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.

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New Year’s Observations: Supporting Educational Change & Teachers

I read an article the other day in Edweek about a recent study of teachers regarding the many educational reforms/changes they have seen and been asked to implement in the last couple of years. The article, Majority of Teachers Say Reforms Have Been Too Much” by Leana Loewis, reports on results from a survey done by the Edweek Research Center. I won’t repeat all the findings, as you can read the article and look at the results yourself, but the gist is there have been a crazy amount of education reforms teachers have been asked to make, from standards, to pedagogy, to assessment, to evaluation, and, frankly, it’s exhausting and they are getting tired. And often these changes happen all at once with results expected immediately. A quote from the article that says it all: “Teachers are incredible. They keep up with it because they have to.”  But – at some point, somethings gotta give. In large part, what teachers need is time and support, and this made me think back to something I wrote in my personal blog about change and how educational leaders can support these teachers who are struggling with so many reforms. I’d like to share my 3 suggestions for supporting teachers and change/reform as we begin this New Year.

Observation 1: CHANGE IS EMOTIONAL – change is hard NOT because we don’t want to change (often assumed of teachers who resist change), but because there is often a lot of emotion behind the change. Teachers may want to embrace new curriculum, or learn new roles and new skills, however…they may have LOVED what they used do use or do still want to do that – and it’s emotionally wrenching to have that taken away or altered. In a sense, teachers may be mourning for what is gone and nostalgic about how perfect it was (which it most likely wasn’t). There may be an emotional road block to educational reforms…one that can be overcome, but it will definitely take time, support, and understanding from leaders, students, parents and other teachers, as well as commitment on a teachers part to persevere.  So leaders – remember this about your teachers when it comes to implementing new educational reforms- it may be an emotional reason vs. fear of new or different resources/strategies. Try to address the emotion and provide relevance and reasoning for change and time and support.

Observation 2: RESISTANCE/RELUCTANCE TO CHANGE IS MULTIDIMENSIONAL – It’s easy to tell someone that if they learn a new skill or strategy, that things will be fine or be better. But learning that new skill/strategy or knowledge might not be the true road block – it could be that they don’t understand the relevancy to what they do, or they have preconceived notions or beliefs that cause resistance, or they are missing some necessary background experience/knowledge.What matters here is again, time to learn, but more importantly, dissemination of background, relevance, and connection to what they do and how these new or different skills/resources/strategies will make things better. Without a reason, a purpose, a connection, learning the how-to won’t ever change the internal beliefs and therefore never change behavior in a lasting, effective way.

Observation 3: SOME CHANGES MAY NOT BE FOR EVERYONE – it’s hard to accept, but not everyone can, will, or needs to change, whether that be a skill, strategy, or knowledge base.  What is important is to understand this, try to provide all the time, information, and support to push change along, but in the end, accept that some folks are not going to change and be prepared to deal with it. Whether this means encouraging them to find another place that fits their needs and interests, providing alternatives or simply accepting status quo, forcing those who are not ready, willing or able to change does NOT lead to success.

In education, we tend to introduce education changes, with little training and little time and expect miraculous results quickly. Real change, with long-term benefits is not quick – so let’s take this new year to really look at what we are expecting from our education reforms and assessing whether we have provided that time, addressed those emotional needs, provided reasoning and support. If you want success, you have to work at it.

Equation App (Pt 2 in series) – Solving Equations – Why Use a Calculator?

Solving equations is a large part of the mathematics curriculum as students move into those upper-level concepts. If we look at the Common Core Standards, students start solving one-step equations for one variable in grade 6, adding on to the complexity as they move into higher mathematics where they have multiple variables and simultaneous equations and complex functions. It is important to help students understand what solving equations really represents – i.e. determining the values of unknown quantities and to help them solve them in a variety of ways (i.e. graphically, using a table, using symbolic manipulation, and yes….using technology such as a graphing calculator). And connecting those unknown quantities to real-world contexts is a big part of this as well. Students should solve in multiple ways and express their solutions in multiple ways so that they really understand the inter-connectedness of the multiple representations (graphs, tables, symbolic) and what all these quantities mean in context.

That said, many teachers are reluctant to use the equation solver that is often part of a graphing calculator because, as I have heard multiple times, it does the work for the students and just gives them the answer. True. But – there are ways to utilize the equation solver so that it supports the learning, not just ‘gives the solution’. The obvious way, and probably the most frequent way, is to have students solve the equation (s) by hand, showing all their inverse operations/work, maybe even sketching a graph of the solutions, and then using the graphing calculator to check their solution. Very valid way for students to both do the work, show their steps, and verify their solutions. But – the reverse is also a great way to try to help students learn HOW to solve equations. Working backwards, so to speak.

