As a homeschool graduate myself, I found it interesting to reflect on my experience in math first as a student and then as a teacher and parent. I would not have anticipated this as a child, but I am so glad that I have been given the opportunity to return to math.

When I was being homeschooled, the Saxon math books were just being published. I did Math 76 and then the following year did Math 65 when it was available. As a student, math was a chore, a thing to be done because it was assigned. I did fine, but I did not enjoy it and I certainly acquired the impression that I was not a “math person”. I am not sure what I expected from math, but I had no intention of doing anything beyond the bare minimum to get a passable SAT score and graduate from high school. 

In college, however, I fell in love with math. I had an inkling that this might happen as I had older siblings who had experienced something similar when they met Euclid. And Euclid rightly deserves this credit. Euclid’s geometric proofs provide a clear example of scientific knowledge. In each proposition he begins with something given and either constructs or demonstrates something. The proof proceeds through a series of statements showing the equalities of specific lines or angles; from each equality Euclid draws attention to a new equality until one reaches the equality that had been sought at the outset. It is a beautiful example of what Fr. Laux is describing in Catholic Apologetics when he says that we come to the knowledge of truths that are not self evident through a process of reasoning, that is, by comparing two things with a third on the principle that two things that are equal to the same thing are equal to each other. Euclid showed that that I could know something. Once I had done a demonstration, I knew with confidence that it was true; and Euclidean demonstrations are particularly satisfying because in most cases one moves from the cause to the effect. Later in college I had the opportunity to see that this is how God knows, but also that math is one of the few places where we, humans, can achieve this type of knowledge. 

This experience with Euclid gave me a new appreciation for math. But it would be naïve to hand children Euclid and expect the same results and one certainly couldn’t expect middle school or high school students to charge into Ptolemy, Apollonius, Descartes or Newton. I knew that Euclid, standing at the beginning in one way, was the culmination of what we as parents and teachers are preparing our children for in grade school, middle school, and high school. I wanted my children and my students to fall in love with Euclid just as I had, but preferably without a disagreeable experience in grade school and high school. I wanted my children and my students to be successful in math from the outset.

For a decade after college I taught other people’s children math, and for the following decade I’ve been blessed to teach my own children math. I still don’t think I am a “math person”, but I think we are all capable of being people who can do math and I think that is key to getting our children to do and to love math.

Building Habits of the Mind

As a teacher, by the time I began working with a student, math was already a struggle. The parents were seeking help because the student didn’t understand, was doing poorly on tests and the parents didn’t have time. A live tutor is certainly preferable to turning on a tutorial video, or flipping back to example problems in the lesson or solutions manual because a live tutor can, if attentive, identify where the student is struggling and adapt directions accordingly. But generally speaking a tutor or teacher is reacting to an existing problem. As a teacher, I can write out the problem for the student, going through step by step to show the student the process. I can direct the student back to an earlier concept that is shaky and lay out concepts that must be reviewed. But it became very clear, very quickly, to me as a teacher, that if the student is going to be successful in math, it is through building up certain habits. Even in a live class where the teacher may have the benefit of being with the student from the start, before any struggle has arisen, success comes through the habit of math. A good teacher, and a good parent, is directing students to those habits.

I am going to step back to Euclid for a moment. There were many things that I learned in Euclid that I might have told you I already knew. I had been told that the angles of a triangle were equal to two right angles. I had been told that the square on the hypotenuse was equal to the sum of the squares on the two sides. I had used these formulas in countless problems in high school and I didn’t have any doubt as to the truth or usefulness of them, but I didn’t actually know them until I had worked out the proofs for myself in college. And it was incredibly satisfying, night after night as I did my homework, to come to each new conclusion for myself. Furthermore, it was satisfying to come to a truth I already knew, such as the Pythagorean theorem, in more than one demonstration, that is, coming to the same conclusion through a different set of reasons. Euclid didn’t show us how to do a few propositions to give us a feel for the experience and then throw out a bunch of conclusions with the understanding that all of these could be proved in a similar fashion. The process of the demonstration is important because it is causal to our knowledge of the conclusion. And I am forever grateful to my college math teachers that when a mathematician like Ptolemy, Descartes, or Newton did skip over steps—perhaps because the steps were so obvious to them—that we were asked to puzzle those steps out so that the steps were clear to us. We were asked to make the knowledge our own.

Euclid and those who followed him were setting up habits of the mind. Every Euclidean proposition you do makes it easier to do the following one. When I first started Euclid, cocky students would share stories about how they had come up with their own proofs that were different than Euclid’s but reached the same conclusion; they didn’t need Euclid’s proofs at all, they could just work it out for themselves from the given. This was initially mind-boggling to me, for it seemed like one would have to wade through so many equalities until one stumbled upon the right conclusion. And that was when I realized that to work out a proof for oneself, without the text, one has to have a good, and as Laura says, docile, imagination, but also a firm grasp on the conclusion that is being sought. Those students already had good habits of mind when it came to finding and forming new equalities. I had to learn those habits of mind and for me that was going to come from patiently following Euclid’s lead through the proofs as he wrote them.

