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Problem-Based Teaching can be difficult to get the hang of, so these tips and tricks are sure to help! We have found consistently, that every teacher can use problem-based teaching with the right tools and training. Soon we will be adding video footage to many of the tips below. If you continue to struggle, please email or phone us for expert advice.
In problem-based teaching the most powerful tool that you have is your ability to be flexible. Every lesson runs differently, and you need to be prepared in order to move flexibly in many different conditions. Sometimes you find that students already understand the concept that you are about to teach and your preparation needs to have included more complex concepts, and sometime you find that your students have major misconceptions that underpin the concept that you were about to teach! Being prepared ahead of time allows you to maintain your flexibility and adapt your lesson to exactly suit the needs of your class.
When preparing to teach a problem-based lesson it is important to ask yourself several questions before you start. Make sure that you consider these questions during your planning time:
Read through the lesson plans for the Journal problem that you are about to use, and look at the misconceptions carefully. Have a think about what questions you could ask students to help them evaluate their wrong ideas and figure out a new mathematical understanding. Also, make sure that you have prepared to alter your question for support and extension students. Suggestions are given in the lesson plans.
You should never teach a new mathematical concept to students without first finding out what they know. Otherwise your teaching and time can be greatly wasted.
When you start a lesson, make sure to keep the following questions in mind, and ask the appropriate ones of your class.
Make sure that you keep monitoring the learning situation throughout the lesson so that you can cater for the needs of individuals. As you are going along ask:
At the end of the lesson make sure that you can answer these:
Remember: Above all be flexible and change the lesson as needed to respond to what the students are doing. There is no point teaching something that they are not yet ready to learn!
Student misconceptions often seem to form when teachers push procedural skills and rote learning at the expense of deep mathematical understanding. Often as teachers we tend to only ask routine-style questions: ones that the students have seen before and that we expect them to know the answers to already. We are surprised when students seem to not understand a concept that we have taught previously, and tend to leap to correct any misconceptions that we do find.
Problem-based teaching takes a fundamentally different approach. For starters, one of the main aims of problem-based lessons is to find out what misconceptions your students already have. Unless we know about misconceptions we can’t deal with them. In order to find misconceptions we need to ask questions that look or sound different from questions that students have answered previously. If we only ask questions that students already know the answers to, we only see what they can remember rather than what they really think.
Misconceptions can occur in all sorts of mathematical concepts. In Back-to-Front Maths we try to set up problems to find out what misconceptions your students have so that you know where to target your teaching. This helps make teaching much more efficient in the long run even though it can take a lot more time to begin with. We have also found that it helps students enormously with NAPLAN questions as often the questions are non-routine, and the answers given contain common misconceptions. This being said, there are a number of principles which will help you and your students in dealing better with misconceptions.
Often as teachers we use oral questioning to test our class, or as a precursor to teaching a new concept. It is tempting to have a “shot gun approach” to wrong answers, saying no and moving on to the next student until we find the one that we want. Unfortunately this tends to teach students a few bad lessons:
It is difficult when we are pushed for time to remember that when we ask a question to our class, if only 2 students have their hands up then there is a very good chance that the other 25 students don’t yet understand the concept. There is no point getting the right answer at this point and moving on, as most of your class doesn’t understand the basics.
Instead of this approach, in problem-based teaching we try to approach every answer, whether right or wrong, in the same manner. We take the answer given, get the student to work with us, and try out the answer in the question to see if it makes sense.
Often during oral questioning I will deliberately choose the student who is likely to give the wrong answer. This gives me a great learning opportunity! I ask the student questions and lead them through the question bit by bit until I see them realise that their answer is wrong. At that point I usually ask, “So do you want to keep your answer or do you want to change your mind?” By allowing students to change their initial answer I can then praise the Thinking rather than the Answer. Doing this helps students to see the reasons for an answer and understand concepts much better. In short, I have turned a bad situation into an opportunity for all the students to learn and understand a concept.
I also use this approach to challenge students even when they have the right answer! It helps gifted students to re-evaluate their ideas, builds confidence in their own thinking and reasoning when they are right, and in their ability to work stuff out when they are wrong. It also encourages kids to “have a guess” because they won’t be rejected and will be praised when they can think it through.
