Applying Prior Knowledge Is About Precise Mathematical Language

In the course I am currently teaching at Drexel University, we have been focused on the importance of addition and multiplication properties as students progress through mathematics. Particularly the idea of inverse numbers (additive inverses and multiplicative inverses),the additive and multiplicative identities, and the commutative and associative properties of addition and multiplication. A strong foundation in these concepts, which starts in elementary school and builds as students progress to more abstract and complex math concepts such as proportional reasoning, solving equations, composition of functions, and working with matrices, is really important. In fact, if we spent more time using precise language and justifying our reasoning with properties consistently, as we model and help students learn and discover, there would be a lot less confusion and much more connection of prior knowledge to ‘new’ concepts. Instead, we often provide a short-cut, or a ‘trick’ (with a cute acronym like KSP (keep, switch, flip) or ‘Cross-multiply-divide’ with no basis in the true mathematics. Students focus on memorizing isolated rules versus connecting new concepts and seeing learning as just an extension of prior knowledge.

Let me try to explain what I mean by providing a sense of prior knowledge and how it connects to more abstract concepts:

Prior knowledge:

  • Additive identity: 0 and the Additive Identity Property:  a + 0 = a (5th/6th grade)
  • Additive inverses create the additive identity – so -b + b = 0 or -c + c = 0 (5th/6th grade)
  • Multiplicative identity: 1 and the Multiplicative Identity Property: b*1=b or 1*b=b (5th/6th grade)
  • Multiplicative inverses create the multiplicative identity (i.e. a 1) = d * 1/d = 1 or -1/f *-f = 1 (5th/6th grade)
  • Addition and multiplication are commutative (switch the order and you get the same solution) (1st/2nd grade) and associative (switch the grouping and you get the same solution) (3rd grade)(this explains why we want to change subtraction to addition of the additive inverse number, and why we change division to multiplication of the multiplicative inverse (reciprocal) – so we can USE THE PROPERTIES!!!

Understanding the above, then makes solving equations easier – and we don’t need to avoid equations with fractions or decimals, because the properties apply to these rational numbers as well.

Example:  -5 = (1/3) x – 8

  1. Change the problem to addition of the additive inverse: -5 = (1/3)x + (-8)
  2. Add 8 to both sides (commutative property – can add in any order) because adding additive inverses (8 and -8) make zero (additive identity property)  -5 + 8 = (1/3)x + (-8) + 8
  3.  Group the inverses (associative property) and solve:  -5 + 8 = (1/3)x + (-8 + 8) which is equivalent to 3 = (1/3)x + 0  equivalent to 3 = (1/3)x
  4. Use the multiplicative inverse property (multiplying by the reciprocal will create a 1) and multiply by 3/1 on both sides:  (3/1)*3 = (3/1) *(1/3)x (commutative property allows us to multiply in either order on both sides).
  5. The multiplicative identity property says 1 times any number is itself, so we end up with 9/1 = 1 *x or 9 = x

*Note – we did not use subtraction or division at all – we used the understanding of inverses, identities, and addition/multiplication properties to explain. No tricks, and working with actual numbers (so fractions and integers) with justification for all steps.

Example: Solve the proportion  3/16 = x/20

  1. This is really an equation where the quantity x is being multiplied by 1/20. Understanding that I can use the multiplicative inverse to multiply by the reciprocal to make a 1, I multiply both sides by 20/1:
    • 20/1 * 11/12 = x/20 * 20/1 (commutative property lets me multiply in either order on both sides)
    • I can even decompose my multiplication and think about making ones through the same understanding: 4*5*3 /4*4= 1*x
    • 15/4 = x/1 or x = 3.75 (multiplicative identity)
  2. Note – the trick we often tell students to memorize is ‘cross-multiply and divide’, but if instead we focused on just applying their understanding of multiplicative inverse and making those 1 pairs, there would be less confusion, less forgetting the ‘trick’, and less applying that trick to other problems where it is in appropriate. 

Obviously I can’t demonstrate a whole course of study in one blog post – what I am really emphasizing here is how important consistent mathematical vocabulary and use of properties is, instead of acronyms, short-cuts, tricks, mnemonics, etc. that we often give students with no basis in understanding. Instead of seeing math as a connected whole, building on to prior knowledge as they move through the grades and topics, we treat it as isolated topics with no connection. It’s no wonder students think every year they are learning something new. If last year when they worked with division of fractions their teacher taught them to “Keep, Change, Flip”, and this year the teacher is talking about Ketchup Covers Fries or KSF….no wonder they are confused. None of these are grounded in the properties and vocabulary of mathematics.

