So it's been almost two years, but I'm
trying my hand at blogging again. I have lots of thoughts and
opinions about math education, and need to make the effort to get
them out there.
As a child, I hated math. I thought
that it was something that I simply wasn't good at, and wouldn't ever
really understand. I didn't really start to love math until I took
classes for my undergraduate degree at Southwestern University. I
learned constructivist teaching methods including Cognitively Guided
Instruction (CGI) and strategies to increase real numerical
understanding in Van de Walle's Student Centered Mathematics.
Through these constructivist approaches, I began to see math not as a
set of isolated procedures, as I had been taught in school, but as a
logical way to navigate a landscape of related figures.
Mathematical understanding is powerful,
and I find it incredibly rewarding to equip my students with this
power. I love being able to make a real difference in my students'
lives, but I've often been frustrated with the constraints of the
system I teach in, and not being able to have a bigger impact. I
believe that increasing students' abilities at math, including
critical thinking and problem solving, is good for them as
individuals, and good for our world. Being adept mathematicians and
problem solvers opens doors for students in careers in engineering,
technology, and science. And more engineers, mathematicians,
scientists, and citizens who are critical thinkers is what we need to
help solve the big problems our civilization is facing.
This blog started out as platform where
I could pitch my ideas about how to impact math education with
emerging technologies. I still want to do that, to some extent, but
I also want to address the other aspects of teaching that positively
impact students' mathematical understanding.
The most effective teaching happens
when I am working one-on-one or with a small group of students. I am
able to engage with each child, discover their understandings and
misunderstandings, and scaffold through purposeful questioning and
modeling. Teaching this way is a craft. It is not easy and often
does not come naturally. As teachers, we want our students to be
successful. It is easy just to show them the steps and teach short
cuts, but this does not lead to lasting understanding. We have to
let them struggle, find out what they know, and help them build
understanding from there. This can be a long, arduous and,
occasionally, frustrating process, but it builds deep and lasting
knowledge. I want to explore questions and strategies around this
type of teaching, especially problem solving, on this blog.
When I am working with a small group of
students, one of the biggest issues is how to best use the rest of
the students' time and attention. They usually work in small groups
on center activities, or on individual worksheets. This creates a
few problems. I cannot give them immediate feedback or help on what
they are doing. Of course, I have modeled the activities, and will
grade their work later, but that doesn't help them in the moment. I
don't understand why we are still using text books and worksheets in
classrooms. Why aren't our students working primarily on computers?
They could be participating in motivating, game-like programs that
individualize for the students' needs, give appropriate scaffolds
(such as models or number lines) when needed, and record data in real
time for the parents, teachers and administrators. Such programs
would not replace effective teachers, but allow them to better use
their time. Instead of spending hours grading worksheets after
school, with little useful data to gain from the effort, teachers
would automatically get data on student understanding. With this
information and with the saved time from individually grading,
teachers would be able to focus their time on better planning for the
next day's lessons. The ed tech industry is incredibly active and
vibrant, with new products and solutions constantly being presented.
I would like to spend time reviewing some of these solutions, and
proposing solutions of my own.
In my last post, I asserted that it is important for students to develop solid mental math skills in order to be proficient mathematicians. Mental math, sometimes called number sense, is an internal representation of numbers and number relationships. The basis for developing these skills is flexible thinking about numbers, where students can easily think of one number in a variety of ways. I demonstrated how this type of flexible thinking about numbers can help students solve problems using mental math strategies that are more efficient than using traditional algorithms. Van de Walle (Teaching Student Centered Mathematics) recognized four different types of number relationships that are important for number sense development. (The following is quoted directly from Teaching Student Centered Mathematics)
Spatial relationships: Children can learn to recognize sets of objects in patterned arrangements and tell how many without counting. For most numbers, there are several common patterns. Patterns can also be made up of two or more easier patterns for smaller numbers.
One and two more, one and two less: The two-more-than and two-less than relationships involve more than just the ability to count on two or count back two. Children should know that 7, for example, is 1 more than 6 and also 2 less than 9.