By this, I mean, use the equation solver to give students the answer first, and then see if they can figure out how to use symbolic manipulation and inverse operations to reach that outcome. As an example, start with a simple linear equation, such as 2x – 5 = 31. Have students plug this into the equation solver and get the solution of 18. Then, in pairs or small groups, have students look at the original problem and try to figure out how they can manipulate the coefficients and constants using inverse operations to get to that solution of 18. So maybe, plug the 18 in for the x.  What would they have to do to the other numbers in order to isolate that 18?  This forces students to use inverse operations to try to ‘undo’ the problem and end up with 18. In doing so, they are discovering the idea that to isolate a variable, you have to undo all the things that happened to it.  Give them a harder problem. Same process….and let them get to a point where they try to solve using their ‘understanding’ of inverse, and then they use the calculator to ‘check’.  The idea here is students are figuring it out by starting with the solution and working backwards to understand the process for solving equations. And they develop the process themselves versus memorizing it.

Rather than thinking of the calculator as a solution tool, think of it as another way to help students discover where those solutions come from.

Here’s a quick video on using the Equation App (solver) on the CG50. The process is the same on Casio’s other graphing calculators. This is another installment in the app exploration series, started last week with the Physium App.

Origami – The Math Behind the Paper Folding

I am about to start teaching an online geometry course, and it has me missing some of the things I use to do with my students to help them discover relationships, and work with angles and symmetry, which was origami. Origami is the art of paper-folding – and using it in geometry is a great hands-on and visual tool to help students discover angle relationships, symmetry, linear relationships.

Origami is something I am sure most of you are familiar with and maybe have even attempted to create some origami art yourself. I have two friends who are origami wizards and often post their creations on FB – and it’s pretty amazing the shapes they create. When I recently went to the Museum of Math in NYC there was a whole exhibit devoted to Origami.

In my class, obviously, we did relatively simple constructs – folding one piece of paper into things like cubes, birds, shapes. The focus being on the folding and shapes created from each fold and looking at the angles and relationships that developed after each fold. But – as I have discovered, there is some really complex math behind origami, and really complex shapes that are created all from one sheet of paper that are simply astounding. I just found this Ted Talk from 2008 by Robert Langdon that discusses the mathematics behind Origami and how because of mathematics, folds that before were impossible are now possible, allowing for origami constructions that are astounding. Those of you who teach geometry, I think this will be very interesting to you, though I think other math subjects as well will find some applications. At the end of the video there is also a link to some templates for folding some more intricate origami constructs.

 

CG50 – What Are All Those Apps?

As many of you know, I post quick videos in the blog to show different things about the Casio calculators or math or teaching. Many of these are posted on my YouTube Channel. I will occasionally get comments from viewers asking questions, and I do my best to answer them. If I can’t answer the question, I find someone who can, or research until I do have a response. Just the other day, when I was asked “how do you use the constants on the CG-50 calculator”, I was not quite sure what was being asked, since I tend to use the calculator from a mathematics teaching perspective, and hadn’t explored using constants (from a science perspective) and wasn’t even sure what was meant by the ‘constants’ in this particular question (as it could mean the constants in a given equation).  Turns out the viewer was asking about the Physium Menu/App on the calculator, and how to get the constants from these tables and values into calculations. This is something I have honestly never used because I am not a science teacher and therefore rarely, if ever, have need for this app. But – it got me curious and seeking out an answer (which I did find and explore so I could give a reasonable answer).

In my ignorance, I realized that there are many apps on the CG50 (and other Casio graphing calculators) that I have never really explored, not just the Physium App. Mostly I focus on the most-used menu items – Run Matrix (to do calculations), Graph (to work with functions and graphs), Table (functions using table representations), Equation (solving equations), and Picture Plot. But there are a lot of other menu items that I need to explore and learn to utilize since they all are useful for different contexts and applications. This is now a goal of mine – to try to learn and explore the basics of the other menu items (apps) of the CG50 (and other) graphing calculator, starting with the Physium Menu/app. Here’s what I have discovered:

The Physium application has the following capabilities (so science teachers, take note!!)

Periodic Table of Elements

  • You can display the periodic table of elements
  • The table shows the elements atomic number, atomic symbol, atomic weight and other info
  • Elements can be searched for by element name, atomic symbol, atomic number or atomic weight

Fundamental Physical Constants

  • You can display fundamental physical constants, grouped by category to make it easier
  • You can edit the physical constants and save them as required
  • You can store physical constants in the Alpha memory and use these saved constants in calculations in the RUN-MAT menu/application

Now, I am still not a science teacher, so this would not be a menu item I will use often, but I wanted to do a quick video of what I discovered in my own exploration.  And – there is a link to the how-to guide for the Physium Menu/App for those of you interested in exploring more. If you have a CG10 or other graphing calculator from Casio and don’t have the Physium menu/app, you can download it here.