What we should be doing instead is focus on applying properties and using the mathematical language/vocabulary/properties right from the very beginning and ALL THE TIME. So instead of disconnected acronyms of KSF or KCF,  they focus on extending their understanding of additive inverse, inverse operations with the inverse number and division of fractions ends up being just an extension of what they did with subtraction of integers – i.e. use your inverse operation with the inverse number. So dividing with rational numbers is just multiplication (inverse operation) by the multiplicative inverse (i.e. reciprocal), similar to subtraction being addition (inverse operations) with the additive inverse (opposite signed number) – same general idea, same vocabulary, and just building on prior knowledge.

Let’s stop dumbing down mathematics and use the words and properties that truly allow students to connect and look for those patterns and develop their own understandings and rules. Let’s get away from tricks and mnemonics as our ‘teaching’ method – instead, let students figure that out themselves through the use of precise math language and application of properties. Let’s start in elementary school. Use precise mathematical language (along with clarifying words of course, but always with (not instead of) proper mathematical language/vocabulary/properties).

Think about it – we wouldn’t change the Spanish word for grandmother (abuela) or the French word for bread (pain) to other words, because then how would we communicate and be understood by others speaking those languages? Why is it okay to change the words or use different words or tricks, instead of the using the math language and properties? No wonder students are often so confused or why teachers think they have to ‘reteach’ things every year – if we are not consistent with students in using mathematical language, we are in fact talking a different language to them. No wonder they so often seem lost and frustrated.


Rethinking Summer School – Equity & Promoting Student Learning

Summer school – I know that it conjures up bad thoughts in most of our minds. Having to go to summer school usually means you failed a course or a grade and you have to make it up.  But – do only the ‘failures’ or the ‘bad kids’ need to go to summer school? Is that what summer school is for? This is what most of us think of when we consider summer school, when in reality, summer school should be a place where all students could go to keep on track, get ahead, or learn some new things. Research shows that the 3-month summer break is often a huge learning set-back for many students, particularly minority students and students living in poverty, causing a widening of the achievement gap, in part because these students are often denied opportunities for summer ‘enrichment’ courses or camps. Summer school options are usually focused on remediation and failures, and not very enticing for students to attend voluntarily, and so we have most students taking a 3 month break from any learning. But what if we approached summer school differently? What if it weren’t a punishment, but rather a place where students were motivated by other students or college student mentors and were engaged in new and interesting topics that kept them learning?

I found this really motivating TedTalk by Karim Abouelnaga, who from his own experiences with school, decided to try to change the way we rethink summer school. It’s not too late, even for this year, for those of you educators out there getting ready for this years summer school to consider making some changes that would make summer school a learning opportunity for all students.

Multiple Representations on the Casio Graphing Calculators

One of the key things we try to help students with when studying functions is the idea of multiple representations – i.e. graphical, symbolic (equation) and table.  Ideally, we want students to be able to discern what the function represents or looks at no matter what representation they are given, and to be able to find patterns and important components about that functions from all representations.  Students should never learn about functions just through graphing, or just through symbolic manipulations or just through looking at data points in a table – they should be able to go back and forth and determine which representation is the most useful for the situation.

Unfortunately, too often, the emphasis is on one representation at a time, or at most 2. Let’s look at the graph and find the minimum, maximum, or intersection. Or, let’s find the roots of a quadratic by factoring, or symbolic manipulation. Or, here’s a table of points, where are the x-intercepts or the y-intercepts? Ideally, we want students to be able to look at all of these representations simultaneously so that they see the relationships between the representations and come to understand what the points represent in the table, in the equation, or in the graph.

Technology is one way to show all these representations at the same time, and then quickly manipulate and explore. There are obviously many technology tools out there, but as I have stated in previous posts, the most accessible technology tool for most students and teachers is the graphing calculator, not only because of it’s affordability, but because it is a tool most students have readily available.  It would be nice if all students had computers or tablets for daily classroom use, but that is still NOT the reality.

I have put together a quick video showing Casio’s three graphing calculators – the fx-9750GII, the fx-9860GII, and the CG10/20 or Casio Prizm, and how they can display the equation, graph and table representations of a function on one screen. No matter which model you have, you can achieve the same functionality, allowing students to work with multiple representations and explore relationships quickly and efficiently.

Check it out:

#CCSS Attending to Precision – Mathematical Practice #6

Precision in words and actions is an important skill.  It helps communicate ideas and understanding. Without precise language and processes, miscommunication, misunderstanding, confusion, and chaos rule. Obviously, in the bigger scheme of things, lack of precision can be dangerous. For example, if a civil engineer designing a bridge is not precise in their measures and calculations, bridge collapse and death are possible. One of the things educators need to do is foster this skill of precision in our instructional practice. Which is why helping students “attend to precision”, is one of the 8 Mathematical Practices in the Common Core State Standards. Teachers should be cultivating precision in their classroom.

