The Language of Math – Consistency to Support Students

I’ve been teaching some courses at Drexel University, and in those course we really focus on the language of mathematics and using students prior knowledge to help them make connections and build on their mathematical understanding.

In a current course, we are exploring integer addition and using manipulative’s to provide both a visual and concrete connection to the idea of creating zero pairs, and then progressing to the more abstract addition of integers without manipulatives and how do you support students understanding and language. What has come up frequently is the terms ‘cancelling out’ and ‘disappearing’ and ‘opposites’ to explain or help students understand that creating zero pairs allows you to use the additive identity property.  But – what’s really happening, mathematically, is that we are using additive inverse (opposites) to create these zero pairs, which are NOT cancelling out or disappearing, but instead, are creating the quantity zero. And, once we have created this quantity zero, the remaining value can be added to those zeros using the additive identity property. Cancelling out or disappearing implies they don’t exist, which, when we expand the idea identity into multiplication where we use the multiplicative inverse to create a 1 (not a zero), cancelling out really seems confusing.

Seems like it shouldn’t make a difference, but think about it – if we use terms like opposites, cancel out, or disappear in one grade, and then the next grade or future courses, the same ideas are referred to as additive inverse, zero pairs, additive identity property, students will be confused and think they have never seen these concepts before. While it is very important to use language students understand to start the process, I think as mathematics teachers, no matter what grade level, we need to model proper mathematical language in conjunction with the ‘student-friendly’ terms we tend to rely on or fall back on. This way, students can relate, but then learn, build on, connect & utilize mathematical language so that we are all communicating on an equal playing field.

Something as simple as -3 being referred to as negative 3, minus 3, or even 3 negatives. That’s confusing. Let’s learn to be consistent with our mathematical language and use the correct vocabulary – both in our own teaching, but also in what we expect to hear from our students. So if a student says that the opposite of 3 is -3 (positive 3 and negative 3), let’s acknowledge that they are thinking correctly, and in mathematics we refer to that as an inverse. This will then help them make that connection when we talk about inverse operations.  Consistent mathematical language supports students understanding as they progress into more abstract mathematics.

We want our students to communicate mathematically with the language of mathematics and become proficient mathematicians. Let’s then make a conscientious effort to use and model correct mathematical language instead of the ‘short-cuts’ or ‘simplified language’ we tend to use.  Again – important to start out with this type of language to help students connect prior knowledge, but then more important to model using and support students use of, correct mathematical language. I think it would go a long way in preventing some confusion students experience as they move from grade to grade or course to course.

Math Magic or Calculators?

I was perusing my news feed trying to find something of interest to write about, and came across an article entitled “The Common High School Tool That is Banned in College” i.e. the calculator. It’s an interesting article, worth a read,  basically comparing the high school perspective on the use of calculators to the college perspective or non-use of calculators. There is no right or wrong answer – I think it depends on the math content, what you want students to do (i.e. basic algorithms to solve problems or using mathematics to solve deeper problems).  Depending on your goals, the use of calculators and technology differs. As with any technology, calculators are a resource that needs to be used appropriately, and we need to be teaching that.  Common Core Mathematical Practice #5 – Using Appropriate Tools Strategically is all about this. Calculators have their place and are important to help explore and expand mathematical understanding, but we have to help students understand when their use is necessary and not a ‘crutch’, as stated in the article.

This was on my mind obviously, when I then ran across a tweet post by Go!Math Videos @gomathvideos that shared a TedX talk by Arthur Benjamin entitled “Faster than a Calculator”, which naturally sparked my interest and seemed related to the question of should we be using calculators. In the video, Arthur Benjamin has members of the audience use calculators while he does calculations in his head. He then goes on to wow everyone with his math ‘tricks’ (what he calls mathemagics). He ends by doing a 5-digit square calculation by thinking out loud as he ‘solves’ a problem. It’s fascinating – he changes numbers to words to help him solve – he is definitely using his own ‘algorithm’. The video does not answer the question should we be using calculators – but it definitely shows that calculators are just one way to get a solution and it may not always be the fastest. Anyway – just some fun for this last post of 2016. Enjoy!