Anchors or “benchmarks” of 5 and 10: Since 10 plays such a large role in our numeration system and because two fives make up 10, it is very useful to develop relationships for the numbers 1 to 10 to the important anchors of 5 and 10.
Part-part-whole relationships: To conceptualize a number as being made up of two or more parts is the most important relationship that can be developed about numbers. For example, 7 can be thought of as a set of 3 and a set of 4 or a set of 2 and a set of 5.
In this post I am going to propose activities that would be much more effective at developing flexible thinking about numbers, than the rote addition and subtraction drills that are currently available on the Khan Academy site. One of the activities that Van de Walle recommended was dot configuration flash cards, to help students quickly recognize sets of objects arranged in familiar patterns, without counting. This not only helps develop spatial relationships, but also with part-part-whole relationships. The following are mock-ups I created for ipad apps using these dot configurations. (The design of the following apps could be improved greatly and made to me much more game like and motivating, but the purpose of the mock-ups were to show the educational objective - not present the perfect design.)
In the above example, a student would be given a dot configuration and would try to identify the amount as quickly as possible. They would practice this until they knew the dot configurations without counting.
This example shows a dot configuration in a ten-frame. Van de Walle used ten frames to develop the relationships between single digit numbers and ten. A variety of different number relationships are obvious when looking at this ten frame. It shows 6 dots, which is made up of 5 and 1 more. It also shows that 6 is 4 away from ten. It is very important for students to be able to quickly identify how far away a number is from the next benchmark number when doing mental math.
The above is another example of an app using dot configurations to build number sense. The students would simply touch the number and move it over the the corresponding dot card.
Once the student has learned the dot configurations by heart, than we can extend the part-part-whole relationships by showing only part of dot configuration and asking the student to tell what is missing. In the above example 4 dots are shown, and just by looking it is obvious that 1 is missing from the traditional 5 dot configuration. This helps the student cement the fact that one way to make 5 is 4 and 1.
I have found an ipad app that does use dot configurations in a creative way to help students learn numbers (though not with ten frames). The following screen shots are of the ipad app MathGirl Number Garden, which I bought for my 5 year old daughter. We both love it, and I recommend both it and the Math Girl Additon House, to anyone with young children (though it is obviously geared towards girls).
The above dot configuration of 9 makes it easy to see that 9 is made up of 3 threes, or even 6 and 3. On this app, the student simply touches the correct number that matches the number of flowers.
On the above dot configuration it is easy to see that 6 is made up of 2 threes.
On this dot configuration it shows 6 made up of 5 and 1. Showing one number in a variety of ways increases the flexible thinking about numbers that is necessary for good mental math skills.
Again, here is another way to display six. Displaying the same number in a variety of ways, helps to build flexible thinking about that number. In this example, using the hands helps build relationships with the benchmarks 5 and 10. It is easy to see that 6 is 1 more than 5, and easy to imagine that 4 more are needed to get to 10.
In this post I hope I have successfully demonstrated the value of dot configuration activities, and how they can build flexible thinking about numbers. I would love to see something like the above activities included in the Khan Academy's math scheme. I think they would be much more effective at building the number sense required by today's mathematicians, than the rote addition and subtraction drills that are currently there.
In my next post, I plan to share some ideas of number line activities that would build the four different types of number relationships that I identified at the beginning of this post.
The people at Gen Yes blog did a very in depth analysis of Khan Academy from a constructivist point of view, which is a good read if you are interested in the debate about the value of the Khan Academy. This is where I found a link to Conrad Wolfram’s TED talk on computing and math education. Wolfram stated that there are really four parts of math: posing the question, translating a real world problem into a math problem, computation, and then turning the answer back into a real world answer and verifying. He said that we spend most of our time in math education teaching the third step, computation, which is the least important step to learn today. Throughout history doing math meant doing loads of computation. Not today. He asserted that people doing math in real life are not working out computations themselves –they have computers to do that. The important thing is to know how to problem solve, which the other 3 steps relate to – and which we spend the least amount of time on in education. I am going to spend much more time going into the problem solving aspects of math education in a later post, where I will talk about my experience with Cognitively Guided Instruction.