Building up the habits of the mind is why math has to be done every day. It is not the sheer volume of data, but rather the mental discipline that is formed by daily math lessons. This is why when we have a very bright little student, we don’t let them do the Kindergarten math book in a month and then stop doing math until next September. Rather, we seek out other similar math books and let the student continue that process of building mental facility with the matter at hand. For the same reason, we don’t just do a few multiplication problems and call the concept mastered, but rather we practice it over and over, first at a simple level and then slowly building in complexity.

Eight Ways to Build the Habit of Math Together With Your Child

When my oldest was little, he loved letters and words. He loved to be read to; he had a tremendous attention span and a fantastic memory. So it was delightful but not very surprising that he learned his alphabet very quickly and was working out letters on signs everywhere we went. Since he had mastered the alphabet it made sense to me to share numbers with him.  We were playing with letters and he joyfully worked through all 26 letters. So I handed him a numeral and said the name “1”. He repeated it and set it down. Then I handed him the next and said “2”. He dutifully repeated it and put it down. But it was clear as we looked at the numbers that these were not registering in the same way. And it was equally clear to me that teaching letters and numbers was not the same thing. In the case of letters we were associating a set of sounds and symbols. He knew they had an order and could reproduce the alphabetical order without difficulty, but we also both knew that his knowledge of any set of sounds was independent of the other letters, while the concept of two depended upon understanding that this was two ones. I could teach him a set of number names, and let him pair a symbol with a word, but I hadn’t taught him anything meaningful about numbers until I had taught him about a unit and the concept of grouping units. That was an eye opening moment for me in education.

It was clear to me that if I was going to teach my son math, I was going to do math with him, and we were going to build those habits of mind together. I think it is important at this point to say that I love doing school with my children. This does not mean that it is fun or easy and there are certainly many moments that aren’t very enjoyable for any of us. But love is an act of the will and math is the place where we have had the most opportunity to exercise that decision. So what does this look like?

First, do math every school day. It is usually the first subject we do because I am of the opinion that one should get the hardest things out of the way at the start when there is the most energy and focus. But I am open to mixing up the order if one of my kids thinks that will be more motivating. And if we have appointments cutting into a school day, math is what gets done in the time we have.

Second, review math every school day. School is not done until the math lesson has been reviewed. In the early grades, math is done with me, so the review happens in the moment. But once my child is doing math independently, he brings the text and notebook to me once it is done and then goes on to other subjects. When I am ready to review math, that child comes and sits with me and we go through the lesson together and rework anything that is missed.

Third, encourage cautious independence. Math, after handwriting, is a good place for students to develop independence. One can start with something more rote like math drills where the student is working on something he already knows. This gives the student an opportunity to work on his own, to build confidence in working without a guide, to accomplish a goal without prompting, and to bring a finished task to mom! In the beginning this might be only a small portion of the student’s math, but as the student responds to this independence, the time and scope increases until the student is able to work through his entire math lesson on his own. But since math is about habits of mind, I mentioned caution. I don’t want my children to learn bad habits. So there are a few things we do alongside this growing independence. Once the student is ready to approach the lesson on his own, I make sure we schedule time to review the new concept in advance. At first, we do this to start the math session for the day. I make sure I have time for that child, we read through the lesson and work through the practice problems together and then he goes off to do math. As he gains in both confidence and mastery, I move this to the end of our review session. So once we have reviewed today’s lesson, we preview tomorrow’s lesson. This allows me to ensure the student has reviewed the new concept and that there is space for him to ask questions if anything is unclear, and I can see how the practice problems go. This allows my child to be more independent the next day, starting math on his own, which allows for more flexibility in his schedule. Finally, keeping in mind that we are working on habits, and that it is better to learn a habit correctly the first time, than unlearn a bad habit, if my child is unsure of how to do a problem, I have him circle it so that we can review it together. This has the added benefit of giving me a quick pulse in our review sessions of just how independently the child is working. If there are only a few problems circled, that is a sign that things are going well. If lots of problems are circled, then I need to be more attentive and more involved in the daily lesson.

Fourth, model the process of learning. When my child has missed a problem, whether it is something he attempted or something he circled because he couldn’t do it, it is an opportunity for teaching. Whether or not I think he should “know this” by now is beside the point. He doesn’t right now, and this is the moment for him to learn. We look back at the lesson together. If it is from that lesson, it is an easy opportunity to stress that this is new and being learned. But if it is from an earlier lesson, it is an opportunity to show him how to find the small lesson number under the problem and look back to that lesson. I review how it was taught and then we redo the problem together. 