Some useful phrases that help with this include:
Pausing is also a very useful skill to be used in combination with this approach, so make sure that you check it out as well.
Often as teachers we are very pressed for time. There is a very large temptation to push students quickly through content to make sure that we get everything covered. This however is a false economy, leading inevitably to the requirement for re-teaching the same content. The art of pausing makes a significant difference to our ability to teach for Understanding.
I use pausing when I ask a question to a class and only a couple of kids try to answer my question. There is no point gathering student answers at this stage, as most of the students have not understood the concept or the question.
There are a few stages in pausing, and I would like to explain them in brief here:
This whole process takes between 2 and 5 minutes, and means that at the end of the time I can always give the response, “Nice thinking!” to a student who otherwise would have been completely lost.
Catering for a very diverse group of learners can be challenging for the most experienced teacher. One simple tool that I have developed to help me cater for multi-levelled classes is a challenge table.
A Challenge Table is a spare table or set of desks somewhere in your class that is set aside as a work space. I use it to invite small groups of students to work with me for five minutes during our problem-solving time. I find that having this dedicated space allows me to maintain a great deal of flexibility within my classroom.
I invite different students to work with me based on what I need to happen within the whole class. Often I invite students with similar communication styles rather than students who are working at the same levels. Inviting all of the students who are dominant allows them to “fight it out”, while freeing up your other groups to think for themselves. Inviting all of the students who are very quiet and just go along with the group under normal circumstances means that one of them eventually has to contribute something to solving the problem!
You can use the Challenge Table in whatever manner best suits your class. Consider some of the following ideas:
Who you choose to invite is up to you, but make sure that everyone gets a turn. Working one-on-one with the teacher is a privilege that most students look forward to!
The language that we use to describe the learning situation in problem-based teaching is very important. Language sets the scene for student expectations about the way the classroom operates, and shows students what is valued by the participants. Students read our metalanguage to find out what we believe is important and unimportant. Often we take care to use the correct mathematical terminology when teaching, but do not consider how the metalanguage that we use to talk about our learning affects the learning behaviour of students.
One simple example of how metalanguage affects the learning behaviour of students is using what I call the “language of right and wrong”. Often we use terms that emphasise the importance of getting the answer correct without even realising that we are doing it. Some terms that emphasise the importance of the answer instead of the thinking process include:
Metalanguage can instead be used to emphasise thinking mathematically by changing our responses to right/wrong situations:
This fairly simple technique is one way of increasing the risk-taking behaviour of your students and therefore their willingness to be wrong. If we tell students that we are looking at their thinking rather than their answers, and yet we keep using the “language of right and wrong” students don’t believe us. Students read into what we say and work out what is really important to us by the metalanguage that we use.
I have never met a class of students who do not need differentiation. The reality of teaching is that in every single class there will be multiple levels in operation at the same time. This means that for us as teachers, differentiation has to be part of our thinking in every single maths lesson, or we are failing to teach the students who have been given to us.
1. Adjusting for support students:
One of the questions that I am most often asked in Professional Development sessions is what to do for support students. I have found that most of the support students that I work with respond exceptionally well to a problem-based approach. In my experience, many students who are in the “support” category have somehow missed out on learning the fundamental mathematical principles and patterns that make mathematical learning practical. They have some fairly major misconceptions, and therefore they tend to see each new algorithm, fact or formula as a stand-alone concept that has no links to other concepts. This makes learning incredibly inefficient, and usually means that the further through school they go the further behind they get. While we try to help, students often learn new concepts slowly and only retain them for a limited period before lapsing back to where they were. Dealing with this in the midst of a normal classroom with 20 other students can however be challenging.
Why what we are doing doesn’t work for support students:
Often we tend to give support students more and more routine-style questions to practice “the basics”. This is fairly ineffective as if the students can actually answer the questions then they aren’t learning anything new, and if they can’t answer the questions then we need to be with them to tell them every step.
