Where do the misfits go?

Algebra Day 1

Some points after reading Connecting Math Ideas by Jo Boaler on creating algebra.  Also a great activity to have with students, or yourself.

Tell your students the following everyday

• Addition, subtraction, multiplication, and division is part of math but math is so much more!  While algebra is a focus in many classrooms, don't forget about geometry, probability, or data analysis.
• You can't make understanding an algorithmic process.  Students need to experience confusion, misunderstanding, and failure as part of the understanding process.
• Being good at math is not something that is easily explainable.  Accuracy and speed in procedural mathematics is only one way.  Other ways include being open minded, logical, discussing conjectures, and trying to answer "why?".  Everyone can be good at math and we need people who can do math in those different ways.
• While a problem may only have one correct answer, it may have many different correct solutions.  While some solutions are more efficient, or effective than others, it is more important that everyone understands at least one way of arriving at the correct answer.  Math must make sense.
• We can get better at certain skills of math by practice, however talking and discussing with your fellow students will allow you to understand mathematical ideas deeper.

When developing algebraic thinking with students, we shouldn't force students do what is the most efficient (as sometimes this is objective anyways), but to do what makes sense.  When presented with a set of data, or a pattern, Thorton (2001) argues that it is less important that students be able to find the algebraic rule than they recognize a rule can be represented in equivalent algebraic expressions.  When "finding the rule becomes the focus" most mathematical thinking is lost.

Lets, for example, look at the following grid.  I want you to tell me how many blocks are coloured in this 10x10 grid without counting the blocks individually.  It is crucial you do this without writing, talking to a partner or counting, as it will force your mind to make generalizations which will be the basis of deeper learning to come.  

As a teacher we should first ask students now to share answers, not strategies, with an elbow partner.  This would then force some to reevaluate their strategy and possibly pick up any common errors he/she may have made.

Now how did you do it?  Here are some strategies in Arithmetic form:

4x10-4

10+9+9+8

10+10+8+8

10x10-8x8

4x8+4

As a reader, are you able to explain how each of the expressions above arrive at the same answer? An important task in furthering one's mind into algebraic thinking.

Imagine if this was done in a classroom.  Jason stands up and explains why he simply went 4x10-4.  There might be some smiles, nods, confused looks.  Some students will have seen the same answer provided a different way.  Does this happen on a worksheet?

Of course now, you have realized that the correct answer is 36.  Would then the expression "30+6" be appropriate?  This is where conceptual differs from procedural.  Yes 30+6 equals 36 but unless a child can create meaning to the 30 and to the 6, I would disagree that 30+6 would be an appropriate response in this context.  This illustrates the idea of solution(30+6) vs answer (36). 

What if the grid was 8x8, how would the above expressions change?  Would a number change in every expression or only some?  This idea of visualizing a problem and solving it, is crucial to the advancement of one's knowledge.   

Going back to your strategy and stretching the square to an unknown length, could you create a verbal description of what would you do?

Now the algebra begins..but first...

Noss, Healy, and Hoyles (1997) point out that somewhere we stop seeing algebra as a tool but instead of the end point of a problem. The confusion starts when we see problems as a way to practice algebraic skills instead of using algebra to explore problems.

When bringing in letters into these expressions we must remember some powerful pitfalls:

• The largest misconception is that letters are labels or initials. (pg 25)
• Changing the variable to a different letter changes the solution.  (Let students pick the letter, and encourage different students with different letters)
• The equal sign simply means "gets the answer". (Try writing 5=2+3 as often as 2+3=5 especially in earlier grades)

Taking these points into account, create a variable for the side length of the square.  Could you create an expression using your variable to show the amount of squares which would be shaded?  Knowing what you have done so far, how could you test your expression?

In a class at this point it would be helpful to ask the class "What will be staying the same? What will be changing? As we bridge the gap from numbers to a variable".   For example if the strategy you are using is 10*10-8*8, then the multiplication and subtraction will remain the same while 10 and 8 will change to x and x-2 respectfully.  I would not tell them this, but instead question it.

If you followed all of these tasks then you have now done the problem in 3 different approaches: Arithmetic, Verbal, and Algebraic.

The crucial part of this example, or if this lesson was to be developed into a classroom, is to ask questions and not simply give answers.  Remind them we are not seeking the solution, but a solution that makes sense to them.  Embrace students' wrong answers since learners who are given competing ideas, engage in cognitive conflict and such conflict promotes learning more than the passive reception of ideas that are always correct and seem straightforward (Fredricks, Blumenfeld, and Paris 2004).

