This idea is all about going just one step further by adding in some creative non-examples. Non-examples are incredibly helpful for students to understand the boundaries of what constitutes a particular idea. In the right angle example, we could do some obvious non-examples such as these:

However, we should also do some close-but-not-quite examples such as these:

The non-examples push on the various aspects of right angles. For example, right angles are angles, so I made some that are not angles because they are rounded or non-intersecting rays. Right angles are also exactly 90 degrees so I made that were super close to 90 degrees (like 89.9 degrees). Have students debate whether or not they are right angles or explain why they aren’t right angles even though some people might think they are. I also like to have the students come up with tricky examples and non-examples. Think about quadratic functions, if a student came up with this as a tricky example and non-example of quadratic equations,

I’d be really excited! Especially if they could talk about what makes them tricky. I promise you’ll have a much more interesting discussion when you get creative with your non-examples.

]]>We should use errors as points of celebration! Errors make for some of the best discussion points. If everyone does everything correctly, there’s not much to discuss. Errors also mean that students tried something! The desire and follow through to try something (even if it doesn’t work) is what many of us hope for from our students. We learn from our errors and reflecting on those errors leads us to more fruitful endeavors.

Although many of you may agree that errors deserve a place of glory in the mathematics classroom, students don’t always see errors as a good thing (shocking I know). One reason for this might be because we tend to ask for correct answers in the mathematics classroom. Also, we tend to only show our final, correct draft solution strategies rather than our incomplete, messy, rough drafts (shoutout to Mandy Jansen).

Here’s one simple idea to exalt the glory of errors: **ask for them**. That’s right, all I want you to do is ask for wrong answers. Here’s an example. Say you show students this equation to solve, *x* + 3 = 5. Don’t have them solve it, though I know you want to. Instead, ask them for some answers they know are wrong and have them explain how they know they are wrong. Here are some things they might say in our example:

- 0 is wrong because if you don’t add anything to 3 you won’t get 5.
- I know a negative number would be wrong because you have to add something positive to 3 to get to 5.
- I know a million is wrong because it would be way too big.

What’s so great about this? First, students feel very comfortable when they aren’t on the hook for a correct answer. Many students fear sharing their thinking because they don’t want to be wrong. Second, we have narrowed down the list of potential answers, we know it is positive and less than a million. This can help students further develop their number sense. Third, this encourages students to share their reasoning, something we can all get behind.

I hope you try out this strategy and let me know how it goes!

]]>As the days went on and unexpected events occurred, I realized that I always planned more than I had time to cover. My schedule got further and further behind and it stressed me out! I realized was that things will always come up and the curriculum is far too vast for the number of days I had with my students. I also realized that I was not beholden to my pacing guide, I was beholden to my students’ learning. Students are not programmable robots; it is not possible to know ahead of time exactly how long it would take to learn a particular concept. I realized I had planned for teaching, not for learning. As my friend and former colleague Debi Hanuscin has said, we should teach at the speed of learning.

Now this all sounds fine and good, but the realities are that many of you may have pacing guides and common assessments, so what can you really do? Well, something that I found incredibly freeing was quite simple in retrospect, I planned in *flex days.* For at least one day every chapter or unit I planned a day of TBA. Mentally, this gave me space to not stress out when a sudden assembly was held or my well-planned one-day lesson stretched over two days. This did not address some of the larger issues, but, in the spirit of 180 Ideas, it was one, easy to implement idea that helped in some small way. I still do this as a professor and I even do this by setting aside an hour here and there on my calendar that is marked as Flex Time. This allows me to have a dedicated time for myself to work on something I am passionate about or to just enjoy a leisurely stroll or coffee during that time.

I hope you plan in a Flex Day in the future. Feel free to share any ideas or variations you have on the flex day in the comments.

]]>As you might imagine, I did not magically step in to the classroom with my mathematics background and transform into a phenomenal teacher. I taught much like my high school and college teachers taught. I lectured as I went through various procedures and when students didn’t understand, I lectured again. This was not effective then and it’s not effective now.

During this time, I took night classes at the district offices in order to become alternatively certified. I don’t remember much about these courses except that they were mainly general pedagogy courses that focused on practices such as using thinking maps or ways to group students. These classes provided me with some new ways to think about classroom management, but I still wasn’t what I would call successful in helping my students understand mathematics deeply.

After earning my teaching certification, I decided to take night classes to learn more about my chosen profession. I enrolled in an M.Ed program at the University of Central Florida and learned there was an entire field devoted to the study of how to teach and learn mathematics: mathematics education. Reading and learning about mathematics education is where I really started to learn new ways to approach mathematics teaching.

Upon completing my Master’s I wanted to know more and so I decided to apply to doctoral programs in mathematics education. Initially, I thought earning my PhD would help me learn to be a more effective teacher, and it did. However, rather than returning to the classroom as a teacher, I decided to teach future teachers and research how to continue to improve mathematics teaching.