What does this mean, to “attend to precision”, in the context of a math classroom?  Here is how the practice is defined in the Common Core:stock-photo-58636092-triangle

Math Practice #6: Attend to Precision

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 

If you look at some of the words/phrases I have highlighted above, you will note that precision focuses on communication, using definitions and symbols accurately and appropriately, labeling and identifying quantities carefully, giving explanations based on facts and definitions. In short – understanding the words and symbols being used in mathematics in order to communicate with others. Being precise has to do with helping others understand what you are doing and saying. It is not enough for the student to understand a mathematical concept, they have to be able to help others understand. Why is this so important? Because, the Common Core Standards are designed to help students become “college and career ready”, and in college and in careers, people must communicate with others to accomplish tasks and solve problems. They must be able to precisely explain what they mean, what they have developed, what they want others to do – which requires common language and clear explanations.  In other words, precision.

So, let’s go back to what this means in the context of the classroom. Teachers, no matter what grade – preK through college – should be using the correct language of mathematics and expecting their students to also use this language precisely and appropriately.  They should expect students to explain their thinking using that language, whether verbally or in writing, as students progress through the grades. I know students always groan when they hear “show your work”, but it is important.  And not just their work, but the why of their work. If students are working with measures, then their solutions and explanations should include units of measure. They should use vocabulary and definitions to explain their thinking. The more we have students talking and communicating with math, right from the beginning, the more confident and precise they will become. As teachers, we need to model this as well, by making an effort to use proper mathematical language and symbols, as appropriate for your students, and helping students do the same.

Below is a chart, based on work I did this summer with teachers exploring the Mathematical Practices, that gives some student outcomes aligned to teacher actions that may be helpful as you think about ways to help students “attend to precision” in your classroom.

Students should be able to…… Teachers support this by….
Use correct math vocabulary Teaching vocabulary, (with visuals, if appropriate), and using precise mathetical vocabulary consistently
Know and use definitions appropriately Teaching definitions and modeling using these consistently and intentionally
Communicate/explain their thinking using words and symbols, both written and verbally Encouraging classroom discourse; use think aloud strategies; establishing a culture of inquiry and communication
Record and label their work Providing exemplar for what precision looks like; setting expectations, modeling expectations and providing consistency
Choose and use appropriate mathematical symbols when solving problems or explaining Teaching appropriate symbols and their meanings and using/modeling these consistently




Graphing Piecewise Functions – Casio Prizm vs. TI-84+ CE

2016-02-04_16-50-11In my explorations of hand-held calculators and how they can support mathematics learning, I want to continually share when I learn new things. Why calculators? Well, the obvious answer is because I am working with Casio. But the real answer is, if you actually go around the country and go into math classrooms, calculators are still the most-used and available technology to students.  I know, I know -we hear about iPads, tablets, laptops, etc. in use in classrooms, but the reality is these are NOT readily available to most students.  I think I did a post already about this (Calculators, A thing of the past?), but from my own personal experiences, teaching and working with teachers (some of these in the last couple of months), most math classrooms are still working with the following technology: one computer with projector/screen (sometimes a whiteboard, most often NOT), and then hand-held calculators.  And, unfortunately, not even enough of those for each student.

So – yes, despite the ‘edtech revolution’ we hear about in the news, in the real, every-day classroom, students are most often using calculators, and this will be the case for quite a while unless there is some funding-miracle, which, as we know, is very unlikely.  It’s a sad reality – as an edtech supporter, I would love more than anything all 2016-02-04_16-22-37students to have access to technology on a regular basis that allows them to quickly research, explore, practice and visualize mathematics, whether that be via tablets or computers or calculators. But as most of us who work in/with schools know, that is NOT what’s actually happening in most math classrooms.  That said, let’s focus on the great technology that is accessible to a majority of students – and if not, should be, since it’s affordable, portable and can do much of the visualization and exploration that students should be doing in mathematics – graphing calculators.

Now another reality is that TI seems to be the go-to calculator found more often in schools, a lot of this due to brainwashing and really good marketing and the old “change is hard” mentality in education. I myself was a TI graphing calculator user the whole time I was teaching in public schools because that’s what we had. What I am now finding more and more, as I learn the Casio and compare it to the TI, is that I can remember what to do on the Casio way more so than I can on the TI.  That’s just one thing, though admittedly a pretty major thing.  And – while many of the steps for using the TI and Casio are often similar, the Casio is often quicker and more efficient than the TI, and can usually provide a visualization on one screen that helps make a connection which might otherwise be impossible to see when having to look at separate graphs (i.e. graphing  y= and r= on one graph).