Wishing everyone a Happy and Safe New Years!

Using Connections to Build Understanding

I am teaching a Geometry & Spatial Reasoning course for Drexel this semester for their math masters program for teachers. Absolutely love it because I am learning so much from my students/peers, but because it really is bringing home the importance of prior knowledge to help build connections and real-world connections in helping students learn versus memorize, and construct and reconstruct based on their ability to make connections.

My students, who are a mix of very new math teachers, experienced teachers, and even some career-switchers still in the early stages of teaching, are having this great discussions on the importance of using prior knowledge to help student make their own connections. Some have been doing this all along, but others, as they themselves struggle with some of the geometric concepts we are ‘learning’ (relearning in some cases), are coming to understand the value in helping students use what they know to build on and connect to new information. Makes it easier to recall, and builds a confidence in students that when faced with an unknown situation/problem, they have the skills and confidence to look at it, break it down or add in things to make the unknown familiar and then look for and make use of structure (see what I did there….Common Core Math Practice #7!) to help reach a solution or develop a new conjecture/conclusion.

As an example, we’ve been doing a lot of work with inscribed angles in circles and how do you help students use prior knowledge to build the idea that an inscribed angle is half the measure of it’s intercepted arc if you don’t want students just memorizing formulas? Basically, the conversations revolve around constantly using prior knowledge to make connections, which might mean you need to add in an auxiliary line to a given shape to ‘see’ something familiar (i.e. a linear pair or a triangle, as examples). A strategy that really helps students look for and make use of the structures they are familiar with to help them make sense of a problem.  Here’s an example of just one way to explore inscribed angles, using previously knowledge about triangles:

• In Fig 1, we have an inscribed angle and its intercepted arc a. How could you show that angle 1 (the inscribed angle) is half the measure of it’s intercepted arc? Here’s where students need to make sense of this structure – what prior knowledge can they use to help them?
• In Fig 2, they add in a radius (auxiliary line), because they know all radii in this circle (any circle are equal – doesn’t change the original inscribed angle….but now – we have a triangle and a central angle (angle 2).  What do they already know? Well, they know the central angle 2 is the same as the measure of the intercepted arc, which is the same intercepted arc as angle 1 (inscribed angle).
• In Fig 3, students are looking at the triangle created and using prior knowledge – we can mark the two radii equal, making this triangle an isosceles triangle, which they already know from prior knowledge has two base angles that measure the same (angle 1 & 3). Angle 2 is an exterior angle to the triangle, and angles 1 & 3 are remote interior, which they know from prior knowledge sum to the measure of angle 2. Since angle 1 & 2 are equal (isosceles triangle), that makes them each half of angle 2 (Sum divided by 2). Angle 2 is equal in measure to the intercepted arc, so angles 1 & 2 are each half of that, so the inscribed angle 1 is half the intercepted arc.
• Fig 4 shows that the relationship holds true even if you change the size of the inscribed angle.

This is of course just one example for an inscribed angle, but they can then use this to show that inscribed angles that are not going through the center of the circle have the same relationships – ie add in auxiliary lines, use linear pairs, or triangles or other known things to help make sense and show new things. Prior knowledge, connections – they really matter.

As teachers, it is our duty to make sure we are modeling and helping students use what they know to build these connections and see the relationships. It takes deliberateness on our part, it requires modeling, it requires setting expectations for students till it becomes a habit (habits of mind) to look for and make sense by pulling in previous knowledge.