Today, however, I want to continue to look at this third step, computation, in relation to Wolfram and the Khan Academy. The Khan Academy provides instruction solely in computation. Both their video tutorials and their practice activities are geared towards learning computation. They teach primarily traditional methods of computation, which were very efficient strategies to use in the past, but are less useful now.
Wolfram asserted that there is one time when it makes sense for people to do calculations themselves and that is when they are doing mental math. What does it mean to do mental math? When people are doing mental math are they doing the same thing they would do on paper - traditional algorithms, like carrying and borrowing, only doing it in their heads? The answer generally is no. People are not simply doing traditional algorithms in their head, when they are doing mental math. Mental math is also not simply proficient fact recall, though people with good mental math skills usually do know their math facts. Mental math is an internal representation of core number relationships. This internal representation makes it easy for someone to solve math problems in their head, and also to estimate. It is great that we have calculators and computers to do the hard work of computation for us now, but we can’t simply type in a problem, and trust whatever comes out. People need to have enough knowledge of numbers and number relationships to know if an answer seems right or way off, so that they can easily identify mistakes, before these mistakes cause big problems.
John Van de Walle (author of Teaching Student-Centered Mathematics, which I highly recommend to anyone involved in math education) asserted that the basis for mental math is flexible thinking about numbers, where students can easily think of one number in a variety of ways. For example, with the number 77 a student might think of it as 70 and 7, or 3 away from 80, or 75 and two more, or one less than 78, or three quarters and 2 pennies, or 23 away from 100. When a student is able to quickly think of a number in a variety of ways than it makes it easier to solve a variety of different problem types quickly and easily – quicker and easier even than solving it with the traditional algorithm. Here is a screen shot from the Khan Academy with a subtraction problem involving the number 77.
I solved it the traditional way, the way that Khan showed in his video tutorials, and the way you were probably taught in school - borrowing. First I borrowed one from the ten's place, since there weren't enough ones to take away 7. Then I computed 14 - 7 = 7, and moved over to the ten's place. There were 0 tens and I needed to take away 7, so I had to borrow from the hundred's place. Then I found that 10 - 7 = 3 and moved to the hundred's place. There were no hundreds to take away, so I brought down the 5 and finally had the answer of 537. A student who had learned to flexibly think about these numbers might solve it in an assortment of different ways, many of which would be more efficient than the traditional algorithm, and could probably be done in the head.
For example, a student might think: I know that 77 is 23 away from 100, and 614 is 514 more than 100, so the difference between 77 and 614 is 23 + 514, which is 537. This thought process is illustrated on the above number line.
Or a student may think: If I take 80 away from 614 than that is 534. 80 is three more than 77 I took away 3 extra, so 614 minus 77 must be three more than 534, so the answer is 537. This thought process is illustrated on the above number line.
Or a student may think: I want to go back 77 from 614. If I just go 14 back I get to 600, and then I still need to go 63 more back. I know that if I took 63 off of a whole hundred than there would be 37 left, so the answer must be 537. This thought process is illustrated on the above number line.
All three of these mental math strategies would be more efficient than the traditional algorithm taught on the Khan Academy. I believe that there are types of math practice activities that could be done on the computer in a format similar to that of the Khan Academy which would be more effective at developing these skills in students than the drills of traditional algorithms that are currently on the site. A student would still get the benefit of immediate feedback, and teachers would still get useful data on student progress. The activities would not simply be the rote drill of traditional algorithms, but activities that would build knowledge of number relationships that would increase flexible mathematical thinking. I hope to follow this post shortly with mock-ups of ideas for such practice activities.
I'm a middle school math teacher. When I saw Salman Khan's TED talk in March about the Khan Academy, I was ecstatic. Finally, technology that would automatically individualize instruction for each student's educational needs, would scaffold their learning with hints and tutorials, and would give me tons of data so that I could be a more effective teacher! I immediately logged in and got started, emailed my principal with the details on the program, and started spreading the news about it to other teachers. Within a few weeks I had my Pre-AP class logged on with accounts and working on it, and the next week my Grade Level class was working on it as well.