An alternative approach: using problem solving to teach basic concepts
I have found consistently that when students work out a mathematical principle for themselves they have a much better retention of the concept. With support students this is crucial in order to build some of the deep mathematical principles that they have missed out on. Instead of using routine questions, try using problem-based teaching to encourage students to experiment with ideas about how to solve a problem. Each learner should be challenged appropriately using a problem that is just out of reach at their current level, but not so hard that they do not have any knowledge to bring to the problem. The lesson plans for each journal problem have suggestions for how to change the content level within each problem to cater for your support students.
Classroom strategies
To manage the multiple levels within your class consider using these strategies:
2. Adjusting for gifted learners or students with high content levels:
In the past 12 months I have enjoyed working with a number of classes with students who are either gifted, or who are sent to maths tutoring so that their content levels are very high. Often these students can be particularly difficult to cater for in an average classroom, and most of us are guilty of leaving these students to look after themselves while we cater to the “lower average” group at some point. There are a number of unique problems that I often see when working with students in this category. Here are some of them, and some workable solutions.
Students who are bored in traditional lessons:
Students are not sufficiently challenged and get bored easily with the lower-level content and therefore make trouble. This is often in the form of them seeming totally innocent while the other students in their group misbehave! I have seen a number of student responses to this situation. These include: finishing all their work quickly so that they can misbehave and then when told to “get on with it” legitimately claim that they have finished, avoiding doing any work because it is so boring and they “already know all this stuff”, refusing to show their “working” because they really did just do it all in their heads, and asking questions to get the teacher to go off on a completely different tangent so that they don’t need to do the work. These problems are often solved by problem-based teaching with appropriate adjustments to the problem to maintain the level of challenge.
Students who don’t want to participate:
Students who suddenly don’t know how to solve a problem in a problem-based lesson, but for whom teachers have assumed a higher level of intelligence, better thinking and better problem-solving. These students tend to act out early when first being exposed to problem-based teaching. They also claim that they “aren’t learning anything” or that this work “is just for babies” in an attempt to avoid solving the problem.
If this becomes a problem in your class try the following: give the students the option of completing traditional worksheets instead, the give them so much “busy work” that they can’t possibly finish it all. When they become “out-worksheeted” give them the option of joining back into a group to do the problem-solving task with a number of provisos. The first is that they need to accept that they will probably answer wrongly at first before working out how to solve the problem. They have to be ok with being wrong. The second is that this work is not “easy” like worksheets, and they can’t just give up when they don’t know what to do. They have to be prepared to try something and to contribute to the group. The third is that they have to be prepared that someone else might offer a better solution than theirs, and they therefore need to work with a group not try to be the leader all the time. When students can see how much fun the groups are having, there are very few who elect to keep going with worksheets after about half an hour!
Students who finish before everyone else:
Students with high content levels often solve the initial problem really quickly because they have and then don’t have anything to do. Try directing these students straight to the manipulation problems as these require much deeper thinking without just having higher content levels. Consider also asking “what if” questions and non-standard questions to your students based on the initial problem. These work to increase the problem-solving requirements without teaching more difficult content. They are also great practice for NAPLAN!
Watch out as well as fast-finishers often pass on their answers to everyone else without giving them the opportunity to solve the problems for themselves! Make sure that you spend some time talking through your expectations with students before starting the lesson. Also, consider using your challenge table to adjust the problem for your students in higher levels while everyone else is just starting the problem.
In problem-based teaching we ask lots of different questions, and also lots of different types of questions. As it is often difficult to track which questions I ask in a lesson I often have someone else record the questions so that I can examine them afterwards and see how effective they were. Below I have included a list of some of the question types that I use in most lessons, their purpose, and an example of a question that fits this category. In most problem-based lessons that I have had observed I ask between 120 and 150 questions in about an hour and a half.
E.g. “What if you could cut the blocks up to make them smaller? Could you do it then? How would you cut them?”
E.g. “What do you mean?” “I’m sorry, I really don’t understand what you are talking about. Can you tell again?” “So you are saying…. (repeat the student’s exact words so that they can hear them and realise that this is not what they meant)? Is that right?
E.g. “Let’s make 23. Now let’s have a look at what you have made. So do you think cutting the blocks in half is still ok?” These can also be more general, “there’s a bit of a problem – can you see what it is? Ok, so why is that a problem?”