Another thing we need to do is make connections to real-world. My students are sometimes struggling with this idea of relating prior knowledge and new ideas to real-world applications, but if you get in the habit, its not so hard to do. Since I am focused on circles and the lines that intersect them now with my class, I pulled up a ready-to-use lesson from Casio’s lesson library that is a great example of a real-world connection to circle concepts that would force the use of previous knowledge.  The lesson is briefly described below:

• – Use coordinate geometry to represent and examine the properties of geometric shapes.
• – Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

This activity uses the Prizm Graphing calculator and picture capability to help build understanding.

The kinds of questions and connections to prior knowledge that can be asked of students just by looking at the image are pretty endless. What relationships do you see (i.e. lots of diameters, or straight angles, lots of central angles, all the angles are 360, are their auxiliary lines we could add to find the areas or relationships or angle measures, etc.).

If you look around, you can probably find a real-world example of most math concepts your are working on with your students. Show them pictures, show them real objects they can get their hands on. Start asking questions. Ask them what they recognize or think they already know. Ask them if they could add something or take away something to see a familiar object/concept. How does that help them? What relationships and connections help them get to something new or interesting?

My Drexel course and student are reemphasizing for me (and them) the importance of prior knowledge to help build connections on a continuous basis, all the time, every day. It helps students think mathematically and consistently use vocabulary and math concepts to deepen and create new understanding and relationships. It also promotes logical reasoning and problem solving – win-win!

Solving Equations with A Scientific Calculator

Solving  equations is a skill that students are expected to be able to do in pre-algebra and beyond. If we look at the Common Core State Standards, these skills actually come into play starting as early as 6th grade, with students expected to solve one-step equations and progressing to systems of equations by 8th grade. An important aspect of solving equations is connecting a real-world context to these and understanding what the ‘solution (s)’ mean in terms of that context.

The use of calculators or technology to help students solve equations is a controversial one at best, and as a math teacher, I do believe that students need to know the processes to solving equations without the use of technology first. But – when we get down to real-world application and problem-solving, the technology becomes a tool that allows students to go beyond just “getting the solution” and to making meaning out of those solutions, and using their solutions to make decisions – which is the ultimate purpose of finding those solutions, right? In these cases, I firmly believe that the use of technology, (more often than not a calculator), is a necessary tool so that students deepen their understanding and are not bogged down in the process of the calculation. Part of the practices – “use appropriate tools strategically”.

As an example, let’s consider a simple real-world context that involves solving a system of equations, something required by the time students reach 8th grade (see Common Core Standards). Let’s say a scientist is mixing a saline solution and has one solutions that is 10% saline and the other 25%. He needs to make a 85 ml bottle that is 15% saline. How much of each of the two solutions should he mix to create the 85 ml bottle of 15% saline? This requires our two equations, with x = the amount of 10% solution and y= the amount of 25% solution.

• x + y = 90 ml
• .1x + .25y = 12.75 (15% of the 85 mL saline)

Perhaps students are actually in science class doing a lab and creating this new solution. While it would be reasonable to do this by hand using substitution, if this is part of an experiment, then using a calculator to get the answer quickly and therefore get on with the experiment might be a more logical step, especially when time is of the essence in classes. I am going to demonstrate on the fx-991Ex how to solve this problem.  I am using a scientific calculator because in middle school, students are more than likely going to have access to these versus a graphing calculator. This video shows how you can quickly solve the simultaneous equations, and also, with the QR code capabilities, also see a graphical representation of the solution.

If a scientific calculator is all your students have access to, remember that they can do a lot more than you might think.  I will explore more features of the ClassWiz in later posts as we continue to explore mathematics and using technology to support learning.

Creating a Classroom Culture That Encourages Student Discourse

I just spent two days earlier this week working with middle school math teachers. Our focus was on the 6 – 8 Common Core Geometry standards, and how they build on elementary geometric concepts and continue to build that understanding that students need when they get into high school geometry. As part of our work, we also focused on the Math Practices, because it’s the intentional alignment of practices and content to create engaging mathematics activities that really help students develop the deeper understandings. By that I mean you shouldn’t be teaching the content standards in isolation – they should be combined with helping students make sense of the problems, choosing appropriate tools to explore and apply the standards, and really explaining and justifying the conclusions they make.  Practices and content go hand-in-hand.