PBL on Math 30-2 Research Project

Here is my attempt to create a PBL for a High School Math class.   In WCNP, in Math 30-2 and 20-2 there is an actual outcome around researching a topic and relating it to math.  Below you will find all resources and timelines.  Please feel free to use, change, alter, as you see fit.

PBL Math Research Project

Creating Discourse-Friendly Classrooms

Some great simple things to create a discourse-friendly classroom from
"Literacy Strategies for Improving Mathematics Instruction" (2005)

• Arrange desks so that students can easily turn to see each other.
• Encourage students to direct questions and explanations to the class, rather than the teacher.
• When recording ideas, use the students' words as much as possible.
• Try not to repeat or paraphrase everything students say.  This teaches the other students that they can simply listen to you.  Ask the student to repeat louder if need be.
• Remind students that a conversation has both listening and speaking skills.
• Stand in a variety of spots in the classroom.
• Remember, students listen harder when a peer speaks than when an adult does!
• Give students time to think.
• Arrange lessons so that students have a product to share as they explain their thinking.  

Even better is if the student takes the lead in the class.  This is done by 

• Asking open questions to stimulate thinking.  "Is this logical?" "What do you wonder about?"
• Honor ideas even if they are incorrect.
• Encourage arguments between students.
• Confusion is ok!  Make sure students know that you want them to be confused, and that you will let them be this way.
• Tangents are great teachable moments.
• When a student brings up an idea ask the rest of the class if they have any questions or ideas.
• Counter questions with questions not explanations. 
• Even with a correct answer, ask if there is any another way this can be done, or if there are improvements to be made.

A great reminder!

Math wars confusing curriculum and pedagogy

Recently, in the media there has been a lot of false statements around the WNCP curriculum, and consequently these attacks fall on the hands of teachers.  Before I continue I will explain two important words in the educational world.

Curriculum: This is simply WHAT a teacher needs to teach.  For example, a Grade 3 student needs to be able to recall and understand up to 5 x 5.

Pedagogy: Is HOW the teacher teaches.  For example, using direct instruction, peer coaching, PBL, etc.

Contrary to social media, presentations, and other means of critiquing the curriculum, the curriculum does not say:

• How teachers need to teach the outcomes.

• Discovery learning is a must. In fact the word discovery does not appear once in the entire document.

• "21st Century skills development", and "experiential learning".  Neither of these phrases appear once in the entire document.

• Students should not be memorizing their basic facts.

Keep in mind that the curriculum is the WHAT not the HOW nor even the WITH WHAT.

Recently, an event, around public education, was held in Calgary at a private, gated school. The event was designed to inform parents of "the best practices in math".  While the event was designed to encourage change at the government level, make no mistake this was a blatant attack on math teachers in Alberta, and other teachers in provinces following the same curriculum.

 Here is one of their recommendations:

Now as you can see "Teaching Strategy" does not fall under curriculum but actually pedagogy.  The irony is that I am sure all teachers, at some point, do some direct instruction.  I have had the honor of being in many math classrooms around the province and I can attest that teachers have balance.  This slide almost paints the picture that teachers are simply sitting around hoping that students will learn math through osmosis.   Also, there is no "imposing of one model of instruction".  Enter a classroom and you will see that teachers truly implement strategies which are the most beneficial based on the classroom make-up and the outcome(s) being taught.

Next we have "Some things to watch out for"

Of course some of these points make sense.  We should be wary of many phrases as their intent could be misguided. Also, in reference to the second point, no one is arguing against memorization and procedures.

For their third point, I have not met one person who suggests that students should not memorize their math facts. The difference is, however, students should memorize these facts out of application and use, not out of necessity.

This means, show students math in a context and for a purpose, and the memorization will occur.  Have students roll dice, play cards, board games, car games, etc, as most (if not all) games have some link to reasoning, logical thinking, and mental mathematics.



"Understanding is not more important than skill"- This is again in reference to the actual art of teaching in the classroom.   I have yet to meet one teacher who denies that skill is useful, but let's remember if we only focus on skill then learning can be disguised with simple memorization.

So why is there such an attack on the curriculum?

I believe because some are confusing the terms curriculum and pedagogy.   Also, because we have a generation (parents) who learned math through memorizing algorithms and are confused around why their own children are not coming home with the same algorithms. Recently, some parents are now seeing the benefit of the change.