We've been working on it for the better part of a month now, and spend at least half of our class time each week on it. The kids' responses have been generally positive. The most common response is that they like going at their own pace and not feeling rushed or slowed down by the rest of the class. I have loved getting the detailed data in real time on my students' progress. It has made it so easy to pinpoint where I am needed, and give appropriate attention. Also, the kids are engaged. I don't have to spend lots of energy making sure that they are paying attention and following along - which is great!
I want to make clear that I am a fan of the Khan Academy. I appreciate what they are doing and the service they are providing. I think it is great that Salman Khan and his associates are trying to tackle some of the real issues that teachers deal with every day in the classroom (student engagement, individualized instruction, targeted teacher support, etc). However, if they are trying to provide a framework for an entire math curriculum - early elementary through advanced calculus - there are a lot of areas for improvement. The practice activities are just rote drills. They aren't taking advantage of the power of their medium - the computer. The activities could be much more interactive, and constructivist. Much of my feedback is going to be focused on these areas of possible improvement, but again this is feedback from a critical friend. I think that they have made a great start, and are providing a great service by giving free access to their curriculum to anyone in the world.
Things I think that Khan has right:
We should use technology to humanize the classroom. Free up time for teachers to interact with students over meaningful content. The most powerful teaching happens when I am interacting one-on-one with a student, or with a small group of students. It might not seem like students working individually on computers would increase the number of meaningful interactions that they had with their teacher, but I have found that it has. I spend less of my class time in whole group instruction and less time disciplining. I don’t have to spend a lot of time and energy making sure the rest of my class is engaged, so I can really focus and help the students who need it.
There should be mastery before moving on. This is especially true in math. I am teaching 6th grade math right now, and most of my grade level class does not know their addition/subtraction fact families (they still count on their fingers), or their multiplication facts. For most of what we cover in 6th grade math – fractions, ratios & proportions, decimals & percents - students need to be efficient with multiplication and division facts. It makes it very difficult to teach new concepts when they are not proficient in the math that they should have learned last year or the year before that or the year before that. I agree with Salman Khan - that students often end up having "swiss cheese gaps" that end up causing them to hit a wall with the level of math they can do. There is a lot of talk about having high expectations for students - but he actually puts a system in place to uphold these expectations. Currently in the education system we can tell students that we expect them to learn something, but when they don't there aren't very many consequences. With the Khan Academy the student actually has to live up to the high expectations in order to advance.
Use technology to differentiate for different learners – in a way that is manageable and realistic for the teacher. I know that the best thing for my students is to be given individualized lessons, based on their individual needs. When you have a class of 25-30 students, this is very difficult to accomplish. It means planning 2-3 different lessons for each class, preparing different sets of materials, and hitting all the different objectives and levels during each lesson. A program like Khan Academy lets students work at their own level and pace, without putting a huge burden on the teacher.
Use technology to gather meaningful student data so that teachers can make educated decisions about who and what they should focus on. The Khan dashboard has been incredibly powerful to use in my classroom. They collect data on the activities that students are working on, and it updates the student data in real time, so I can constantly know what my students are working on, and if anyone is having difficulty. This has been made it so easy to target the students who need my help, and give them my individual attention, instead of trying to make the whole class listen to me teach something that most of them might not need or not be ready for.
Students love the individualized pace of the Khan Academy and in many ways it is safer than the traditional classroom environment. Students can go back and get hints on their practice activities as many times as they need. They can watch the video tutorials as many times as they want. This is much safer than asking for clarification of a topic in front of your peers. Some of my kids have told me that they like Khan Academy because they don't feel rushed and can take their time to really understand the concept. Other students have told me that they like it because they aren't forced to wait for the rest of the class - they can move ahead and feel less bored than in class.
Suggestions to Improve the Practice Activities:
Concepts should be first practiced in more concrete ways, then move toward abstract algorithms. For example, the concept of addition and subtraction should not just be drilled with simple algorithms. Though students should eventually become efficient at knowing these facts by heart, there are many things that they can do to build this fact knowledge instead of rote problem drills. Software could be such a powerful way to provide other models for learning instead of just worksheet problems on computer. They should have activities with dot configurations, number lines, math bars, hundreds squares, arrays, base ten blocks, word problems, and scales to name a few. Activities should start as more concrete (i.e. students choosing which amount of dots is more, or which side of a scale should be heavier) and then move towards abstract activities such as algorithms.