E.g. “Is 23.7 going to be bigger or smaller than 23? Is it going to be bigger or smaller than 24? So let’s make 23 from MAB. Now let’s make 24. Now let’s see what you have made. Is it bigger than 23? Is it smaller than 24? Ah, so we have a problem. What is the problem? Ok, so cutting your blocks into halves didn’t work because when you put them all together it was bigger than 24? So how else can we try cutting them? Ok – try that. Remember to check once you’ve drawn it that it is between 23 and 24.”
E.g. “Which buttons would you press on a calculator to check your answer? Can you write them for me in these squares in the order that you would press them?”
E.g. “How many ones are there in this ten? How many tens are there in this hundred? How many hundreds are there in this thousand? So how many thousands do you think there might be in the next place? How did you know? What is the pattern?”
E.g. If during the day I bought lunch for $5 and a hat for $10 and then I found that I had $2 left, what did I start with?
E.g. “What if there were 5 students instead of 4 coming for dinner? How much of each ingredient would we need now?” “What if we also had to pay for a bus?”
Make sure that when preparing for a lesson you ask yourself these questions:
Varying question formats is a great way of promoting student thinking and also of catering for students who are working at different levels. There are several different ways that this can be done. Several that I use regularly include:
1. Change the numbers in the question and ask students to adapt what they have done to solve the new problem. Look to see that they use what they have already done to work out the answer to the new problem rather than going all the way back to the start every time. For example:
a) 26 students went to the zoo. The admission was $5 each. How much did it cost?
b) What if two teachers went with them? What would it cost now? (do they start from scratch
c) What if three students were away that day and missed out? What would it cost now?
d) What if the admission price was $7 instead of $5? What would it cost for each of the scenarios above?
e) What if the admission price was $7.50? What would it cost for each of the scenarios above?
f) What if they also had to pay $1 each for a bus? What would it cost for each of the scenarios above?
2. Ask students a non-standard problem with a similar principle. These are particularly prevalent in NAPLAN tests. For example:
a) 2 o x 5 = 115. What goes in the box? (filling a gap)
b) The total admission cost for a class to go to the zoo was $115. 23 students went. How much did they each pay? (non standard multiplication and division)
c) Which of the following shapes is a hexagon (give a regular octagon, pentagon and an irregular hexagon)? (non standard shapes)
d) What number has 23 ones and 42 tens? (non standard place value)
e) If 1/3 of the group was 5 students, how many students were there? (working backwards, non standard fractions)
f) I need to be at school at 3:00. It takes 45 minutes. I need to set my alarm for 5 minutes before I have to leave. What time should my alarm be set for? (working backwards, multistep)
g) Change 1.23 cm into m. Change 1498m into mm. (non standard conversions)
3. Ask students a problem with multiple answers or an open-ended problem.
4. Ask students a problem with missing information, and have them ask you for the missing information rather than solving the problem.
5. Give students a number of wrong answers to a problem and ask them which one is right. Make sure that you include common misconceptions in your answers.
For most concepts in Back-to-Front Maths you will find Backwards Problems and Manipulation Problems. These are designed as non-standard questions to extend student thinking.
One of the most difficult problems to overcome when establishing a problem-based classroom is encouraging risk-taking behaviour on the part of the students. Unless students are willing to try out an idea and risk being wrong, there will be very little progress. I have found that the further through school students go the less willing they are to take a risk, and the more avoidance behaviours they have developed. The most common avoidance behaviour is simply to do nothing in the face of a problem, and to wait until the teacher breaks down from time-pressure and explains how to use the formula/algorithm.
Risk-taking taking can be encouraged using a number of strategies. Below I have outlined some of the most common things that I do when implementing problem-based teaching for the first time. These include the initial discussion and scene setting, engaging students and provoking a response, deliberately displaying a misconception, using metalanguage and combating right/wrong thinking.
The initial discussion and setting the scene:
It is important to discuss your expectations with students up-front so that they know what to expect. I always begin with a class discussion that goes something like the following. I apologise for it being so wordy, but it is difficult to explain without demonstrating.