Also there has been use of the drop in PISA scores, however there has been no actual evidence that this drop has been caused by curricular changes.

The confusion, for a child, might start when a child is learning one way at school, and then coming home to hear that the strategy is not right.  We must also realize that teachers are trained professionals around education.  These professionals implement effective instruction based on the individual needs of the students.  It is unfortunate that some want to see the art of teaching go to a procedural task of "tell students what to do, ask students to imitate the learning, repeat".

If you have a question around the math your child is learning, phone the teacher.  Social media, news, and other hands not in K-12 education, have a way of distorting the truth.  Keep in mind that teachers are trained to teach your child math in a way that is meaningful, and creating a passion towards numbers.

I remember back when I was in school and how there was an immense number of people whom hated math.  It seemed as if math was the number one hated subject in school. (No research simply guessing here).  Isn't it time this changes?  Isn't it time we cultivate passion and number sense?

Math class needed a change, and this change is healthy.  There is now balance.  Before there was a focus to teach it one way and all students were required to learn that one way.  Finally, alternative efficient strategies are not only accepted but encouraged!  We are allowing students to not only learn math, but actually like it!

When is the world going to be full?

A possible project which could be used in a math class.  I have designed this to fit in the Alberta Math 30-2 curriculum.  Feel free to change or alter as you see fit.

Pose the question:

Introduction: When is the world going to be full?
Give students 5 minutes to talk in groups.  After, have a debrief and share possible hypotheses.

Next, ask for possible information we will need to answer this question more accurately.  Some other possible questions might be

• How many people can fit on the plant?
• How many people can the plant sustain? (Different question than the first)
• What is our current growth rate?
• What is our current population?

Some of these questions will be easy to solve, while others might be more difficult.  I would ask students, in groups, to research the answers to the questions they asked.  Some of these will have an answer all will agree with, such as "What is our current population?", while other questions might have a range of answers depending on the website found, such as "How many people can the planet sustain?"  Create answers that everyone in the class can agree with, or even commons answers in different groups as long as each student in each group agrees.

Body:

Creating an extrapolation of our population.

You can google our current growth rate, however this number, most likely, has no meaning or even understanding of how it was derived.  As a class, possible growth rates will be explored.  Below is a chart of recent population numbers on our planet.  (You might want to use different years, or more years).

Year	Population (Billion)
2016	7.4
2015	7.3
2010	6.9
2005	6.5
2000	6.1
1995	5.7
1990	5.3
1985	4.9
1980	4.4
1975	4.0
1970	3.7
1965	3.3
1960	3
1955	2.8

Now the question becomes, "What sort of data is this?"  This is when I would pause and teach exponential, sinusoidal, linear, quadratic, cubic, and logarithmic functions.  Using this data, I would create an equation of each type of function.  Here is a graph of all the functions displayed together...(I used 1970 as year 0.  This occurred as, after deciding this, I needed more data points. I also added a logistic curve to show what would happen if population growth become 0)

Now is the discussion time.....

Ask some general questions such as:

• Any general thoughts?
• Any similarities?
• Could you create a possible scenario in which each graph would be accurate?

You could also embed the critical learning of each function into this:  For example you could ask:

• What is the amplitude, period, and median value for the sine function?  What does that mean in this context?
• What are the zeros of the cubic function, what does this mean in this context?
• What is the growth rate of the exponential function?  What does this mean in this context?

And any other questions which may come to mind.  Now you may want to remove any functions in which the class agrees might be inaccurate.  Such as the cubic, sinusoidal, and/or linear.

Using the remaining graphs determine when the Earth would be full, using the number the class researched earlier.  The answers might be earlier, or later, than the class first hypothesized.  

Conclusion:

Have the discussion "Should population growth be addressed? Why or why not?"

Do not rush this project, take your time.  Stop, throughout, to teach certain skills, or concepts.  Also allow students to research on their own and pose their own questions throughout.   

Math as a language

If math is a language then we need to treat it as such and the best way to learn a second language is immersion.

Focusing on math in grades 1-6, children, when they come home, should be speaking this language.  Parents should ask their children what "words" they are learning.  These "words" could literally be English words such as "prime, composite, factor, multiplication" or it could be what Schwartz and Kenney, 1995, suggest are mathematical nouns such as numbers, measurements, shapes, spaces, functions, patterns, data, and arrangements.

