Helping students Master Facts

Coming from Junior grades I know that facts are important for students to help them with math.  In addition, I know that learning facts also helps students with solving problems.  However, whenever you talk to anyone this is such a bone of contention.  Some feel that facts are the most important parts of math and some feel they will be learned through problem solving and inquiry.  I tend to lie in the middle of these groups leaning moreso to the inquiry approach.  Don't get me wrong facts are very important to learn and are a critical part of mathematics.  They do help students; however, I also see the other part where students only know facts and cannot apply them to problems.  IN this case facts are harmful to students development because they keep trying to apply rote learning with no understanding.

So with this in mind what is a teacher to do?  I recently came upon some great advice from van De walle's book, Elementary and Middle School Mathematics: Teaching Developmentally.  In his book he has a whole chapter on mastering basic facts.  Van de Walle offers three components to learning facts and none are through strict drill and or quantity of drilling facts.

His components are:

1) Help children develop a strong understanding of the operations and of number relations.


2) Develop efficient strategies for fact retrieval

3) provide practise in the use and selection of those strategies

This is great but what does this look like in a classroom.  I can't say for others but in my classroom this is how I have interpreted these components.

Number sense is beyond just learning algorithms or memorized facts.  You need to understand how numbers work together, their significance, decomposing and composing, and other mathematical reasoning.  All of these help you with mental facts, which in turn helps you with mastering basic facts.  In my classroom, we do a variety of things:
                              
                       a) String lessons: this is fifteen minutes before the problem where we practise mental facts.  These strategies relate to the problem and I hope that students start to apply them in the problem.  This might be adding by tens, using friendly numbers, adding with doubles, etc.

                      b) Problem Solving: chosing a proper problem is just as important to helping students learn math facts.  The problem you chose should allow students to practise their fact recall and not just a traditional algorithm.  In addition, when you debrief the problem there should be some talk about efficiency and using these facts.  This will promote student thinking in this area and see why its important to learn and use their facts.                       

                      c) Teacher Talk: Often when students talk about a strategy I will articulate with certain math talk.  So what you are telling me is this.... Your use of vocabulary will always assist student learning.  I also sometimes do think alouds of my thinking, to help student conversations.  This always is accompanied with talk about what students think I did.

                         b) Math games that focus on these skills.  All of our games in the classroom focus on certain skills.  It helps students practise their facts and learn about numbers beyond just pure memorization.  It also brings out talk among students and teacher.

In addition to this we also do math fact Mondays and Math game Friday.  During Monday my students do a "mad Minute" type of activity.  Though it is not truly a mad minute as it is more about practice of facts then of fast recall.  Students do have a time limit but it is more that it happens at the end of the day.  I will also like to say that my students asked for this activity and relish the moment when they can show me how much they have learned from the week before.  I give my students ten minutes to answer about 60 questions.  We also graph our results over the weeks and set goals for the next.  The emphasis is on goal setting and improving their individual learning.  Results are never shared among the students.  On Friday we do a whole period on math games.  This is important as it give students time to play and practise.  Even though that after finishing a problem they do get to play games not every child gets the same amount of time, this way they do.

Furthermore, Van deWalles chapter there are many great suggestions on the type of strategies that these things can bring out and is a read I recommend all teachers doing.

This only some of the things that I do in the classroom to assist in fact recall.  It is important but how you do facts is just as important.  How do you help with facts?  What type of activities do you use?  Love to here from you.

Accountable Talk in the Classroom: Practical Advice for the Classroom

I have recently finished one great book and one great article on Accountable Talk and Classroom Discussions. 

Stein, M. K., Engle, R., Smith, M. & Hughes, E,  Orchestrating productive mathematical discussion: Five practices for  helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10, 313-340. 

Chapin, Suzanne, O'Connor, Catherine, & Anderson, Nancy. Classroom Discussions: Using         math talk to help students learn. California: Scholastics. 2009.

Accountable talk is one of my passions as I have spent the last four year studying the impact it has on my classroom.  I highly reccomend these two readings for anyone interested in learning more about accountable talk.  However, I also know that in teaching we really don't have time to sit down and read.  For this reason I thought I would summarize them for you and include them in my blog (I appologize in advance as this will create a rather long post).  These ideas come from the two resources above and my own thesis work.  I hope they are practical advice for anyone in their teaching practice.

