A Better Calculator

A company called Desmos (that I know nothing about), has launched a free online graphing calculator. What’s beautiful about this one, aside from its ease of use and powerful graphing abilities, is that it uses HTML 5. I know nothing about HTML 5, other than it makes things work on iPads.

This is a screenshot from my iPad.

Their calculator is at www.abettercalculator.com . Try it out. Play around with it.

Making sliders for students to play with is simple. Just type in “a=1″ and a slider appears. Now write a function that has “a” in it, and your slider just works.

Here’s one I made with four sliders for a trig function.

Here’s another nice one that Scott Seitz made for a piecewise function.

It doesn’t work on my iPhone (yet?), but the iPad looks great. Manipulating the sliders on the iPad takes a little getting used to, but after that, they work beautifully.

This one is well worth the price.

A Nice Little Story

Much has been discussed on this blog about the pedagogy of the revised curriculum. Some people (mostly in Manitoba) want to debate about whether the pedagogy of discovery, exploration, and constructivism actually works. This week, I had a conversation with a colleague, who shared the story of her two daughters with me. I hope my colleague isn’t a reader of this blog. I didn’t actually ask her if I could share the story. But it is too relevant not to.

Daughter 1 – High School Student

Daughter 1 is a product of the old math curriculum. She receives very high grades (in the 90′s). My colleague, a math educator, fears that these grades are earned by memorization and imitation, and have very little understanding behind them.

Daughter 2 – Junior High School Student

Daughter 2 is in her fourth year of learning with the revised curriculum. Her grades are decent, and she loves math. She has been fortunate to have four years of teachers who embrace the philosophy of the revised program of studies.

The Story

While on holidays over Christmas, my colleague’s husband took to reading alcohol content labels on the beverages he was buying. In their hotel room, Daughter 1 noticed a bottle with 6% alcohol and another with 12% alcohol. She commented that if mixed, the alcohol content would be 18%. Daughter 2 jumped in, and tried to correct her older sister. The older sister wasn’t understanding the explanation, so Daughter 2 used all the strategies we want kids to use. She drew pictures. She experimented (hopefully not by sampling). She convinced her older sister that the alcohol content of the mixture would lie between 6% and 12%. She even extended it the next morning at breakfast when comparing 1% and 2% milk.

Which daughter is better equipped to handle the rigours of University math?

Happy New Year

Resolution: Write shorter blog posts.

Common Ground with WISE

My post yesterday has generated much discussion. I’m trying hard to understand where WISE is coming from, so I sought out more information. One of the people behind WISE is Dr. Robert Craigen.

Dr. Craigen appeared on QR77 with Rob Breakenridge this week to talk about their movement. The full conversation between Breakenridge and Craigen can be found here. It’s about 16 minutes long, so let me summarize some of what I heard.

First of all, I should say that Rob Breakenridge impressed me. He seemed aware of what the revised program of studies is about, and he asked good questions to facilitate the conversation without making apparent any of his own biases. It was a good interview.

Secondly, Dr. Craigen did a good job explaining the WISE position. In fact, he presented his position in a much better way than it is presented on their website. I disagree with much of what is written on their site, but if Dr. Craigen truly believes what he said on the radio, then we have some common ground.

Dr. Craigen states that it is important for young kids to have algorithms to perform calculation because those algorithms will help them in later courses. I agree. I suspect we disagree on how these algorithms should be developed. I suspect Dr. Craigen wants teachers to show the students the algorithm that he thinks is best. I want teachers to help students develop their own (correct) algorithms so that the students understand the algorithms better. Students who can’t develop their own (correct) algorithms need to be steered to one. We are not leaving kids completely on their own. I have never suggested to teachers that we send kids home with no algorithms at all.

Dr. Craigen states that there is a false dichotomy between skills and understanding. He says that both skills and understanding are important. I agree with this statement. The struggle we face, though, is that when we teach in a more traditional way, for most of our students, all we get at are the skills. Dr. Craigen states that some students get the understanding in spite of this method. For years, I taught kids to factor polynomials. They could do it, and I could prove it because they could do it on tests. I’m not as confident they understood factoring. The revised curriculum and its more constructivist approach helps get at this understanding. I believe that kids can demonstrate math skills without understanding. I don’t think the converse is true. I don’t think they can demonstrate understanding without the skills. I agree that both are important, and we need to teach so that both happen.

I have less common ground with Dr. Craigen’s beliefs about automatization. He states that repetition and exercises are important so that students no longer need to think about the math. He wants students to be able to do the basic calculations and practice them so they become automatic. I would argue that too much automatization leads to imitation without understanding, particularly in the higher grades. If students can be made to truly understand their division algorithm, they don’t need to practice it 100 times. Some practice is important. Excessive practice is not worth it. The problem we face with asking kids to practice in K-12 education, that Dr. Craigen may not experience in his setting, is that students who would really benefit from practice don’t do it. Students who don’t need the practice faithfully do every single exercise we assign.