Next, as a language it is important to learn it in a context.  Imagine if I asked you to conjugate verbs and nouns of a language through worksheets and repetition, how much of this practice would stick?  This is similar to children learning "naked numbers"; sitting and simply reciting their times tables.   Suppose there was one child learning math through flash cards and forced to recite their multiplication tables, while a second child plays board and dice games, counting activities while you shop, or even helps you count your change at the grocery store. Which child's experience would "stick" more?  Which child would grow to enjoy numbers and which would grow to hate it?

Keep in mind that the way a child learns a new concept is extremely important in how effective the learning truly is.

One common technique to learning math is through mnemonics or jingles.  However, I want to relate this back to learning a language.  I can sing the "Canadian National anthem" and "Happy Birthday" both in French due to the nature of how I learned it; through the song and rhyme.   Other than these two songs and some simple phrases, I remember nothing else in French.  Do I understand French?

Learning, or essentially memorizing, through jingles without connection to deeper meaning will not allow the child to retain the understanding needed to store this knowledge in their long term memory and ultimately not allow the child to extend this thinking for a new purpose later on.  (I can think of many multiplication songs, which don't really teach the idea of multiplication at all.)

Next, as we read math we must understand that students, not only have to attach meaning to previous knowledge, but also decode the language itself.  Barton and Heidema (2002) say:

In reading mathematics text one must decode and comprehend not only words, but also signs and symbols, which involve different skills.  

See the problem occurs that the symbols in math truly represents the unique alphabet of the language.  Not only does a child have to learn "plus, addition, more" and other words synonymous with addition but also the symbol "+".  Also, some mathematical concepts have multiple symbols associated with them.  For example multiplication could look like "x,X,*, ∙ " or even simply a set of brackets.

For students to truly comprehend math, and find value in this language, we need to show how these symbols translate to English words.  This is where more confusion might occur as math word problems truly combine literacy with numeracy.  I am not suggesting this practice stops, but just pointing out the fact that students (especially ESL) could face another issue of dealing with a mathematical word problem.

The simple fact is how you read a math problem (or really any sort of reasoning, rational, logical, etc problem) is much different than how you would read a fiction text or novel.  The amount of information found in one sentence of a math problem is drastically higher than the amount of information found in a sentence in a novel.  Lastly, most math textbooks (and I would argue most school textbooks) are written above the grade level they are intended to be used in.

So a child/student is struggling with a word problem now what?

I remember back when I started teaching I would give hints such as "total means you should add" or "difference means you should subtract", however this is more procedural work for the child.  Memorize and output the math.  What is more important is for the child to actually hear the thought process out loud.  The first time a child encounters a math word problem, the adult (or teacher) could verbalize the actual thought process which is occurring as he/she reads the problem and truly illustrates how to translate the language of English to the language of Math.  In gradual release of responsibility model, this is called "I DO".

What we do not want from our students is simply knowing what "tricks" to employ when they see certain key words.   After hearing a child read a problem, a question which could be asked is "Are you unfamiliar with any of the words in the problem?"  Keeping in mind the meaning of a word, when it is used in math class, could imply something drastically different than when it is used in a different context.

When encountering an unknown word, simply giving the definition or asking the child to "look it up" usually is not sufficient in securing the understanding needed.

One great strategy is to use a Frayer Model, where the child would write a definition (in their own words), and provide examples and non examples of the words.



Next, ask the child if he/she is clear on what the problem is asking and to possibly read the problem out loud.  The idea of reading aloud, slows the child down and forces to not only see the words but also to hear them.  The worst thing you could do is simply tell your child what to do.  The best teachers have bite marks on their tongue to stop them from speaking.

Lastly, students need to know that certain ideas may have implied constraints.  For example a common question could be "How many ways can you arrange 3 books on a shelf?" with the common correct answer being 6.   If we named the books A,B and C then the arrangements would be

ABC   ACB   BAC   BCA   CAB   CAB

However, in reality these 3 books could be arranged in many more different ways; stacked horizontally, vertically, slanted to the left, to the right, forming shapes such as A, etc...

It may sound like difficult work, and it can be, but this work is valuable in advancing the knowledge of a child in mathematics.  The child must grow and learn to read math in a way they can internalize and ultimately distinguish information.  Simply telling a child the process is not only ineffective but can actually be detrimental to the development of his/her understanding of the concept being presented.