“You know last year when you did a maths test, were the questions usually completely different from anything that you have ever seen before, or kind of like ones you had done in class? So do you think that your teachers were looking at your “working out a new idea completely for yourselves”, or do you think they were looking at your” remembering what you had been taught”? Well, today I’m not looking at your remembering. I’m looking at your working stuff out for yourselves. That means that today our maths lesson is going to be a bit different to what you are used to.
For starters, I can’t ask you a question that you already know how to answer. Then you would just be remembering stuff instead of working out something new. So today I’m deliberately going to try to confuse you! I don’t want to see all the stuff you know – I want to see what you can work out. Who in maths is usually a bit confused? Well you guys have a real advantage today: you know what it feels like to be confused. Today everyone is going to be confused for at least some of the time – even the kids who are usually pretty good at maths. Oh, and if you put your hand up to say, “Mrs Kennedy I don’t know what to do”, I’m going to respond with, “That’s great! Now you have something to work out!”
Now to show you that I’m serious about looking at your thinking I’m not even going to mark your answers. I honestly don’t care if they are right or wrong, and I probably won’t tell you if they are right or wrong. I’m looking at your thinking, and that is far more important. You know sometimes people can get the right answer, but have totally wrong reasons or no reasons at all. I’m going to mark that as totally wrong. And sometimes you can have really great thinking, but you just added up a bit wrongly so your answer ends up being wrong. I’m going to mark you as totally right. I’m marking the thinking and the reasons, not the answers.
Let me explain. Say I ask you a question, and remember I am deliberately tricky and weird, and you have no idea at all and you come up with something that is totally wrong. Then I ask you some questions, and you do a bit more thinking, and you have another idea. This one is wrong too, but it’s getting closer and you’ve done some great thinking. Then I ask you some more questions and you do some more thinking and try out something else. You’re still wrong, but you’re getting really close now. Then you have this moment where you think, “oh, now I get it!” and you work it out. Do you really need me to tell you that you are right? And can you see how much great thinking you did along the way to get there? That’s what I’m looking at.
This brings me to one of my rules: you can’t use a rubber today. Imagine what would happen if every time you tried something that turned out to be wrong you rubbed it all out! How would I see all that great thinking? Then I would only have an answer, and I don’t care about the answer anyway. Today you are allowed to change your mind as many times as you want. Just don’t rub it all out. Instead, just rule a line and start again underneath. Once you’re happy with your final answer put a big star next to it. Also, I need to explain that I don’t have xray vision – I can’t see what you are thinking unless you explain it to me or show it to me on paper. Now I don’t care about your spelling, or how beautiful your writing is, but I do need to see your thinking. So show me your thinking on paper. Write down all your ideas no matter if they are right or wrong.
I have another rule too. Who does their best thinking with a friend? Well today you are allowed to work together, as long as you are both thinking. In fact I don’t even care if you just copy down exactly what they have written down. It’s not really going to help you though – am I marking your answers? No, that’s right, I’m looking at your thinking. So if you and your friend have the same idea, then write that down. But if you have a different idea, then try that instead. That way you can try out both ideas and learn together.
You know how sometimes when teachers say that they want you to work it out using any way you want but they really mean that they want you to work out what’s in their heads? Well, when I say I want to see your thinking I actually mean it. I really want you to try out your own ideas. If you don’t have any ideas, that’s ok, just have a guess. That’s trying something and that counts as thinking. I don’t care if it’s wrong. I care that you tried, and you know you might just work it out by guessing!
One more thing that I need to let you know. You know how sometimes teachers ask you questions like, “Are you sure about that? Can you just check that one for me please?” and what they really mean is, “That one is totally wrong”? Well I’m going to ask you questions like that even if you’re right! I actually want to see what you think, so I am deliberately tricky. I can even use my voice to make the wrong answer sound like it’s the right one: “When you add two and three together, is the answer five (sound uncertain) or is it four (sound certain)?” I really do want you to have to think about it. Oh, and if you already know the answer please let me know so that I can make it trickier for you, or else you’ll miss out on a chance to show me your thinking.
Ok, so now that’s all out of the way, are you ready to be confused? Great! Let’s get started.”