Fifth, pay attention to the cues. As a student I found it very hard at times to figure out how a specific problem fit into the formula from the lesson or the examples. I think the process of modeling that application from the lesson with later problems is very important. Sometimes I find that lesson after lesson after lesson it is the exact same type of problem that is being circled by my child. It is tempting to blame the student for not paying attention, but this is a clue to me that either Saxon or I am not teaching it properly. I need to think about what is not clicking. Sometimes it just takes time; sometimes we’ll go back through that same type circled in the last 5-10 lessons so that he can see them all side by side. 

Sixth, model the steps. When a problem is missed, the error is in the steps. I am adamant about having the steps written out. When a student tells me that he can do the problem in his head, I say, “That is great, but I can’t see inside your head, so I need you to write down what you did here.” Math is about the process and I don’t want just the conclusion; I want to see how the student got there, primarily because I want to ensure that the student knows how he got there. But as both a parent and a teacher I have found that students often don’t know what we mean by writing out the steps. So when we are doing circled problems or redoing missed problems I write out just what I want seen. Half the time my child will figure out what he missed before I finish. Now, I do want to add here, that this is a place where you need to be attentive to the student in question and attentive to where accommodations might need to be made. As a rule, I want to see the steps worked out, and it can be particularly important on our assessment assignments that those steps be written out. But Laura has shared in previous talks that she would have one of her children do every test twice. He was good at math, and because of his dysgraphia had developed habits of working out as much as possible in his head. Rather than write down all the steps, which was physically uncomfortable, he worked through every problem twice and if he did not get the same answer for any particular problem he would work through that problem again in order to determine which was the answer he wanted to submit – in other words, which answer he was confident was right. This is a different process, but brings about a similar end result. I have my child write down the steps, so that if an error is made, we can return to those steps and find it – allowing for a form of doing the problem twice. My brother-in-law, by doing each problem two to three times, was still working through the steps, albeit mentally, but had a check by comparing answers, to confirm his work. In both of these cases – and in other acceptable accommodations – the key is that the student sees that there is a right answer, and that one must be able to confirm that he has confidently proceeded to that right answer. Finally, I do want to stress that any student, no matter the method of testing, should always be capable of producing the steps used. Even if those steps were done entirely within the head, the student should be able to articulate them. Otherwise all one has in the answer is a lucky guess.

Seventh, engage the student in the process of correction. When we are busy, it can be easy to just circle what is missed and hand it back to the child so that he can redo the missed problems. There is a time and place for this and this is particularly true when the student is older. But when the student is young, when he is almost certainly genuinely unsure, and before he gets frustrated I think it is better to redo these with the child. But that doesn’t mean that I just do the problem, which is a very easy mistake to fall into. Rather, I talk through how to properly set it up. Sometimes that is enough and my child can finish it from there. Sometimes we break the problem into stages and the student can do the computation, but I have to guide him to connect the stages in the right order. As the student gains in maturity, confidence, and independence, then there is space to hand back the test with circled problems and let the student redo them on his own. A few cautions. First, make sure the student is ready for this emotionally; if handing back the lesson is going to set off a tantrum or despair, step back and go through the problems together – we don’t want the habits of mind associated with math to be associated with these negative emotions. Second, make sure the student has confidence in the process. Have you already seen in the earlier stages the ability to refocus, reset the problem, check the formula used, pay attention to the signs, and work with confidence to a new, hopefully correct answer? If the concept in the problem is too new and the student can’t figure it out, then handing it back to be redone solo will lead to frustration and not success.

Eighth, celebrate! This is really important. As adults, as parents and teachers, we can see the whole. We can see the end toward which the child is heading, but they have no idea how far off the terminus is. We can see how much they have learned because we have a frame of reference, but they are doing this for the first time and only have their own isolated experiences. I regularly point out to my children—and I think I should do this even more—how far they have come and how much they have accomplished so far. We were decluttering recently and my oldest suggested that I get rid of his used math books to make space for new books. But those books are a reminder for both of us. When he bemoans how easy his younger brother’s math is, we can look together at how he was doing the same problems at the same age. It is easy now, but it was challenging then. They are a beautiful reminder that what is hard becomes easy with patient practice.

When Math Leads to Frustration

Now, I would love at this point to say that I do all of the above with the utmost patience and sweetness and grace. But my children keep me honest. I have a short temper, I get frustrated, I despair of certain concepts being mastered, I get emotional. Math isn’t just about habits of mind, but habits of the will; remember, I “love” doing math with my children. This is something that I have chosen; this is something that I think is worth doing both for them and for me. And just like the math problems my children did in the past, this process of doing math together is going to become easy only through patient practice. When I fail—and I often do—I have to remind myself and them that tomorrow is new and a fresh start. 