Concepts should be broken down into finer granularity. This is closely related to the previous point. The steps between activities are too big. For example, students should be able to practice multiplying positive fractions, instead of having to multiply negative and positive fractions right away - even if they have already "mastered" multiplying positive and negative numbers. A new concept should be introduced in the simplest possible way, and then subsequent activities could become more challenging.
More and better scaffolds should be provided. Students should be encouraged to use them – not penalized, as they are now when their streak is erased. Perhaps if they get a hint then the problem wouldn't count towards their streak, but wouldn't knock them all the way back, either. Students should have choice into what type of scaffold they would like - number line, base ten blocks, math bar, etc.
Eventually it would be great if these activities were more game like. The badges are a good first step. My students kept wanting to play games on another math site that didn't have the tutorials or support because they felt more like video games. I'm not saying it has to be incredibly flashy - but right now the activities are just like worksheets on a computer. They could definitely be more interesting. I know, Khan Academy is a non-profit - so they probably can't afford flashy game development. It's just a thought.
Mastery should vary based on the complexity of the task. Mastery for basic facts needs to be very high – and data needs to be taken on individual facts, not just the whole objective. Instead of data just on one digit x one digit – data on each table and each individual fact. The facts that give the most trouble, should occur more often.
Get rid of multiple-choice questions. It wouldn't be that hard to make all the questions have to be answered with a number. This would make the activities more challenging, and a better test on whether they had actually mastered a concept, or just got good at eliminating wrong answers.
For process skills that take many steps, like multi digit addition and subtraction with carrying or borrowing, or multi digit multiplication or long division, there should be a grid where students enter answers to each step – if they get it wrong the program should identify the step in which the mistake was made. If a student continues to make mistakes in the same area they would be diverted to a different activity to review that specific skill. Many of my students got stuck for long periods of time on some of these long process skills. It was difficult for them to write neatly with the scratch pad - so it was easy to make a simple mistake. When I worked in British schools, all 'maths' was done on grid paper notebooks, with only one digit written in each box. This really supported the students in writing out their problems clearly. I notice that my students here in America make many more mistakes because their work in not laid out clearly. If the practice activity provided a grid for students to type into, than they would probably make less process mistakes and it would also allow the program to identify where the mistakes were made.
Suggestions to Improve the Tutorials:
The math videos available are extensive and great, but not all students learn from lecture. There should be other types of videos as well. Instead of trying to make them all themselves, Khan Academy should let other people submit them. They should vet them and then post them for users to access and rate. The highest rated videos should appear first. Maybe eventually, the program could point you towards other videos that learners like you found helpful.
Suggestions to Improve the Structure:
Schools do not have the highest speed internet connections. Lots of students working on this website at once – which needs to be constantly reloaded – really slows things down. It would be much better if they could access this program as an app – which would periodically (every 10 min or so) send data back to a website so that teachers could still get real time information on students’ progress.
Teachers need to have more controls on what students do in practice exercises – it shouldn’t be totally open. I had some students jumping to activities that were still gray (had not yet come up as suggested activities) and way too hard, just to goof off. I think teachers should be able to control if students should be able to pick any activity or just the ones that are highlighted. Also, it would be useful if teachers could choose an area that students needed to focus on - fractions or division, for example.
Teachers should be able to group students into classes – so they can view groups separately on the dashboard. I have two different classes, and I see them all at once. This is confusing when I'm trying to focus on students in the particular class that I am teaching.
Teachers should be able to enter names of students to go with accounts, so that they aren’t always trying to remember who “dragonscotdude” is. (This is the real user name of one of my students.)
There are age restrictions on both Google and Facebook accounts. We lied about ages to get around this – but this is not really a good policy. There should be another way to login. Also – almost universally in the U.S. facebook is blocked on school campuses.