In our many collaborative discussions these past two days, as we really dived into both practices and content, what was very apparent was how important it is to create a collaborative, safe, classroom. Mathematics classrooms should constantly focus on vocabulary use (by both teacher and students), modeling, discussing your thought process (in many ways – spoken, written, pictorally), explaining and clarifying your thinking, asking questions, and really focusing on all types of communication. Mistakes or misconceptions that students have should be expressed freely, without fear of embarrassment, and students should be free to try multiple pathways to solutions and multiples ways to express their understanding. Students are not going to talk about mathematics if they feel they will be laughed at or considered ‘stupid’ – and that requires a classroom culture that fosters real communication between students that involves listening, ‘arguing’ against someones responses in a constructive, polite way, and a sense that it’s okay to make mistakes because we are all in this together, learning.

What the teachers expressed as their “ah-ha” from our days together was that in order to create these types of classrooms and math learning, you have to start the process right at the beginning of the year, during those first weeks of school, when students are new and class is unknown, and math concepts are relatively familiar since we are starting, in theory, where we left off at the summer. Those first couple weeks of school are the perfect time to create that collaborative classroom culture.

Do a little bit each day – change the groups up, do it whole class, do it with partners. The idea is that you are helping students learn to talk to each other constructively so that when you get into the real learning of new math concepts, they are already comfortable with each other, with some of the learning tools they will be using, and understand that in this math class, we work together and listen to each other and support each other.

Learning is not an isolated activity – we, as teachers, are there to facilitate learning and help students become active, productive, problem-solvers. This happens in classrooms where it is okay to communicate, it is okay to make mistakes, it is okay to have your own approaches to problems but that requires justification of those approaches so others can learn from them.  The more you create this type of learning environment, the more your students will persevere in tackling those tough learning moments.

Statistics – Understanding vs. Deceiving

Someone posted a Facebook link to a blog post w/pictures from Ben Orlin‘s blog, Math with Bad Drawings, entitled “Why Not to Trust Statistics”. If you click the link you will see the post is several drawings with stick figures quoting statistics, and then a mathematical representation of those statistics showing how deceiving statistics can be if you don’t in fact have all the data yourself. The point being, if you are given statistical comparisons without the data, the ‘words’ you hear or the graphs/pictures you see may in fact be incredibly deceiving and not accurate.

Here is one illustration from Orlin’s post to clarify this (see rest of his post for the remaining ones):

The warning here – don’t trust statistics without understanding where the data comes from and what the data actually represents. Not something I think is done generally – just listen to the news or read the papers/magazines.  Heck – look at our politicians and the ‘false’ or deceptive statistics (among other things) constantly quoted or visually shown in their speeches. A lot of statistics (verbal and/or visual representations) come with none of the background data or context, so imagine how much deceit – intentional and unintentional – is occurring. (Always fun to check out the ‘fact checks’ of political speeches to realize the spin put on many statistics).

A major problem here is that many people do not understand the statistics – what’s a mean? what’s a range? what’s a mode? what’s a median? Lack of understanding, lack of data, combined with a deliberate spin on the statistics either verbally and/or visually leads to confusion, misrepresentation, bad decisions, believing something to be true when it’s not, and so much more. It’s scary. And only with education can this “lack of understanding” or maybe it’s better to say “willingness to believe what we see/hear” be combated.