So, why the long story? Since starting in my current position I have been thinking about all I have learned. In doing so, I always think back to 22 year old Zandra and try to come up with advice for her. I cared so much about my students but I was under-learned (probably not a real word) and overwhelmed. I had common assessments and pacing guides to adhere to and a seemingly never ending list of emails, grading, and lesson planning to complete. Looking back, were there some small practices I could have quickly taken up to improve my instruction? The answer is undoubtedly yes.

Realizing this has led me to this multipart project called 180 Ideas. In this project I will give 180 ideas that any mathematics teacher can try out in the tomorrow. The idea relates back to a diet book I read some time ago that encouraged the reader to take up a new practice each day. Each one of these practices were really small and easy, like eat an apple a day. However, by the end of the book, those that had enacted each practice and kept them up would be following a vegan diet. The book could have just as easily said, in one sentence, become a vegan. However, that’s not easy or practical. Instead, these little steps culminated in a large change. Even if someone only did a portion of the practices, he or she would still be healthier than they were at the start.

180 Ideas is meant to give 180 little steps that you can take up. 180 because that would be one per day for a typical school year (thanks to Sam Otten for the idea to make it 180). Also, doing a 180 means turning around, and I hope this helps you to turn around some aspects of your practice you are not yet satisfied with.

If you were to consistently enact all or most of the 180 ideas, you would likely transform your practice. Some of you may just take up a handful. I would still argue that I think that both you and your students will benefit. Some of you probably already do many of these ideas regularly. It might still be helpful to read through these ideas and think about them as you do them. This might be especially helpful as you mentor new teachers into the profession.

I hope you give some of the 180 Ideas a try. For you, the teacher, these ideas are meant to invigorate your practice and give you small practical ways to try something new. In terms of your students, these ideas can help get your students talking and thinking deeply about mathematics. I hope that some of you may also contribute some of your own ideas as the project continues.

]]>I think we’d all agree that classrooms should be welcoming and safe places for students. We would probably also all agree that they are also places where learning occurs. I am not sure that the spaces we design for ourselves to learn would mirror those many of us have designed for students. In my own home office, I often feel a bit overwhelmed and anxious when there are too many items out and there are too many things around that are irrelevant to my current projects. I know not everyone may feel this way, but many people do. Visit your local library and you’ll see the numerous titles related to minimalism and mindfulness. There aren’t many books on maximalism or eclectic design. One reason for this is probably because the notion of maximalism (if that were a thing) would likely not result in a calming environment. Also, humans tend to like rationale arrangements. Students spend nearly as many of their waking hours in the classroom as they do at home. If our home environment is more peaceful when we are mindful in our placement and procurement of items, might it stand to reason that our classrooms may as well?

Many classrooms (my former classroom included) have a number of elements that are not tied to anything currently happening. I’m not just talking about the random Garfield posters, I’m also talking about numerous old papers and supplies. What would it be like if the things that are out and about in our rooms were just those things students can use or relevant to the present lesson? What if we were purposeful in putting out just what is needed when it is needed?

Many of us may already do this, I’m not talking to you. I’m talking to those of us who have so many things on walls for so many years that we couldn’t even name them all if we were not in the room. I’m talking to those of us that walk by a random object daily that has just become another dust collector. I used to have a number of origami figures in my classroom all the time. They were just kind of always there. Would the students have been more intrigued if I just brought them out when we were about to make them? Then, maybe could we have just shown and tell with a gallery-walk type activity rather than keeping out the 60 new figures my students made for the next few months?

I’m not saying you have to have a cold classroom, we already agreed classrooms should be warm and welcoming. I’m advocating for a purposeful classroom. For example, if we are doing a unit on quadratics, I could bring in some examples of quadratic functions in the real world and put them up in my room to use as part of an activity during the unit. I might also set up some hands-on activities that illustrate parabolic motion the students can play with before the bell. The things around my room could then relate to what we are actually doing. Many of us teach multiple classes, so perhaps there could be particular areas for the different classes, but it would be purposeful. For elementary teachers, this might mean putting aside anchor charts when we are not talking about that topic or having out only the manipulatives that students could use for the particular lesson we are on.

The notion of a purposeful classroom goes beyond the visual environment, it extends to the seating arrangement as well. In future posts, I’ll write about other aspects that might benefit from a more purposeful consideration.