I love Dr. Craigen’s observation that in the old curriculum, there was a great deal of teaching without understanding going on. This comment is certainly true. Even with this new curriculum, it is still true in a lot of places. It’s true in nearly every University mathematics course I took. It isn’t a new problem. I fail to see how going back to more of that kind of teaching, as Dr. Craigen advocates, will alleviate this problem.

Dr. Craigen and I both want our students to come out with all the skills, understanding, and problem solving abilities that will benefit them in future studies and in life. He admits that a discovery and exploratory approach is good, and should be maintained, but this approach should be balanced with some direct instruction. I agree with this statement entirely. I see a typical math lesson following a patter that goes like: Exploratory activity, students share what they learned, teacher fills in gaps, students practice what they learned. I’m starting to wonder if teachers are leaving out the important third step. Are they sending students home without filling in the gaps? Are they sending students home with no understanding or skills? If so, then they have misinterpreted the intent and philosophy of the revised curriculum.

What Dr. Craigen doesn’t address is a passion and love for mathematics. I have it. He has it. The students he teaches mostly have it. The students I teach mostly don’t have it. A discovery and problem solving approach can show these kids that mathematics can be interesting, relevant, and fun.

A stand and deliver approach whereby proofs and algorithms are explicitly presented works well for him, me, and about half of the students he teaches. It works for far fewer of the students I teach. I’m not convinced that from his world of academia, Dr. Craigen understands my world. The bottom line is that in my world, I need to be a better teacher than the one he suggests I should be.

Improving Math Education

Yesterday I came across these guys. The Western Initiative for Strengthening Education in Math (WISE) is a movement organized by some University Mathematics Professors. On their front page, they state:

We began this initiative because we are experts in mathematics and we care deeply about the education of Canadian children. Children who do not receive the strong education in math that they deserve may ultimately be excluded from many careers in trades, technology, science, engineering, business, and economics, to name a few. Our ultimate goal is to ensure that all children have the opportunity to achieve their potential in math so that they may enjoy lives free of innumeracy, may experience the beauty in math, and so that they may have a wide range of career opportunities.

It seems like a noble goal. Digging deeper into their site, I discovered that they believe the following things will improve mathematics instruction in Western Canada.

• Only “math specialists” should be permitted to teach math, at all levels from K-12.
• Math curriculum should be written by “professional mathematicians.” They should decide on both content and pedagogy. Only those who use and teach math at such a high level are qualified to make decisions about math curriculum.
• Standard algorithms must be taught.
• Students must be given lots of practice.
• Calculator use should be minimized.
• “Mathematicians” should review all resources including texts, teacher guides, and all other tools.

Their site asks people to sign a petition to lobby governments to make sure these things are addressed. Many parents have responded indicating support for this initiative, mostly because their kids can’t make change. Regular readers of this blog (both of you) will probably know where I stand on this one. For any new readers who happen across this post, let me tell you how angry this made me.

The sheer pompousness of this group of mathematicians, who seem to think that everybody should learn mathematics the way that worked for them, astounds me. I guarantee that this group of professors has absolutely no idea how to deal with a range of struggling learners. Their classes are full of only the best math students, and those who love math. They deal with students who had over 80% in high school math, and yet I suspect they still can’t (or won’t) differentiate adequately to help students who struggle in University math. Despite their ignorance of what math looks like in the trenches, they assume that their methods would work for all the diverse learners we have in K-12. Rigour is the answer. Algorithms are the answer. Real mathematicians in front of kids is the answer. Give me a break. Not one of them would last more than a week in a K-12 math classroom. And even if they did survive, only the best and brightest students would have learned anything. The rest would have been shuffled out to a shop class.

Despite my anger, I wrote a comment that respectfully disagreed with them, and tried to post it on their blog. They claim to encourage debate on this issue. My comment was rejected, so I’ll share it here. I’d love to hear from some Math Education Professors. They must deal with this kind of thing from their colleagues in the Math department. How do they handle it?

My comment to the folks behind WISE (?):

You guys are attributing an old phenomena to a new curriculum. It’s misguided and actually ironic in a lot of ways.