Ben Orlin’s illustrations made me think about how we teach statistics – usually with a group a data points or just a list of numbers with little context, which students then calculate the statistical measures and graph the results. But do we spend enough time comparing these different measures ( I am just thinking about measures of central tendency here) or really work with outliers and how these impact the measures (see example above for the ‘mean’ salary). Do we put enough context to these numbers so that the meaning of these measures truly makes sense? Do we provide context, real data, and real opportunities to look at visuals and verbal representations of statistics and make sense of them, in order to help students make informed decisions? My personal experience is no….though with new standards such as The Common Core, I think this might be changing as there is definitely more emphasis on statistics, real-world context, and interpreting and making sense of data. That’s encouraging.

Thinking about students and teaching, here are some visuals (using the Casio Prizm) and the data from Ben Orlin’s example (i.e. 8 salaries from a ‘company’: 7 at \$30,000 and 1 (CEO, of course!) at \$430,000.  These can really get the conversation going on what is a measure of central tendency, how can the same data reveal different numbers of be perceived differently, and how do outliers impact data and data reporting, how do visuals distort or reveal?

Here are the 1-variable statistics – as you can see the mean (in the illustration above, and the one used to give the “average” company salary of \$80,000) is a distorted statistic, since the CEO’s salary (max value) is so much larger than all the other salaries. A better ‘measure’ to use would have been the median or mode, as those are more realistic to this set of data, where all but one person makes that salary.

Visually, if we look at two different versions of a box plot, one with \$430,000 as an outlier, the other as part of all the data, you see some funky looking box plots (which would be a conversation all to itself…where’s the box? where’s the whiskers?).  But – in outlier mode, you can see the red outlier is significantly different than all the other data (which is all the same).

If we look at the data with a pie chart, bar graph, or histogram, we also see visually how the \$430,000 is an extreme data point.  All of this leading to the question of how an average is sometimes NOT the best statistical measure if you know what the data is and how it is spread.

Statistics is so important and prevalent in so many areas of our society, so let’s make sure we are helping students not only know how to find these statistical measures, but more importantly, help them to question what they see and hear, make sense of the data and understand the potential discrepancies, distortions and misuse of statistics so that they are making informed decisions based on real data and not swayed by a pretty picture or a scary number that is meant to deceive or sway.

Math – Always Something New or Different

If you hadn’t heard, a group of Georgia Tech Mathematicians have proved the Kelmans-Seymour Conjecture, a 40-year old problem. Here is a an article that describes the conjecture and its proof in more detail for those of you interested: Georgia Tech Mathematicians Solve 40-year old Math Mystery” Now, I personally had no idea what this conjecture was till after reading the article – Graph Theory was not something I spent a lot of time on in college or in my teaching career.  What struck me was that this conjecture has been out there for 40 years with people trying to prove it, and it took a collaboration of over 39 years between six mathematicians to prove it:

“One made the conjecture. One tried for years to prove it and failed but passed on his insights. One advanced the mathematical basis for 10 more years. One helped that person solve part of the proof. And two more finally helped him complete the rest of the proof.”

Elapsed time: 39 years.” (Ben Brumfield | May 25, 2016)

Here’s what I love about this – it shows that math is a collaborative endeavor, that takes time and different approaches and insights and that something new can always be discovered or proved. Which is what we should be focusing on in K-12 math education, instead of the idea that there is one answer to a problem.  The standards for mathematical practice (part of the Common Core and based on NCTM Principles to Actions) are all about this collaboration, problem-solving, communication. It’s slow to take hold, and politics is working against it, but look at what can be accomplished when mathematicians, i.e. students, work together to problem-solve?

Math is not a single-solution, one-way only, or  learn-in-isolation. Let’s support the practices, let’s support teachers, let’s support students and create mathematical learning experiences that promote collaboration, real, relevant problem-solving.  It requires teachers being willing to accept multiple approaches and multiple methods of explanation (verbal, written, visual). It requires noise – collaboration is not sitting quietly at your desk.  It requires “mess” – using whatever tools or resources help students think about problems. It requires time.  But think about the new and different math that students will create and explore – and think about how much better prepared they will be for the mess that is the world.  That’s ‘college and career ready’ in my opinion.