 

Elevators and Number Sense

Number sense should develop early, and what simpler way to do it then to start with elevators?

Elevator, Vicenza, Italy

Why elevators you ask? Well, I just returned from 2 weeks in Italy. Partly for work: training elementary math teachers in Vicenza, Italy on College & Career Ready Standards for UT Dana Center International Fellows and Department of Defense Education Activities; and partly for leisure: touring Venice, Cinque Terre, Florence, Tuscany and Rome with my husband, sister, and brother-in-law. The first thing I noticed was the elevators have negative numbers to indicate those floors below ground zero (i.e. what we usually call floor 1 or Lobby in the U.S.)   It’s not the first time I’ve noticed this – in England, in Paris, in Germany – all these other countries indicate on their elevators the ground floor to be 0, the floors above ground 0 are 1, 2, 3…. and the floors below ground zero are -1, -2, -3….

This way of numbering elevators makes sense. Much more sense than Floor 1, or Lobby and then Basement, Basement2 (or LL1, LL2) – which is our typical way of indicating the ground floor (1) and the floors below ground level (Basements/Lower Levels). If you were a young child living in these countries and taking the lifts (or elevators), you are regularly exposed to integer numbers – with a contextual connection that the ground floor of a building is ground 0, and the floors below the ground are negative numbers, and the floors above the ground are positive numbers. It may not even be explicitly explained to young children, though they would be using the terms ‘negative 1’ or ‘negative 2’ to go down below the ground floor. They will have this repeated exposure so when they are ‘officially’ taught about negative numbers in school, they have an immediate connection to prior knowledge about the numbers in an lift/elevator and can make a real-world connection. Negative numbers won’t be new or hard to understand because it’s just the numbers in the elevator. Or – the numbers of the temperature, because let’s not forget, these countries also use the Celsius temperature scale, where freezing is 0, and anything above 0 degrees is above freezing and getting warmer (positive) and anything below 0 degrees is getting colder (negative). The further from 0 in either direction, the warmer or colder you are – again, real-world connection and a contextual understanding of integers.

Number sense. Number lines. Integers. Real-world connections. Just from elevators and temperature scales.

This repeated exposure, informal as it may be, is developing an intuitive understanding of numbers and their real-world meaning. And when students are then exposed to number lines and positive and negative numbers more formally, in a school setting, they already get what that means because it is familiar to them. They can apply what they already know to ‘mathematics’. The formalization makes sense, and connections make sense, and understanding is that much deeper.  This is different in the U.S., where students often struggle with the idea of ‘negative’ numbers and number lines and the distance from zero because we are teaching them something new.  We don’t have a real-world exposure to negative numbers because we use LL or B1 to represent lower than 0, our ground floor is never called 0, it’s 1 or Lobby or G (ground). Our temperature doesn’t have 0 as the freezing mark – it has 32 degrees Farenheit. Think how much easier it would be to connect negative numbers (those numbers smaller than zero) to negative floors or negative temperatures. Freezing makes sense at 0. Negative temperatures are colder than freezing. Positive temperatures are warmer than freezing. 32 degrees – not quite the same one-to-one connection to a number line, is it?

Anyway – my point is that something as simple as changing the numbers on an elevator to integer representations would go a long way in helping young children develop number sense early on so that by the time they get to school, they already have a natural understanding of positive and negative numbers. Early on they would be exposed to the idea of 0 being the ground level, positive numbers mean higher floors or farther away from ground zero, and negative numbers mean lower floors, below the ground, and the further you go below ground, the more negative you get, the farther away from zero you are. Number lines would then be ‘recognizable’ because there’s a contextual connection. (If we could change our temperature scale to Celsius that would be great too, though that one is a lot harder to do).

Relabel elevator buttons to reflect numbers on a number line – a simple change that could go a long way in developing informal number sense in children.

 

 

Equity, Equality, and Access to Quality Education – Part 1

Back-to-school is already upon some, and for many will be starting up in the next few weeks. With that in mind, and especially with the very public conversation around school choice and ESSA and accountability for schools, I’ve decided to do a 3-part series on equity, equality, access and quality education. These are ‘buzz’ words that are thrown about in news stories and education settings, but I think often times these words or terms are used incorrectly, or interchangeably, with many people not really understanding what is really being said or what the meaning behind these terms actually might be. With that said, this first part in my series is going to focus on defining these three terms so that we are all on the same page and have a common understanding in which to move forward.