1. Parents who hated math in school are supporting this movement and effectively asking us to teach their kids in the same manner that didn’t work for them.
2. This site was set up by some University Math professors. Twenty years ago, when I took first year calculus, it was these same people who told us on the first day to look left and right at the people beside us. They told us only 1 in 3 would pass. They expected a 66% failure rate. They wore that failure rate proudly, like a badge. This happened in a time when we had the kind of curriculum this site advocates. That kind of curriculum didn’t produce good thinkers who are successful in University math. It produced good imitators. Now those same professors want us to keep producing imitators rather than thinkers?
3. University math professors were, in fact, involved in the writing of this new WNCP curriculum. The ones I talk to are excited about the prospect of getting University math students who are deeper thinkers. I encourage this site’s group of Math professors to talk to the ones who were involved in the writing of the new curriculum. Or do they think they know better than their own colleagues?
4. Not one student has graduated from the new WNCP math curriculum yet. The first set of graduates come out next year. Any perceived deficiencies in students’ mathematical abilities right now is based on the outgoing curriculum, which was the kind of curriculum this site is advocating. You folks seem to want to go back to the very system you have suggested was broken.

Nursing – University of Alberta

In an earlier post, I discussed post-secondary acceptance of Math 30-2 in Alberta. One of the disappointing things at the time was that the University of Alberta was not going to accept Math 30-2 for Nursing.

I am happy to report that yesterday I received an email indicating that they have decided to accept Math 30-2 for entry into Nursing at the University of Alberta. I am confident that we will continue to see more and more acceptance from post-secondary institutions for this middle stream. I hope it will become a viable option for many of our students.

I will update this post with links from the University of Alberta and the ALIS website, once they have been updated to reflect this information.

Facilitating Assessment Sessions

Last week, two different people both mentioned how tricky it can be to facilitate assessment sessions with teachers. Peter Liljedahl was in Edmonton doing an assessment session, and on the way to the airport, he and I talked about the challenges of doing assessment sessions. That night, I read a Shawn Cornally blog post, where he mentioned the same thing.

The challenge with assessment sessions is that you typically have a diverse group in the room. There are usually people who have been told to attend, and who aren’t open to looking at assessment practices. There are usually people in the room who are so far ahead of me in their assessment journeys that I fear looking like an archaic fool with outdated ideas.

Most of the time I start an assessment session with a bit of a rant about how our high schools have been turned into nothing but sorting machines. We sort kids for the Universities and Colleges. We sort kids for scholarships. We sort kids for streaming into classes. When you believe you work in a sorting machine, you can justify some pretty poor assessment practices. It suddenly becomes important which student understood the material first, and which student turned in the work on time. You want to reward people who meet deadlines, and punish those that don’t. The challenge in an assessment session, particularly at the high school level, is to break down this belief that we need to sort kids.

Peter handled this challenge absolutely brilliantly in his session. He shared his 4 Purposes of Assessment, which are:

1. Not to rank
2. To make students the primary consumers of assessment
3. To evaluate what we think is valuable
4. To report out (“Because three times a year, we have to dock with the mother ship”)

When Peter started, I wasn’t sure why #1 was on the list. It’s not a purpose of assessment. It’s a non-purpose. He spent quite a bit of time explaining why our job in assessment isn’t to rank students. At first, I didn’t see where it fit.

He made sure that the teachers in the room understood that one before moving on to the others. And the brilliance is that by having them buy into that one first, he could answer all the, “yeah but” questions that always come up in assessment sessions.

Examples

Teacher: Don’t you think it’s unfair to the students who are ready to write their test on time if you give other kids extra time?

Peter: No. Because we’re not ranking kids with assessment.

Teacher: If I give a kid a second chance, doesn’t that hurt those that got it the first time?

Peter: No. Because we’re not ranking kids with assessment.

Teacher: If I give all these kids re-dos, my average will be over-inflated. All my kids will get A’s.

Peter: It doesn’t mater because we’re not ranking kids with assessment.

Conclusion

Well played, sir.

Peter Liljedahl on Problem Solving

Last week, I had the pleasure of attending a workshop with Peter Liljedahl from Simon Fraser University. The session was on assessment, but he opened with his take on problem solving. I’m going to discuss the assessment stuff in a later post, but I really wanted to talk about his problem solving process while it was fresh in my mind.

Peter grouped us randomly by having us select cards, and then gave us the following problem.

1001 pennies are lined up in a row. Every second penny is replaced with a nickel. Then every third coin (might be a penny or a nickel) is replaced with a dime. Then every fourth coin is replaced with a quarter. What is the total value of the coins in the row?

He gave each group a whiteboard marker, and a window or small whiteboard to work on. We were to solve the problem on the space we were given. Peter circulated and encouraged us to look at what was written on the other windows if we were stuck. In the end, most of us were satisfied that we were right. Peter never told us the answer or revealed it in any way.

Here’s what interested me about Peter’s take on problem solving, which is very similar to what I have advocated as Learning Through Problem Solving.

