harold Jacobs Elementary Algebra text with a friend of his who planned on going to high school the next year. As such, his friend's goal was to complete the course, and my son committed to the same goal as long as he is able to keep up. We met twice a week to learn concepts and work problems, and they did homework between our meetings. As my son was familiar with the math history topics from the younger levels, we blended in the advanced reading materials and some activities as they match the ideas in the algebra course, vs. strictly following the lesson plans.

One activity suggested in the Harold Jacobs text to demonstrate the idea of a direct variation function involved experimenting with dropping bouncy balls from various heights, and recording the data in tables to generate an algebraic formula. It was an extremely effective activity for these middle school boys. We completed the experiments and generated formulas to describe the direct variation between the height we dropped the ball from and the bounce height. I observed that this experiment was similar to one I had done in the Galileo lesson in our math history studies, but it was different in that we were measuring the bounce height, rather than the time. I realized this was easier for younger children to measure.

Removing the more abstract aspect of the formula, I realized I could do this with my fourth grade daughter's math group. The girls were working on multiple digit multiplication, easy division and easy fractions / proportions in word problems. We were using a Singapore word problem book to provide a sequential framework for them to work on these skills between our meetings. So whatever activity we did, I emphasized the math skills they were working on, even as other math skills, and many logical reasoning skills may have come into play.

One day the girls were doing some Hands On Equations work which involved solving equations with "x" and "(-x)." One of the girls wanted to know, is there such a thing as a "y" or "z"? What a great lead-in to the bouncy ball experiment I had already planned I thought. I could say, yes, we'll get a "y" in there today.

So the girls did the same activity the boys did – dropping the balls, recording the heights, and finding patterns. They estimated the relationships between the two different balls – one ball bounced on average about 2/3 of the way up, the other bounced three fourths of the way up. If we were careful with our measurements, the relationships were strikingly accurate. We converted the bounces into percentages of the original drop height using calculators at first. They have not technically learned percentages, but we've encountered them many times in activities, and I put percentage ideas in terms of cents and dollars which they do understand – i.e, three quarters is the same as 75 cents out of a dollar. We put up a table of their results where they could see that no matter what height they dropped the ball from,the bounces were about the same fraction of the drop height, two thirds for the first ball, three quarters. I made sure we were rounding the figures to significant numbers they could understand.

When presented this way, my 9 year old daughter could easily answer the question: If the ball is bounced from 10 feet, how high will the bounce be? Initially she said 7 feet, drew the picture up to three fourths, and then realizing it was 7-1/2 feet, corrected her answer. She also could figure out that if I dropped the other ball from 9 feet, it would bounce up to 6 feet high. She could do this if I kept the numbers round and simple. Now she has another concrete "hook" to continue to refer to as we work on these skills.

The key with activities like this with younger children is to keep the numbers simple and intuitive, so they do not have to rely on more complex algorithms such as long division to get the answers, confusing the lesson to be learned. When children can begin to comprehend simple relationships such as basic fractions of halves, quarters and thirds (and many children can begin to comprehend these in terms of dividing up food or items by age 5 or 6), and can do simple addition, they can begin doing these activities, and the parent need not worry about the fact they can't complete them.

I allow younger children to use calculators to complete activities when the math is beyond their comprehension, to again facilitate what they are to learn without confusing it with what they aren't ready for. When we did the Cheops Pyramid activity in the Thales lesson (from Mark Wahl's Mathematical Mystery Tour), an activity I have done successfully many times, the math can become complicated for all but middle schoolers on up. But if I give younger kids a calculator they can complete it. It gives them an idea of how to use a calculator, the importance of a decimal point, and experience with rounding. For these kids, the learning objective isn't how to do division in repeating decimals. It's to see that mathematical relationships can be built in spatial objects. Once the calculations are done, they compare the results and see that the numbers are very similar.