]]>This is not incorrect, but it is also not the only way to solve this system. Students could just as easily solve the first equation for 3*x* and substitute that in for the 3*x* in the second equation to get 2*y* – *y* – 4 = 16. I think this equation is actually easier to solve than the first. You may be thinking, okay so there are two ways, big woop. Well friends, there are other ways. You could rewrite the second equation so that you can more easily see 3x + 4 like so, 2y = 3x + 4 + 12 and then replace the 3x + 4 with y so you end up with 2y = y + 12. Again, I think this is easier than the textbook’s way. It also encourages students to understand the substitutions they are making and have more flexible reasoning. I hope you try different substitutions out with your students to find as many different as you can think of. It’s super fun, give it a try!

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Discussing fractions without discussing their associated whole (also called referent unit) can be problematic. The quantity is assumed to refer to some referent unit. However, I can show students a picture such as this

and ask them to show me . Students might produce this

or this.

Each response is possibly correct. If the first student was thinking of one of the bars as the whole, than she is correct that the shaded region is of that whole. If instead, a student perceived both bars together as a whole, then 2 shaded boxes would represent of that whole. Thus, it is incredibly important to clearly define the whole when discussing fractions in school mathematics.

Typically, teachers are not very clear on what the referent unit is when discussing fractions. This may lead to some student confusion. For example, if I ask a student which is bigger, or , it is typically assumed that they are referring to the same whole and so is indeed less than . However, if I add in a context then this may not actually be the case. For example, I might compare of a mouse’s weight and of an elephant’s weight. This might cause some confusion because students are typically given blanket statements that is always less than without stating that this is assuming that they are referring to the same whole.

Another reason why understanding the whole is essential is that some operations with fractions involve consistent referent units (addition and subtraction) while others do not (multiplication and division). For example, in the equation *x* + *y* = *z* (where *x*, *y* and *z* are fractions) *x*, *y*, and *z* each refer to the same referent unit. However, in the equations *xy* = *z*, the referent unit for *x* differs from *y* and *z*. I will explore this further in part 3.

The short answer is because fractions are complicated! This is the first of a multipart series on the basics of fractions. In this first part, I’ll break down the two common ways of thinking about fractions.

The first way that people think about fractions is what others have referred to as “*m* out of *n*.” Using this meaning of fractions, someone would think about 1/2 as 1 out of 2. So to put this in context, if I cut a piece an orange into two pieces, one of those two pieces is 1/2.

This way of thinking about fractions is not wrong, but it is a bit problematic. First of all, it doesn’t emphasize the size of the parts in relation to the whole very well (more on that later). Secondly, and super problematically, it makes understanding improper (when the fraction is greater than 1) fractions difficult. What does it mean to have 3 out of 2 parts of an orange? Nothing, actually. So this is not actually the preferred definition of fractions, but it is where many students and teachers start.

Consider the first reference to fractions in the Common Core.

Partition circles and rectangles into two and four equal shares, describe the shares using the wordshalves,fourths, andquarters, and use the phraseshalf of,fourth of, andquarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

This standard seems similar to the m out of n definition. Students are expected to cut circles into two parts and describe the whole as two of the two parts (2/2) or one part as one of the two parts (1/2). Most people are surprised to learn that this is actually a first grade standard. That’s right, kids first encounter the dark lord, ahem, I mean fractions in first grade. Even more interesting is that this standard is in the geometry strand, not in number and operations as one might expect.

The definition of fractions I prefer can also be found in the Common Core Standards (from the third grade number and operations strand).

Understand a fraction 1/bas the quantity formed by 1 part when a whole is partitioned intobequal parts; understand a fractiona/bas the quantity formed byaparts of size 1/b.

I’ve worked with a lot of people who’ve seen this and tuned out after the first sentence (you may have been one of those people!). However, if you are able to stick with me, you might come to love this definition as much as I do.

What this is saying is say you have something and you cut it into *b* equal parts. Each individual part will be 1/*b* of the original. So back to our orange, if you cut it into two equal parts, each part is 1/2 of the original orange. If you instead took that orange and cut it into three equal parts, each part would be 1/3 of the orange. With me so far?

Now, the next part of the definition says, if you take some number, let’s say *a*, of those 1/*b* parts, your total amount would be *a*/*b*. Back to the oranges, if I cut the orange into four pieces, each piece would be 1/4. Now, take two of those 1/4-sized pieces and what do you have? Two, 1/4-sized pieces or 2/4. That’s not too bad, right?

So why is this preferable to the other definition? Think about 5/4 of that orange. With the m out of n definition, I’d have to try to make sense of 5 out of 4 pieces of orange (nonsense!). With this definition, I’d take to think about a 1/4-sized piece and then think about five of those pieces to make 5/4 (sense!). So the real power of this definition lies in the ability to construct any fraction and the emphasis on the pieces in relation to the whole. If everyone learned this definition well, fractions would be a lot easier for everyone!

Stay tuned for part 2 of my fraction exposé where I delve into the whole and why it is so darn important.

Consider the system of equations below.