But when we start fresh the next day, and when I fall again, in order to persist, I have to keep looking at my expectations and their abilities. Am I frustrated because I am expecting more independence of my child than is reasonable? Are we getting bogged down in the problems and the lesson is taking too long? Is the setting too distracting for my child to focus on what is an already difficult subject? Am I demanding perfection rather than mastery or even competency? Can I, as the teaching parent, tell the difference between errors due to new material vs errors due to a lack of understanding? Can I adapt my teaching method or the manipulatives available to where my child is struggling?

If I can be honest with myself about these questions, then I will be a better teacher and a better parent. I can refocus my attention on those habits of mind and habits of will—both for me and for my child. And when we keep those habits of mind and habits of the will foremost, it becomes easier to approach adaptations and modifications where we might encounter learning or behavioral issues. Will this modification help to accomplish the end we are seeking?

The single most important thing we are doing in math is setting up those habits of mind. The student is learning how to find a problem in the words on the page, find the formula to use for that problem, apply that formula to the problem, and work out the problem. This is our goal. This opens the door to modifications such as creating a sheet of formulas the student can reference, or taking a test open book. Certainly it is better if a student can memorize the formulas or take tests closed book. But if the student cannot, then we need to see that the modifications allow for the most fundamental aspect of math to be accomplished. And depending on the student and the situation, I find that the process of looking up formulas, for example, leads to the very memorization we were initially seeking.

Similarly, being attentive to the habits of mind, provides insight when it comes to using a calculator. If a student knows how to identity and apply the correct formula, it is okay to allow the student to use a calculator for the more rote process of calculation. We want to know that the student can add, subtract, multiply, and divide. Those should be practiced as they are taught, and facility should be encouraged, particularly when it comes to setting up these processes with decimals, fractions, and exponents. But we should not lose sight of the fact that the concept is more important than the calculation.

Best Practices Surrounding Test-Taking and Answer Keys

Now, I would be remiss in any conversation about math not to address two key areas of concern: proctoring and keys. For the sake of your child, more than anything else, keep the key—or anything key-like, such as internet resources—locked up. There is a world of difference between the parent providing the student with a sheet of formulas, or authorizing the use of a calculator and the student sneaking these on his own. Be attentive to the needs of your child and be clear about what is authorized. If the need is there, authorize it and document it.  Finally, proctor tests. It is good to practice the proper setting for test-taking, but ultimately this is as much for your child’s protection as anything else. We love our children; we pour our time, energy, and attention into them. We are guiding them through the incredibly challenging process of growing up. It is understandable that we as parents would not expect our children to cheat, and even to be very confident that our children have not done so. But we need to pair this confidence and trust with a few important points. First, sin is shocking. We are shocked when our children fail, with good reason, because we have raised them not to and their general behavior confirms that they usually do chose the virtuous path. But we all sin and we have to be attentive to that possibility in our children. We are doing our children no favors when we leave them in a near occasion of sin with the temptation in front of them. Proctoring removes those near occasions and temptations. But proctoring also provides our children with a right understanding of the testing process. Our children are young and learning how things work. They don’t know what is expected in a test unless we lay out those boundaries for them. The majority of children who copy from a key, or a passage from a text, do not realize that they cannot do this. And, in fact, they may have been directed to do these very things in the past. If you, as a parent, have checked the key as part of working out a problem with your child, then recognize your child is following that same process when he “checks his steps” against the key. By proctoring, you remove that confusion and clarify that a test is taken under different parameters. Finally, when you, as a parent, have proctored the test, then if any questions arise, you can speak with confidence to the setting in which your child completed the test and the materials or tools accessible.

A Skill for Every Area of Life

To close, I want to reiterate: if you can, do math with your children. I know, as a parent, that we have limited time and there are so many demands. We are already trying to read with them, to discuss religion with them, to find their hearts. These are important, but math is as well. When we do math with our children, we highlight the importance of math and we learn it—or relearn it—ourselves. We as parents build confidence and are better able to guide our children through those tough concepts. We don’t need to do every problem—though some parents do! But we should work with them through the new concepts. We’ll gain that facility and model for them how much we value math, but more importantly, how much we value them as children. We do math with them because we love them! We want them to form these habits of mind that will carry into not just high levels of math, but every area of life.

Just last month, my youngest sat back and said, “I love doing math.” Sometimes I know that he genuinely enjoys it and delights in the satisfaction of working out the problems. But sometimes it is equally clear to me, as his mom, that it is not easy and he would rather be off playing. And yet, after years of working on these habits of mind and will, he stays and does math. He is consciously choosing to discipline his will and do math. This is our goal! 

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For more on the importance of the study of math, check out our recent Light of the Faith talk on the place of mathematics in a liberal education!