To realize how powerful some of these lessons can be in terms of retention of ideas, my oldest who entering his junior year in high school still recalls many of the concrete lessons he learned. He homeschooled for 9 years before entering high school two years ago. Recently he saw the pyramid my daughter made and commented, "Oh, that's the pyramid that has pi built into it, because it's basically a half sphere, right?" He was 11 years old when he first did the pyramid activity in one of our co-op classes. He tells me that he recalls the formula for the circumference of a circle because he remembers our Egyptian rope-stretching activities. The circumference of a circle is the diameter times three and a "little bit" – a funny idea from his beloved Murderous Maths - and he routinely expresses that in the abstract form, C = d pi or C = 2r pi, while recalling its meaning, it's not just a formula.

I have done the rope stretching activity twice now with my 9 year old and will likely do it again before she gets to this point. Each time we've done it, she enjoys it and learns something more from it. In the last instance we did this activity, we practiced division factors of 12 to get the proportions of the right triangle in place. We also practiced multiplication when finding Pythagorean triples. My 12 year old now has a series of these memorized from doing the activity and then extending it to a chapter in Ed Zaccaro's Challenge Math where he solved a number of right triangle problems that used Pythagorean triples to keep the answers in whole numbers.

Another rather obvious example for early elementary incremental learning is reading math readers or playing around with manipulatives for fun and exposure, but filing away in your mind what the lesson of the activity is if they aren't ready to master it. My youngest daughter was working on addition with regrouping when she was 7 years old. At 5, she read "A Fair Bear Share" MathStart reader which focuses on regrouping, and has read it many times since then. We also worked quite a bit with an abacus at one time. While she could follow a year ago or so, she couldn't reproduce the process if given a problem in a workbook.

We later encountered regrouping again reading "Mr. Base Ten Invents Mathematics." She exhibited more conceptual understanding in following it on the page, but no interest in attempting to do it herself on paper. Then, at 7, she encountered regrouping in a Singapore workbook. We brought out the Fair Bear Share book, Mr. Base Ten and the abacus again, and used these old friends to help learn the concept as it was presented in her Singapore book. The books turned out to be more effective than abacus for her at this point, as she is a print oriented learner. We could refer to the characters and objects in the books when going through the idea and developing a process for her to figure out her answer. She quickly developed her own personal notation to make sure she does not lose track of her ones to be carried over, and in a matter of a couple days she had this idea fully mastered. A month later, she mastered regrouping of tens and hundreds, realizing the same idea applied as she had learned for her 1s. She moved out of numbers and quantities she could concretely understand to applying to a more abstract numbers and quantities.

The idea here is that when we first pulled out these materials in the context of the ideas presented in the math history lessons, I was not attempting to teach her the lesson to mastery, nor did I wait to introduce materials to her because she wasn't ready to master the concepts. She was having fun and enjoying the ideas presented. When she was ready, we pulled out these same materials she was already familiar with, and the lesson was very quickly learned to mastery. I filed away in my mind that her next logical step could be multiple place values of regrouping, which she herself discovered in the Singapore book a few days later. One more lesson to show her that the same process applies to other place values and she understood it, in large part because she really does understand place value concretely through many hours of exploration with base 10 blocks. Therefore she understands that she is carrying tens, or hundreds, not ones, as many children get confused when taught the regrouping algorithm.

If she did not appear to be ready to understand this, I would have waited and kept her supplied with other math activities. Her "next" learning objective might be what she encounters in her Singapore book again,or it might be what she is learning with Hands On Equations, a program that teaches algebraic ideas in a logical sequential manner. I am prepared with the materials that will blend well with what I see her working on next.

My goal, and what I hope will be a benefit to others using these materials, is to become a better and better math mentor to my children with this constant exposure to math in contextual and interesting applications that are far off the page of what I was taught. For many parents, elementary math concepts are no longer routine, but can become exciting and interesting in these context, giving us a fresh and new perspective to share with our children.

If one wants to teach incrementally using activities, the Primary Level readers and lesson plans contain multiple activities for all basic concepts in early elementary. One would need to separate, however, the activities from the readings to present them incrementally. This is fine, the plans are written as guides, not strict methodology. In fact, the only setting that the plans really would be strictly followed would be in a classroom setting wherein everyone needs to be on the same page. In a home setting, you have total control over how to use the materials.