2*x *+ 3*y *= 14

-2*x *− 5*y *= -2

As long as the lines are not parallel or the same line, they will intersect at just one point (*x*,* y*). Solving this system means that you are searching for that particular coordinate point that satisfies both equations. This is the goal of solving a system of linear equations, to determine whether or not there are points that satisfy both equations.

Elimination is one method to do this. With this method, you would add the two equations together in order to eliminate one of the variables so that you are left with a single equation with a single variable. So in the example above, you would get the following:

(2*x *+ 3*y) + (-2x − 5y) *= 14 + -2

-2*y* = 12

*y* = -6

You can then substitute in this y value into either of the original equations to determine the related value of *x*, like so.

2*x *+ 3*y *= 14

2*x* + 3(-6) = 14

2*x* + -18 = 14

2*x* = 32

*x* = 16

Thus, the solution to the system is the point (16, -6).

Ordinarily, you cannot add a term from one equation to another because they are not indeed like terms. That is to say, an *x* in one equation is not necessarily the same *x* from another equation. For example, consider *x* + 3 = 5 and *x* − 2 = 7. The value of *x* in the first equation is 2 and the value of *x* in the second equation is 9. Therefore, you can’t simply add the equations together to get *x* + 3 + *x −* 2 = 5 + 7 because the *x*‘s are not like terms.

Technically, you have just added 7 to both sides of the *x* + 3 = 5 equation, however, then you would also have added *x* + *x* but, as stated above, the *x*‘s don’t represent the same unknown value. If you ignore this fact, that would mean, that you would have 2*x* + 1 = 12, but that doesn’t make sense because you just added an *x* with a value of 2 (from the first equation) to an *x* with a value of 9 (from the second equation). If you add those *x’*s, you should have gotten 11 (2 + 9), however, the 2x means it should be double the value of x, so if you take it as the x from the first equation, this would mean 2x should be 4, but if you use the value of x from the second equation, this would mean 2x would be 18. You cannot have 2x simultaneously equal 18 and 4, and this is indeed the wrong answer based on the actual values of each of the *x*‘s. It would be okay to add these together if you, for example, keep the *x* from the first equation as *x* and then rename the *x* from the second equation to something else like *y*. This would result in the following: ** x** + 3 +

Back to elimination, we do indeed keep the *x*‘s the same and add them as if they were like terms because they are indeed like terms. This is because we are looking for the solution for which those *x*‘s are equal. If this were not the case, we could not add the *x*‘s together. This is an important point that should be emphasized with students.

So in summary, elimination works because we are looking for the values of *x* and *y* that are the same in both equations. When we add the two equations together, we are really adding the same value to both sides of one of the equations, which is allowed. If we were to annotate the steps in the process above it would look like this:

To solve this system of linear equations

2*x *+ 3*y *= 14

-2*x *− 5*y *= -2

Add -2 (in the form of -2*x *− 5*y *on the left*)* to both sides, which is okay because we are looking for the point where *x* and *y* are the same for both equations.

(2*x *+ 3*y) + *(-2*x *− 5*y*) = 14 + -2

Simplify the new equation and solve for y.

-2*y* = 12

*y* = -6

Substitute the value of *y* into either of the original equations to determine the corresponding *x*-value.

2*x *+ 3*y *= 14

2*x* + 3(-6) = 14

2*x* + -18 = 14

2*x* = 32

*x* = 16

Thus, the solution to the system is the point (16, -6).

Teachers should encourage their students to discuss these steps and the rationale as they solve to make sure they understand why this works. I hope this was helpful!

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If I have not grappled with these ideas, it is not likely children have either. This is problematic for many reasons. The most important of these reasons (IMO) is that mathematics is fun. When I say mathematics, I don’t mean simple computations, though they can be, but rather trying to understand challenging mathematical ideas is simply fun. People often fail to see mathematics as fun (or useful), probably because it is hard to get excited about repetitious procedures and rote memorization – the unfortunate cornerstones of mathematics education in the U.S. In this blog I will try to uncover some aspects of mathematics that you may have missed in school.

Currently, I am a mathematics educator. Basically, this means I teach people how to teach mathematics and I also research the teaching and learning of mathematics. When I was a teacher I had no idea that a number of people had researched some problems I was encountering in my day-to-day. In fact, many really intelligent people have been studying the teaching and learning of mathematics for a long time. Unfortunately, many of these papers are extremely inaccessible to those outside of academia, both in terms of cost and readability. I am going to try to make some of this work more accessible through this blog. As a former public school teacher, I will try to point out the practical applications of some of this work to help connect these ideas to the people who might actually put the ideas into practice.

So to recap, this blog is about really digging into to some things that will help people become more mathematically educated. I will bring up a number of fun mathematical ideas that I have thought about (and continue to think about) over the years and I will also try to translate research into a more useful format. I welcome feedback, questions, and conversation as you read through my thoughts.

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