Implementing Classroom Discussions

Establishing and Maintaining a Respectful, Supportive Environment:

·         LAY DOWN THE LAW (in a collaborative manner):

o   that every student is listening to what others say

o   that every student can hear what others say

o   that every student may participate by speaking out at some point

o   all have an obligation to listen

·         neither student or teacher will participate in bad environment.  Everyone needs to feel comfortable.

·         Emphasize the positive and forestall the negative

·         Establish classroom norms around talk, partner work, and discussions (what does it look like, sound like and what should we be doing)

·         everyone has the right to participate and an obligation to listen

Focusing Talk on the Mathematics:

·         During the discussion time you need to focus the talk on math:

o   plan your questions carefully

o   Have good formative assessment happening at all times

o   Make a plan as to what big ideas you want to cover

o   Anticipate problems and possible solutions

Providing for Equitable Participation in the Classroom Talk:

·         Here are some strategies that will assist you in making it all equitable:

o   Think-pair-share

o   Wait time

o   Group Talk

o   Partner Talk

o   Debates

o   Random  Choice on who Talks

 

Types of Talk Moves:

Talk Moves That Help Students Clarify and Share Their Own Thoughts

·         Say More:

o   Here you literally ask the student to explain more.  "Can you tell me more?", "Tell us more about your thinking.  Can you expand on that?"; or "Can you give us an example?"

o   This sends the message that the teacher wants to understand the students' thinking.

·         Revoicing:

o   It is sometimes hard for students to clearly articulate what they are trying to say by revoicing or having a student do this it allows the original student to check and make sure what they said is true or to hear it in a new way

o   It is not just repeating but more of paraphrasing the students ideas

·         Model students thinking:

o   This is not so much a talk move as it is a way to help talk

o   As students talk record what they are saying without comment.  When they are done ask them , is this what you meant?

o   This allows students to reflect and think about what they said in comparison to what was written

·         Wait Time:

o   Wait time is so important.  I cannot stress this enough.  The longer you wait the better responses you will get.  It allows students to process what you or another student asked and be able to formulate their thinking

Talk Moves That Help Students Orient to Others' Thinking

·         Who can Repeat?

o   I would classify this under the first category but it also helps students with understanding what their peers are saying

Talk Moves that Help Students Deepen Their Reasoning

·         Press for reasoning

o   Here you are basically asking students to think about why they did this.  This can be done by asking:

§  Why do you think that?

§  What convinced you that was the answer?

§  Why did you think that strategy would work?

§  Where in the text is their support for that claim?

§  What is your evidence?

§  What makes you think that?

§  How did you get that answer?

§  Can you prove that to us?

o   Not only are these excellent talk moves but excellent questions that push students beyond their thinking and make excellent mathematical connections.

Talk Moves That Help Students Engage with Others' Thinking

·         These are excellent questions that help students build upon their own thinking and the thinking of the community

·         Do you agree or disagree...and why?

o   This really brings students into direct contact with the reasoning of their peers

o   You can do this by:

§  Thumbs up or thumbs down

§  Why do you agree or disagree?

·         Who can add on?

o   When you ask this question make sure that you wait for answers as this may need time to develop connections.

Five Practises suggested by Stein, M. K., Engle, R., Smith, M. & Hughes, E.

1: Anticipation (P.322)

The first thing is for the teacher to look and see how students might mathematically solve these types of problems.  In addition, teachers should also solve them for themselves.  Anticipating students’ work involves not only what students may do, but what they may not do.  Teachers must be prepared for incorrect responses as well.

2: Monitoring students' work (P. 326)

While the students are working, it is the responsibility of the teacher to pay close attention to the mathematical thinking that is happening in the classroom.  The goal of monitoring is to identify the mathematical potential of particular strategies and figure out what big ideas are happening in the classroom.  As the teacher is monitoring the students work, they are also selecting who is to present based on the observations that are unfolding in the classroom.

3: Selecting student work (P.327-328)

            Having monitored the students, it is now the role of the teacher to pick strategies that will benefit the class as a whole.  This process is not any different than what most teachers do; however, the emphasis is not on the sharing, but on what the mathematics is that is happening in the strategies that were chosen. 

4: Purposefully sequencing them in discussion (P. 329)

With  the students chosen, it is now up to the teacher to pick the sequence in which the students will present.  What big ideas are unfolding, and how can you sequence them for all to understand?  This sequencing can happen in a couple of ways: 1) most common strategy, 2) stage 1 of a big idea towards a more complex version or 3) contrasting ideas and strategies.