Your own comfort level in working activities that contain ideas you yourself never really learned may be a factor. My ability to teach with these materials has improved dramatically over the years because I myself understand them well. I did not have anything more than a typical math education myself until a few years ago. I never heard of Fibonacci numbers, Pascal's Triangle, or Pythagorean triples before embarking on this study with my children. I could not naturally and comfortably present this material to my kids unless I had read, investigated and understood it myself to some degree. In understanding it myself, I could see the underpinnings of the math ideas – that Pascal's Triangle is built on a very simple addition process that a first grader can understand up to a certain level, and that I can go even further if we make it into an art activity, because visual representations of relationships in the triangle become apparent. But if I don't understand it myself, I can't see these underpinnings.

So just as with any study, learning ahead of your kids will make you much more comfortable presenting to your children. As you know your own child, you'll see connections they are likely to make based on their current development. And likely they'll make many more connections you won't expect them to make if you do not limit them by not exposing them to ideas beyond where you assess their development to be. If you enjoy the material, you will be much more likely to inspire them with your own enthusiasm.

Knowing your children is also instrumental in how much of what type of resource to use and the timing to use it. For wiggly children who do not have long attention spans, abbreviated readings make sense, and possibly you may linger more on hands on activities, or reserve the more challenging history readings for bedtime when they can become quite attentive, especially if it means they might be able to go to bed a little later :o) If readings are too challenging, consider putting the book away for six months or a year, rather than allowing them to develop a negative attitude toward the book that would mean future exposure will be resisted. Focus on the kinds of readers or activities your child is enjoying. In homeschooling, it seems to me, timing is everything. Seasoned homeschoolers will tell you, what is "wrong" for a child now may be totally "right" a year later.

Middle School / Pre-Algebra Level


Middle school often tends to be a period where curriculum and classrooms keep kids in the pre-algebra territory until they seem ready for algebra, recycling concepts in progressively more difficult settings. This can be a wise strategy in terms of delaying formal algebra instruction until they have all the tools necessary to complete a full course, but it can be boring for a child that has essentially learned basic mathematics, but who has not yet fully developed the level of "proportional reasoning" needed to move to the abstract level formal algebra requires.

This is a level that the Living Math Plans lend themselves quite well to use as written. The course provides a review of all basic pre-algebra ideas from counting and place value on up, but in contexts they've likely never seen before. Many of the activities are generated to exercise pre-algebra skills in real contexts. When algebra is referred to, it is usually possible to get the answers without it. Decimals, percents, and ratios are used extensively in activities. Exponents, radicals and other important concepts for algebra success are woven in. Links between geometry and algebra are brought in to give students more of an idea where all this math they are learning is heading.

The plans can just as easily be used, however, in a similar way as Primary Level if a family desires, especially if a child is borderline for the level, or highly asynchronous with their reading and math skills. Historical readings can be scheduled while the activities can be done in a different order based on the child's skill level. Readings can be done through the week and a family can schedule an activity day, since the amount of time required to complete these activities increases with the skill level.

After a year of co-op classes, my middle schoolers grazed on reading multi-concept books such as the Murderous Maths series by Kjartan Poskitt and The Penrose series by Theoni Pappas. My oldest son had read these before, but understood the math in them much better after having gone through the math history activities.

Advanced Level and Up:  Algebra and Beyond

This level offers up a number of ways to use the Living Math materials as well. If an advanced level student has never encountered the ideas in the plans, as is the case for many parents, then working through all the material at a comfortable pace is beneficial. There are many opportunities to learn algebraic ideas that are embedded in the plans. It can be used as a pre-cursor to a formal algebra course, as in fact this level was for my oldest son who in his pre-secondary years was more language oriented than math oriented (this has changed in his high school years to being evenly balanced in skill and interest). He completed a formal algebra course with ease after spending nearly two years with the Living Math plans. The materials can also be used as a conceptual review after an algebra course is completed, as the contexts will be different from most algebra texts, and this was the situation with some of my co-op students as well. Finally, a challenging algebra book is suggested through the lessons (Gelfand's Algebra) for students wanting to learn algebra in a problem-solving framework that is not a typical textbook. Even students who have had an algebra course may find this challenging, it is recommended by the Art of Problem Solving staff for gifted students and students who really want to understand algebra, vs. learn it procedurally.