Engaging students and provoking a response:
I find that playing a character is often important in getting students to “lighten up” and enjoy maths. In many maths situations in primary school playing is low on the agenda, and doing things “the right way” is emphasised. It is essential to turn this around if you want your students to really engage with a problem and experiment with ideas. Often I am deliberately overly theatrical or a little weird in order to provoke a response from students who would otherwise try to ignore me.
Deliberately displaying wrong thinking or a misconception:
Students are much more likely to show you their own misconceptions if you pretend to have one yourself. I will often tell students that what they are doing with that formula doesn’t make any sense at all, and then demonstrate a misconception. This challenges students to check and see if their thinking does make sense, and to prove that my misconception is actually wrong. This is especially helpful for engaging both students who have a misconception, and students who really know that I’m wrong. It’s always fun proving the teacher wrong!
In the same manner, I always try to get students to have a guess at what they think the answer might be before we get stuck into serious problem-solving. This usually takes the form of a class vote. I always put my hand up for every option and make sure that they all sound equally valid. For more help on misconceptions, follow the link.
Using metalanguage:
This is such an important concept that it has its own section. Follow the link to find out about metalanguage.
Combating right/wrong thinking:
Many students firmly believe that there is really is only one way to solve a problem, and that it is the way that the teacher is about to show them. All other methods are wrong, or if not wrong at least not as valid as the teacher’s method. They see it as a total waste of time trying to work out something themselves instead of waiting for the teacher to show them how to do it. To combat this thinking, we need to help students think about maths as something other than the duality of right/wrong.
Most students don’t think about how mathematical ideas and theories are formed in the first place. Formulas don’t just exist – people work them out by experimenting with different ideas and seeing what works. Formulas are a summary of the mathematical thinking that someone did. They are certainly useful, but are in no way what true mathematics is really all about. True mathematics is about using what you already know to work out something that you don’t know yet. It is about finding patterns and connections between ideas. It is about experimenting and finding out what works, and then summarising the principles involved so that they can be used in other situations and by other people. If we could only use existing formulas, then nobody could work out anything new.
I am often asked in professional development sessions what model of group work to use in problem-based lessons: mixed ability or like ability groupings. My answer to this question is somewhat more practical. I have found that in most classrooms we actually have to use behaviour groupings: students are matched up according to who won’t kill each other. This is a pretty workable solution in problem-based teaching lessons as long as you are able to maintain some flexibility through a strategy such as a challenge table.
Another good idea is to consider letting students choose their own groups on the proviso that everyone is included and that everyone is working. Students often choose to work hard as long as they can work with their friends. Explain to the class that if anyone is not included or the students aren’t working you will break up all of the groups and choose them yourself. This usually ensures good working and good behaviour.
I do have one more tip for managing group work: don’t have groups bigger than three students to start with. When groups get bigger than three I usually find that someone is letting everyone else do the thinking. Pairs or threes usually work well.
Firstly, ask yourself which groups you tend to teach to. We all do it, partly because it is impossible to actually get to every student in our class, and partly because we have some students who seem to demand far more than their fair share of our time.
Secondly, ask yourself which students in your class are actually being challenged at an appropriate level in your lesson. I have found that most of the behaviour problems tend to come from students who are not in this group. (1) Students for whom the problem is too difficult and they act out to avoid looking stupid, (2) Students for whom the problem is too easy and they are bored. When you have students acting out, look firstly to their learning to check that it is appropriate.
Thirdly, in problem-based lessons it is very important to treat students as important people with valid ideas. Be deliberately respectful towards your students to cultivate a culture of respect as well as to encourage students to try out ideas and experiment.
Fourthly, in problem-based teaching you cannot remain in one location throughout the lesson. In order to monitor student learning you will need to move between the groups, not stay within a few metres of the board.
Finally, try using maths as a reward and getting positive peer pressure on your side: “If you are not working with me then you will have to miss out on this maths today and I’ll find you something else to do instead.” Exclude students who are misbehaving from your lesson, and give them routine questions on worksheets to do instead. These need to be completed silently, in isolation. Also, you need to have a formidable stack of these worksheets so that students understand that they will never run out and therefore have the excuse of having nothing to do. I usually find that within half an hour students are begging to be included in problem-based maths again.
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