5: Helping students make mathematical sense (P.330-331)

As the students share their strategies, it is the role of the teacher to question and help  them draw connections between the mathematical processes and ideas that are reflected in those strategies.  Stein et. al. suggest that teachers can help students make judgments about the consequences of different approaches. They can also help students see how the strategies are the same even if they are represented differently.  Overall, it is the role of the teacher to bridge the gap between presentations so that students do not see them as separate strategies, but rather as working towards a common understanding or goal of the teacher.

 

Trouble Shooting Talk in the Classroom

My Students won’t Talk:

v  First ask yourself: our my students silent because they have not understood a particular question? --> sometimes they need to hear the question a few times and have time to think

§  if this is the case then give students time to think  (wait time is very important)

§  also revoice it or have another student revoice the question

v  Second they may be shy or unsure of their abilities:

§  If this is the case you may need to revisit strategies for talking

§  Think-pair-share is an excellent way to get kids comfortable to talk

§  it will also take time to get kids comfortable.  Wait time again is important as it holds students accountable.  Also making them feel comfortable and that mistakes are okay will assist with this difficulties

The same few students do all the talking:

v  Wait-Time:

§  I know that I say this a lot but it allows the other students to think and then participate while making the ones who always participate  (it will feel awkward at first but wait as long as you can)

v  Have students Revoice:

§  This is good strategy to bring validity to students answers and encourage others to talk

v  Conferencing with the ones who talk a lot:

§  You also don't want to ignore the ones who talk  all the time.  You can talk to them and let them know that you are not ignoring them but are just trying to allow others to participate.

v  Turn-Taking/ Random presenters/ group discussions:

§  These are all roughly the same strategy.  It allows you to have certain presenters share their thinking without offending or allowing others to take over the conversation

Should I call on students who do not raise their hands?

v  there is research to suggest that students will learn by listening but you will also hinder the class progress in discussion.  To help try creating a positive space that allows all students to feel comfortable and willing to participate.

v  "right to pass": 

§  allow students at the beginning of the year the right to pass.  You'll notice that they may do this at first but as you build the community they do this less and less

v  Call on reluctant to students after partner talk:

§  Often when you give them a chance to share first they are more willing to share or at least have a response from their partner

My students will talk, but they won’t listen

v  Set the classroom Norms:

§  remind each students that they have the right to be heard but that this also means an obligation to listen

v  Students Revoice:

§  When students need to revoice then they have to listen

Huh?” How do I respond to incomprehensible contributions?

v  The temptation is to simply say, "Oh, I see.  How interesting...." and quickly move on to another student.

v  Try Revoicing or repeating what they have said.  After you have done this ask them is this what you meant?

v  Record their strategy on the board and ask them is this what you meant?

Brilliant, but did anyone understand?

v  Repeat what they said, then have another student repeat what they have said (if really important have many students repeat)

v  Break the explanation up into small chunks and revoice or have the students

I have students at very different levels

v  Pair students in ability groups:

§   Similar abilities with similar abilities.  This allows students to contribute at their level and to also struggle at their level.  In addition, it allows you as the teacher to differentiate as needed.  When you scaffold you can do so by group not by individuals

v  Parallel Tasks:

§  Give students similar tasks but with varying degrees of difficulty (still around the same big idea)

What should I do when students are wrong?

v  First ask yourself is there anything wrong with having the wrong answer?  Sometimes wrong answers provide rich and meaningful discussions

v  Need to establish Norms around respectful discourse and discussion with wrong answers

v  Mistakes are always an opportunity for learning to happen

This discussion is not going anywhere or Students’ answers are so superficial!

v  This may be happening because you are asking to many students to share or revoice the ideas that are happening in the classroom or in the case of superficial classroom  norms have not been established or the types of questions have been simple and direct

v  Use the working on phase as an opportunity to direct your bigger discussion:

§  As you are walking around and looking at work, look for the progression your students are taking.  This will lead you to a group discussions.  What questions are the students asking themselves?  What problems are occurring?  What big ideas are they trying to work out, have worked out or are struggling with?

v  Look at the type of questions that you are asking:

§  As teachers we are comfortable asking questions but do our questions already have responses?  Are we leading the kids to OUR thinking or our we allowing the students talk to LEAD the thinking.  Yes you are very much in control of the discuss and have to lead but it is not YOUR thinking but THEIRS that should be articulated.

§  Higher order questions build-upon or go beyond the thinking that is being presented.  As a teacher we need to help with the connections in mathematics.  Compare student work?  Compare strategies, Pros and Cons, naming and identifying.  We need to go beyond just show and tell