If a student is ready for algebra and will be working through an algebra course concurrently, the pace will need to be slow enough to make room for the time required to learn the algebra material. One way to accomplish this is to make the math history lessons the basis of social studies, and treat the readings and activities as that subject. It might mean that the algebra course would take more than the usual year if you wish to get the full benefit out of both programs. Homeschooling allows us to pace a course this way.

My middle son went through the Intermediate Level math history materials a couple years ago, and even that was his second round, as we'd done quite a lot of reading and activities since he was 5 or 6 years old. He has been doing an abbreviated version of the Advanced Level plans, going through materials such as String Straightedge and Shadow that he never read before. He is picking up ideas he did not fully understand, or ideas he forgot, and the familiarity of the previous exposure makes them feel like old friends. His choice to work on formal algebra course last year with his friend was due to the opportunity created by his friend's goal to be ready for high school next year, his own realization that he is enjoying algebra after going through Hands on Equations the past few months, and his learning style which is less print oriented than his brother, he learns better with me teaching than trying to teach himself. After half a year of formal algebra, we decided to table that for next year, and he picked up Ed Zaccaro's Challenge Math and 25 Real Life Math Investigations books for the rest of his 7th grade year.

So in his situation, we used a fully sequential math textbook as the basis of his math learning for part of the year, laying over it the math history reading and activities that match up to it. The Harold Jacobs Algebra was a good choice for this, since Jacobs does bring in a lot of number patterns and other tools for being able to learn algebra in an analytical way, vs. simply learning via rote practice of processes introduced. Taking the time to work on the binary system worked well with understanding the difference between exponential group vs. pure multiplicative growth or additive growth – and these ideas are presented early in the text as they learn to differentiate between different sorts of functions and their graphs. If you would like a look at how I've blending these in this fashion, I posted a tentative syllabus here: http://www.livingmath.net/JacobsAlgebraYear/tabid/1000/Default.aspx

I could do the same thing with a geometry text. Two years before my oldest son took Algebra I, he took a high school Euclidean geometry class. This provided structure for his math learning that we laid our math history studies on as well.

Incremental Learning Objective Tagging

It is a goal of mine to go through all activities and "tag" them with the concepts they focus on. This is very time consuming, but the project is moving along. I have a concern that activities might be tagged as only being beneficial in teaching certain concepts. In reality, numerous activities can present ideas a kindergartner can learn as well as a high schooler if they have never been exposed to the idea (the Egyptian rope stretching is a great example, or the King's Chessboard, etc.). Parents exposed to these ideas for the first time can understand this.

In the mean time, the Primary levels have grown considerably since I originally wrote them to provide a suggested rotation of all primary level concepts through a Cycle of lesson plans. It is not possible, however, that every child will be working on the same concepts at the same time. So it is up to parental discretion as to how much of the various concepts they cover with the child and how they do it.

Book lists are posted for each unit, which include extensive reader lists by concept. So even if you only purchased the first unit, if your child is working on skip counting, going through all four units of reading lists for skip counting resources is fine. If they are working on division, look at the Unit 2 list of readers and incorporate those in your living math studies. You won't ruin a scheduled reader by reading it ahead of time as most Primary children enjoy reading math picture books multiple times. A to-do of mine is to include a list of basic concepts included in each unit, while it can be derived by looking at the book lists, it would be easier to see if the rotation were seen visually on the website. It's one of the project I am fitting in as I can with my own homeschooling.

Julie Brennan
Summer 2008