1. The problem wasn’t written down any where. He just told us the story, clarified, and put us to work. This isn’t a textbook problem.
2. He grouped us randomly, rather than letting us sit with the people we came with. Throughout the day, he kept on re-grouping us. His feeling is that this frequent changing of the groups leads to more productive classes, and helps include all students. Kids are more willing to work with someone they don’t like when they know that it won’t be for long. This process helps include even the shy and “outcast” students in the classes. These students are included because the people working with them know that the groups will change again very soon, and so they can be patient for that long. Peter also has a way to assess group processes, which I will discuss in the subsequent post on his assessment material.
3. Writing on non-permanent surfaces is critical to Peter’s process. He has observed that when kids work on paper, they take a long time to start. When they work on something like a window or a whiteboard, they start right away, because they know they can erase any mistakes easily. Groups that are stuck can step back and see what the other groups are doing as a way to help them get started. When finished, each group’s work  is displayed prominently around the room so that students can share solutions with each other in a non-threatening way. Students could do a gallery walk and discuss the different solutions, or sit down and write down one that works for them individually.
4. This is the third time I have had the pleasure of working with Peter. He has never once told us the right answer. This drives teachers nuts. I’ve tried it in classrooms, and it drives kids nuts. The message, though, is that they have to figure out the answer. The teacher isn’t going to let them off the hook. An even more important message is that the teacher is less concerned about the answer than she is about the process. These are two pretty nice messages.

Peter’s problem solving process fits really nicely with the Learning Through Problem Solving discussed frequently on this blog. There’s always a “yeah, but” when I share some of this with teachers. They have one whiteboard, no windows, and nothing else to write on in their classrooms. Let me solve that problem for you.

Superstore has $20 whiteboards that are just the right size. Staples has tons of different sizes available for less than $30 each. I’m confident you could get enough whiteboard space in your classroom for 15 groups to work for less than $300. Your principal will find that kind of money in a hurry, especially when you explain that you are going to use it to enhance mathematics instruction by engaging students in group processes that will lead to better problem solving skills. I’ll write you a letter if you need one.

Gummy Bear Extension

The good folks at Vat19.com just keep making the Gummy Bear problem better for me. Thanks to John Burk for noticing this one and throwing it out on Twitter.

My favorite Learning Through Problem Solving activity right now is the Giant Gummy Bear problem. You can read my post on it here. I have used this problem with students and teachers, and it is always a favorite. Teachers who have used it have emailed me to tell me that they received Giant Gummy Bears from their classes as gifts after doing this problem. It really does go over well in class. Students tend to wonder how many small gummy bears make up the 5 lb gummy bear. They wonder about the dimensions of the 5 lb gummy bear. It’s fun, and it leads to good math. I’ve had trouble coming up with extensions for the problem until now.

Then the nice people over at Vat19 made me one. And I didn’t even ask for it. Check this out.

The beauty of this one is that I don’t even have to edit it. It works exactly how it is. It provides just enough information, but still leaves lots of math questions students could explore.

It’s an extension to the original because the math involved is going to be different than the math involved in the 5 lb problem. The  34 fluid ounce tummy throws a nice 3-dimensional wrench into the calculations. Students will have to compensate for this hole in the belly of the 26 pounder.

Questions I see them having include:

• How many regular gummy bears make up the 26 pound one?
• How tall is that 26 pound gummy bear?
• How many small gummy bears would fit in the 26 pounder’s belly?
• Is the cost of the 26 pounder proportional to the cost of the small ones and to the 5 pound one?

Question I have:

• Can somebody with an extra $200 send on of those my way?

The Lottery

I was in a meeting this morning, and we were discussing how to connect literacy across the curricular areas. I flashed back to high school, and a great short story we read. I started wondering whether I could use Shirley Jackson’s “The Lottery” in a math class. Then I began to wonder if a 3773 short story would fit with Dan Meyer’s 3 Act Mathematical Story Telling.  Here’s what I would try with this story.

Act I

Have students read The Lottery, by Shirley Jackson. Ask them what they wonder about. They will probably wonder about lots of things non-mathematical. Eventually they might wonder (Spoiler Alert!) what Tessie Hutchinson’s chances of winning the lottery were.

Act II

Ask the students what information they require to be able to answer the question. If they wonder how many families were in the first draw, you can have them look back through the story and count, or tell them that there were 16. They will also need to know that there are five members in the Hutchinson family in the second draw.

Act III

Students work it out. I still need to come up with a better way to reveal the answer, which is that Tessie had a 1 in 80 chance of winning the lottery.

Sequel

If this lottery has been going on all of Old Man Warner’s life, what is the probability that he survived to age 77?

Edit

Kendall reminded me that I started with connections to English class, and I meant to close with connections to English class. I would totally do this in collaboration with my school’s English teacher.