My goal here is to point out places where Casio has an advantage over TI (and I am comparing the Casio Prizm and TI-84 CE, which are the graphing calculators most similar and also both are accepted on standardized tests). Obviously, my opinion is probably considered biased – though I am speaking as someone with over 26 years experience, one who has used many different technologies and only ever taught with the TI (Navigator included). I honestly find the Casio more fun and easier.  More intuitive. I just can’t remember where things are with the TI – it’s frustrating! As they say with many things – once you go Casio, you’ll never go back! But – I don’t think I would feel this way if I wasn’t constantly comparing the two side by side, something most teachers never get the chance to do.  With that said, here is another side-to-side comparison of the Casio Prizm and the TI-84 CE showing how to graph a piecewise function, something I believe Algebra II teachers are probably getting into about now, that helps illustrate my preference for the Casio over the TI.

Prime Numbers & Poetry

I will admit – it’s the holidays and I am in a quandary of what to write about. I’ve decided to cheat a little, and look for inspiration elsewhere – i.e., searching Ted Talks for a math-related topic that I find interesting or inspiring. In my perusing of Ted Talks, which I do every couple weeks since there are so many interesting topics and people to learn from, I found one that I had watched a couple months ago.  It struck me as a great one to share for two reasons. 1) It involves poetry & math, so it’s a lovely cross-curricular exercise if you were to use it with your students; and 2) the math poem, when you listen to his subtle innuendos and wording, teaches quite a lot about prime numbers. It’s really very clever.

I am sharing the whole video, though only the first poem is about math – prime numbers to be exact. I’d suggest listening to it a couple of times in order to really catch the very clever way in which Harry Baker brings in understanding about prime numbers through his love poem called “59”.  If it helps, you can use the transcript of the video to see the words of the poem.

I hope you enjoyed this fun poem. If you have never listened to a Ted Talk, I highly recommend doing so.  I used these with students because there are many out there that are relevant to many subjects and are engaging and thought provoking. They are great conversation starters or reinforcement or introductions to new ideas.

QR Codes, ClassWiz & Expanding Limited Technology

While in Japan (see my first post) the R&D folks at Casio were showing the new EDU+ app for smartphones that reads the QR codes from the ClassWiz calculator. My first reaction was “cool!”, my second reaction was “why?” since, as I thought at the time was why would you need a QR code to get to an online graphical representation of the data from the calculator when you could just use a graphing calculator?

But – a light bulb did go off as I played with both the calculator and the app and thought about schools I’d been to. I realized the whole purpose of the QR code is for those students and teachers who do not have graphing calculators, for whatever reason – i.e. grade level (they are in elementary and early middle school for example), cost prohibitive, or just not an option. The ClassWiz calculator, a scientific calculator, is new to the U.S. market this August, and is very cheap (about $27), easy to use, and can create & display graphical representations via QR codes, so an added feature that teachers and students can utilize. It’s a nice option for showing graphical representations quickly when other tools are not available.

Let me demonstrate how it works using the ClassWiz Emulator and some real-world data I got from the eeps Data Zoo (a fun place to get some interesting data to use with students). I thought the Roller Coaster Data below was interesting. I am going to do a very simple example, so I created a table to compare the largest drop to the length of the coaster.  I then chose the QR code button, which generated a QR code. Since I was in the emulator, I could just click the QR code and go directly to the visual representation on the internet. But, if I’d had my smartphone and the hand-held calculator, I could have used the app to scan the QR code and create the URL for the visual representation.

Look at the short video clip below to see how the process works:

You might be asking yourself why go to all this work if you are going to have to go on the internet anyway? Why not just use an internet graphing calculator? True enough – you could do this.  However, the reality is, most students in classrooms do not have access to computers and internet (only about 1/3 of schools have regular access to mobile devices such as laptops & tablets, for students)(see previous post). Students at the younger grades usually don’t have access to graphing calculators. The majority of the time, classrooms have the teachers computer with a projector set up, relying on whole-class demonstration. We want students hands-on, collecting their data and entering their data, which means students with the calculators.  And then, yes – have them plot their points and sketch their graphs. But – how great, if the teacher and students can quickly generate a QR code right on their calculator and the teacher can pop up a visual of the data right away and have a meaningful class discussion about the relationships students see, what might be the best type of fit for the data, should the graph go through zero and what’s the meaning of that (just to name a few questions)? Students can change the data, or compare different data, generate new QR codes and compare all these different graphs. The QR code functionality of the ClassWiz is just an example of how to expand the capabilities of the technology you have in your classroom. Another resource that allows students to explore and understand mathematics.