Quality Education

This term is loaded. Everyone wants a quality education for their child and schools and states strive to provide quality education for all their students. But what does this mean? What does this look like? I am going to define it here and in later follow-up posts we will dive more deeply into this.

There are many definitions out there for what quality education means. I actually had a hard time finding an ‘official’ definition, but found the term ‘quality education’ used frequently in vision/mission statements from many education organizations and school districts. Which is interesting – we use the term, yet we don’t define it, so how are we ensuring that students are indeed getting a quality education?

Here is a definition of Quality Education from ASCD (Association of Supervisors of Curriculum Development) and EI (Education International) which I think provides a strong common understanding that will connect to equity, equality and access.

A quality education is one that focuses on the whole child—the social, emotional, mental, physical, and cognitive development of each student regardless of gender, race, ethnicity, socioeconomic status, or geographic location. It prepares the child for life, not just for testing.

A quality education provides resources and directs policy to ensure that each child enters school healthy and learns about and practices a healthy lifestyle; learns in an environment that is physically and emotionally safe for students and adults; is actively engaged in learning and is connected to the school and broader community; has access to personalized learning and is supported by qualified, caring adults; and is challenged academically and prepared for success in college or further study and for employment and participation in a global environment.

A quality education provides the outcomes needed for individuals, communities, and societies to prosper. It allows schools to align and integrate fully with their communities and access a range of services across sectors designed to support the educational development of their students.

A quality education is supported by three key pillars: ensuring access to quality teachers; providing use of quality learning tools and professional development; and the establishment of safe and supportive quality learning environments. (retrieved from http://www.huffingtonpost.com/sean-slade/what-do-we-mean-by-a-qual_b_9284130.html)

Equity and Equality

The definition of equity in the dictionary is “the state or quality of being just or fair”. The definition of equality is “the state of being equal, especially in status, rights and opportunities”. So what does this mean in terms of education, especially as these two terms are often used interchangeably, when they are very different when it comes to education? Let’s look at each separately in terms of education.

Equality in education would mean that all students are treated the same and are exposed to the same opportunities and experiences and resources. This is deemed as fair because everyone is getting the same instruction, the same assessments, the same resources, the same access to teachers. However, if students are coming into a classroom with different capabilities and different backgrounds – which is the reality no matter where you are – (this means educational knowledge, socio-economic status, family support, etc.), then treating them equally is going to disadvantage most students. No one will get what they truly need to learn – most will not get the appropriate supports and opportunities they need to be successful and to learn to their full potential (as examples, those with special needs would not get the additional supports needed and ‘gifted’ students would not be exposed to more challenging learning experiences they might need).  Everyone gets the same and so everyone suffers to some extent.

Equity in education means that all students get what they need from education, meaning instruction, assessments, resources are distributed so that every students individual needs are met in a fair way so all students can be successful. This relates to the statement above, under quality education, that students have access to personalized learning so that their educational needs are supported, allowing them to be prepared for future success, whether that be a career, college or some other aspiration. So unlike equality in education, equity in education is not the same for everyone, rather it supports everyone with what they need. A students socio-economic status, gender, race, or ability level do not prevent their access to education resources and opportunities. Equity does NOT mean equal. Equity implies an education for each child that meets their specific needs,  both pedagogically and developmentally, so they can be successful in their future endeavors no matter where they live or what their economic status might be.

Access

Access to education is closely tied to equity and equality. I almost didn’t separate it out, but I do think it is a key component behind why many students do NOT get equitable education opportunities. The goal of providing quality education to all students means we are providing them with equitable access to resources and learning opportunities – i.e. students with learning disabilities are getting the extra services and supports they need to be able to learn; students from low-income areas are getting the technology and materials and qualified teachers needed to address their instructional needs; students who excel at math or science are provided with technology and resources that allow them to explore and expand their understandings; students who are artistically or musically inclined are provided with teachers and courses that let them learn and create.

It was hard to find a ‘definition’ for access, because it’s really a process of ensuring students get what they need. I found this nice summation of access on the Glossary of Education Reform that I am going to use to inform our discussion going forward:

 “The term access typically refers to the ways in which educational institutions and policies ensure—or at least strive to ensure—that students have equal and equitable opportunities to take full advantage of their education. Increasing access generally requires schools to provide additional services or remove any actual or potential barriers that might prevent some students from equitable participation in certain courses or academic programs”.

As you can see, all these terms and ideas are related, and it is often hard to think of them in isolation. Hopefully now you have a better understanding of each, and in our follow-up posts, we will explore issues surrounding these using our common understanding.